Active Learning for STEM and Science Education

#### **Chapter 1**

## Emergent Chemistry: Using Visualizations to Develop Abstract Thinking and a Sense of Scale within the Preschool Setting

*Karina Adbo*

#### **Abstract**

This chapter is a summary of 5-years of research regarding children's emerging abstract concepts. A longitudinal study focusing on children's conversations during a series of activities with a chemistry focus was designed and implemented. Results show that practical experience with magnifying glasses, microscopes, and the deconstruction of several items did not provide enough backdrop for the children to imagine what an even smaller world would look like. Instead, the children applied their experiences from the macroscopic world to describe what they saw. It was not until aminations, zooming in from the macroscopic to the atomic and molecular levels were used that the children's concept of small began to develop. Results show that the next stage of concept development, besides using descriptions from everyday experiences was the realization these were new experiences, that it was in fact something new they were seeing. Animation technology also helped the children realize that atoms and molecules are everywhere in everything, suggesting that the time elapsed between the transition from the macroscopic level to the submicroscopic level also provided the children with a sense of scale.

**Keywords:** preschool education, abstract thinking, scale, natural science, chemistry

#### **1. Introduction**

Preschool education in all countries has one general goal in common. That goal is to provide children with a broad experience base. This is done since experiences are seen as the foundation of learning, creativity, and imagination. Learning can be seen as a change in previous experiences and Vygotsky proposed that imagination is based on experiences [1]. A proposition suggests that it may be difficult for us to imagine something which we have no experience of. Creativity was as also described by Vygotsky as, new ways to combine experiences. Indeed, learning, imagination, and creativity are some of the intended outcomes of preschool education.

#### **2. Preschool education**

Learning can be viewed from different perspectives. A general perspective was provided by Piaget who thought of learning as a change in previous experiences. Piaget [2] then continued to describe learning as being possible within different stages while Vygotsky used another perspective and described learning as more situationally bound. Vygotsky saw learning as not only occurring in all instances but also dependent on what was possible in every situation. He saw learning as occurring in the space between the real and the ideal form. Today researchers use the word the present form instead of the ideal form to emphasize that the social world surrounding a child may not in fact be ideal [3]. The real form is the child's current experiences and the term present form is used to describe what the social environment surrounding the child contributes.

How experiences are formed was also described by Vygotsky with the use of the concept of perezhivanie. The concept of perezhivanie [4] provides a perspective where we all are seen as viewing the world through our own personal prism. A prism that is formed by our previous experiences the prism determines what we notice and how this information is then interpreted. Suggesting that our current experiences affect our learning of new experiences, and our new experiences will affect our current experiences (see **Figure 1**).

Those experiences that are seen as having the greatest impact are those who are emotionally connected. If the theory is placed on a time scale such as a lifetime, then it suggests that emotional experiences will impact what we learn but also that our learning becomes more personalized as time passes. An interest in birds will cause us to focus more on birds something that will make us learn more about birds, something that in turn will make us notice more details regarding birds. If this is the case, then this perspective provides us with three important conclusions regarding preschool education: the first one is that the present form must include a manifold of new experiences, and the second is that it is important to intentionally create positive experiences since they will affect lifelong learning and the third is that activities should be designed for individuals. This research takes inspiration from a culturalhistorical approach [5] that involves both cultural and individual aspects.

What is/should/could be included in a broad experience base within preschool education is today argued in the research literature. Most if not all authors agree

#### **Figure 1.**

*A general overview of the difference between the social and individual perspectives when the concept of perezhivanie is used.*

#### *Emergent Chemistry: Using Visualizations to Develop Abstract Thinking and a Sense of Scale… DOI: http://dx.doi.org/10.5772/intechopen.105216*

on the fact that the content of preschool education should meet the children's own interests and needs. For some, this approach means to follow the children's lead and expand on their own interests [6] while others see it as actively introducing children to new experiences by conveying content indirectly through play [7]. In practice, most preschool education is of course a mixture of these two counterparts. As it would be difficult to only follow the children's lead since they all have different experiences so what is following the experiences of one child is staging a new experience for another child.

The content of activities in preschool education is also under scrutiny, there are those authors who argue that preschool education should be free of any academic content [8] while others see preschool as an opportunity to provide children with play-based activities with content that are designed and could well be seen as academic depending on our definition of what is included in academic content. The one thing that most authors agree on is that preschool education should at all costs avoid schoolification. Schooloification is a term used for describing a more direct transfer of academic school knowledge to the preschool level. The key to avoiding schoolification is to analyze children's activities and allowing their questions and current focus, that is, to be the force that drives experience building [9] regardless of whether the content is deliberately introduced by the teacher or not.

#### **2.1 Development as a case of emergence**

When researchers describe learning in the preschool context is often described in terms of emergence [10]. When using the term emergence focus is placed on children's own version of content. It is a version that is not evaluated for its correctness but instead because it provides clues that help us provide a more supportive preschool environment. The use of the word emergence also recognizes that what research can describe are only small parts/glints of the process of learning.

#### *2.1.1 Content of preschool education*

Today natural science is a part of the experiences that most preschools provide. It is in fact difficult to avoid natural science content in preschool as children have a natural tendency to explore their surroundings [11]. And as natural science is found in all aspects of the immediate world surrounding the child, i.e., in the material-, biological- and cultural aspects it is a content that is difficult to avoid. The goals for this content can be found on both individuals as well as societal levels. On the individual level, the goal is of course to meet individual children's own curiosity and to build positive experiences of science [12]. Positive experiences in science are important for creating self-confidence, that is, positive subject identities so that children feel confident that they are good at science. Indeed, motor active hands-on activities contribute positively to positive attitudes toward science [13] providing a stronger positive subject identity. Another goal is to create cultural motives [14] and thereby making science a natural part of the child's everyday life. On a societal level, the goals include creating interest in the natural environment to develop children's environmental awareness to build for a more sustainable future [15].

From the perspective of learning a natural science content, an early introduction to the concepts and words that natural science is comprised of is seen as beneficial, since many of them take a long time to get familiarized with [16]. Indeed, research on children's science learning on a general level can be separated into categories:

development of abstract thinking [17], development of words, concepts, and their content [18], developing understanding of processes and transfer of knowledge/ideas between contexts (near- and far transfer) [19] as well as the process of science inquiry [20]. Most of these studies include topics that could be categorized into biology and physics. Very few studies have focused on chemistry and how a more chemical content could be transferred into a preschool environment [21–24]. This is even though so many of our chemical methods can be found in the child's immediate surroundings and explored through hands-on experience. One example is separation methods such as filtering something that we do on a daily basis while: separating pasta from water, ions from water, tea leaves from the water, ground coffee from water, preventing pieces of food from entering the pipes in our kitchen or strands of hair from pipes in the bathroom, light from entering a room, butterflies from the air, fish from the water, or preventing sound from entering our ears. Another reason why chemistry should be a part of the preschool environment is because the single most important feature of chemistry is imagination. Chemists have spent centuries trying to imagine what an abstract world, the submicroscopic level could possibly look like and how its different parts combine and dissembles. When taking this perspective, it becomes interesting to explore children's abstract thinking and how they imagine the submicroscopic world, and what sort of circumstances we need to create in a preschool environment to begin this abstract and imaginative journey.

#### **2.2 The development of abstract thinking**

Abstract thinking is something that has been defined by many authors. Some examples of these definitions are; the process when children can represent reality by using a representation, for example, an object, that is, symbolic representations, "thoughts that are not immediately connected to the environment" [25]. Or, detaching from the concrete world while still maintaining a connection to it through representations. The first one to define what a representation is may in fact have been Aristotle who stated that the world and the world as we describe it are not the same things. The way we can view the emergence of children's abstract thinking is through their use of objects and symbolic representations in play, where a stick can become a sword or the symbol X on a map can represent a hidden treasure.

The development of abstract thinking, that is, real-world experiences to abstractions thereof has also been described by a few authors. These descriptions vary in specificity from the general to the more specific and from suggesting causes for development to suggesting how this development occurs. When turning toward causes for the development of abstract thinking research suggest that children have a natural tendency to seek patterns in their real-world experiences to generalize the experiences [26]. Suggestions for how this development occurs date as far back as Piaget (1953). Piaget described the development as including three parts: action, symbolic mediation, and then later abstract thought. More specific description has also been suggested. One of these descriptions separates this development into four different parts: i) motor, ii) symbolic representations, iii) functional dependency, and iv) thought. Functional dependency [27] is here used as a category to describe, children practicing making connections between reality and abstract representations as being one of the first steps toward abstract thinking. A stage in development that becomes specifically interesting to further explore as it affects how the representation is used and gives us clues to how natural connections between the representation and the real world are formed, and what they may look like.

*Emergent Chemistry: Using Visualizations to Develop Abstract Thinking and a Sense of Scale… DOI: http://dx.doi.org/10.5772/intechopen.105216*

#### **2.3 Abstract representations**

Here it is not the children's own representations of experiences that are in focus but instead how children being to understand the representations that we use in our daily lives, our cultural tools. When turning to the line of research that focuses on exploring already made representations, they are viewed in the perspective of the dual representation strategy [28]. Were a representation is seen as both a representation of something and at the same time as an object, in itself. The different types of representations that have been described in the early-years literature include representations such as maps, images, and scale models [29], or interactive representations, such as gaming or educational TV programs [30]. Research on young children's use of representations shows that the connection between reality and representation is difficult for young children. Something that has been shown through experiments where children have been presented with scale models of a room, a toy was hidden in the room as well as in the scale model, and the quest given to the child was to find the hidden toy in the real room. For children as young as two years old, the success rate of this quest increased when the idea of a shrinking machine was introduced. The shrinking machine gave the children the impression that the scale model was in fact the real room that had just become smaller [29]. The same result was described in other studies where the scale model was replaced by a video of someone hiding a toy. The rate in success of finding the toy was higher when a window frame was placed around the image of the video giving the children the impression that they were looking into the room itself. It has also been shown that children's use of representations improves if the child does not get to play with the object itself before it is used as a representation. A result suggests that the child's familiarity with the object causes them to place their focus on objects instead of the object as a representation [28]. This result suggests that "removing the duality" makes representations easier for young children to understand.

Research result deriving from young children's use of interactive games and activities adds additional levels of complexity to the use of representations. The more sophisticated interactions become less useful in terms of content if the actual interactive part is difficult to handle as the child's focus is then placed on the object of interaction instead of the intended outcome. Similar results have been seen in for example laboratory work for much older learners where unfamiliarity with equipment changes the intended focus of the laboratory from the intended phenomena to the material itself [31]. When the representation comes in the form of a photo, then young children seem to notice that there is a difference between photographs and real objects [32] but they do not seem to understand this difference something that can be seen when children try to remove items from a photo [33]. When maps are used or scale models as above scale become an additional problem. Young children have difficulties seeing the lines on a map as roads since the lines are much too small to fit a car [34].

Research results also show that it is important for a continuous transformation between the concrete and the abstract, to facilitate functional dependency and abstract thought [35]. It is in fact not until the last decades that technology has made it possible to provide more realistic representations than simple images and molecular models, on a macroscopic scale. Some researchers see visualizations as being powerful since they have the ability to make us think in visual terms instead of in abstract terms, [33] thereby providing us with a kind of immediate reproductions to better support imagination.

So what do we know about children's descriptions of the abstract world? Research also tells us that children use metaphors, analogies, and similes (it looks like… something that they recognize) to describe unknowns [36]. These expressions can in themselves provide us with information to better support our youngest learners [37] as they are highlighting specific aspects, show us interpretations and become means for sharing feelings.

#### *2.3.1 Project design*

The results presented here are a summary of a longitudinal project, an intervention, designed to explore the development of children's abstract thinking. The intervention was designed as, an educational experiment where activities were designed and analyzed before the next activity was designed to ensure that it was the children's own interest that was the driving force behind the activities. These precautions were made to ensure the children's own interests were in focus and to avoid schoolification. The development of the concept of small was chosen although several other topics could just as easily have been applied. The reasons for this choice were emergence. Exploring children's emergent science and here, especially children's emergent chemistry means that no evaluation of the scientific correctness of the content was made. Studying emergence then means simply exploring children's own versions of scientific content. Using the development of the concept of small as an entrance also meant that we studied actual emergence since none of the children had previous experiences of levels below the macroscopic lived world, something that was also established in the first set of activities. Means that our findings would be a result of our activities and results would show the first-time experiences of the submicroscopic world. Also transfer between different contexts could also be studied as the interventions initially were the only source of information.

To provide the children with a backdrop for the science activities the activities were framed within a story. The story of the king and his royal family was derived from visits to the preschool and by observing the children's free play. The story began with the king's birthday where he was given a magnifying glass and began to explore the world around him [21].

#### **3. Results**

After the first activity, it was clear that none of the children had the experience of small things on a level below the macroscopic level. The smallest thing that they could imagine was on the level of baby bugs. When the interventions moved along it was clear that the story of the king quickly became superfluous. When the researcher entered the preschool, all children attending ran up to her and asked what are we doing today? Suggesting that the items and the researcher were enough to catch the children's attention [21]. This result may not seem significant, but it may also help preschool teachers to make science activities more easily accessible since the results support the idea of the design of play-based activities with an intended content. As the intervention proceeded the magnifying glasses were replaced by microscopes, but the children's descriptions remain at the level of similes. The size difference in children's descriptions moved from baby bugs to pea flour (which was the word they used to describe the content of peas). Mortared sugar was described as ice blocks and the connections made were toward lived experiences, such as brushing teeth and why

#### *Emergent Chemistry: Using Visualizations to Develop Abstract Thinking and a Sense of Scale… DOI: http://dx.doi.org/10.5772/intechopen.105216*

sugar is harmful to your teeth. All interventions so far did not bring about imagining a world that could not be seen. The choice to introduce zooming-in videos was made to provide as real experiences of the atomic-molecular level as possible. Zooming-in videos of a range of different everyday items were introduced to the children. An experience that was appreciated by all the children. When the molecular level was reached, similes were used once again by the children "look at the meatballs." After a few zooming-in videos that ended in meatballs, the children realized that the meatballs were everywhere, in everything and that it may not be meatballs at all but instead something else. At this point, the researcher labeled them as molecules and atoms [22]. These results support the perspective of the dual representations. The animations were as realistic as possible something that may have assisted in the development of this particular form of abstract thinking as results suggest that removing the duality of the representation makes them more easily understood by children [28]. When using the perspective of functional dependency to look at the actual connection that the children made between the representation and the real world. The connections were based on similes and the change did not begin until the children had been exposed to several animations. A result that also supports the idea of continuous transformations between the lived world and the representation is required [35]. There is no doubt that animations provided these children with new experiences of a world that normally is beyond sensory experiences. New experiences that may be the base for future imagination and creativity. This is a very important result for educational settings as images of atoms and molecules naturally are macrolevel representations and moving animations this may so far be the only way to provide relatively realistic experiences of the submicroscopic level something that may support learners at all educational levels. If the concept of perezhivane and its effect on lifelong learning is valid, then experiences like this may have long-lasting impacts on a personalized learning process.

#### **4. Conclusions**

The results show that developing abstract thinking from motor interaction of the macroscopic world with magnifying glasses and microscopes did not induce children's imagination of what came next. Providing visual experiences of the submicroscopic level did show the next stage in concept formation: the realization that it looks like something … but it is not… brought gave rise to a concept by the researcher labeling the experiences. This concept together with "they are every was in everything" suggests that the children to some extent understood the scale of the atoms and molecules, at least to the extent that they are small enough to be inside things. A result that supports Vygotsky's idea that experience is a base for imagination. This is an example of functional dependency in the way the concept has been interpreted in this context. The fact that the children saw several data animations of different items supported their connections between the animation and reality since it was not until several animations had been seen that the separation between their experiences and the animation began to show. Suggesting that connections need to be made on several occasions.

#### **Acknowledgements**

This research was funded by the Crafoord Foundation (Refnr: 20200712).

### **Conflict of interest**

There are no conflicts of interest.

### **Author details**

Karina Adbo1,2

1 Faculty of Education and Society (LS), Department of Natural Science, Mathematics and Society (NMS), Malmö University, Sweden

2 Faculty of Health, and Life Science (FHL), Department of Biology and Environmental Science (BOM), Linnaeus University, Sweden

\*Address all correspondence to: karina.adbo@lnu.se

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Emergent Chemistry: Using Visualizations to Develop Abstract Thinking and a Sense of Scale… DOI: http://dx.doi.org/10.5772/intechopen.105216*

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#### **Chapter 2**

## Integrating Analogy into Scientific Modeling for Students' Active Learning in Chemistry Education

*Song Xue and Danner Sun*

#### **Abstract**

Although active learning places more responsibility and emphasizes the learners themselves, as compared to the traditional learning approach, teaching guidance is still essential in the active learning environment. In most chemistry classes, students are provided with limited opportunities to explore the atomic structures at the sub-microscopic level or participate in learning activities. To address these issues, an effective teaching approach enabling students' active learning called "Analogy integrated Scientific Modeling" (AiSM) has been proposed with the aim of facilitating students' learning of abstract chemistry concepts. This chapter introduces how AiSM approach is integrated into chemistry class and promotes students to understand the atomic structure. The chapter starts with a theoretical background, which consists of analogy in teaching the atomic structure, the scientific modeling, and the AiSM approach. Subsequently, two lesson exemplars illustrate how the analogy and the scientific modeling can get integrated into a chemistry class. Last, the initial results of a pilot study are discussed to demonstrate the effects and benefits of the AiSM approach on promoting students' active learning in chemistry classes.

**Keywords:** active learning, analogy, educational paradigm, scientific modeling

#### **1. Introduction**

One of the primary purposes of science education is to improve students' conceptual understanding of science. Eliminating student misconceptions is always the priority of science educators and researchers. The scientifically inaccurate understanding of conceptions referring to misconceptions has become an obstacle to students' learning process [1, 2]. Moreover, it is widely perceived that the students taught by a typical, teacher-centered approach are unable to integrate their knowledge and thinking critically or creatively, resulting in their lower learning achievement and common misconceptions [3–5]. Consequently, the traditional approach, according to which the teacher acts as an information provider and the students as passive recipients, appears to be antiquated. Recently, the active learning approaches, which enable students to participate in their classes actively, have begun to gain traction to assist students in becoming meaningful learners [3, 6].

Being involved in the active learning process, the students can effectively develop their new understanding and make connections with the previously gained knowledge, which will assist them in gaining further understanding of certain issues [7, 8]. This is accomplished through active learning, which requires higher-order thinking skills and engages students in the learning activities [9]. Therefore, it is meaningful to facilitate students to develop an accurate and comprehensive understanding of the scientific concepts through active learning.

Some chemical concepts, in particular, are regarded as abstract and difficult to comprehend, which may lead to misunderstandings. One of these chemistry concepts is the "atomic structure." Many studies have shown that students found atomic concepts challenging to grasp, especially for the concept of atomic structure, and tend to develop alternative ideas instead [10]. It is mainly because the atom is a type of matter in the sub-microscopic world whose structure cannot be observed directly. Furthermore, students have few opportunities to investigate the atomic structures at the sub-microscopic level, and also the strategies which could assist them in learning the properties of atomic models are few and far between [10, 11]. Thus, building an active learning environment will be valuable for the students to explore these abstract concepts and enhance their scientific understanding by eliminating common misconceptions.

Scientific models and modeling are significantly and pedagogically beneficial in chemistry learning [12–15]. Scientific models are usually used to represent a scientific phenomenon employing different forms, such as analogies [16]. Scientific modeling is crucial in scientific inquiry for generating and evaluating hypotheses and describing natural phenomena. It also plays a crucial role in developing new scientific knowledge [17]. Scientific modeling has been largely neglected in the traditional school curriculum, being reduced to either an explanatory tool for strengthening lectures [18] or simulations, in which students have very little authority for developing their models [4]. Teaching with analogies is an approach, in which teachers help students conduct analogies for understanding the specific content by recognizing the analogy's coherence with the target concept. However, according to some observations, an analogy was sometimes inappropriately designed and implemented, thus failing to generate the intended outcome [19, 20]. Due to these reasons, a new pedagogical and instructional design framework, namely the "Analogy integrated Scientific Modeling" (AiSM), has been promoted as a more comprehensive way of integrating analogies into the scientific modeling process in chemistry classes.

This chapter focuses on introducing and discussing the theoretical background of the AiSM approach. And a pilot study is presented to verify the effects of AiSM on students' conceptual understanding of the atomic structure in chemistry classes. The study will inform the design and development of effective teaching approaches for promoting active learning in science education.

#### **2. Theoretical framework**

#### **2.1 Using analogies to teach atomic structure**

According to the outlook on the chemistry education adopted by Sarantopoulos and Tsaparlis [21], an analogy could be defined as a system of relations between the parts of two particular domains, namely the analog and the target (i.e., the familiar

#### *Integrating Analogy into Scientific Modeling for Students' Active Learning in Chemistry Education DOI: http://dx.doi.org/10.5772/intechopen.105454*

domain being the analog, and the unfamiliar one being the target). In chemistry education, using analogies may help students understand the new concepts by building upon their familiar experiences or prior knowledge. For example, one commonly used analogy is "the atom being like a tiny solar system." In this case, the spatial and dynamic features of the sun and the surrounding planets are analogous to those of the atomic nucleus and the electrons. To unfold this relationship, an atom is composed of a nucleus and electrons, which travel outside the nucleus rapidly. This is identified as a target concept in the learning process. In the analogy of the solar system, the sun is being compared to the nucleus, while the electrons are being compared to the planets. This analogy helps students understand the relationship between the atomic components and their respective locations.

Most analogies relate to the empirical phenomena, among others, the key functional relations involving causes and their consequences [22]. The analogy presents a new explanation for the occurrence of various phenomena by transferring knowledge about the causal relations, which then enables the further transfer of knowledge when applied to the new situations [22, 23]. According to the principles of constructivism, students could learn new, complex, or even abstract concepts by incorporating the new information into the pre-existing knowledge by use of analogies. This should help students to develop a conceptual understanding and, in turn, reduce their cognitive load. Since one of the challenges of learning about the atomic structure is that the atom is sub-microscopic matter and cannot be observed directly, teachers and educators usually develop analogies to illustrate and visualize the key features of atomic concepts using the reference to the real-world scenarios [23–25]. This teaching method entails drawing on students' prior knowledge and experience to facilitate their understanding. In the secondary education curriculum, the atomic concepts usually include the atomic and quantum theories, the atomic structure, the periodic table, and the chemical bond (in the curriculum of some countries). **Table 1** presents some examples of analogies used in the teaching practices.

However, a closer look at these analogies reveals that they may have their limitations as teaching tools. Each analogy has certain elements that the teacher intends for the students to focus on and use to understand the new concepts. However, some elements of the analogy are inappropriate for the target concepts. For example, everything external to the solar system is driven by its central star, whereas everything external to the atom is driven by the outermost orbitals and their electrons. If the students do not distinguish the difference between the analogy and the target concepts, they may develop new misunderstandings regarding these concepts. Therefore, it is suggested students use only appropriate portions of the analogy during the teaching process [21]. It is also necessary to highlight the relationship between the target and the analogy with a proper scientific explanation [22].

Recently, Gray and Holyoak summarized five principles of an analogical approach based on the research in cognitive psychology and cognitive neuroscience [22]. These five principles include [1] capitalizing on the prior knowledge, [2] highlighting the shared structure, [3] explaining the connections between the semantic information and the mathematical operations, [4] considering the cognitive load, and [5] encouraging the generation of inferences. This approach was proposed to maximize the benefits and minimize the problems encountered in the analogy instruction [26]. Moreover, it provided a framework for better guiding the application of an analogy in the classroom settings, to foster greater conceptual understanding and transfer.


#### **Table 1.**

*Using the analogies to teach the unit of atomic structure.*

#### **2.2 Mental models and GEM scientific Modeling**

#### *2.2.1 Mental models*

According to Johnson-Laird [27], learners construct internal cognitive representations, or mental models, during their interaction with the environment, artifacts, technology, or communities. A mental model can be defined as a form of organizing

#### *Integrating Analogy into Scientific Modeling for Students' Active Learning in Chemistry Education DOI: http://dx.doi.org/10.5772/intechopen.105454*

knowledge employing representing objects, states of affairs, sequences of events, ways of the world, or the social and psychological actions of daily life [28]. There is growing recognition that mental models are the reasoning tools for both scientists and students in science [29]. The mental models help us make predictions and develop causal explanations between variables [30]. For instance, when students were asked which model was more appropriate for describing the structure of an atom, they chose the Rutherford model and the Bohr model, which could both be considered mental models. This example shows that a mental model is not always a scientific one - it can be analogical, partial, or fragmentary [31], but it can be changed and revised through the learning process.

#### *2.2.2 GEM scientific models*

One of the objectives of science education is to make students think about the natural world as a scientist [32]. Therefore, a teaching approach based on the scientific modeling was proposed and developed with the aim of urging the students to build, contest, and ultimately change their knowledge of how the world works, as scientific modeling is one of the main approaches adopted by scientists to investigate and explain the natural world. This approach requires selecting and identifying the relevant aspects of a given, real-world situation and subsequently developing different types of models for different goals to better understand, manipulate, or predict a particular phenomenon [33, 34].

Several scientific modeling-based strategies reported in the literature have involved students in the interactive modeling processes [35–39]. This chapter adopted the GEM (Generate, Evaluate, and Modify) approach [28], designed to develop students' scientific understanding of chemistry through scientific modeling. At the beginning of the GEM modeling process, with due consideration given to the students' pre-existing knowledge, the teacher provides background knowledge or a set of information and asks students to *generate* the relations between variables in the mentioned context. In the *evaluation* process, the teacher provides some additional evidence or new information to the students, who would then evaluate and explain the reasons for the relations, which developed in the *generation* process. Finally, based on the evaluation results, the teacher encourages the students to *modify* the relations between the variables or solve the new cases, if necessary. The GEM approach is applicable to numerous empirical studies [28, 40], in which students' mental models are represented by some external forms, such as drawings, concept maps, computer simulations, or animations. Furthermore, these studies suggest that students' prolonged participation in the GEM cycle could help them achieve the goals of the critical learning process in chemistry.

#### **2.3 AiSM approach for promoting active learning**

Some recent work has focused on the role of analogies in building and revising mental models [28, 41, 42]. AiSM is a teaching approach for promoting students' active learning through the integration of analogy into GEM scientific modeling. In AiSM pedagogical approach, the analogies could result in the mental models being manipulated and transformed as a part of the GEM process. The analogy, treated as an external representation of a mental model, can provide explanatory power for making sense of the familiar (analogy) and the unfamiliar (transfer target).

The analogy can be represented in various forms, such as texts, pictures, videos, verbal examples, and computer simulations [28].

In this section, the interaction between the analogies, the mental models, and the modeling is further clarified. **Figure 1** shows the mechanism of AiSMT. In the process of developing mental models, students have to evaluate and integrate the new information into their existing metacognitive framework. When the analogies are used at the beginning of the teaching process, they may help students *generate* an imperfect preliminary model (M1), which is later "evaluated (M2) and modified (M3)". An analogy may also be used at critical points in the subsequent modeling process (GEM) to provide the missing aspects for the target concepts or the scientific phenomena.

Regarding the scientific modeling process, the literature has shown that scientific modeling can be enacted through different processes [42]. These include other modeling processes, such as model construction, model use, model revision, model comparison, and model validation analysis [43], all of which are interactive and do not require the presence of all the other processes in each cycle/iteration [42]. However, the empirical studies reported that novice modelers encountered numerous challenges when taking over a modeling task or other related activities [42, 44]. **Figure 1** presents a relatively simple way of including the three modeling processes: a linear teaching sequence following the GEM or an interactive or cyclical process.

Selecting a pattern for the scientific modeling is determined by the complexity of the modeling task or the target concept. It is important to stress that the modeling, including at least three of these processes, could ensure a meaningful construction of the students' mental models [44]. Thus, active learning may happen when using an internal mental model of a construction process stands in contrast to the direct transmission process. Active learning requires learning by mentally developing or processing information, using construction and criticism rather than listening, and is reflected in an integrated knowledge schema. Teachers could judge whether the students' mental models are scientific through an external representation, such as the analogy, and thus help students construct knowledge by providing them with new information. The theoretical framework proposed in this study is hopefully applicable to the lower educational level or less complex modeling tasks, as well as the novice teachers without modeling experience.

#### **3. The application of AiSMT and AT into chemistry learning and teaching: Lesson design and Pilot study**

#### **3.1 Lesson design**

Based on the above literature review, this section presents two lesson exemplars centered around analogy-based teaching (AT) and analogy-integrated scientific modeling teaching (AiSMT), both based on a case of the atomic structure taught in the local Taiwan chemistry curriculum. The learning objectives are (1) the scientific knowledge (learning about the history of atom discovery and understanding the concepts of the atomic structure, the spectrum, the atomic orbital, the electronic configuration, the periodic table, and the chemical bond formation following the lessons) and (2) the scientific abilities (for 10th graders, students should have the ability to "build models based upon the scientific problems or by means of a group discussion and can use, for instance, "analogous or abstract" representations to describe a systematic scientific phenomenon, and then understand the limitations of the model (p. 32)" [45]. Both lesson exemplars had some common features, e.g., both teaching exemplars were based on constructivism, they employed analogical examples (see **Table 1**), and taught the same scientific concepts supported by digital technologies (e.g., PhET).

**Figures 2** and **3** demonstrate the process of the AT and AiSMT respectively. The main differences between these two exemplars are (1) AiSMT engaging students in the scientific modeling process (GEM) of atomic structure by using the analogies with promoting multiple interactions between the students and the teacher. Therefore, the development of students' mental models occurred during the process of collaborative construction between the students and the teachers. Regarding the AT, it is more straightforward, as it involved students compiling the information provided by the teacher to select the proper analogies, which they later used to construct the mental models. (2) The other difference is that the AT asked students to consider the relationship between the analogies and one single target concept (atomic structure), whereas the AiSMT encouraged students not just to think about a single target concept, but also to make a connection and a comparison between the other related target concepts (energy level, electron leap). (3) After the actual teaching for the whole unit of the

**Figure 2.** *The flow of the analogy-based teaching (AT).*

#### **Figure 3.**

*The flow of the analogy-integrated, scientific modeling teaching (AiSMT).*

atomic structure was implemented, it has been found that the AT took less teaching time than the AiSMT (7.5 h vs. 8.25 h).

#### **3.2 Pilot study**

A quasi-experimental design was implemented to compare the effects of the two instructional approaches, mainly the AT and the AiSMT, on the high school students regarding their understanding of the atomic structure, as presented in the Taiwanese chemistry curriculum. Two groups of participants, the AT group (n = 69), and the AiSMT group, (n = 68), with an average age of 15.6, were selected and engaged in a three-week teaching intervention. The two teachers, a female teacher (T1) with eight years of teaching experience in chemistry taught the AT group, and a male teacher (T2) with six years of teaching experience in chemistry taught the AiSMT group. Both teachers did not engage in any specific modeling-based teaching training. However, they had prior knowledge and experience of modeling-based teaching acquired during their teaching career.

To examine the effects of two instructional approaches on students' conceptual understanding and compare the learning differences between the two instructional approaches, a pre, post, and delayed post-test design has been conducted. Meanwhile, to collect the data about teachers' respective perspectives on the differently designed teaching approaches as well as their views on the students' learning techniques, the teachers both observed each other's classes during the experiment, and they were invited to participate in the semi-structured interviews after teaching (30 minutes per teacher).

A detailed description of the research design and the results can be found in the work of Xue et al. in 2022 [46]. In this book chapter, the key findings were briefly presented. The quantitative analysis of the pre-and post-test results has shown that, generally, both instructional approaches could significantly facilitate students' content understanding in the field of atomic structure with a large effect size. This finding revealed the value of both instructional approaches in facilitating students' active

#### *Integrating Analogy into Scientific Modeling for Students' Active Learning in Chemistry Education DOI: http://dx.doi.org/10.5772/intechopen.105454*

learning. By comparing the results at the post-test stage, it has been found that there was no significant difference in the content understanding between the two groups. However, the delayed post-test results provided evidence for the significantly lower effectiveness of the integrated modeling in terms of facilitating the content knowledge retention, as compared with the effect of the AT. The MAI group remembered and recalled the atomic concepts thoroughly and did better in the delayed post-test. This finding may result in the increased value of incorporating scientific modeling, which could further assist in the retention of scientific knowledge.

Both teachers agreed that the two instructional approaches were applicable and effective. They also shared their thoughts on each other's approaches. T1, commenting on the AT, concluded that it was important to emphasize the analogy's correspondence with the target concept in teaching, while not neglecting the analogy's limits. She highlighted her positive attitude towards this innovative teaching method when she observed the AiSMT class. She found most students to be engaged in the conversation as well as the activities between the teachers and their peers. This has encouraged them to promote cognitive processing and activated their learning interest as a result. However, she also expressed some concerns. Firstly, she was concerned about the timing of the class, worrying that there would not always be enough teaching time to incorporate the modeling activities. Secondly, due to the fact that the modelingbased teaching entails a lot of questions and interactions, it may be stressful for some students and hard for some low-achievers, who have never engaged with any modeling or scientific inquiry. Thirdly, the application of modeling-based teaching requires a high level of competence, such as carefully developed instructional design, and providing relevant supporting resources, such as computer simulations and background information, which may be a challenging task for some novel teachers. T2, who implemented the modeling-based teaching, said that adapting the GEM approach is beneficial for the students who need to constantly reflect upon, analyze, and revise their mental models. In terms of the teacher's role, he said, the following: "I need to have a deep and comprehensive understanding of the scientific concepts which I taught. In addition, I should keep in mind related concepts students may have some misunderstandings about, in order to eliminate them by means of the modeling process". What was worth noting is his remark on the modeling activities, which apparently did not take up the entire class time but were rather directed at the selection of the suitable teaching elements or concepts. He ended up by summarizing the three criteria for selecting concepts to be used in the modeling-based teaching: target concepts, which have constituent aspects or variables; the possibility of integrating them with the students' life experiences; and the history of the used development and/or evolution being a scientific one.

#### **4. Discussion, conclusions, and implications**

This study presented a literature review that discussed the related theory and principle of using analogy and scientific modeling in teaching the atomic structure. A new pedagogical approach, AiSM, has been promoted with the aim of integrating the analogy into the scientific modeling process (GEM) in chemistry education. Two lesson exemplars were designed to show the uses of the AT and the AiSM in active learning. The empirical study has shown that the two teaching approaches proved to be effective in promoting students' content understanding. Both teaching approaches had an equal effect in enhancing students' understanding. These results are aligned

with the studies, which reported the positive effects of analogy and scientific modeling and the equally positive effect on students' learning improvement [21, 36, 47]. However, the result, which has not been previously described is the students who engaged in modeling outperformed the non-modeling group [40, 48].

The new finding was found that modeling-based teaching could maintain longer memory and a better understanding of the content being taught among the students. Hofstadter's interpretation of the cognitive processes explains the reason behind the effectiveness of enhancing retention of content understanding by the use of modeling [49]. According to Hofstadter, cognition is mediated by continual processes until a long-term memory node is accessed. Once this is done, cognition gets transferred to a short-memory node where it is unpacked to some degree. This allows for the new structures to be perceived, and the ensuing high-level perceptual act activates the further nodes, which, in turn, are being accessed, transferred, unpacked, etc. (p. 517). Modeling-based instruction is very much in line with the cognitive processes described above. Following the GEM approach implemented in this study, students were leveraged with prior knowledge and experience to develop models of the target concept. They were subsequently evaluated and revised the initial model in accordance with the newly provided information. The modified model was applied to the new contexts and situations, so it could be further improved upon. These constant processes of reflection and improvement could mediate the development of the mental models, which in turn could enhance their fitness and correspondence to the outside world [50].

From the data collected in the interview, the two chemistry teachers who participated in this study recognized the value of analogy and scientific modeling in the development of students' mental models. The dilemmas and challenges of implementing the modeling-based teaching, such as being difficult to control and timedemanding, were found to be consistent with the earlier research [47]. However, the teachers' reflections on the integrated modeling are another worthwhile aspect of this study. The previous research presented the complexity of modeling-based teaching and the strategies for promoting competence using it [28, 51]. This study resulted in the implications of the integrated modeling being learned from the evidence of teachers' empirical application. It could be concluded by listing the four positive aspects of applying modeling-based teaching in the formal school curriculum. These would include selecting the targeted scientific concepts or phenomena along with variables, elements, or factors, combing them with learners' life experiences, and finally choosing concepts with the history of scientific evolution and reasoning.

This study adds to the ongoing conversation on the use of analogy and scientific modeling in chemistry education and contributes empirical evidence in order to justify the use of analogies and modeling as a means of improving the content understanding among students. It is advised that integrating the analogies into the scientific modeling can be achieved in the future classes, provided that the combination of the more science-specific content would be used. Moreover, the appropriate and cautious design for use of the analogy and modeling should be considered by the teachers. Future research could focus on the teachers' professional development of lesson design for analogy-based scientific modeling in science education.

*Integrating Analogy into Scientific Modeling for Students' Active Learning in Chemistry Education DOI: http://dx.doi.org/10.5772/intechopen.105454*

#### **Author details**

Song Xue1 \* and Danner Sun2

1 School of Education and Social Work, University of Dundee, Dundee, United Kingdom

2 Department of Mathematics and Information Technology, The Education University of Hong Kong, Hong Kong, China

\*Address all correspondence to: 2393774@dundee.ac.uk

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 3**

## About the Notion of Inverse Problem in STEM Education

*Victor Martinez-Luaces, José Antonio Fernández-Plaza and Luis Rico*

#### **Abstract**

Inverse problems play an important role in STEM disciplines; although this concept is not well-defined in STEM education. For instance, Mason considers inversion as 'undoing', whereas Keller observes that if two problems are inverses of one another, then one of them has been studied extensively, while the other is newer and the former is called 'direct', while the latter is called 'inverse'. Groetsch observes that if *y* is the effect of a given cause *x* when a mathematical model *K* is posited (*Kx* ¼ *y*), then, two inverse problems arise: causation (given *K* and *y*, determine *x*) and model identification or specification (given *x* and *y*, determine *K*). This last view is an adaptation of the IPO-model, taught in Computer Science. During the last 5 years, we designed and put in practice and experience based-on inverse problems and their utilization in teachers training courses. This area is strongly connected with active learning, since as Kaur observed, an effective mathematics instruction begins when the instructors take the role of designers with the aim of facilitate active learning activities. In this chapter, we reflect on these experiences to construct a wider theoretical framework for inverse problems in STEM education.

**Keywords:** inverse problems, STEM education, mathematical models, didactic analysis, problem posing, task enrichment, active learning

#### **1. Introduction**

There exists a wide range of learning activities, commonly included under the term active learning (AL). Among them, it should be mentioned the computer-assisted learning, project work, role play exercises, small group discussion, individualized work schemes and collaborative problem-solving.

As Lugosi and Uribe [1] observe, there is no definition of AL, used and accepted by everyone, although it is regarded in general as 'classroom practices that engage students in activities that promote higher-order thinking' [2], which includes the analysis, synthesis and evaluation of the information presented rather than receiving it passively [3].

One of the most used definitions is the one proposed by Freeman et al. [4]: 'Active learning engages students in the process of learning through activities and discussion

in class, as opposed to passively listening to an expert. It emphasizes higher-order thinking and often involves group-work.'

A similar vision is given by Good & Brophy [5], who argued that 'active learning involves providing pupils with an opportunity in which they raise their own questions and use teachers and other resources to pursue self-defined goal'.

In the same way, Kyriacou [6] states that AL 'can be described as the use of learning activities where pupils are given a marked degree of ownership and control over the learning activities used, where the learning experience is open-ended rather than tightly predetermined, and where the pupil is able to actively participate and shape the learning experience'.

Independently of the definition of the concept, in the last decades an increased importance of the role of AL pedagogies was observed, which includes several reported benefits in teaching STEM disciplines [4, 7]. For instance, Freeman et al. [4] conducted a meta-analysis of 225 studies which reports the positive impact of AL pedagogies on student learning in all the STEM disciplines.

On the other hand, as Wieman [8] stated, mathematics has a different behavior, since it is a very traditional discipline in terms of its teaching and, consequently, it shows more resistance than other STEM areas in adopting teaching methods based on research results.

This is a very important issue, since as Kaur [9] mentioned, an effective mathematics instruction begins when the instructors take the role of designers, and they facilitate the AL activities. Then, in order to change teaching practices, it is necessary to start by working with prospective teachers in their formation courses.

For this reason, during the last 5 years, we designed and put in practice an experience based on inverse problems in teachers training courses.

It is important to mention that despite the essential role of inverse problems in many different disciplines, they were almost forgotten [10] and unfortunately, in the traditional Mathematics Education the situation was the same as mentioned in different works [11–13].

One of the consequences of this fact is that an elaborated theoretical framework for this theme was no developed. In an attempt, by Groetsch [11, 12], the IPO model—which is commonly taught in in Computer Science courses and textbooks [14]—was adapted for this purpose. In this article we propose to construct a wider theoretical framework for inverse problems and their utilization in Mathematics Education and so, the adaptations of the IPO model can be considered just as a starting point for this purpose.

This paper reflects on some examples previously described [14–16] and analyses data obtained in a doctoral thesis fieldwork [17], which concern several dimensions of the Didactic Analysis. Nevertheless, in this first attempt for building a theoretical framework for inverse problems and their role in Mathematics Education, we focus only on the cultural/conceptual dimension of Didactic Analysis [18]. It is important to remark that the chapter objectives, methodology and results interpretation are based on this theoretical framework.

Our main purpose is to consider this chapter as a starting point for future deeper developments about inverse problems in Mathematics Education.

#### **2. Inverse problems in mathematics education**

In a recent paper by Mason [19], the following excerpts can be read: 'the pervasive mathematical theme of inverse also known as doing and undoing … ', ' … can be used to get an answer (a doing), it is useful to consider the undoing … ' and 'The pedagogic point here is to emphasize the power of formulating and considering "undoing" or inverse problems'. These excerpts show that—at least for some researchers such as Mason—inverse problems are more or less a synonym of 'undoing'.

As Kunze and collaborators mentioned, 'this distinction is not well-defined' [20] but later, in the same paper they state that 'however, in general, a direct problem involves the identification of effects from causes' in full agreement with Mason's viewpoint.

Long time ago, Keller [21] proposed a different point of view when he says that 'We call two problems inverses of one another if the formulation of each involves all or part of the solution of the other'. After that, he argues that due to historical reasons, ' … one of the two problems has been studied extensively … while the other is newer … ' and then, he states that 'In such cases, the former is called direct problem, while the latter is called the inverse problem'. So, in this original point of view, a problem is classified as direct or inverse depending on its own history, more than the inputs and outputs of the proposal.

Outside Mathematics Education, it is possible to find a wider conceptualization, for instance, Groetsch [22] says that 'The type of direct problems we have in mind is that of determining the effect *y* of a given cause *x* when a definite mathematical model *K* is posited: *Kx* ¼ *y*' and then, he adds ' … two inverse problems may be immediately posed. These are the inverse problems of causation (given *K* and *y*, determine *x*) and model identification (given *x* and *y*, determine *K*)'. Then, it is possible to outline the resolution of a conventionally so-called direct problem, as it is showed in **Figure 1**.

In a previous paper [14], it was observed that there is a strong connection with the IPO model (input-process-output), usually taught in computing and information technology courses. The IPO cycle can be represented as in **Figure 2**.

Now, as Groetsch mentioned, in the causation problems the model and the effect are well known, and the question is about the cause, so this situation can be schematized as can be observed in **Figure 3**.

In model identification problems, both cause and effect are known, and the main question consists in determining the model that gives the expected result. This situation is schematized in **Figure 4**.

It is important to remark that in other works [11, 12], the model identification inverse problems were called 'specification problems'.

#### **Figure 1.**

*Scheme for an initial direct problem, following Groetsch ideas.*

**Figure 2.** *IPO model.*

#### **Figure 3.**

*Scheme for causation inverse problems.*

**Figure 4.** *Scheme for model identification inverse problems.*

In a previous paper [13], three different types of inverse problems were described: causation, specification and double-inverse problems. The first two categories correspond to those described by Groetsch, whereas the last one corresponds to those proposals that can be considered at the same time, causation and specification inverse problems. As it was mentioned, the last kind of problems were called 'double inverse problems' by Martinez-Luaces [13].

An interesting example appears in a problems list which was released by PISA a few years ago [23]. One of them is the so-called 'Apartment Purchase' problem which proposes to measure the room sizes of an apartment, as a first approach to calculate the total floor area. Then, the statement adds the following: 'However, there is a more efficient method to estimate the total floor area where you only need to measure four lengths. Mark on the plan above the four lengths that are needed to estimate the total floor area of the apartment'. Since the problem statement asks to measure four lengths in order to obtain the same result of the first approach, it should be considered as a causation inverse problem. On the other hand, in order to choose the correct four lengths, the student must know what to do with those lengths, and so, implicitly it should be considered also as an inverse specification problem.

In the previously mentioned doctoral thesis research [17], it was observed that in several proposals, the student needs to solve a direct problem as a first step, and in a second step, use the obtained result as an input for solving an inverse problem. An interesting example is described in the next paragraphs.

In the fieldwork of this doctoral thesis, the participants—prospective teachers of Mathematics at Secondary School—were asked to reformulate the problem proposed in **Figure 5**.

A prospective teacher proposed an inverse problem (see **Figure 6**), where the field geometry and the stake position are the same of the original statement.

In this proposal, the rope length *<sup>R</sup>* <sup>¼</sup> *<sup>L</sup>* <sup>3</sup> is given, and the statement mentions that the sheep eats all the grass in the accessible area. The problem has several questions, the first one being about *R*<sup>0</sup> , the rope length that allows the sheep grazing the same amount of grass. After that, other questions ask for the rope lengths corresponding to the third and for the fourth day, once again, in order to eat the same amount of grass of the first day. Finally, the author asks on which day the sheep will not have the same amount of grass to eat?

**Figure 5.** *Direct problem: Accessible area for a sheep grazing in a field.*

**Figure 6.** *Sequential inverse problem.*

It is easy to observe that this proposal is as a 'mixed inverse problem' with a structure direct-inverse, since firstly it is necessary to solve a direct problem (for the first day) and then, the order of the variables involved is inverted for the following days.

Also, it is not difficult to pose an inverse problem with the opposite structure, i.e. inverse-direct, for instance: For the rope length such that the sheep can eat 50% of the grass, which is the perimeter of the accessible area?

Then, we can describe two subgroups of mixed inverse problems: direct-inverse and inverse-direct.

So, considering the type of inversion, we can describe four groups: causation problems, specification problems, double (both causation and specification) and mixed (direct-inverse or inverse-direct).

It is important to comment that posing inverse statements from an original statement is an important way of enriching mathematical problems. In fact, it is not the only possibility, since modeling can also be considered for task enrichment and of course, both can be combined in the so-called inverse modeling problems, firstly described in a book chapter [24] and a journal paper [25].

Despite the importance of these strategies, inverse problems were almost ignored. For instance, Bunge [10] remarks that 'inverse problems are so difficult and have been so discriminated that the first international congress on the subject was held as late as year 2002' and 'the treaties on the subject can be counted on the fingers of one hand'. In a certain way, this can be considered an unexpected result, since—according to Groetsch [11, 12]—the first mathematical inverse problem was posed almost five centuries ago (a ballistics inverse problem studied by Tartaglia in 1537, which consisted of determining the elevation angle θ for reaching a given shot range *R*).

The same situation happens in mathematics education, where inverse problems have been ignored, at least in traditional courses [11, 13].

Because of these facts, it is not surprising to observe the absence of a theoretical framework adequate to study the inversion of a statement as a task enrichment strategy in mathematics education. In order to construct this theoretical framework, firstly we need to analyze three related topics: problem posing, task enrichment and Didactic Analysis, the last one being the tool we have chosen for developing this frame.

#### **3. Theoretical framework**

As we already mentioned, there are three topics that deserve to be considered to analyze the inversion as a strategy of task enrichment in mathematics education, with'problem posing' being the first one.

#### **3.1 Problem posing**

Problem posing is a traditional research area, where important authors such as Brown and Walter [26, 27], English [28, 29], Kilpatrick [30], Silver and Cai [31] produced several milestone papers. When those authors use the term 'problem posing', they include new problems formulations but also reformulations of given statements. These problem posing activities are developed in different ways, with more or less structured formats [28, 29, 31–33].

An interesting case takes place when students need to pose a different problem as part of the solving process of the given one [34]. This situation already appeared in the works of Polya [35], who proposes this possibility as a possible strategy which consists of establishing variants, such as discarding one or more of the original statement conditions.

It is important to remark that problem reformulation is not always a strategy for problem-solving. In fact, in some works problem invention starts from an experience or a situation previously given [32, 33].

It is also possible to combine the last previous approaches, for instance, the students are asked to solve a certain problem, where one of the conditions was changed, and for this reason, it should be considered as a new problem [32].

Brown and Walter [26, 27] proposed a different approach for obtaining new problems, in a strategy that they called 'What if not?'. Their approach consists of changing the problem conditions or restrictions to create a new statement. This strategy may lead to interesting causation/specification inverse problems.

In the work of Stoyanova [36], problem-posing activities are classified into three groups: free, semi-structured and structured. In free problem posing activities, there are no restrictions. In the case of the semi-structured activities, problem posing is based on a previous experience and/or some quantitative information. Finally, in structured problem posing, the original statement problem is reformulated (or some conditions are changed) in order to obtain a new one. In our research a direct problem was given to the participants, and they were asked to propose an inverse reformulation. As a consequence, our fieldwork constitutes an example of structured situation, if the previous classification is followed [36].

#### **3.2 Task enrichment**

In our research, problem posing is carried out for tasks enrichment as the final goal. It is important to remark that Lester and Cai [37] stated ' … teachers can develop worthwhile mathematical tasks by simply modifying problems from the textbooks'.

Other authors, such as Santos and Barmby [38], observed that 'the question of what is meant by enrichment has been an ongoing question for researchers'. Moreover, Barbe [39] says 'an aura of vagueness and confusion seems to surround the term' and 40 years after that, Feng [40] concluded that 'no overall consensus has yet been reached on the definition and nature of enrichment'.

Taking into account the previous comments, what is a rich task that consists more of a description than a precise/exact term definition. An example of this approach is given by Grootenboer's works [41], since he makes a description of 'the key aspects of rich mathematical tasks', like the following:


After discussions with several teachers, Clarke and Clarke [42] suggested their own list of characteristics for 'rich assessment tasks'. Some of the selected characteristics are the following:


As it was mentioned, this list is about 'rich assessment tasks', although it can also be used for 'rich tasks' characterization. Obviously, those characterizations are not a definition of rich tasks, but they reasonably describe what can be expected in task enrichment activities.

As it was mentioned, in our research we consider inversion as another tool for task enrichment, so there is a strong connection between Section 2 and sub-Section 3.2 of this work.

Finally, as Bonwell and Eison [2] remarked 'Though the term "active learning" has never been precisely defined in educational literature, some general characteristics are commonly associated with the use of strategies promoting active learning in the classroom' and particularly, they mentioned the following ones:


It is easy to observe a strong connection with the characteristics of rich tasks described by Santos and Barmby [38] and Clarke and Clarke [42], and most of these characteristics are present in our fieldwork, described below, in Section 4.

#### **3.3 Didactic analysis**

This sub-section is devoted to briefly describe the characteristics of the Didactic Analysis. The didactic Analysis considers 'the structure, levels, dimensions, categories, and components to perform and organize the didactic school mathematics content analysis and how it makes possible the design, implementation and evaluation of teaching and learning activities, corresponding to any specific mathematics subject' [18, 43, 44].

'This description gives rise to a cyclical structure, where the information obtained in each analysis will be essential for a new implementation of the didactic analysis' [45].

The system of components, categories and contents for the didactic analysis of the mathematics school curriculum is briefly summarized in **Figure 7**. **Figure 7** is a remake of Table 2 of a previous paper written by one of the co-authors of the chapter [18].

In our research, didactic analysis was the main tool for the design of the fieldwork and also for the prospective teachers'productions analysis. However, due to length constraints, in this chapter, we analyze only the conceptual content analysis of the proposals, so our focus is placed on the meanings analysis (i.e. the second column of **Figure 7**).


#### **Figure 7.**

*Components, categories and contents for the didactic analysis.*

It is important to comment that the meaning concept in this work based on the socalled semantic triangle, which corresponds to an interpretation of Frege's works [46– 48]. Following this frame, the mathematical content analysis can be organized into three categories: the contents'structure, the representation systems and the senses or modes of use.

The conceptual structure takes into account the relations of the concepts and the procedures which are involved in the studied content, paying attention to the mathematical structure that includes them.

Representation systems are related to the different forms of 'representing a mathematical content, which can be expressed through signs, graphics, symbols, rules, relationships, conventions, along with their translations into other concepts and conversions according to different procedures'.

Finally, the sense considers the modes of use of the content, which includes phenomena, situations and contexts that give meaning to the mathematical content.

#### **4. Research objectives and methodology of a related study**

The general objective of this work consists of characterizing and identifying prospective teachers'strategies to propose inverse problems and reflect on those experiences, in order to clarify and construct a wider theoretical framework for inverse problems in STEM education.

The specific objectives related to the general objective are:


For this purpose, the fieldwork of this research was carried out with two groups of prospective teachers at the University of Granada, Spain, during the academic year 2018–2019. The first group of prospective teachers (Group A) had 32 students and the second one (Group B) consisted of 33 students.

It is important to mention that both groups' professors (Moreno in Group A) and Ruiz-Hidalgo in Group B) collaborated with this research, which was developed in two sessions. In the first one, a brief explanation about inverse problems was given to the participants, and also a few examples were discussed. As a homework of this first session, an inverse reformulation of a given problem was requested. That problem was about a swimming pool filled with water, and the reformulation should be considered as a task enrichment of the original statement to be utilized in Secondary School courses.

In the second session, the prospective teachers' reformulations corresponding to the swimming pool problem were analyzed with the whole group. After the swimming pool reformulations discussion, a new problem (the sheep problem, **Figure 5**) was provided for reformulation to the participants as a new homework. There were several differences with the first homework, since the participants were asked to solve the given (direct) problem and then propose an inverse reformulation and solve their own proposal.

It is important to remark that only a few reformulations were discarded because they corresponded to ill-posed problems, or the reformulation cannot be considered an inverse one.

It should be mentioned that several participants proposed two or even three inverse reformulations. Then, there is not a single proposal for each participant, and not all the proposals were considered for this study. As a consequence, the reformulations were coded by responses rather than coding by author.

For the classification of the prospective teachers' reformulations corresponding to the sheep problem, several criteria were taken into account. One of the most important elements analyzed was the kind of inversion, but also possible changes in the geometry, the use or not of external variables, among others criteria, were considered.

As mentioned, firstly, inverse problems were classified in terms of the type of inversion (i.e. causation, specification, etc.). Nevertheless, there are several variants that deserve to be considered, for instance, a general inversion of the given function can be proposed, or an inversion just for a particular value and even an interval inversion, among other possibilities. A similar situation happens with specification inverse problems, where the prospective teacher proposal may ask about parameters interpretation, graphical representations or even a more creative solution without using integrals. Lastly, double and mixed inverse problems were also found among the proposals.

A similar analysis can be made for the item 'changes in geometry', since they can include changes in the shape of the field and/or the stake position as well as other obstacles, such as fences, not included in the original problem.

Considering all these variables, a first classification was made, in order to summarize the information about the reformulations (and their authors) in a table which is partially shown in **Figure 8**.

For this first classification (see **Figure 8**), a list of the variables considered are as follows:


Finally, considering this classification, it was possible to observe three groups of problems, which will be deeply analyzed in the following section.

#### **5. Results**

In the previous analysis, we used the following nomenclature: firstly, the capital letters 'PT' indicates that the participant is a prospective teacher. Secondly, the letters are followed by a two-digit number (which corresponds to the student number), and


#### **Figure 8.**

*Example of a first classification of the proposals.*

finally, in some cases there is a small letter, which refers to the number of reformulation considered, if it is applicable. As an example, the code PT23c corresponds to a proposal authored by 23rd prospective teacher, and it can be observed that this participant proposed at least three reformulations (here, the one considered is the third).

As it was mentioned, the analysis of the proposals allowed us to identify three groups of problems.


#### **5.1 The first group of problems**

The first group of problems is formed by prospective teachers' proposals based on the procedural content knowledge. An interesting example is given by the production PT31a, which considers a square field 0, ½ �� *L* ½ � 0, *L* , where the sheep is tied at ð Þ *L=*2, 0 . The situation is sketched in **Figure 9**.

*About the Notion of Inverse Problem in STEM Education DOI: http://dx.doi.org/10.5772/intechopen.106479*

**Figure 9.** *Sketch corresponding to the proposal coded as PT31a.*

The author mentions that *<sup>r</sup>* <sup>¼</sup> *<sup>R</sup> <sup>L</sup>* represents the ratio of the rope length to field side length, and he/she gives the following data: *<sup>L</sup>* <sup>¼</sup> <sup>20</sup>*m:* and *<sup>f</sup>* <sup>¼</sup> *<sup>A</sup> <sup>L</sup>*<sup>2</sup> ¼ 0*:*877 with *A* being the area accessible for the sheep. The required output of this proposal is the radius *R*.

In his/her solution, the prospective teacher divides the area *A* into three parts (a circle sector and two right triangles) as in **Figure 10**.

Since the circumference is given by ð Þ *<sup>x</sup>* � *<sup>L</sup>=*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>¼</sup> *<sup>R</sup>*<sup>2</sup> for *<sup>x</sup>* <sup>¼</sup> 0 it is easy to obtain *h* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>R</sup>*<sup>2</sup> � ð Þ *<sup>L</sup>=*<sup>2</sup> <sup>2</sup> q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>R</sup>*<sup>2</sup> � <sup>1</sup> <sup>4</sup> *<sup>L</sup>*<sup>2</sup> q , then each triangle has an area equal to *A T*ð Þ¼ <sup>1</sup> 2 *L* 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>R</sup>*<sup>2</sup> � <sup>1</sup> <sup>4</sup> *<sup>L</sup>*<sup>2</sup> q ¼ 1 <sup>4</sup> *<sup>L</sup>*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffi *<sup>r</sup>*<sup>2</sup> � <sup>1</sup> 4 q . Moreover, the angle *θ* in those triangles can be obtained as *<sup>θ</sup>* <sup>¼</sup> cos �<sup>1</sup> *<sup>L</sup>=*<sup>2</sup> *R* � �, and then the angle of the circle sector can be written as: *<sup>α</sup>* <sup>¼</sup> *<sup>π</sup>* � 2 cos �<sup>1</sup> *<sup>L</sup>=*<sup>2</sup> *R* � � . So, the circle sector area is *A CS* ð Þ¼ *<sup>α</sup>* <sup>2</sup>*<sup>π</sup> <sup>π</sup>R*<sup>2</sup> <sup>¼</sup> *<sup>π</sup>* <sup>2</sup> � cos �<sup>1</sup> *<sup>L</sup>* 2*R* � � � � *R*<sup>2</sup> , which can be written as *A CS* ð Þ¼ *<sup>π</sup>* <sup>2</sup> � cos �<sup>1</sup> <sup>1</sup> 2*r* � � � � *R*<sup>2</sup> , and then a non-linear equation is obtained: *f r*ð Þ¼ 1 2 ffiffiffiffiffiffiffiffiffiffiffiffi *<sup>r</sup>*<sup>2</sup> � <sup>1</sup> 4 q <sup>þ</sup> *<sup>π</sup>* <sup>2</sup> � cos �<sup>1</sup> <sup>1</sup> 2*r* � � � � *<sup>r</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:*877.

**Figure 10.** *Sketch corresponding to the solution of the proposal coded as PT31a.*

The prospective teacher solves this non-linear equation in *r* by using Bisection Method, resulting in *r* ¼ 0*:*925 and finally *R* ¼ 18*:*5 m*:*

As it can be observed, the prospective teacher solved his/her proposal without using integrals. As a final remark, it should be mentioned that the problem solution mainly requires procedural knowledge.

#### **5.2 The second group of proposals**

This second group is very homogeneous and contains only proposals based on the use of different representation systems. The production PT03 gives an interesting example. This proposal utilizes the same geometry of the given direct problem and

adds a different input. The participant proposes that e *<sup>f</sup>* <sup>¼</sup> <sup>1</sup> *L*2 Ð *L* 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>R</sup>*<sup>2</sup> � ð Þ *<sup>x</sup>* � *<sup>L</sup>=*<sup>2</sup> <sup>2</sup> q *dx*, gives the fraction of area where the sheep may graze, where *R*, *L*>0. A sketch of the situation descripted in the previous statement is the required output of this reformulation.

In his/her solution, the author observes that *y* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>R</sup>*<sup>2</sup> � ð Þ *<sup>x</sup>* � *<sup>L</sup>=*<sup>2</sup> <sup>2</sup> q , is the equation of a semi-circumference centred in the point *<sup>L</sup>* <sup>2</sup> , 0 � � with a radius *<sup>R</sup>* and *<sup>L</sup>*<sup>2</sup> is the area of a square with a field side length *L*.

From these observations, the requested region can be obtained, and the corresponding sketch is the same as in **Figure 9**.

#### **5.3 The third group of inverse problems**

The conceptual content knowledge is fundamental for solving the third group of proposals. For instance, the production coded as PT23c is a representant of this third group. In his/her proposal, the geometry of the field remains the same and the participant asks for the stake position and radius *R* to maximize the accessible area for the sheep with an extra condition: the animal should not reach the boundary of the field.

Obviously, the stake should be in the centre of the field, i.e. at the point *<sup>L</sup>* <sup>2</sup> , *<sup>L</sup>* 2 � � and the radius proposed by the author is *R* ¼ *L=*2, as illustrated in **Figure 11**.

It should be mentioned that the proposed solution—sketched in **Figure 11**—is a supreme, not a maximum. The other important remark is that the solution basically requires conceptual knowledge and no geometrical, analytical and/or numerical procedures are necessary to solve it.

#### **6. Discussion**

In this first attempt to construct a theoretical framework for inversion as a task enrichment strategy, we focused only in the cultural/conceptual dimension of the didactic analysis. Although we have concentrated on this single dimension, we have observed that there several different ways to classify inverse problems according to the type of inversion, input/output, type and difficulty of the solution and external variables/context changes, among others.

As it was mentioned, considering the type of inversion—which is related to the inputs and outputs of the proposal—we can consider four types of inverse problems:

*About the Notion of Inverse Problem in STEM Education DOI: http://dx.doi.org/10.5772/intechopen.106479*

#### **Figure 11.** *Sketch corresponding to the solution of the proposal coded as PT23c.*


Nevertheless, this is not the only possible classification. In fact, this classification does not explain the results obtained in Section 5, where three groups of proposals were observed. Then, in order to explain the previous result, another classification based on the semantic triangle should be considered. Then, we can also classify inverse problems into three groups:


These classifications fit well with the three groups observed in Section 5 and at same time, have a stronger connection with the framework utilized (the didactic analysis).

Another option is to classify inverse problems in pure or combined with another task enrichment strategy, such as modeling. This fact was already observed in previous papers [24, 25] where inverse modeling problems were studied as a different group from other inverse problems.

It should be remarked that in the fieldwork, it was observed that several participants decided to add other variables to the original problem and/or make changes in the context. The external variables can be physical, chemical, biological and economical. One proposal asked for the amount of herbicide and other asked for the amount of fertilizer for the grass, which are chemical external variables. Other participant asked for the velocity of a sheep, which is a physical variable, other one asked for the cost of a fence per unit of length (economical variable) and finally, there was a proposal which requested about the kilograms of grass the sheep can eat per day (i.e. a biological parameter).

About the context changes, for instance, one of the participants (PT22b) considers a goalkeeper that throws the ball in a handball field and which can cover a certain area (instead of a field where a sheep is grazing). Another one (PT27) proposed considering a bush fire without wind. This fire evolves covering concentric circles and the proposal asks for the point where the fire started, knowing that *R* ¼ *L* and *f* is 95*:*60%. It must be remarked that PT22b and PT27 are just a couple of examples where the context is different from the original proposal.

So, the inclusion or not of external variables and/or the context changes can be used as criteria for a different classification into four groups:


Another option considers the inputs and outputs of the proposal. These inputs and/or outputs can be particular values (like in PT33, where the area accessible for the sheep is *<sup>A</sup>* <sup>¼</sup> <sup>433</sup>*:*<sup>36</sup> *<sup>m</sup>*2) or general ones (like in PT27, where *<sup>R</sup>* <sup>¼</sup> *<sup>L</sup>*), a croquis (like PT03, where a sketch of the region where the sheep may graze is requested) or even a process (like PT34, who asks for a criterion that allows distinguishing among different geometries in terms of the grazing area).

Lastly, it should be mentioned that the problem may be classified by analyzing its possible solutions. Then, the proposals can be classified in terms of the difficulty (trivial problems, low, medium and high difficulty) and the mathematics branches involved (analytical, trigonometrical, algebraic and numerical solutions, among others). These possibilities constitute more 'classical' options for classifying the prospective teachers' productions.

As a final comment, it is important to observe that these different criteria do not contradict previous works of other authors such as Groetsch [22] or Mason [19], or our own previous works [14–16]. On the contrary, we believe that they complement those previous works, giving a broader vision of inversion as a strategy for task enrichment.

*About the Notion of Inverse Problem in STEM Education DOI: http://dx.doi.org/10.5772/intechopen.106479*

#### **Author details**

Victor Martinez-Luaces\*, José Antonio Fernández-Plaza and Luis Rico University of Granada, Granada, Spain

\*Address all correspondence to: victorml@correo.ugr.es

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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*About the Notion of Inverse Problem in STEM Education DOI: http://dx.doi.org/10.5772/intechopen.106479*

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#### **Chapter 4**

## Mathematical Creative Model: Theory Framework and Application in Mathematics Learning Activities

*Subanji Subanji and Toto Nusantara*

#### **Abstract**

One of the most crucial abilities in the face of global issues is creativity. Because of this, research on creativity is always intriguing, especially in the context of mathematics education. Two investigations pertaining to original mathematical models are presented in this paper. The first study had 72 primary school instructors who took part in enhancing mathematical learning. Participants given open challenges with cube net material. Out of 137 high school students that consistently responded, 124 were included in the second study. Two descriptive statistical issues affect secondary school students. After a thorough interview to understand the subject's thought process, the results of the problem-solving process were used to support the subject's stance using an original mathematical model. Impersonation, adaptation, and building make up the creative model's structure. The findings demonstrate the consistency of mathematical creativity model properties. The subject only imitates a successful routine method due to the level of imitation. According to the characteristics of the level of modification, the subject is able to alter the methods discovered in order to meet the difficulties at hand. The hallmarks of the creation level are that they develop novel, rational, and methodical approaches to problem-solving. Based on the findings of this study, a new method of learning mathematics was devised. The open-ended activities that have been packed into the form of a mathematical tree are utilized, together with the strategic approach to problem-posing.

**Keywords:** mathematical creative model, mathematics learning activities

#### **1. Introduction**

In order to prepare future generations for the challenges of a globalized society, creativity is crucial. Pioneers and entrants to change and progress can only be creative people. The foundation of artistic, technological, and scientific advancement is creativity. The advancement of information technology as a creative human endeavor that can make solving problems easier. Numerous academics have looked into the value of creativity, notably in the teaching of mathematics. This is consistent with the essence

of mathematics education, which entails learning by fostering critical thinking and reasoning in order to become an effective problem-solver. Consequently, it is applied in mathematics to develop higher-order thinking skills (HOTS). One of HOTS's most important components is creativity. The greatest degree of cognition, in accordance with the revised Bloom's taxonomy, is creativity. The ability to be creative makes one an excellent problem solver and innovator.

Studying creativity is always fascinating, particularly when it comes to the realm of math instruction. The importance of creative research in mathematics education and the expansion of the scope of study were emphasized by Joklitschke et al. [1]. This supports a study by Schindler & Lilienthal [2] that emphasizes the importance of student creativity processes in mathematics education research. The development of creative mathematics is followed through open challenges to show how creative research progresses from creative outputs to creative processes. Multiple Solution Tasks (MSTs) were employed by Schindler & Lilienthal [2] in order to foster and assess students' mathematical creativity, particularly the creative process.

In addition to the usage of open assignments, problem solving can be used to stimulate creativity. Problem-based research (PPI) is a method used by Leikin & Elgrably [3] to examine how creative solutions to mathematical issues are produced. According to their research, a stronger creative process does not always result in a stronger creative product and a higher degree of strategic originality. But it relates to the end product of creativity. Here, the two distinct aspects of cognitive processing connected to creative problem-solving are the creative outcome and the creative process. Three aspects of creativity—fluidity, flexibility, and creativity—are the foundation of their research. Their findings indicate that problem generation and problem solving cannot be separated when someone completes PPI creativity challenges.

Schoevers et al. [4] investigated into the connection between mathematical problem-solving in elementary schools and creative thinking. There are four closed routine issues, six closed non-routine problems, and four open non-routine problems using geometry tests (multiple solution problems). The findings indicate that creativity plays a significant role in predicting how well children will succeed on various geometry problems. However, it was more closely connected with how well students performed on open tasks that were not routine, which shows that kids with more creativity were more successful at solving geometry problems.

Numerous research, spanning from the classification of creative thinking components to the level of creativity, have been conducted as a result of the significance of creativity. The classification of creativity in this study is not based on the cognitive process by which it is formed, though. The mathematical creative model is a development of Subanji et al. [5] and is based on the mental process of creative formation. The context of issues involving mathematical thinking is examined in their research. Having unique qualities like reason and logic, mathematics is a fundamental science. The rationale behind using the framework of mathematical reasoning is that it calls for higher-order thinking, which is necessary for creative thinking. On the basis of the properties of the mathematical content, a study of argumentation in the field of mathematics was conducted. It is known as covariational reasoning in the context of building graphs, algebraic reasoning in the context of solving comparative problems, proportional reasoning in the context of solving analogy problems, analogical reasoning in the context of solving controversial problems, and controversial reasoning in

#### *Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

the context of solving controversies. In this study, a problematic mathematical problem—one that defies the preexisting framework of thought—is the focus of the mathematical problem that is being used. The controversial question was picked because it was good at causing cognitive conflicts, or dis equilibration in Piaget's terminology, which pushes for higher order thinking, particularly creative thinking. The development of mathematical creative models is based on the cognitive process of the formation of creative thought in solving mathematical problems. This creative model is important for research because it can be used to develop mathematical thinking in the correct way based on cognitive processes. The learning of mathematics is the learning of reason and logic, especially the formation of HOTS, which includes: analysis, assessment, and creation (creative). According to Bloom's taxonomy, creativity is the highest level of thinking. Whoever can reach the level of creative thought will become a good problem-solver, initiator and innovator. The importance of creativity has led to various studies, from the classification of elements of creative thinking to creative thinking levels. Researchers have assessed the originality, fluidity, and adaptability of pupils' problem-solving to assess their creativity [6, 7]; Additionally, Kattou et al. [6] discovered a link between creativity and mathematical aptitude. Four stages—preparation, incubation, illumination, and verification—were identified by Sriraman [8] after studying the creative processes of five mathematicians.

Learning mathematics requires a lot of creative thinking. Numerous academics looked at the elements of fluidity, flexibility, and invention in addressing open questions to determine how creatively kids thought [6, 7]. The number of alternative solutions to a given problem is considered in the assessment of fluency. The capacity to modify several concepts to generate various means of completion is related to flexibility. Originality is the generation of fresh approaches to challenges. Additionally, the structure of the connection between mathematical prowess and creativity was examined by Kattou et al. [6]. According to their research, pupils fall into one of three categories based on their mathematical prowess: those with low, medium, or high mathematical prowess. Sriraman [8] conducted a study of five mathematicians to ascertain the characteristics of the creative process, and the results showed that the creative process of mathematicians followed the four stages of the Gestalt model, namely the preparation-incubation-illumination-verification. However, the mathematical creativity of the three categories also varied, so it was found that students with the highest scores on the math test were also the most creative. In this situation, learning mathematics, especially statistics, can involve the creative process.

The basic and secondary mathematics curricula include statistical content on data analysis and probability, according to the National Council of Teachers of Mathematics [9]. This demonstrates the significance of statistics in mathematics. The mathematics curriculum in Indonesia similarly includes statistical components from primary to higher education. This is because statistics have a wide range of applications in the fields of law, medicine, agriculture, and economics. The sciences of data collection, analysis, presentation, interpretation, and decision-making are known as statistics [10]. There are two types of statistics: descriptive statistics and inference [11]. Measurements of concentrations and dispersion fall within the purview of descriptive statistics, whereas hypothesis tests that can extrapolate from samples and make generalizations about population features go under inference statistics. In this study, descriptive statistics were utilized to present data visually. The kind of qualitative or quantitative data also affects the type of diagram [12]. For instance, histograms, line graphs, stem and leaf charts, and pie charts are frequently used to depict the

distribution of qualitative data (on a nominal or ordinal scale) (interval or ratio scale). In this situation, understanding information requires the capacity to read graphs and diagrams.

Sharma [13] examines and discusses how pupils comprehend the information in graphical forms such as tables and bar graphs. According to his research, a lot of pupils employ instinctive and experience-based tactics. Further research by Aoyama [14] into the hierarchy of students' interpretations of graphs revealed several challenges younger students face when considering open-ended questions due to their lack of prior learning. According to Mann & Lacke [10], descriptive statistics include techniques for gathering data, showing it, and summarizing it using tables, graphs, and summary measures of concentration. Since there are so many graphs used to depict data in written and electronic media, it is important to comprehend how the graph might be used to interpret numbers. In order to solve the measure of concentration problem, students used the descriptive statistics they had learned in this course to portray data as line graphs and bar charts. This study's problem is an open one that has to be solved with original thought.

To address common difficulties, creative thinking is necessary. Mathematical exercises can foster the development of creative thinking [8, 13, 15–19]. Logic-based concepts, structures, and interactions are a foundational part of mathematics, according to this theory. Using logical and methodical justifications, the truth of mathematics is established. Numerous mathematical tasks are performed through logical and methodical thought processes, including as formulating and testing hypotheses, seeking parallels, drawing connections, creating representations, creating generalizations, proving theorems, and ultimately solving problems. High-level thinking calls for pupils to exercise both critical and creative thought when completing these mathematical assignments.

Creative thinking occurs in mathematical activities, called mathematical creative thinking, and is often associated with problem solving. [9, 15, 20, 21]. The National Council of Teachers of Mathematics [9] proposes to give students difficult problems that can promote their mathematical creativity. This can be done because problem solving enables students to improve their creativity skills through various solutions. Baran et al. [15] discovered that mathematical creativity can be seen in problemsolving abilities, especially in open mathematical situations. Chamberlin & Moon [20] discovered creativity in the thinking processes of mathematicians related to the solution of non-routine problems.

Mathematical creative thinking, which happens when performing mathematical tasks, is frequently linked to problem-solving [9, 15, 20, 21]. Giving pupils challenging tasks that can foster their mathematical creativity is a suggestion made by The National Council of Teachers of Mathematics [9]. This is possible because problem solving gives kids the chance to develop their creativity through a variety of solutions. Mathematical creativity can be evident in one's ability to solve problems, particularly in scenarios involving open-ended mathematics, according to research Baran et al. [15]. Chamberlin & Moon [20] found that the way mathematicians solve non-routine problems involves creativity. Therefore, one part of mathematical creativity is problem solving. Additionally, Beghetto & Karwowski [22] contend that teachers might accomplish this balance by turning some normal tasks into nonroutine difficulties. Routine practice must be matched with innovative and creative approaches. Can be achieved by teachers by changing some routine tasks into nonroutine problems.

Researchers have investigated the use of mathematics to foster creativity [15, 19, 23, 24]. To predict students' creativity in solving mathematical problems, Lin & Cho *Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

[24] created a model of creative problem-solving abilities. According to gender, Baran et al. [15] discovered a correlation between creativity and mathematical aptitude. Voica & Singer [19] looked at the creativity of math-gifted children and discovered that those who had a strong grasp of the subject had good inventiveness. Using Model-Eliciting-Activities, Coxbill et al. [23] developed and tracked students' mathematical inventiveness (MEASs). Sheffield [25, 26] has conducted research on the value of fostering and enhancing pupils' creativity as they study mathematics. Understanding mathematics can aid in the development of students' creativity [27]. Some of these studies highlight how crucial it is to research students' mathematical creativity.

The other aspect of the study of creative thinking is examined in this article, specifically the creative model, which is founded on the cognitive process of creative formation.

#### **2. Creative model framework**

General creativity includes creative mathematical thinking. It is possible to think of creativity as a process of thought that involves original concepts and ideas [28, 29]. Numerous methods, such as studies of creative outputs and creative processes, can be used to examine creative mathematical thinking [3]. The cognitive process that results in creative thinking is referred to as the creative process. A type of creativity whose focus is mathematics is known as mathematical creativity. Therefore, the development of mathematical creative frameworks is founded on general creative frameworks that consider the properties of mathematical structures. In general, creativity occurs in daily life and can be divided into three categories: creation, modification, and imitation.

When someone wishes to create a product that replicates an already existing product, they start at the lowest level, which is later referred to as a creative model of imitation. The process of invention in this instance is restricted to product imitation. Even if they just copy, imitators nevertheless engage in the creative process since they consider ways to make their creations more affordable than the originals. Simple cognitive processes are used in the creative model of imitation. This imitation creativity model in mathematics is influenced by the learning process, which only prioritizes procedures. Students can solve a problem if the procedures are known.

The Creative Model of "modification" refers to the second level, which is a change. This level is reached when a product is transformed into a new one by looking at its "functions, advantages, and forms." The highest level is creation, which is referred to as the "creation" model. This occurs when someone creates a new product without first considering the associated items that already exist.

A mathematical creative model is one that can be created from a general creative model in the context of mathematics. The general creative, mathematical creative models categorize creativity into three levels (three): imitation (imitation), modification (change), and creation. **Table 1** below provides the conceptual framework for the construction of general creativity-based mathematical creativity models.

The core of the imitation level creative model is the principle of making products with the same design and functionality but at a cheaper cost. You must cut or substitute other less expensive materials in order to lower the cost of the product. A creative imitation model produces a varied grade of goods as its outcome. The definitions of "original product," "super replica product" (quality 1), "medium replica" (quality 2), and "low replica" (quality 3). Models for creative modification are based on the idea of modifying items to make them more cozy, lovely, alluring, and practical. The

process of modifying a product's function, benefit, or form results in a new product with additional functions, one that is more useful, and one with a more appealing shape. This process is the basis for the creative modification model. Finding new ideas, thinking, and solutions to issues is the foundation of creative creation models.

In the context of mathematics, imitation levels, i.e., imitate only the facts or techniques obtained when completing problems/tasks. The methods he was able to find in this situation varies from the most basic to the most sophisticated. The technique of generating new challenges or tasks by adjusting the data, graphics, and procedures obtained is known as modification level mathematical creative modeling. The process of constructing or resolving issues by gathering data, making graphs, or coming up with novel problem-solving techniques constitutes the creation level of the mathematical creative model.

#### **3. Method**

The research develops a model of creativity based on the idea that creativity is seen as a cognitive process of creative formation. The first research was carried out on teachers of elementary schools attending cube net materials training, and on students of secondary schools undergoing descriptive statistics training. The paper describes only part of the two studies of the creative model. The first study was a creative model for 72 primary school teachers who participated in mathematics learning courses jointly held by Universitas Negeri Malang and the PT. Pertamina. This teacherpowering cooperation lasted 6 years, but the data sources used in the study were only last year. The training is conducted every year at three strengthening stages and in two classroom exercises. The first and second strengthening phases took 10 days (80 hours) to complete. The third phase of strengthening was 5 days (40 hours) long. Learning practice is conducted in intervals between strengthening stages. There are materials to solve mathematical problems at each stage of the reinforcement. Among the problems-solving materials, the most interesting are cube net materials. This material is always chosen by the participants as the most interesting material, because it is related to real life, different from what is usually presented to students, the problem is open, makes sense, and is very challenging. Then, the process of solving the problem of cube net material and conducting an in-depth interview was used as a data tool to investigate creative models.


*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

The second study was the creative model of 137 secondary students to solve problems in descriptive statistical material. Students received two open problems relating to descriptive statistical materials, the tea sale problem and the math test results problem. To assess the consistency of the level of student creative models, two problems were presented. Of the 137 students, 124 were re-examined (59 boys and 65 girls). From the process of solving statistical descriptive problems, followed by tracing the process of cognition through in-depth interviews, and used to justify the position of the subject based on the creative model.

#### **4. Results**

The results of the research are presented on the basis of two studies, namely the mathematical creative model for solving the problem of cube nets and creative mathematical model to solve descriptive statistical problems.

#### **4.1 Creative model of cube net problem solving activities**

This study included 72 primary school teachers participating in mathematics education training. One of the materials presented is a mathematical problem solved with a cube net. The topic was given the following problem about applying cube networks.

#### **Cake box company**

*Pak Romli is the head of the company "MAKMUR" which produces cake boxes. Pak Romli always thinks how to get big profits. Several attempts have been made: first, to make various models of boxes that may be liked by buyers and secondly to make savings, namely to produce as many boxes as possible from the same material. Sir. Regarding the first attempt, Pak Romli has conducted a survey of 200 cake sellers, the result is that 70% of the respondents liked the cube-shaped cake box on the grounds that it could contain more. To save money, Pak Romli conducted an experiment by making a box and opening it with each side still intertwined (not separated from each other), which are often called cube nets, as shown in* **Figure 1***.*

*To make an efficient cube design, Pak Romli prepared a "cube material" (***Figure 2***) in the form of a sheet of paper that has been completed with unit squares. Then Pak Romli*

**Figure 2.** *Cube material.*

*thought about how to use a piece of paper to make as many cubes as possible (and as little paper wasted as possible). You are asked to help Pak Romli efficiently design cube nets from* **Figure 2** *material (the cubes produced are the most and the paper is wasted the least)! Determine the maximum number of cubes that can be made!*

Research subjects participating in mathematics learning training had the characteristics of 11 early-career teachers (1–3) and 61 high-career teachers (4–6). Of the 72 subjects, 9 (13.50%) were imitation levels, 34 (47.2%) were modifications levels, and 29 (40.28%) were creativity levels. Each level's description is given as follows.

#### **4.2 Imitation level**

At the level of imitation, the object only imitates Pak Romli's cube nets and immediately places the cube nets on the cube material sheet.

One can see from the subject's response on the level of imitation that the subject only copies the information provided in the question and keeps the relevant cube materials structured. The subject does not take into account various cube net configurations; rather, what is noteworthy is the subject's cognition, which copies the provided material and simply modifies its position to produce consequences. To further understand the subject's position at imitation level, the researcher conducted an interview.

*Q: What is your comment on this matter?*

*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

*S: The problem is contextual and very challenging.*

*Q: Have you ever had a question like this?*

*S: Never.*

*Q: What do you think when you face a problem like this?*

*S: Since there are already nets of the cube, then I make the same nets with different positions so that the solution is immediately obtained.*

*Q: In your opinion, are there other forms of nets?*

*S: There should be, but I do not remember because all this time I have been teaching the lower class (grade 2), so I do not know the cube nets by heart.*

From the interviews, the subject seems to "imitate" the net only in different horizontal and vertical positions. This may occur because the subject is taught in a lower class and the net material of the cube is given in 5' class.

#### **4.3 Modification level**

At the level of modification, the topic changes the shape of the problem's cube net by changing the position of one unit square. In addition to (1), you can change one step of the left square to (2). Modification from (1) to (3) is done by shifting the topmost square to the bottommost position. Next is the position adjustment.


To further explore the creative model of the subject in modification level, a taskbased interview was conducted.

*Q: What is your comment on this matter?*

*S: This problem makes sense but is not easy to solve.*

*Q: Have you ever had a question like this?*

*S: Never. Usually the problem of cube nets is limited to their type.*

*Q: What do you think when you face a problem like this?*

*S: I am curious. At first I thought it was simple, but after thinking about it, it's not easy, even though I've got this answer, I am still curious.*

*Q: How do you get the nets of cubes (2) and (3)?*

*S: I changed it at the same time wondering whether the result of the conversion is still a net of a cube. And after I imagined the results of the conversion were cube nets. Next, I arrange like that position and get three closed cubes.*

From the interview, it appears that in solving the problem, the subject begins by "changing" the cube nets that are already known in the problem into several different cube nets and then arranging them in the cube material provided.

#### **4.4 Creation level**

At the level of creation, integrative thought focuses on the shape of the 7x5 cube. The subject is "How to cover a cube material as many cube nets as possible and ignore the nets in the problem. The topic was explored and was successfully created by making a "new" cube net that allowed for more cube material configuration.

To further investigate the characteristics of creativity models at the creation level, task-based interviews were conducted. The interview focused on the process of constructing answers for the subject.

*Q: What is your comment on this matter?*

*S: I never thought of a problem like this, because when teaching cube nets, we only showed several alternatives for cube nets.*

*Q: Have you ever had a question like this?*

*S: Never. This problem inspired me to connect math material with life.*

*Q: What do you think when you face a problem like this?*

*S: I realized that to teach mathematics it is necessary to think about the use of mathematics in life. Therefore, so far, I teach cube nets only procedures, so I feel guilty.*

*Q: How would you construct an answer like this?*

*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

*S: At first, I thought this problem was simple, but it turned out to be complex. I started with a 7x5 cube material, while one cube only needed six unit squares. I suspect that five nets of the cube can be made.*

*Q: Why do not you use the cube nets that have been given?*

*S: I suspect that the exemplified web cannot produce many cubes. It was a hindrance, so I ignored it. And I created new cube nets alternatives.*

*Q: What are your next steps?*

*S: I arrange cube nets into cube materials with various alternatives. It turns out that the maximum number of cubes that can be made is four. Then I was still thinking again, the rest, it turns out that there are some that cannot be made into cubes anymore, but there are also some that can be made into open cubes. I think that the open cube is still of some use, too. Finally, I conclude that the most efficient arrangement is four closed cubes and one open cube, there are still six unit squares left but randomized, no more cubes can be made. I am also thinking further, what if the size of the cube material is different (not 7x5), but I have not tried.*

The results of the interviews show that the creation level topics continue to think about the creation of the most effective cube. In fact, the topic does not stop with the solution he is given, but thinks he wants to change the size of the cube material so that it's a little wasted.

#### **5. Mathematical creative models on descriptive statistical problem-solving activities**

This section provides students with the level of creative model to solve statistical descriptive problems [5]. The study was conducted with 137 secondary students and provided two open problems with descriptive statistical materials, namely problem 1 (tea sales) and problem 2 (mathematical test scores). To see the consistency of the level of creative models of students, two problems have been given. Of the 137 students, 124 (59 boys and 65 girls) were consistently re-recognized.

#### **Problem 1. Tea Sales**

*A shop sells two soft drinks "tea bottle" and "tea box" from 2000 to 2009. The graph below shows the sales of the drink "Teh Botol" for 5 years. Make a line graph of the sales results of "Teh Botol" and "Teh Kotak" in one graph so that in 2009 the sales of the two soft drinks were the same.*

#### **Problem 2. Math Exam Score**

*A Mathematics teacher teaches in two classes, namely classes A and B.* **Table 1** *below is the result of the Mathematics score for class A. The number of students in class B is 35 people. Make a bar chart of the Grade B Mathematics grades (Scores 0–100) if the mean, median, and class B mode scores are greater than the class A mode.*

### **Figure 3.**

*(a) Distribution of creative models; (b) creative model by gender.*

One Hundred and Twenty four people responded consistently to questions based on creative model levels. 25 subjects (20.16%) are classified as imitation levels, and their problem solving processes are characterized by imitating data display, graphics, or strategy. Fifty six subjects (45.16%) are characterized by modifying data and graphics display processes to solve problems.; and 43 creative subjects (34.68%) are characterized by problems solved by the construction of new data and graphics. **Figure 3**(**a**) shows the distribution of the creative model for solving the open problem.

**Figure 3**(**b**) shows that the distribution of the creative model of male subjects: level of imitation 12 (9.7%), level of modification 26 (21%), and level of creation 21 (16.9%). Creative models of female subjects: level of imitation 13 (10.5%), level of modification 30 (24.2%), and level of creation 22 (17.7%). In carrying out open-ended problem solving activities, both male and female students are mostly on the modifiedlevel creative model. The following presents a creative model of the subject in solving open-ended problems for each level.

#### **5.1 Imitation level**

The creative model of imitation level is characterized by imitation of the context of problem solving strategies. In the first problem, the creation model at the imitation level is reflected in the process of imitating graphic shapes and numbers to make them similar to existing graphics and numbers. In the second problem, the creativity model at the level of imitation is reflected in the process of imitation only of the available data. This shows that the creative models of the level of imitation of the subjects in problems 1 and 2 are consistent. The student's activities in the mathematical creative model of imitation level are presented in the following.


**No Problem solving activity Subject answer Imitation form** the graphs coincide. Meanwhile, the sales of tea box from 2000 to 2004 experienced an upward trend every year, similar to the pattern of increasing sales of tea bottle. Translate: *Sales results of tea bottle and tea box* 2. • students determine grade B math scores by imitating exactly the same as grade A math scores; • create a bar chart based on the results of imitating class A Mimic the grades of class A that are already available. The subject does not change the existing values and is used to show that the mean, median, and mode are greater than class B.

*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

In problem 1, the subject only imitated the shape of the graph and the trend of the sales of bottled tea so that the two soft drinks had similarities both in the form of graphs and the increase in sales results which experienced the same increase every year. The results of interviews with researchers (P) and students (S) are as follows:

*Q: What is your process for completing problem 1?*

*S: I saw the difference in the increase in tea bottles by six, then I followed it until 2009, it went up by six until I got 64. Then for the tea box graph, the value must be the same in 2009, so I started to make it from the back first from 2009. From my score of 64 less 4 continues every year until 2000, sales results 28*

*Q: Why did you choose the difference 4, not any other number?*

value data. Translate: *Mathematics score of Class B*

*S: because you are free to choose ma'am*

*Q: Why not make six to make it the same as tea bottle*

*S: Therefore, that the graphs do not overlap, mam*

In problem 1, the subject who is at the imitation level in determining the sales of tea bottles in 2005–2009 by "imitating" is exactly the same as the increase in sales in 2000–2004, which is Rp. 6 million every year. The subject used the Rp. 6 million to continue the pattern of increasing sales of tea bottle in 2005–2009 so that the sales of tea bottles in 2009 were Rp. 64 Million. Furthermore, the subject made a graph of the sales results of tea boxes starting in 2009 and imitated the pattern of increasing sales of tea bottle which the increase was the same every year. This subject makes an increase in the sales of tea boxes by Rp. 4 million every year. From the sales in 2009 of Rp. 64 million, the subject made a downward trend by counting backward from Rp. 4 million every year so that the sales of tea box in 2000 amounted to Rp. 28 million. In this case, the subject makes a pattern for the sale of tea bottles in the form of an arithmetic sequence with a difference of Rp. 6 million. By using the concept of the arithmetic sequence, the subject imitated it to determine the sales of tea boxes with a difference of Rp. 4 million. The imitation level creative model that the subject did in problem 1 is reflected in the process of imitating graphic shapes and imitating number patterns.

In problem 2, the creative model of the imitation level is reflected in the subject's activity in determining the score of mathematics in class B which "imitates" the values of mathematics in class A. Although in Problem 2 it is stated that the subject is free to determine the range of mathematics values for class B (values 0–100), but the subject chose to imitate all grade A math scores. To create a bar chart, the subject made the horizontal axis the same as the math score of class A. In other words, the subject used all the scores of the class A, i.e., grades 40, 45, 60, 70, and 85 to make a bar chart of grade class B math scores. Next, the subject made the frequency on the vertical axis so that the mean, median, and mode of class B were greater than class A. The subject's position at the imitation level was strengthened by the following interview results.

*Q*: *Why did you choose the value on the horizontal axis like that? (pointing to the bar chart).*

*S: I made the value the same as the score of the class A, mam.*

*Q: What is your reason for making the values the same and not choosing another value?*

*S: I will just make it the same, ma'am.*

*Q: In the question from being free to choose of class B math score from 0 to 100, did you not read.*

*question*?

*S: Read ma'am,*

The question reveals that although he read the question, he can freely choose from 0 to 100 the Class B score. Furthermore, the subject has made the class B score equal to the class A score. Subjects can determine class A average values of 65, median values of 60, and mode values. Therefore, if the subject determines 70 as the highest frequency of 11 students, the pattern and average of class B are larger than class A. The subject determines that 26 students got a score of more than 70 and as many as nine students got a score of less than 70, so the subject believed that the average value of class B was greater than class A.

The descriptive statistical activities for problems 1 and 2 depend only on the information shown in the task. This is in line with the opinion of Mecca & Mumford [30] which states that imitation occurs if there is an object to be imitated or imitated so that imitation depends on how people work with examples. For example, in problem 1, students create a positive trend in the sales of tea box because there is an example of a graph in the form of a positive trend in the sales results of tea bottle which is displayed in problem 1. Likewise, in Problem 2, students scored mathematics in class B because students observed that class A contained mathematics scores. The math scores in class A can also be used for students in class B. In this case, students can get information about something that they

*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

experience directly and from what is around them. Buttelmann et al. [31] state that children are more likely to imitate reliable models than unreliable ones. While Okada & Ishibashi [32] state that imitation is the core of the learning process, someone imitates the attitudes of others not only superficially but also at the level of deep cognitive processes.

#### **5.2 Modification level**

The Creative Model of the Modification Level occurs when the subject changes and combines components, data, and strategies to complete the task. In problem 1, subject changes some or all of the existing graphics and numerical models to construct the graph. This is consistent with problem 2 completion, which is used to modify the scores of the A class students and construct the completion of the B class data. Examples of mathematical creative models at the level of modification are given below.

In the creative model of the modified level, subject changes the pattern of increasing the sale of tea bottles so that they are different from tea bottles and that the graphics are different from tea bottles. The results of research interviews with students confirm the position of modification level as follow.

*Q: How did you complete problem 1?*

*S: In making the tea box from 2005 to 2009 ma'am, I followed the multiples of six ma'am like.*

*year 2000–2004,*

*Q: Why not make another multiple?*

*S: I think from 2000 to 2004 the increase was regular, so I also made the increase regularly for 2005-.*

*2009*

*Q: Why not make up and down like a tea box chart?*

*S: I just continued the previous one, mam.*

*Q: Why do the lines on the tea box chart go up and down?*

*S: Because I made it different from tea bottle mam.*

In the first issue, the subject made a series on tea bottles sales from 2005 to 2009, continuing the previous year's series. Between 2000 and 2004, tea bottles were sold by Rp. 6 million every year, so the students added Rp. 6 million to the sales of the last year. In this case, the subject made modifications by changing the trend of the sales of tea bottles so that the sales of tea box experienced an up and down trend. The student made the sales of tea box that are different from the tea bottle, so that the shape of the trend line on the graphs of the two soft drinks is also different.

In problem 2, the subject uses some of the score of class A; namely scores 45, 60 and 70 and modifies them to 50, 65, and 90. These values are used on the horizontal axis on the bar chart of grade B math scores. While on the vertical axis, the subject determines the number of students so that the mean, median, and mode of class B are greater than class A. The subject uses a score of 85 in class A and modifies the other values to 90, 95, and 100. These values are used by the student to make a bar chart for class B. The position of the subject at this modified level is supported by the following interview results:

*Q: Why did you make a bar chart like this (while pointing to the student's answers).*

*S: Because judging from the problem, mam, from the questions obtained in class A there are 30 students, from various grades from 40 to 85, right, the order is to make a bar chart for class B whose value is up to us, sir, from 0 to 100. The average value, the median and mode of class B must be greater than class A. The first score is recorded for class A, in class A the average is 65 books, the mode is 60 and the median is also 60. So, how do I find the value for class B, I choose First the numbers are high so that the mean, median and mode of class B are greater than class A.*

*Q: Why were the scores 85, 90.95 and 100 chosen for class B score, why not other grades?*

*S: I choose the largest value in class A, which is 85, then I choose a number greater than 85 so that the average, median and mode exceed class A.*

*Q: Are you sure?*

*S: Sure, because the value is bigger than the grade A class bu.*

The interview showed that the subject merged the highest score of class A 85 and other values of more than 85 and modified the data. Students do so without mathematical calculation, so that the average, average, and mode score of class B is higher than class A.

The creative modification level model is an electronic data modification by merging and synthesizing several objects and concepts to generate new objects and concepts. This is in line with Batanero et al. [33] which states that through synthesis, such as combining the concept of a measure of concentration with the concept of a measure of dispersion, a new concept emerges, namely the distribution of data as a fundamental concept.

*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

#### **5.3 Creation level**

The level of creation of creative models is an activity performed by a person in the development of new information. The level of creativity in solving descriptive statistical problems is marked by the subject developing the existing line graph into a new line graph to solve the problem of presenting data in graphical form. The subject creates a new bar chart based on the table to solve the centering size problem. Subject activities on problem 1 and problem 2 on the creative level of creative model are presented in below.

To further study the status of the subject in the creation phase, the researcher conducted the following interviews.

*Q: How is your process to complete problem 1?*

*S: I made a graph of tea box and tea bottles from 2005 to 2009, some of which went up and some went down, mam, because I think that every sale does not always increase or decrease, there are times when sales also increase and decrease, that's why I made a graph, some are up and some are down.*

#### *Q: What are the initial steps taken?*

*S: First, I made the sales of tea bottle from 2005 to 2009 go up and down, mam. Then I made the sales of tea box starting in 2000, the value was 17, then in 2001 it decreased to 10, and in 2002 it increased again to 26, in 2003 it increased again to 23, in 2004 it became 43. decreased to 25, in 2006 it rose again slowly to 29, in 2007 it rose to 35, in 2008 it rose to 40, and finally in 2009 the sales of tea box and tea bottle were equal to 50. So in the middle of the year the sales had decreased drastic.*

*Q: Is there a number pattern that you make in determining it?*

*S: No mam, I'll take anything.*

*Q: What is the reason for taking any?*

*S: No, mam, just create your own.*

In completing problem 1, the subject of the creative level of model creative is not affected by the shape of the pattern or the shape of the trend in the graph in the problem, the subject makes or creates his own pattern and shape of the trend on the graph. It is believed that sales results should not continue to rise, and sometimes they must also rise and decrease. In this case, the subject is able to connect the problem in the problem with real life, the subject is not fixated on what is shown in the problem, but students think realistically and logically to complete tasks associated with everyday life. The pattern of numbers made by the subject is irregular, meaning that the amount of increase in sales results does not form a line that has a regular pattern so that it results in a different graphic form from the graph shown in the assignment.

In problem 2, the creation-level, subject creates a class B mathematical score bar chart by designing its own class A mathematics score. In problem 2, students can use values in the range 0–100, the range of values used by the subject to create new score of class B. The activity carried out by the subject in the creative level creative model is designing new scores so that the math scores of students in class B are different from those of students in class A. The student chooses scores of B student's score by 30, 35, 50, 65, 90, 95 and 100 where all these values are different from the scores of students in class A. These values are used by students to make a bar chart, where the horizontal axis contains student scores and the vertical axis contains the number of students who got these scores. The subject determined that more students scored 90, 95 and 100, i.e. there were 27 students and eight students scored lower than 90. The results of the following interviews reflect the creative process of a creative subject.

*Q: How is the process of answering problem 2? (while pointing to the student's answer paper).*

*S: The first step is to determine the mean, median and mode of class A. The average of class A is 65, the median and mode are 60. To make it bigger than class A, I make high marks in class B, there are 90, 95 and 100, I made the stems high, class B has a mode of 95 because there are 10 people, while in class A there are low scores and high scores, but the score is not up to 100. Because in the question class B has to be bigger in average, median, and mode, so I make a few few who score low and many who score high.*

*Q: Did you use formulas to solve for mean, median, and class B mode?*

*S: No, mam.*

*Q: How can you be sure that the mean, median and class B mode are greater?*

*S: I made a lot of students get high marks from class A mam.*

*Q: Why did you choose the score in class B like this? (pointing to the horizontal axis of the bar chart).*

*S: Because in a class usually there are those who get low scores, there are also moderate, and some are high and very high, I made it so that the grades of classes A and B are different.*

*Q: Any other reasons?*

*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

#### *S: No, mam, so that the values are different.*

The subject first determines the mean, medium and model of Class A before selecting the mathematical score of Class B so that the average, medium and model of Class B are greater than Class A. The average, average and class A mode values are 65, 60, and 60. Subjects can ensure that the mean, median and mode of score of class B are greater than class A without performing mathematical calculations by placing more students with scores higher than 65. Activities of these topics are classified as creative. This is in accordance with Sheffield [34] opinion that student solutions are considered creative if the student can produce something unique and new to what is in their environment.

#### **6. Creative models in mathematics learning assisted by mathematics tree media**

The mathematical problem that can promote students' mathematical creative thinking is usually open-ended, allowing students to create new ideas and ideas freely. Students should participate in activities that allow them to explore the problems, ideas, and ideas necessary to solve the problems [17, 18]. Problems that stimulate creativity are usually addressed in several ways [8] or open-ended problems. In addition, creative thinking can be built by problem posing. In the development of the level of creative models, the learning of mathematics must use open-ended or problem posing [35].

The importance of open ended in learning mathematics activities has been studied by several researchers [36, 37]. Hitt & Dufour [37] examined students' mathematical activities when they completed an open-ended task related to speed and found that students used different representations in the process of modeling the situation when they were solving open-ended problems. Chan & Clarke [36] provide an open-ended problem in collaborative group work activities. Mathematics learning activities using open ended are able to develop problem solving skills and negotiation skills in collaborative learning groups. This shows that open-ended assignments can challenge students to think higher and ultimately be able to improve problem-solving skills.

Many experts believe that creativity can be developed by open-ended activities and problem posing. The problem is how to package open ended and problem posing in learning mathematics. This paper offers mathematics learning with open ended activities and problem posing which is packaged in the form of a mathematical tree. Mathematics learning is assisted by mathematical tree media, hereinafter referred to as the Mathematical Tree Learning Model. The mathematical tree learning model is intended as a learning model that facilitates students to: (1) pose a problem whose answers are known or (2) answer questions from open-ended problems.

Learning with the mathematical tree media is a form of learning with the following syntax: (1) the teacher models/explains the material, (2) the teacher presents problems and students solve them in groups, (3a) the teacher gives answers on the twigs & students construct the appropriate questions on the leaves. Or (3b) the teacher gives open ended questions on the twig & students determine all possible answers on the leaf, (4) the teacher asks students to exchange and correct other students' answers, (5) the teacher asks students to rate other students' answers, and (6) the teacher provide reinforcement to the problems or answers made by students. In this case the teacher has prepared media in the form of a mathematical tree, which consists of stems, twigs, and leaves. The stem contains the subject matter, the twig contains

open-ended problems or answers, the leaves contain answers to open-ended problems or problems whose answers are already known. Students make leaves (compose problems or determine answers) as much as possible. The more leaves produced, the more fertile the tree is. On the other hand, if the leaves are made incorrectly, they will become INSTRUCTIONS. Therefore, in determining the assessment, the leaf (the correct answer/problem) is scored 3 (three). When the parasite (answer/problem made) is wrong, then the value is –1 (negative 1).

For example, in teaching students about the application of definite integrals, the questions usually given by the teacher are determining the area of a region bounded by some curves or determining the volume of a curve that is rotated around an axis. While learning with the mathematical tree media is done by determining the tree, namely, the integral and the branch is the area or volume whose value has been given. Next, students are asked to make leaves (find as many problems as possible) whose answers are on the branch. The integral tree can be made as shown in **Figure 4** below.

In the integral tree, students are asked to construct a leaf (i.e., a problem whose answer is already known) - the definite integral will result in 12. The definite integral form whose result is 12 is very large, therefore students can arrange as many definite integrals as possible, the important thing is that the result is 12. In this case, it is not enough for students to just remember the procedures exemplified by the teacher, but students must be creative in determining as many alternatives as possible. Therefore, learning with this mathematical tree media can develop students' creative models.

Another example, learning straight line equations. The tree is the equation of a straight line. The stick is to determine as many equations as possible the line that passes through the point (1.2) and determine as many equations as possible the line parallel to *y* ¼ 2*x*–3. The line equation tree is presented in **Figure 5** below.

The thinking process of students who are built in learning with the media tree of mathematics can be described as follows.

In determining the equation of a straight line that passes through ð Þ �2*:*1 , the students' thinking process is built by determining any line equation. For example *y* ¼ 2*x*, then students will be able to think that if *x* ¼ �2, then y must be worth one. Whereas when *x* ¼ �2, the value of y is 2 ð Þ¼� �2 4. To get a value of 1, you must add five. Therefore, so that *y* ¼ 2*x* through ð Þ �2*:*1 , it must be changed to *y* ¼ 2*x* þ 5.

**Figure 4.** *Integral tree.*

*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

This thought process will be able to empower students' reasoning. If the students' reasoning is well developed, then when students are asked to solve other problems, for example the equation of a line through ð Þ �2*:*1 with a gradient of –3, students will

easily determine by thinking, if the gradient is–3, it means the equation of the line is *y* ¼ –3*x* . If *x* ¼ –2, then y must have a value of one. Whereas when we substitute *x* ¼ �2, we get *y* ¼ �3 ð Þ¼ �2 6. Therefore, in order to fulfill 1, the equation of the line is *y* ¼ �3*x*–5. Therefore the equation of the line through ð Þ �2*:*1 with a gradient of�3 is *y* ¼ �3*x*–5.

Mathematical trees can also be developed in elementary schools in various materials: integer operations, fraction operations, perimeter and area. Here's an example of a math tree in elementary school about the number operation tree (**Figure 6**).

When students are asked to make pairs of numbers that meet 3 � *a* ¼ *b* þ 10, students will think, one of which is if *a* ¼ 2, then the left side is equal to 1, and so on the right side the value is 1, then *b* ¼ �9. If *a* ¼ 3, then the left-hand side is zero and so that the right-hand side is zero, b must be �10. In this case, students are doing creative thinking activities, where students make many answers and finally students can find a pattern if an increases by 1, then b decreases by one. Mathematical tree learning will build HOTS and creative thinking.

#### **7. Conclusion**

Mathematical creative models are based on cognitive processes of creative thinking in mathematical activities and are divided into three levels: imitation, modification, and creation. Levels of imitation are characterized by cognitive processes in which people can only imitate provided strategies/processes problem solving methods. The level of modification is characterized by a cognitive process in which one hopes to change strategy/procedure, and the problem-solving process is more effective or simple. The creation level is characterized by the cognitive process, i.e. the ability to construct new strategies/procedures for solving problems.

Mathematics learning emphasizes only procedures and only forms creative models at imitation levels. Students who can only remember procedures can apply them only to problems similar to solved problems. Students face problems if they face new problems or modified problems. Therefore, mathematics must be learned to stimulate creative thinking, such as using the mathematical tree media.

Mathematical tree learning was developed based on problem posing and open ended. The stimulus is given in the form of open ended, requiring students to determine several alternative correct answers. Problem posing in a mathematical tree has a special characteristic, namely the stimulus given is in the form of an answer and students are asked to construct a problem with the answers already provided.

*Mathematical Creative Model: Theory Framework and Application in Mathematics Learning… DOI: http://dx.doi.org/10.5772/intechopen.106187*

### **Author details**

Subanji Subanji\* and Toto Nusantara Department of Mathematics, Universitas Negeri Malang, Indonesia

\*Address all correspondence to: subanji.fmipa@um.ac.id

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[30] Mecca JT, Mumford MD. Imitation and creativity: Beneficial effects of propulsion strategies and specificity. The Journal of Creative Behavior. 2014; **48**(3):209-236. DOI: 10.1002/jocb.49

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#### **Chapter 5**

## The Language That Grade R Students Use to Achieve the Envisaged Mathematics Outcomes, a South African Perspective

*Shakespear M. Chiphambo and Nosisi N. Feza*

#### **Abstract**

In South Africa, the debate on the Language of Teaching and Learning of mathematics in Grade R has been underway. Amid the robust debate on the teaching of mathematics in home language, the students have no voice. This paper explored the language that Grade R students use to navigate mathematics space to achieve the policy's envisaged outcomes. The theoretical premise for the study is based on Vygotsky's theory of social constructivism. The qualitative approach guided by the case study design of Grade R 8 students were employed in the study. It is anticipated that this paper provides valuable insights into the understanding of the Grade R students' acquisition of mathematical language and contributes to the increased awareness in the field of Grade R mathematics teaching to achieve active learning. The findings revealed that (i) students failed to establish that the number of objects does change with the changed arrangement of the same number of objects and (ii) students were able to understand questions asked in their home language, yet their responses were in English.

**Keywords:** active learning, grade R, home language, second language, students

#### **1. Introduction**

In South Africa, schooling system ranges from Grade R to 12. Grade R is the reception year. This is part of the Department of Basic Education's National Curriculum. It is aimed to give children a firm foundation in preparation for grade one (it is preprimary phase). Grade R is not mandatory; however, the child who skips it is more disadvantaged than the one who goes through it.

Language plays a major role in the teaching and active learning of mathematics to any grade. Many students find it difficult to excel due to the language of instruction, which acts as a barrier to their learning. Several studies [1, 2] acknowledge that English proficiency heralds mathematics proficiency, particularly when English is the medium of instruction. As English second language (ESL) students struggle to understand mathematics concepts that are being taught the English first language

(EFL) speakers feel more comfortable because they understand the medium of instruction. Research confirms that ESL performs low in mathematics compared to their counterparts EFL speakers [3]. Numerous research studies in language acquisition and mathematical learning have developed along mostly discrete trajectories, for example, studies examining the links between linguistic and mathematical literacy [4], the functions of language in the math classroom [5] established the framework defining the four stages of mathematical learning: Receiving, Replicating, Negotiating meaning and Producing. However, few studies have investigated the language that Grade R students use to navigate the mathematics space to achieve the envisaged outcomes of the policy. The two questions that still need to be answered in this study are the following:


Research shows that home language is important and it is acquired during early childhood, commonly before the age of three [6], it is at this stage that the child's development and acquisition of mathematical concepts is critical. Learning a language is not just a simple thing that happens effortlessly. Research argues that learning the first language is one of the unexplainable daily mysteries surrounding us [7]. Many people think that children put no effort to learn the language, but the truth is that there are several stages that a child must go through to learn a language. If learning mother tongue language is that difficult what more adding a foreign language to the child's mind? Across multiple contexts, research on the language of instruction reveals that where native language is used for mathematics learning, teaching and assessment, native speakers of the language of instruction achieve higher scores than the non-native language speakers [8–10].

In [11] behaviourist theory argues that the acquisition of language can be observed. Skinner believed that children are born with a blank slate of mind or tabula rasa. Children acquire the first language by responding to stimuli given to them and they respond through conditioned reinforcement. The innatist theorists including [12] believed that children have a blueprint for language acquisition called Language Acquisition Device (LAD). The LAD is responsible for swift mastery of the language, and this makes it possible even if a child is exposed to the abstract language. The innatist theorists refute Skinner's theory [13] theory also supports the innatist theorists' claims indirectly though, by arguing that the conversations that children engaged in constitute the origins of both language and thought, where thought is fundamentally internalised speech and speech develops in social interaction.

Even though many people advocate for the use of Home Language (HL) as the only medium of instruction, research heightens the importance of learning a second language as well. It is argued that bilinguals have more advantage in performance than monolinguals [14–16]. Second language acquisition increased working memory, and this enables second language learners to achieve higher in mathematics because maths achievements are influenced by the enhanced working memory [17, 18]. The acquisition of the second language is associated with the increase in the density of the grey matter in the brain, which enhances performance [19]. Piroozan et al. [20] study in Iran found out that the children who acquired a second language outperformed in mathematics tests their counterparts who were monolingual and knew their mother tongue only.

*The Language That Grade R Students Use to Achieve the Envisaged Mathematics Outcomes… DOI: http://dx.doi.org/10.5772/intechopen.105446*

#### **2. The South African curriculum and assessment policy statements**

In South Africa, the Department of Education introduced the National Curriculum and Assessment Policy (CAPS) for the subjects listed in the National Curriculum Statement for Grade R-12. The CAPS was designed to shed more light on teachers on what and how they should teach students. According to [21] the National Curriculum Statements Grade R-12 envisaged producing students who can:


To achieve the intended outcomes, the designed activities are expected to be minds-on and hands-on to develop diverse mathematical skills in students. Asmal [21] emphasises that the designed activities must not be 'keep busy' activities but must focus on meaningful mathematics content. The Grade R curriculum is designed in a way that the time a student exits Grade R is fluent in number sense and the four basic operations. The aim is to ensure that students are competent and confident with numbers and calculations [21].

For the deliverance of mathematics content meaningfully, the Language of Learning and Teaching (LoLT) plays a vital role, with this understanding, the DBE made a provision through [22], which is discussed in the section below.

#### **3. The South African language in education policy on education**

In this section, we present the LiEP and how it tends to drive education. In South Africa, during the Apartheid regime, access to education was not equal for all, especially black people were offered low-quality education, for example limited resources, overcrowded classrooms, insufficient infrastructure and ill-equipped teachers [23]. To redress such racial inequality in education, the department of education developed LiEP which stipulates the aims of education as:

• to promote full participation in society and the economy through equitable and meaningful access to education;


The [22] advocates for all languages to be considered equal and for students to learn in the language that they are comfortable with. Research argues that even though the government advocates for the use of mother tongue, if there is no support, students can still suffer because every medium of instruction needs adequate support to attain the intended outcomes [24].

The next section discusses mathematics as a language and how it must be presented to the students for easy access.

#### **4. Related work on mathematics as a language**

Even though there is LoLT, mathematics is a language that has its own syntax and symbols. Asmal et al. [21, 25] describe mathematics as a language that uses symbols, terminology and notations for describing numerical, geometric and graphical relationships to communicate information. To provide students with multiple opportunities to learn worthwhile mathematics, mathematics teachers need to understand the specialised language of mathematics learning and teaching [26]. For the students to pass mathematics, they must first understand basic concepts which are the building blocks of this subject. It is from those basic concepts that the bigger mathematical ideas emerge.

In the true sense of the matter, many African countries have been considering the language of the colonisers as more superior to their own native languages, such connotations are misconceptions because early mathematical concepts are easily understood at a young age when learnt in their mother tongue. The literature argues that forcing students to learn mathematics in the second language poses a threat to their ability to 'thrive' mathematically and, subsequently, undermines their interaction with the wider mathematics community [24, 27]. For years the government has been pushing children to learn mathematics in the ESL, but without adequately supporting them with resources to develop the required mathematical competencies [24]. Any language of learning and teaching needs adequate support for the students to access the information without any setbacks.

Language allows the possibility to link from one concept to another, it is from the link that the meaning of concepts is derived, if the language is weak, the ability to learn is negatively affected [28].

*The Language That Grade R Students Use to Achieve the Envisaged Mathematics Outcomes… DOI: http://dx.doi.org/10.5772/intechopen.105446*

Likewise, [4] argues that language performs at least three critical roles in the classrooms:


Failure to put much attention to the language issues simply means failing the society. The medium of instruction must be accessible to the students without putting much effort. Students individually have their own ways of learning mathematics. Each person is unique, and this is what teachers must take into consideration when designing learning activities. The section below horns on ways Grade R students learn mathematics.

#### **5. Grade R students' ways of learning mathematics**

Research highlights that the first 1000 days of child are critical to the child's future, that is where the child's foundation for healthy behaviour and learning is determined [29]. This implies that once the foundation of learning mathematics concepts is not well-grounded, the whole school life of the child is doomed. In support of this, [30] emphasised that the foundation for lifelong learning concepts, skills and attitudes is acquired during the early years. It further explains that even the development of emotional intelligence namely, confidence, curiosity, purposefulness, self-control, connectedness, capacity to communicate and cooperativeness is acquired during the early years. Feza [31] highlights the numerical abilities of young children prior Grade R possess that exceed the Grade R curriculum expectations supporting literature on the attainment of numerosity before formal learning.

The learning and teaching aspect in Grade R should focus on the holistic development of the child [21]. The teaching aspect should aim to develop the child's emerging numeracy through activities that develop cognitive (problem-solving, logical thinking and reasoning), mathematical language, perceptual-motor, emotional and social aspects. According to [21] the aspect highlighted in this paragraph can be learnt and developed through:

*Stories, songs, rhymes, finger games and water play, educational toys including board games, construction and exploration activities (mass, time, capacity, measurement, etc.), imaginative play, outdoor play and 'playground games'. Many kinds of games and play could include aspects of numeracy, for example measuring during cooking or counting during shopping (p. 14).*

'Play' is undermined as one of the ways which children learn, yet it is essential to the development and strengthening of the child's creativity, imagination, dexterity, cognitive, social, physical, healthy brain and emotional well-being [32, 33]. The studies by [34] have shown that for young students from low-income backgrounds, their numerical knowledge can be promoted by playing a simple number board

game. Furthermore, [35] explored cultural games' contribution to early years of mathematics, discovering their strength in developing number sense and sequencing. Barnard and Braund [36] argue that Grade R teachers continue to allow free play with no purpose although literature advocates for meaningful play. With intervention, this practice can improve towards meaningful goal-oriented play [24, 37]. This implies that Grade R teachers must be creative to instil the culture of learning through play in their centres of learning.

When children play, it means they are in contact with the environment. For mathematics to make meaning to the students, the teaching approach must be more immersed in context-based problems [38] that are meaningful and applicable to their background experiences [39]. Background experiences are important because when children learn to count, the newly acquired symbolic representations of numbers are made to fit onto pre-existing non-symbolic representations [40]. It is worth noting that Grade R come to school with the knowledge of informal numeracy, which needs to be expanded, enriched and developed through appropriately designed learning activities [41]. Learning and teaching that does not include play, rob students of their potential to learn. Games or songs need to be well planned that as they play or sing, mathematics concepts are acquired and developed.

Research highlights that a child learns mathematics informally in home environment; much of their learning is social in nature [42]; for example, it takes place with parents during time of meals, chores and shopping through one-to-one correspondence. The home environment is rich with numerical information [43]. Mathematics teachers must know that when planning for class activities, they plan for students who have some mathematical ideas from home. Home experiences must be taken as the framework upon which formal mathematics can be built.

To have minds-on and hands-on activities as envisaged by the DBE means Grade R teachers must be dedicated and creative in their preparations. As the mediator of learning the teachers must be proactive in everything to ensure that all students are catered for despite their differences. From what is presented, it can be emphasised that schools need well-trained qualified Grade R teachers who are creative enough in teaching mathematics.

#### **6. Conceptual framework**

This study is situated in a social and cultural context, and the sociocultural perspective provides a wider lens of how both the teacher and the school provide opportunities for students to learn mathematics. It also provides an opportunity to explore how students navigate through the language to access the intended mathematical content knowledge. The theoretical premise for the study is based on [13] theory of social constructivism. Ref. [13] argues that learning occurs when an individual internalises a social experience through interacting with a peer or an adult. In Vygotsky's cultural-historical theory, play is an essential part of early childhood. Vygotsky believed that play promotes cognitive, social, and emotional development in children. In mathematics education, students are expected to construct their own mathematical knowledge from previous experiences as they interact with peer or adult.

*The Language That Grade R Students Use to Achieve the Envisaged Mathematics Outcomes… DOI: http://dx.doi.org/10.5772/intechopen.105446*

#### **7. Methodology**

The study employed a qualitative approach guided by the case study design. The study's sample comprised eight Grade R students of mixed gender purposefully selected from the five primary schools in the Queenstown district of South Africa. Students in these schools belong to the same cultural group and speak the same home language. The data was collected using the video camera, and the video clips were then viewed to elucidate how Grade R students navigate the language space in the lesson. The data was coded and analysed thematically and reported in themes.

#### **7.1 Instruments**

The data was collected from the video clips which were captured in different schools when students were being engaged in different mathematical activities, for example, to identify the number of items per group as shown in **Figure 1** below, in **Figure 2** students were to identify numbers arranged in mixed order and to match the number of items on the right-hand side with the correct numerical value on the left-hand side. The questions were asked in learners' HL (IsiXhosa) one of the South African native languages.

Activity 1.

Researcher: Jonga ezi zam zisemfanekisweni iziciko (Look at my bottle-tops in the pictures).

Question 1: Researcher: Zeziphi ezininzi? (Which ones are many?)

Questionn2: Researcher: Khazibale sibone (Count them).

Question 3: Researcher: Xa uzidibanisa zonke zingaphi? (When you add them all, how many are there?)

After this activity students were further engaged in Activity 2 below, which sought to examine their language and numerical proficiency.

Activity 2.

Before the students were to match the number of items on the right-hand side with the correct numerical value on the left-hand side, they were asked to identify the numbers on the left-hand side arranged in mixed order.

Researcher: Khawutshatise inani ngalinye nomfanekiso walo? Sebenzisa icrayoni. (Match each number with its picture. Use a crayon).

#### **Figure 1.**

*(a) and (b) The two groups of five bottle-tops arranged in a different order.*

**Figure 2.** *Identification and matching activity.*

#### **8. Ethical issues**

All the instruments used to collect data for the study were ethically cleared by the Central University of Technology.

The permission to collect data from the schools was sought and granted by the Queenstown district education office. Subsequently, the school principals of the schools in the project permitted the researchers to collect data without any hindrances.

Considering the age of the participants, the consent forms were distributed and signed by the parents of all the participants involved in the research project.

Participants' parents were assured of the anonymity that no real names of the participants were to be used when reporting the outcomes of the research project. The *The Language That Grade R Students Use to Achieve the Envisaged Mathematics Outcomes… DOI: http://dx.doi.org/10.5772/intechopen.105446*

following codes were used: School 1: Student 1 (S1S1), School 1: Student 2 (S1S2), School 2: Student 1 (S2S1), School 2: Student 2 (S2S2), School 3: Student 1 (S3S1), School 3: Student 2 (S3S2), School 4: Student 1 (S4S1), School 4: Student 2 (S4S2), School 5: Student 1 (S5S1), School 5: Student 2 (S5S2).

#### **9. Findings**

When students were asked to compare the two categories of bottle-tops and identify the group that has many bottle-tops. Students' responses varied as shown in **Table 1**.

**Table 1** shows how each of the eight students responded to the three questions of Activity 1. S1S1, S1S2, S2S1, S2S2 and S5S1 identified the number of bottle-tops in **Figure 1b** as greater than those in **Figure 1a** while S3S1, S4S1 and S4S2 considered the bottle-tops in **Figure 1a** to be more than those in **Figure 1b**. Responding to the second question, all other students counted the bottle-tops in English except S1S2 who counted in HL, but not in an orderly manner, the last number mentioned was 'shumi' (ten).


#### **Table 1.**

*How students responded to questions 1 to 3 of Activity 1.*


#### **Table 2.**

*How the students identified and named the given numbers (3, 1, 5, 2 & 4).*

Responding to Question 3, S1S1 and S5S1 said that bottle-tops in **Figure 1a** and **b** combined together are 'ten' while S1S2 said that 'shumi' (ten). S2S2 and S4S2 responded that the bottle-tops combined are 11. S1S1 and S5S1 said that the total number of bottletops is five (5). S3S1 did not give any response to the question.

**Table 2** illustrates how each of the students named the listed numbers. Of all the students in the sample, only S4S1 and S4S2 managed to identify the given numbers correctly. S2S1 managed to identify all other numbers except one which was identified as six. The responses given by S1S1 were in IsiXhosa (Home language, HL); even though HL was used, the students could not get the question correct. S2S2 could not be able to identify the given number correctly. S1S2 and S5S1 both students were not able to identify the first four listed numbers while the fifth number was identified by S5S1 as five, yet it is four and S1S2 gave no response to that.

The next question required students to match the number of items on the right-hand side with the correct numerical value on the left-hand side as shown in **Figure 2**. **Table 3** below illustrates how each of the students responded to the question.

The data presented in **Table 3** above illustrate students' responses to the matching item question. S2S1, S4S1 and S4S2 managed to match the given number of items to the correct responding numerical values.

Another group of three students, S2S2, S3S1 and S5S1 matched five items to a numerical value of 4. S3S1 provided unique responses as follows: matched two and three items to numerical values of five and two, respectively. No match was given to numerical values of three and one; one and four items.

S2S2 matched three and four items to a numerical value of five and no match was made for the numerical value of three. S5S1 matched four and one items to the numerical value of one while three items and a numerical value of three were not matched to anything.

Another unique response was given by S1S1 who straight matched the number of items on the right-hand side to the numerical values on the left-hand side as explained; three, one, five, two and four items were matched to the numerical value of five, two, three, four and one, respectively.

The last student S1S2 when asked to match the items to the corresponding items could not match any.

*The Language That Grade R Students Use to Achieve the Envisaged Mathematics Outcomes… DOI: http://dx.doi.org/10.5772/intechopen.105446*

**Table 3.** *How students matched the group of items to the numerical value.*

#### **10. Discussion of the findings**

The findings are discussed in this section. The discussion is based on how each of the students responded to the questions and what that means to the research community as far as learning and teaching early childhood mathematics is concerned.

The findings reveal that some of the students are comfortable with the use of HL to learn mathematics, i.e., S1S1 proved that by counting in HL as shown in **Table 1**. Even though the student was not accurate in the counting, but it showed the student had a bit of an understanding of the mathematical terms in HL, such ideas just need to be supported. This finding is consistent with [24] assertions that the principal use of HL represents an immeasurable fund of knowledge and an essential cognitive resource for mathematical sense-making for ESL speakers [24].

The ability to match the items to the given numerical values shows students' proficiency in counting. The researchers asked the students in HL, but S2S1, S4S1 and S4S2 responded in English and got the question correct, this reveals that some students are bilingual. From the findings of this study, it is also revealed that most students use both languages (English and HL, IsiXhosa) to navigate the learning concepts. This leads to suggest that students must be supported in being proficient in HL which can be used as a resource to learn the second language. Literature reveals that bilinguals benefit from advanced inhibitory control skills compared to monolinguals [14–16] as a result they perform better in mathematics than their counterparts (monolinguals).

For students like S1S1, S1S2, S2S2, S3S1 and S5S1 who had problems in matching the items to the correct numerical values, such problems emanate from a misunderstanding of mathematical language. It is argued that communicating mathematically requires an in-depth comprehension of the mathematical language and the multiple illustrations employed in communicating mathematical concepts [5].

Some of the students, for example, S1S1, S2S1, S4S2 and S5S1 when counting the bottle-tops in **Figure 1a** and **b** counted in English, listening carefully to their counting, skipped some of the numbers which is an indication that they are not yet conversant counting in English. Literature highlights that students in the advanced stage of language native-language acquisition can easily extend vocabulary, and good comprehension of the second language [5]. This implies that students need to be thoroughly developed in HL to have access to the Second Language (SL). Researchers argue that HL must be viewed as 'resources' if it needs to benefit the bi- or multilingual education system [9, 10].

The researcher used HL in all the questions, but surprisingly most of the students responded to the questions in English, for example, the counting of the items. Out of eight students involved in the study, one used HL to name the given numerical values, and the rest responded in English. For such instances, there is a need to support bilingual abilities because learning a second language assists in the growth of the density of grey matter in the left inferior parietal cortex of the brain, which leads to an improvement in performance. Research highlights that the acquisition of the second language at a young age, the denser the grey matter gets which is an advantage to the students [19].

#### **11. Conclusion**

The findings of this study led to the following conclusions:


#### **12. Recommendations**

Based on the findings of this study, we recommend the following:


#### **13. Limitations**

These research findings cannot be generalised but can be transferrable to other contexts to strengthen the findings. It is advisable that other researchers should interpret the findings with caution as they are based on the data collected from the *The Language That Grade R Students Use to Achieve the Envisaged Mathematics Outcomes… DOI: http://dx.doi.org/10.5772/intechopen.105446*

same cultural group that speaks the same home language. The replication of the same research with a different cultural group may yield different findings.

#### **14. Suggested areas for further studies**

We suggest that further studies can be conducted on:


#### **Acknowledgements**

The authors would like to thank all the students who participated in the study to make it a success.

#### **Declaration of interest**

No potential conflict of interest was reported by the authors.

#### **Author details**

Shakespear M. Chiphambo1 \* and Nosisi N. Feza2

1 Walter Sisulu University, South Africa

2 University of Venda, South Africa

\*Address all correspondence to: schiphambo@wsu.ac.za

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[28] Harrison D. Language Can Reshape our Economy. *News 24*; 2014. South Africa. Available from: http://www.news24.com/archives/ city-press/language-can-reshape-oureconomy-20150430

[29] The South African Early Childhood Review (2017), Ilifa Labantwana, The Children's Institute at the University of Cape Town, the Department for Planning, Monitoring, and Evaluation (DPME) in the Presidency, as well as Innovation Edge

[30] Asmal K. South African Government. Education White Paper 5 on Early

Childhood Education 2001, Meeting the Challenge of Early Childhood Education in South Africa. Pretoria, South Africa. [Accessed: 15 June 2021] Available from: https://www.gov.za/documents/ education-white-paper-5-earlychildhood-education

[31] Feza NN. Self-regulation in early years of learning mathematics: Grade R observed self-efficacy skills shared and aligned. In: Feza N, editor. Metagonition in Learning. United Kingdom; 2019:83- 95. DOI: 10.5772/intechopen.788929

[32] Tamis-LeMonda CS, Shannon JD, Cabrera NJ, Lamb ME. Fathers and mothers at play with their 2- and 3-year-olds: Contributions to language and cognitive development. Child Development. 2004;**75**:1806-1820

[33] Ginsburg KR. The importance of play in promoting healthy child development and maintaining strong parent-child bonds. Pediatrics. 2007;**119**(1):182-191. DOI: 10.1542/peds.2006-2697

[34] Ramani GB, Siegler RS. How Informal Learning Activities Can Promote Children's Numerical Knowledge. Psychology, Cognitive Psychology, Educational Psychology, Developmental Psychology. Oxford Handbooks Online. United Kingdom. 2014

[35] Feza NN. Black students' rich mathematical experiences: Mathematics concepts and Xhosa cultural games for reception class. In: Farland-Smith D, editor. *Early Childhood Education*. 2019b:151-163. United Kindom. DOI: 10.5772/intechopen.73467

[36] Barnard E, Braund M. Strategies for the implementation of mathematics in grade R: Teachers' beliefs and practices. South African Journal of Childhood Education. 2016;**6**(1):a409. DOI: 10.4102/ sajce.v6i1.409

[37] Hazell E, Spencer-Smith G & Roberts N. Improving Grade R Mathematics Teaching in South Africa: Evidence from an Impact Evaluation of a Province-Wide Intervention. Durban; 2019.

[38] Van den Heuvel-Panhuizen M, editor. Children Learn Mathematics: A Learning-Teaching Trajectory with Intermediate Attainment Targets for Calculation with Whole Numbers in Primary School. Rotterdam/Tapei: Sense Publishers; 2008

[39] Ginsburg H, Galanter M, Morgenlander M. Big Maths for Little Kids Workshops. New York: Teachers College Innovations; 2004

[40] Batchelor S, Keeble S, Gilmore C. Magnitude representations and counting skills in preschool children. Mathematical Thinking and Learning. 2015;*17*(2-3):116-135. DOI: 10.1080/10986065.2015.1016811

[41] Papadakis S, Kalogiannakis M, Zaranis N. Improving mathematics teaching in kindergarten with realistic mathematical education. Early Childhood Education Journal. 2017;**45**(3):1-10. DOI: 10.1007/ s10643-015-0768-4

[42] Ramani G, Siegler R, Hitti A. Taking it to the classroom: Number board games as a small group learning activity. Journal of Educational Psychology. 2014;*104*(3):661-672. DOI: 10.1037/ a0028995

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#### **Chapter 6**

## A Method to Improve Comprehension and Learning in Science Education – A Case Study in Systems Engineering and Automation

*Sergio Velázquez-Medina and Pedro Cabrera-Santana*

#### **Abstract**

Difficulties are commonly detected in students with respect to the acquisition of certain specific competencies in a particular topic. One strategy to optimize the assimilation of knowledge and improve the learning results of students in a specific topic is through the use of the active learning process. Active learning can serve to facilitate autonomous and collaborative learning in specific topics as a complement to in-person classes. In this chapter, a method to improve comprehension and learning is developed and applied, using for this purpose both autonomous and collaborative works. The case study presented is undertaken for one of the subjects in the area of systems engineering and automation in one of the public universities of Canary islands (Spain). Different specific topics of the subject were selected. To check the effect of the application of the proposed method, a statistical analysis was performed. For this objective, *t*-test and the p-value statistical were used. As results, it was found that 100% of the students who presented some difficulty in relation to the general subject obtained higher relative results in the specific topics that they worked on when employing the proposed method, compared with their global result in the subject.

**Keywords:** integrated active learning, improve comprehension, autonomous learning, collaborative learning

#### **1. Introduction**

It is not uncommon to find that, for one reason or another, it is difficult to transmit certain knowledge of some subjects to the student, or that, because of the nature of such knowledge, its transmission entails certain difficulties [1–5]. In consequence, the skills derived from such knowledge may not be correctly acquired. In [3], a study of this question in relation to the particular case of students with difficulties in acquiring knowledge in topics related to energy conservation was undertook. They detected

that students had difficulties in acquiring skills because of the presence of complex scientific concepts and because they found the topic to be monotonous. In [1], it was identified how the active learning process improves the cognitive engagement with concepts in the classroom.

Autonomous learning, accompanied by a close follow-up on the part of the teacher, is one of the strategies that can be used to achieve better learning results in certain specific topics and to optimize skills acquisition [2, 6–11].

In [11], the importance of autonomous learning was established, defining it as a process in which the individual takes the initiative, with or without the help of others, in diagnosing the learning needs, formulating the learning targets and identifying the human and material sources required for the learning process. In [7], the importance of the developing skills in autonomous learning by engineering students is pointed out, as a way to support life-long learning. In [8], the feasibility and functionality of the implementation of autonomous learning in Pakistan are analysed, studying for this purpose the particular case of the teaching of English in four universities in the country. The professors concluded that the implementation of autonomous learning techniques helps to make students more self-sufficient. In the implementation process of the techniques they used, they observed different obstacles that arose as result of particular features of the education system in their country, cultural precedents and certain psychological aspects of the students.

When autonomous learning is assimilated by a student, it can subsequently be transmitted to the rest of the students as a collaborative learning strategy [12] and as a complement to the classes of the teacher. The benefits of collaborative learning include**:** development of higher-level thinking, oral communication, self-management, leadership skills, promotion of student-faculty interaction; increase in student retention, self-esteem and responsibility and a better understanding of diverse perspectives [13]. In [14], the collaborative problem-solving games into spatial dialogues in a virtual cellular biology game is explored.

Various techniques found within the framework of active learning processes can be used in collaborative teaching as alternatives to traditional processes [12, 15–19].

In [12], a particular case of the teaching of mathematics is studied as part of the Business Administration Degree at the International University of Catalunya (Spain). They had noted a certain lack of motivation on the part of the students with respect to this subject and began to use game-based learning (GBL) techniques with the aim of stimulating interest in it. According to the authors, the results obtained were very promising. In [19], how collaborative inquiry learning offers educators a context within which support can be found for the understanding of scientific-technological concepts is analysed. In their case study in the United States, they used both GBL and problem-based learning (PBL) techniques.

From the analysis carried out on the scientific publications found, it can be deduced that all of them make exclusively a qualitative analysis from their experiences. In addition, in their studies they use a single technique in the collaborative learning process, generally game-based learning.

An interesting and useful technique for the transmission of acquired knowledge to the rest of the students when the number of students is high, and/or when insufficient time is available in the in-person class, involves the making of mini-videos (less than 10 minutes long) in which each student transmits the knowledge they have acquired [20–23]. Supervised and monitored by the teachers, these videos can subsequently be

*A Method to Improve Comprehension and Learning in Science Education – A Case Study… DOI: http://dx.doi.org/10.5772/intechopen.106477*

published in a virtual environment, open to the other students, so that they can view them and use them as an educational complement.

In the study presented in the present paper, a specific method is developed within the framework of the active learning process. The method is applied to the particular case of the learning of a topic related to systems engineering and automation. The original contributions of this study are as follows:


#### **2. Methods and materials**

#### **2.1 Method**

Among others, the Center for Teaching Innovation of Cornell University [13] defines different basic aspects that need to be considered when establishing an innovative method related to active learning. With this in mind, and for the particular case study presented in this paper, the method developed by the authors is structured into four stages (**Figure 1**).


The activity that is developed will serve as a complement to the classroom teaching, and the active learning during the autonomous and collaborative work stages will serve as reinforcement for the students in the assimilation of competencies.

#### *2.1.1 Activity preparation stage*

At the beginning of the term, a clear description and explanation of the activity and its purpose are given to the students. The rules for participation in the activity and the marking criteria are explained (**Figure 1**).

For the case study, the activity was obligatory and the mark given to it corresponded to 5% of the total mark for the subject.

The teacher, after looking at the historical data on the topics and/or concepts that have commonly presented certain assimilation difficulties on the part of students, designs related work activities. The random formation of groups with a maximum of two students is proposed. Each group is then randomly assigned the task of preparing a series of concepts linked with one of the specific topics (hereinafter referred to as STO). Work on concepts related to the same STO can be assigned to different groups.

#### *Active Learning - Research and Practice for STEAM and Social Sciences Education*

*Method applied to the case study.*

#### *A Method to Improve Comprehension and Learning in Science Education – A Case Study… DOI: http://dx.doi.org/10.5772/intechopen.106477*

Initially, in the case study, tasks linked to six STOs of the subject were assigned, ensuring that each STO was tackled by a minimum number of students.

For the development of the collaborative work stage, each group is assigned one of the following techniques: PBL directly in the classroom, the application of GBL directly with the rest of the students or the making of mini-videos with a maximum duration of 5 minutes which, after inspection by the teacher, are uploaded to the virtual environment of the subject in question to be shared with the other students who signed up for the subject.

#### *2.1.2 Autonomous work stage*

For the follow-up of the work of each group, the professor arranges voluntary tutorials at specific times and dates outside the teaching timetable.

For the case study, 4 h were allocated each week in the virtual environment for 15-minute sessions with each group should they require assistance with the task.

Each group can request a date and time through the virtual environment of the subject in question. In this stage, the professor monitors the work of each group to ensure the subsequent success of the collaboration stage, helping the students in the development of the required skills, the approach to typical problems, reflection techniques, etc. Any group that does not implement this stage in an appropriate way is removed from the activity and is given a mark of 0. The groups that perform this stage correctly advance to the collaborative work stage. Prior to commencement of the collaborative stage, the groups have to prepare and submit a document explaining the work that has been developed along with the guidelines they will follow for that stage. These documents are uploaded to the virtual environment of the subject in question which the other students will have access to.

#### *2.1.3 Collaborative work stage*

In this stage, each group has to transmit to the rest of the class the knowledge that they have acquired in the STO they have been working on. For this, they use the PBL, GBL or mini-video strategy. The mini-videos must be viewed individually by the other students in the class. This latter strategy is particularly useful when there are time restrictions and/or a large number of students have signed up for that particular subject.

#### *2.1.3.1 Marking the activity*

The student must have advanced to the collaborative work stage for the work developed to be evaluated and awarded a mark. The weight of the mark given is evenly split between the assessment of the professor and the assessment of the other students in the class (except for the second member of that student's work group). The mark given should take into consideration the document presented in the autonomous work stage as well as the activity developed in the collaborative work stage (see **Figure 1**).

For the assessment made by the students themselves of the work of their classmates, a questionnaire has been developed using a five-point Likert scale [24]: 1: Very poor; 2: Poor; 3: Sufficient; 4: Good; 5: Excellent. Aspects such as the following should be evaluated:


#### *2.1.4 Assessment stage of the success of the method*

Finally, for the evaluation of the success of the method, the results from three of the six STOs were used. The groups which participated in the three STOs which were discarded had not performed the task in the prescribed manner (**Figure 1**). The students who worked on the selected STOs are given the name of final test students (FTSs).

The data considered for the assessment of the success of the method applied to the case study are the results obtained in the official examinations and on the official dates for the subject in question. For the purposes of the assessment, this official examination should clearly differentiate between the different STOs worked on in the learning activity, allowing a mark for all students for each STO as well as a global mark in the subject.

The following data are analysed in the assessment of the success of the method:


#### **2.2 Materials**

One of the subjects taught in the School of Industrial and Civil Engineering as part of the Industrial Organization Engineering Degree is Control Engineering. Within the structure of the degree course, it is considered part of the 'Automation' material. The degree comprises four academic years of two terms each, with a total of 240 ECTS (European Credit Transfer and Accumulation System) credits [25]. The Control Engineering subject is given in the second term of the third year of the course.

The degree meets all the official teaching regulations in Spain and has been subjected to and passed an evaluation process in accordance with the protocol established by the corresponding State and Autonomous Community agencies. It additionally complies with the quality criteria and standards established by the European Association for Quality Assurance in Higher Education (ENQA) [26] and is registered in the Spanish Registry of Universities, Centres and Qualifications (Spanish initials: RUCT) [27].

The so-called Verification Report (Memoria de Verificación in Spanish) is the reference document for each official qualification in the Spanish state. The document includes the objectives, competencies, academic structure, available resources, etc. for *A Method to Improve Comprehension and Learning in Science Education – A Case Study… DOI: http://dx.doi.org/10.5772/intechopen.106477*

each qualification, including the reference degree of the present case study. In point 2 of the corresponding Verification Report, the competencies associated to the degree in question are listed, the following of which can be highlighted:

• Transversal competency G6: 'AUTONOMOUS LEARNING. Detecting deficiencies in one's own knowledge and overcoming them through critical reflection and choosing the best option to extend this knowledge'.

This competency is associated to the reference subject in its corresponding Teaching Programme.

The case study considered in the present paper was undertaken for the aforementioned subject of Control Engineering during the 2018/2019 academic year. The number of students included in the initial stage of the trial was 37, with a total of 20 groups being formed (see **Figure 1**). As an additional tool, the virtual work environment of the reference university was also used for the management of the activity.

#### **3. Discussion of the results**

Following the criteria established in the methodology (see section 2.1.4), a total of 17 students advanced to the collaborative work stage and were thus considered FTSs. These 17 students worked on three of the STOs that were initially proposed in the activity. **Table 1** shows the STOs that were finally evaluated and the corresponding number of FTSs.

**Figure 2** shows the results of the comparison of the marks obtained by each FTS in the global examination of the subject with the mean mark obtained by all the students. It can be seen that some of the FTSs had general difficulties in assimilating concepts of the subject, while others appear to have found it easier.

**Figures 3**–**5** compare the mean grade obtained by each FTS in the specific parts of the global exam concerning the three STOs that were worked on with the mean grade of all students in the same STO. In general terms, it can be seen that whereas a total of 7 FTSs scored below the mean grade in the global exam of the subject (see **Figure 2**), for the case of the results in the STOs, only two of these (FTS-1 and FTS-5) were awarded a grade below the mean value of all students.

To better evaluate the improvement in the results of each FTS as the result of the application of the method developed in the present paper, the so-called relative grade (RG) metric was used (Eq. 1):


#### **Table 1.**

*Description of specific topics (STOs) worked on by the final trial students (FTSs).*

#### **Figure 2.**

*Comparison of the grade of each FTS in the global exam with the mean grade of all students (minimum grade of 0 and maximum of 10).*

#### **Figure 3.**

*Comparison of grade awarded in STO-1 by each FTS with the mean grade of all students ((minimum grade of 0 and maximum of 10).*

#### **Figure 4.**

*Comparison of grade awarded in STO-2 by each FTS with the mean grade of all students ((minimum grade of 0 and maximum of 10).*

$$RG = \frac{G\_{-}FTS\_{i} - \overline{Grade}}{\overline{Grade}} \tag{1}$$

where *G\_FTS*i is the grade obtained by each FTS, and *Grade* is the mean grade of all the students who sat the exam.

*A Method to Improve Comprehension and Learning in Science Education – A Case Study… DOI: http://dx.doi.org/10.5772/intechopen.106477*

#### **Figure 5.**

*Comparison of grade awarded in STO-3 by each FTS with the mean grade of all students ((minimum grade of 0 and maximum of 10).*

#### **Figure 6.**

*Comparison of relative grades for each FTS.*

**Figure 6** compares the RGs for each FTS obtained in the global exam (**Figure 2**) with those obtained in the different STOs (**Figures 3**–**5**). It can be seen that 13 of the 17 FTSs improved their relative results as the result of the application of the innovative educational method developed. It should be highlighted that all the FTSs who had difficulties with the assimilation of concepts of the general subject (FTSs with values of RG<0 in the official global exam) improved their RGs in the STOs that they worked on.

Statistical analysis is performed to check the effect of the application of the proposed method. Since we want to evaluate the degree of association or independence between a quantitative variable and a categorical variable, the inferential statistical procedure resorts to comparing the means of the distributions of the quantitative variable in the different groups established by the categorical variable. Being this dichotomous, *t*-test with a significance value of 0.05 was used. Further, the parametric requirements are met, such as the normal distribution of the quantitative variable in the groups that are compared and the homogeneity of variances in the populations from which the groups come [28].

First, it was evaluated the difference between the results obtained for the RG in the worked STO with those in the official global exam (**Figure 6**). These values were compared with a null hypothesis where that difference was less than or equal to zero. So, it obtained a *p* value of 5.4E-3. This indicates that student improve their results when the proposed method was applied.

#### **4. Conclusions**

After application of the method developed in this paper, student assimilation of competencies which had historically presented certain difficulties improved in general. A total of 88.2% of the final test students (FTSs) who used the proposed method obtained higher grades in the specific topic they were working on than the mean grade of all the students. In addition, 76.5% of the FTSs obtained a higher relative grade (RG) in the specific topic they were working on with the proposed method than the RG they obtained in the global exam of the subject, with this value rising to 100% in the case of FTSs with the highest difficulty in the subject (with an RG<0 in the global exam).

The p value obtained in the statistical analysis guarantees the results obtained with the proposed method. It can be applicable to subjects in any branch of science, as a way to improve the assimilation of competencies that students tend to have difficulties with.

#### **Acknowledgements**

This research has been co-funded by ERDF funds, INTERREG MAC 2014-2020 programme, within the ACLIEMAC project (MAC2/3.5b/380). No funding sources had any influence on study design, collection, analysis or interpretation of data, manuscript preparation or the decision to submit for publication.

### **Conflicts of interest**

The authors declare that there are no conflicts of interest.

### **Author details**

Sergio Velázquez-Medina1 \* and Pedro Cabrera-Santana<sup>2</sup>

1 Department of Electronics and Automatics Engineering, University of Las Palmas de Gran Canaria, Canary Islands, Spain

2 Department of Mechanical Engineering, University of Las Palmas de Gran Canaria, Canary Islands, Spain

\*Address all correspondence to: sergio.velazquezmedina@ulpgc.es

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*A Method to Improve Comprehension and Learning in Science Education – A Case Study… DOI: http://dx.doi.org/10.5772/intechopen.106477*

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### **Chapter 7**

## Perspective Chapter: Active Learning Strategies in the Veterinary Medicine Programme under the Think4Jobs Project

*Rita Payan-Carreira, Hugo Rebelo and Luís Sebastião*

#### **Abstract**

Active learning has been introduced in the Universities to reinforce the students' skills development and increase their motivation and engagement while also fostering the transferability of knowledge into the profession, contrasting with a classical approach, where passive knowledge transfer occurs, and students act as sponges for information. Albeit not completely conceptualized, active learning demands the student's involvement with the learning activities, the analysis and ability to respond to specific situations, and a critical reflection on the learning process. In Health Sciences, case-based and cooperative learning are among the most used active learning strategies. They present multiple configurations and vary greatly in terms of implementation. Students' adherence to active learning depends on the perceived utility, level of effort requested by the activities, and self-confidence in the quality of achieved learning. Under the Think4Jobs, an Erasmus+ project, a University-Business collaboration was implemented to design work-based activities for pilot courses of the Veterinary Medicine program aiming to increase the students' adherence to active learning strategies while reducing any mismatch in students' competencies at graduation. In this chapter, we propose discussing how the collaboration was conceptualized and implemented. We also present some activities jointly designed to foster students' clinical reasoning/critical thinking and decision-making.

**Keywords:** skills development, competency-based education, work-based learning, university-business collaboration, learning activities, clinical reasoning, critical thinking

#### **1. Introduction**

Active learning has been the center of attention for researchers and teachers looking for less traditional meaningful learning activities. Active learning is an instructional approach that centers the learning process in the student. It uses different activities and situations designed to promote the acquisition of knowledge and skills, through the application of higher-order cognitive processing abilities,

to construct new knowledge, train skills and competencies, and further reflect on their performance in the learning process [1].

During active learning, students are immersed in meaningful experiences, designed to conduct them toward the learning outcomes proposed, either in the domain of cognitive knowledge or skills development. The most interesting aspect of active learning is that it pushes the student to go beyond the factual knowledge to grasp conceptual and procedural knowledge, an "in-depth understanding". That will allow transferring factual knowledge into multiple and different situations.

The active learning concept embeds in the constructivism framework of learning since it empowers the student as the conscious constructor of his/her knowledge, allowing him/her to regulate the learning process [2] accommodating personal traits while also coping with the academic demands.

The interest in active learning has grown in the past decades. It contrasts with the classical teaching approach, where a passive knowledge transfer occurs, and students act as sponges for information [2], while the measurement of knowledge acquisition is usually focused on factual knowledge, focusing on memorization [3], without deep thinking of the content. In the classical passive learning, memorized knowledge is lost more quickly, making it more demanding for the student to translate it to solve new challenging situations, which command independent thought [4, 5].

Policymakers worldwide, including the OECD [6] demands educational systems thriving for more than factual or unreflective learning, fostering in the students the competencies need to understand the modern world and succeed in their profession. Those competencies encompass higher-order and complex thinking skills along with procedural knowledge and the ability to work in multidisciplinary teams. Two of the utmost sought skills in a broad scope of occupations are creativity and critical thinking., allowing the future professional to cope with the expanding digitalization and automation of the job market, and the multicultural aspect of the profession.

Active learning has been adopted by universities to reinforce the development of students' skills, and increase their motivation and engagement while also fostering the transferability of knowledge into the professional context [7]. Although not completely conceptualized [7, 8], active learning includes multiple learning strategies, as it also grows to adjust to the transformations occurring worldwide at multiple levels. Active learning demands the student's involvement with the learning activities, the analysis and ability to respond to specific situations, and a critical reflection on the learning process to be successful.

The closer to the professional context, the most motivational and enthralling the activity will be for students. Moreover, the need and importance of the course content become more explicit to students, which may be central in foundational courses [9]. It is generally accepted that active learning is a good way to train and enhance the students' abilities and competencies needed for the workforce. Besides, policymakers worldwide advocate the need to strengthen critical thinking in learners across all levels of education, preparing them as active and participative citizens, able to transpose into the labor market the competences developed during learning [6].

In the case of Veterinary Medicine graduation, Universities are constantly challenged to provide day one, job-ready graduates into the labor market, while also keeping pace with the rapidly expanding technical and scientific knowledge and meeting the business expectations about the minimum competencies mastered by day-1 graduates [10]. This means that Universities must ensure the development of cognitive knowledge (or hard skills) as well as inter-personal, social, and communication abilities (or soft skills) [11]. Albeit the universities often focus on the former to grade

#### *Perspective Chapter: Active Learning Strategies in the Veterinary Medicine Programme… DOI: http://dx.doi.org/10.5772/intechopen.105969*

the students, the latter should not be forgotten when assessing students' performance. Besides, it is noteworthy to stress that not all the active learning methodologies will similarly develop specific cognitive or soft skills. Therefore, it may not be possible to use the same strategies to enhance soft skills and hard skills, and it may be necessary to select different active learning techniques, adapting them to the skills to be strengthened in students or the goals established [11].

Under the Erasmus+ project Think4Jobs (2020-1-EL01-KA203-078797), several focus groups (FG) were organized to gather the opinion of professionals' stakeholders and academics about an eventual mismatch in critical thinking competencies they found in recent graduates during traineeships. The FG showed that stakeholders have a different conceptualization about the skills and dispositions deemed as crucial in professional contexts than the academic representatives, even though they consider the trainees to possess a good level of cognitive knowledge [12]. Recently, it has been demonstrated that in most cases, the attendance of a university using the classical learning approach, is insufficient to promote students' creative and CT skills [13], somehow supporting the perceptions of the interviewed stakeholders.

Taking those findings into consideration a University-Business collaboration was implemented to design some work-based activities to be included in piloting course curricula for the master's Program in Veterinary Medicine at Univ of Évora.

In this chapter, we propose discussing how the collaboration conceptualized and implemented activities using a case-based strategy to reinforce critical thinking skills and dispositions in students. We also present some activities jointly designed to foster students' clinical reasoning, critical thinking, and decision-making.

#### **1.1 Case- and problem-based learning as active strategies**

In health sciences, case-based and problem-based learning (CBL and PBL, respectively) strategies, often developed cooperatively, are among the most common approaches to active learning. Both these strategies are based on problem or scenarios, but they present multiple configurations and varies greatly in terms of implementation and the content of the course. At their base, CBL and PBL present the students with situations that mimic real-life challenges or problems that need solving using core knowledge. By using a real work context in the activities, and requesting the students to engage with real-life tasks, would foster the transferability of the outcoming competences [14]. The problematization focus on the development of high-order thinking skills, teamwork, communication and other interpersonal skills, and the ability to act upon an informed, strong decision-making process. CBL and PBL strategies share common features and present also important differences. Still, in some situations developed in medicine and nursing, the format of the activities identified as CBL and PBL became similar and the limits between the conceptualization of the two strategies overlap.

Both strategies rooted in a constructivist approach to knowledge construction, allowing the students to combine theory with practice, to mobilize acquired knowledge and skills into the construction of a solution for an ill-defined situation issued from critical issues or case-scenarios faced by the workforce [15]. With time, CBL and PBL conceptualization suffered multiple adaptations; nowadays, in some situations is not always easy to distinguish between them. Still, in their purest forms, two main differences persisted [10, 16]: 1/ the possible solutions to the problem provided are roughly defined at the beginning (that is, the situation is somehow structured) and the student uses core knowledge previously acquired to solve the situation in CBL, whereas in PBL the student is requested to identify and acquire new knowledge that

is necessary to reach a solution, which is usually not previewed at the beginning of the activity (i.e., the situation is less structured); 2/ PBL extension is usually longer (up to one semester/term)than in CBL, which generally demands a solution in a short time frame (few hours). **Table 1** summarizes the major similarities and differences between CBL and PBL.

In general, the students' perceptions about the use of PBL or CBL strategies in learning are positive [1, 17]. However, in the practice, it is often debated that students' adherence to PBL or CBL, as in other active learning strategies, is not always similar, and may depend on their perceptions of the components of the strategy used (including the relationship established with the teacher or tutor), the perceived utility,


*Perspective Chapter: Active Learning Strategies in the Veterinary Medicine Programme… DOI: http://dx.doi.org/10.5772/intechopen.105969*


#### **Table 1.**

*Main differences between case-based and problem-based learning as used in health sciences.*

#### **Figure 1.**

*Summary of the main competencies endorsed in PBL and CBL.*

the level of effort requested when engaging in the activities, or their self-confidence in the quality of learning achieved [18, 19].

In addition to the acquisition of cognitive knowledge, PBL and CBL allow the student to develop multiple competencies, some of which are illustrated in **Figure 1**. Ideally, when assessing the activity outcomes, and considering that the student must be graded at the term end, the achievements in both cognitive and ex-cognitive competencies should be foreseen. **Table 2** presents available tools to be included in PBL and CBL activities that will be useful to assess the students' achievements.


#### **Table 2.**

*Tools available to assess the students' performance in active learning strategies.*

#### **2. Business-university partnership in the designing of active learning activities**

Stage one of the Think4Jobs project established the recommendations for the Veterinary Medicine Program and allowed to identify the crucial competencies stakeholders deem as essential at the entrance of the final traineeships and in young graduates. Briefly, the competencies identified to fit the skills and dispositions already identified under the critical thinking concept [23] including the ability to question, the analytical skills, a structured way of reasoning, decision-making, autonomy, and self-correction [12]. These recommendations were used as a starting point for the design of the work-based activities, adequate for the course content, to be implemented in piloting courses on the master's Program in Veterinary Medicine. The ultimate purpose was to increase the students' adherence to active learning strategies while reducing any mismatch in students CrT skills and competencies at graduation,

*Perspective Chapter: Active Learning Strategies in the Veterinary Medicine Programme… DOI: http://dx.doi.org/10.5772/intechopen.105969*

as perceived by stakeholders. Moreover, the activities aimed at representing situations currently found in the routine of the profession, bridging the course content directly to the critical reflection and decision-making often requested from a Veterinary Surgeon in a wide scope of occupations.

#### **2.1 Conceptual aspects to consider when designing the activities**

The activities were designed considering the following conceptual views:


For the design process of the active strategies, and because most courses in the clinical area share common goals (the need for the critical assessment of a medical situation that needs solving through a corrective medical or surgical intervention decision making), the HEI and LMO partners decide on proposing a common framework that could be used across multiple courses. By sharing the same framework, the differences in the clinical case used for stating the activity would provide wider flexibility to the process, promote multiple repetitions of the framework in different

contexts which would likely foster the sought transferability of skills, and would also cope with the individual content of the syllabus of each course (which has been validated by a national entity that evaluates the programs' curricula).

In the veterinary daily practice, the professional intervention involves either wellstructured situations (e.g., for vaccination or sanitary procedures, and some routine or elective surgeries) or ill-structured challenging situations (e.g., during emergency situations). The partners decide on the use of a moderately ill-structured situation for starting the proposed activities, allowing to use different difficulty levels to match the level of difficulty to the students' program level as well as to the topic at hand in a particular course or apprenticeship. Moreover, the activities aimed at the development of factual and conceptual knowledge and high-order thinking so in the end, they would enhance the student's autonomous decision-making.

The learning scenarios designed will support CrT skills and dispositions development and will allow students to switch their reasoning from a disease- or systembased thinking into patient-based thinking conductive of a successful medical decision-taking for problem-solving [12].

#### **2.2 Proposed activity framework for courses in the clinical areas**

The framework for the learning scenarios aims at engaging students in the analysis of clinical condition-issued from everyday practice of a veterinary hospital and to decide on the best intervention possible for the clinical condition considering the animal and owner context, resourcing to high-level reasoning, scaffolded by questions. Overall, this approach will contribute to the enhance students' CrT skills.

The case scenario used as starting point is purposely left blank allowing the teacher to select particular situations, adapted to the course content, considering the targeted species or the medical specialties (e.g, Gynecology and Obstetrics, Infectious Diseases, Ruminants Clinics, Small Animals Clinics, Surgery in Companion Animals). The elements to be included in the scenarios and the activity steps follow a medical model [27].

These activities can be implemented during classes, but they may also be transposed into a traineeship context, and to improve the discussion of clinical situations during the clinical rotations. Through the activities, students are encouraged to develop their autonomy in core medical procedures (e.g., collection of clinical history, deciding on the complementary exams needed to direct the diagnosis, discussing available therapeutic approaches with a third party, and engaging in smaller entrustable professional activities mimicking those performed during external traineeships). In the later, students will train communication, empathy, inquisitiveness, systematicity, autonomy, and self-confidence, as well as self-assessment.

The proposed activities are limited in time; they were designed to be completed in 3 h or 2×2h, and they can repeat twice or thrice during the semester. The framework proposed herein was devised to be developed in the courses of the 4th and 5th level of the master's program in veterinary medicine, and in a face-to-face format, even though some support can be provided in a blended approach. For the activities, students will be grouped in cohorts of five elements, which will also reinforce the development of teamwork competencies.

The steps of the framework are detailed in **Table 3**, along with the skills and dispositions they address.

*Perspective Chapter: Active Learning Strategies in the Veterinary Medicine Programme… DOI: http://dx.doi.org/10.5772/intechopen.105969*

#### **2.3 Additional support during implementation**

The implementation of the activities requests the production of supporting materials, to be developed by the teacher and made available to students some days in advance, for preparing their participation. Among the course material to be provided



#### **Table 3.**

*Framework to be used in the case-based activities developed for courses in the clinical area in the veterinary medicine program.*

to students, we include a list of recommended scientific or academic publications, to be used during self-study to construct background and specialty-specific knowledge; a regulatory document, setting the topic, the learning outcomes, and requirements, with an estimation of the time spent in preparing the activity, and defining the outputs to be used for assessment, and also presenting the rubrics to assess the competencies set for the activity.

During the activity, it is critical to have all files related to the case in analysis: the case vignette, the complementary information pertinent to the case analysis (to be provided upon the students' specific request and justification), and the file containing the guiding questions.

*Perspective Chapter: Active Learning Strategies in the Veterinary Medicine Programme… DOI: http://dx.doi.org/10.5772/intechopen.105969*

#### **2.4 Assessment of the activities**

The assessment of the students' performance during the activities must match the proposed outcomes, meaning that the rubric to be used should address both the cognitive knowledge and the CrT skills and dispositions. The evaluation will use the documents produced by students. These documents represent a critical assay of the reasoning process the students engaged with across the activity; scaffolded by the questioning provided by the teacher, the rational underlying the analysis of the situation and data from complementary exams, and the decision made regarding the solution proposed for resolution of the primary condition and the proposed schedule for follow-up interventions should be detailed, in an approach resembling the "think aloud" method.

The questions provided to support the analysis, and the corresponding rational, are categorized as factual, conceptual, procedural, and metacognitive knowledge, for scoring using a 4-point rubric for evaluation of the quality of reasoning. The knowledge categories have different ponderations: 1.5× for the factual; 2× for conceptual knowledge; 2.5× for procedural knowledge and 3x for metacognition [28].

#### **3. Implementation of the activities and anticipated challenges**

These activities have been implemented in the current term in the course of Gynecology, Andrology, and Obstetrics (8th semester of the Veterinary Medicine Program), in a piloting test. At this very moment, two activities were developed (namely for the themes andrology and gynecology) with 45 students distributed in five-elements groups.

From the application of this framework, we found that the perception of the closeness of the requested tasks to the ones Veterinary Practitioners perform daily in a clinical context engages and motivates the students to learning experiences that both increase their cognitive learning and empower them with other skills or dispositions requested in the workplace. Students themselves recognize that the activities are good strategies to increase their learning.

Besides, the involvement of LMO representatives in the construction and modulation of the activities brings work-based learning opportunities into the classroom. Based on Business-University Cooperation, these activities allow bridging the distance between the academy and the workplace, without the need to send students to extramural workplaces during their academic track.

Still, some constraints must be anticipated. Both the students and teachers are strange about these kind of learning activities. They request the students a different level of investment in the self-study routines and are critically dependent on their self-regulation skills; moreover, they will not guarantee the same performance (grading) as a memorizing-based assessment of student's cognitive skills, since the focus on students' reasoning competences will be placed on the assessment and accounts for their learning outcomes; thereby, including the CrT skills in the final assessment might impact the grades the student are used to obtain. The major positive effect—not always clear for students at this stage—may be a softer transition into the demands of the workplace, better integration into larger teams, and an improved quality in decision-making and animal healthcare.

As it is viewed by some as a liability, the proposed activities will be more timeconsuming to teachers in comparison with the traditional teaching and learning

styles, may compromise long syllabus teaching, and certainly increase the teachrelated workload, while also imposing the need to be self-confident, open-minded, non-conformist and not be afraid of being challenged. Moreover, it will request constant contact with the routine and reality of the profession, to build networks with LMO along with coping with the rapidly changing workplace and the ever-growing advancement of science and technology.

#### **4. Conclusion**

This chapter describes the rational and the design of learning activities based on clinical case scenarios, as conceptualized under the Think4Jobs project. The activities were designed to reinforce students' clinical reasoning and critical thinking, recreating work-based experiences that can occur in the everyday activity of a veterinary practitioner. The current study presents some limitations: the conceptualized model is not yet tested for veterinary medicine courses, and only has been piloted during the current semester; therefore, no results are available yet to analyze its suitability or its success. Despite that fact, the current research contributes to the discussion of the topic of active learning strategies and provides the first steps for a framework that can be applied longitudinally and transversal to the courses in the clinical area. Therefore, the proposed framework is still pending validation, which will be performed in the near future.

Finally, some issues may be anticipated when introducing new and different activities that might impact the students' grades or the teacher workload. On the one hand, activities like the one described herein will request the students to develop new and unclear (unperceived) skills, such as self-regulation and autonomy, as well as to embark on a different way of studying, driving a resistance toward active participation. On the other hand, teachers may need further support, either regarding their own empowerment and confidence, as well as to be able to respond in time to the added responsibilities to develop the activities and provide timely feedback to students.

Still, using activities like the ones contemplated in this framework will contribute to better preparation of graduation students to the labor market expectations.

#### **Acknowledgements**

This work has been supported by the "Critical Thinking for Successful Jobs— Think4Jobs" Project, with the reference number 2020-1-EL01-KA203-078797, funded by the European Commission/EACEA through the ERASMUS Programme.

"The European Commission's support for the production of this publication does not constitute an endorsement of the contents, which reflect the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein."

*Perspective Chapter: Active Learning Strategies in the Veterinary Medicine Programme… DOI: http://dx.doi.org/10.5772/intechopen.105969*

#### **Author details**

Rita Payan-Carreira1 \*, Hugo Rebelo2 and Luís Sebastião2

1 Comprehensive Health Research Centre and Department of Veterinary Medicine, University of Évora, Évora, Portugal

2 Research Center in Education and Psychology, University of Évora, Évora, Portugal

\*Address all correspondence to: rtpayan@uevora.pt

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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### Section 2
