**4. Developed materials for the module "understanding of place value"**

The materials for the module "understanding of place value" focus on the professionalization of teachers and facilitators through massive open online courses (MOOCs), materials for professional development programs, and accompanying didactic manuals for students' materials (Section 4.1). The research-based course design was grounded on a variety of evaluated course concepts for teachers and facilitators especially focusing on inclusive mathematics teaching, done by the authors before (for example, see [19, 20]). For the concrete course design, relevant content was selected, taking into account previous research activities on understanding place value (for example, see [8]), as well as considering the time, as the length of the MOOC was set as a systemic guideline. Student learning is pursued by activities and tasks in printed materials, explanatory videos, and web applications (Section 4.2). During the development process of the teacher and student materials, the whole material was discussed intensively in the development team. In addition, some of the printed materials for students were discussed in a larger team of mathematics educators.

#### **4.1 Professionalization of teachers and facilitators**

#### *4.1.1 Teachers*

To assist teachers in their work with students on understanding place value, a massive open online course (MOOC) was developed and initially offered.

#### *Fostering Basic Mathematical Competencies: Concepts and Materials for Teachers and Students… DOI: http://dx.doi.org/10.5772/intechopen.113257*

The implementation was carried out along the three steps of identifying, diagnosing, and supporting students' basic mathematical competencies regarding place value. For example, the principle of bundling and the principle of position were identified as central for understanding place value (see Section 2). With regard to diagnosis, typical difficulties of students on place value were considered (e.g., unfamiliar order of place values). In particular, different possibilities were presented to determine students' understanding of place value, both standardized instruments, which are available in Germany [for example, see [21]], and non-standardized tests [for example, see [8]] (see also the information on non-routine tasks in Section 2). Subsequently, as one way to promote students' understanding of place value, common manipulatives and representations were introduced and analyzed (e.g., base ten blocks). Especially, this involved a critical reflection against the background of the principle of bundling and the principle of position. Moreover, concrete activities and tasks for students to foster their understanding of place value were discussed (see also Section 4.2).

The MOOC took place three times with up to 400 participants in each case. In the implementation, special attention was paid to actively involve the huge number of participants in an online setting during the time span of 2 hours. In total, there were five activities, which were related to the three steps of identifying, diagnosing, and supporting, and realized differently in terms of both content and method. For example, at the beginning, the participants were asked to analyze a set of five non-routine tasks in terms of requirements as well as difficulties and concrete erroneous solutions to be expected (identifying). They could note their ideas in the accordingly prepared columns of a digital board (see **Figure 1**).

This introductory activity was followed up in further activities. Later in the MOOC, for example, a student's solution to these non-routine tasks was to be analyzed in terms of existing competencies and difficulties (diagnosing) or appropriate manipulatives were to be selected to address the student's previously indicated difficulties and to promote his understanding of place value (supporting). The latter activity on reflecting on manipulatives was realized as a kind of voting in the digital board. The participants could select the material(s) they considered the most useful for promoting this student's understanding of place value. In another activity, the participants were asked to reflect on the mnemonic "The comma separates the units", which can be found in some German textbooks in the context of measures. In Germany, the comma (,) is used as decimal separator, for example 2,45 m for 2 meters

**Figure 1.** *Introductory activity.*

and 45 centimeters. The teachers' thoughts regarding this mnemonic for different measures (length, weight, time, and money) could be entered into four columns in the digital board and were discussed in plenary.

Based on the questions and discussions of the participants in the three MOOCs, the presentation on understanding place value was revised and is now permanently available as open educational resource after having registered (see **Figure 2** for an exemplary slide, and https://maco.dzlm.de/stellenwertverstaendnis-bei-natuerlichen-zahlen for the complete presentation in German).

In addition to the slides, the recording of one MOOC is permanently available on the website. Moreover, numerous materials that teachers and support staff can use in their work with students in order to foster their understanding of place value were developed (see Section 4.2). Additionally, there are related didactic commentaries explaining the background and the objectives of the students' materials as well as possible difficulties and expected solutions of students.

#### *4.1.2 Facilitators*

In order to support facilitators to provide professional development courses for teachers, various materials and offers are available. First of all, the above-described slides including activities for teachers can be used in professional development courses given by the facilitators. Similar to the materials used in the MOOC, the developed materials focus on the identification of basic mathematical competencies, diagnosis, and support with regard to place value. Each slide contains notes, for instance, with information on underlying theoretical as well as didactic aspects regarding place value, or hints on the implementation of the activities, especially on expected answers of the participants. Additionally, a profile of three pages was created, including information on the basic idea of the materials, the target group and the goals, the background regarding place value, the structure, and especially the main activities, a schedule, and references.

#### **Figure 2.**

*Exemplary slide from the presentation for the teachers.*

#### *Fostering Basic Mathematical Competencies: Concepts and Materials for Teachers and Students… DOI: http://dx.doi.org/10.5772/intechopen.113257*

The facilitators had the opportunity to participate in an online workshop and were provided with the abovementioned materials in advance. A digital board was set up for the workshop, and during their preparation, the facilitators could share their ideas on these four aspects: What I liked very much/What I did not understand/What I would like to talk about in the workshop/These ideas I do have.

At the beginning of the workshop with about 30 participants, central elements of the materials for the professional development course were once again discussed along the three steps of identifying, diagnosing, and supporting the basic mathematical competencies with regard to understanding place value. In this context, questions documented in the digital board were addressed, but the facilitators also had the possibility to ask further questions. For example, the question whether numbers are always and exclusively written from left to right or whether an inverse notation can also be tolerated (with respect to the irregular structure of naming numbers in German language, see Section 2) was formulated in advance in the digital board and discussed with the participants. Subsequently, with regard to the mnemonic "The comma separates the units", a case vignette was presented in which a teacher states in the context of a professional development course: "I only use tasks in which the mnemonic works." The facilitators should discuss in breakout rooms how they would react to this statement. In the following plenary phases, fundamental decisions with regard to the implementation of the principles for sustainable learning, such as conceptual focus or longitudinal coherence [18] (see also Section 3), were discussed. In another breakout session, the focus was on planning a professional development course on understanding place value based on the presented materials. Against the background of the material adaptations to be made, the following points were discussed in accordance with Drake and Sherin: What content would you *add*? What content would you *omit*? What content would you *replace*? [22].

This time, the breakout rooms were set up with regard to the different target groups of the professional development course (e.g., one-shot course at one school, course for support staff), and the facilitators could choose the suitable room themselves. The exchange on the above points could also be used for the formulation of take-home messages at the end of the workshop, for example, that the adaptation of the material to the target group of the professional development course is central or that adaptations should always take into account the maintenance of the central contents on place value understanding. On the basis of the facilitators' questions and ideas within the workshop, the materials for the professional development course on understanding place value were revised and are now permanently available as open educational resource (see **Figure 3** for an exemplary slide and https://maco.dzlm.de/ stellenwertverstaendnis-bei-natuerlichen-zahlen for the complete presentation in German).

#### **4.2 Materials for students**

In addition to different materials for teachers and facilitators, materials for students have also been developed. First, some insights into printed materials for diagnosis and support of place value are given (Section 4.2.1), followed by ideas for supporting place value understanding through web applications (Section 4.2.2) and explanatory videos (Section 4.2.3). All the tasks and activities put focus on basic mathematical competencies for understanding place value, such as the principles of bundling and position and use non-routine tasks as well as different representations.

**Figure 3.** *Exemplary slide from the presentation for the facilitators.*

### *4.2.1 Printed materials*

The printed materials for students cover four different topics for supporting students' understanding of place value: *patterns and structures*, *number line*, *connecting representations*, and *digit cards*. All materials are available as open educational resources (see https://maco.dzlm.de/stellenwertverstaendnis-bei-natuerlichen-zahlen for the complete materials in German). In the following, some basic ideas for each printed material are illustrated. Afterward, some concrete examples of the tasks in the different printed materials are given.

The printed material *patterns and structures* uses tasks on flexible counting and sequences of numbers for fostering basic competencies [23]. Furthermore, students identify given and create own patterns by using counters and the thousand book. Within further tasks, students analyze operational changes of numbers, for example, by adding 1, 10, 100, and their multiples to given numbers. Also, different representations like base ten blocks and a place value chart are used. The printed material *number line* consists of tasks that focus on locating numbers on different number lines with different interval sizes [24]. Some tasks focus specifically on switching tens and ones or on marking ranges of numbers with the same tens but different ones and the other way round to address the principle of position. The printed material *connecting representations* supports students' flexibility in using different mathematical representations [25]. Therefore, the same numbers have to be represented with, for example, the base ten blocks, the hundred dots field, money, a number line, and a place value chart. Further tasks focus also on finding different ways of representing one number with the same material or representation. This includes reflecting on the concrete representation of hundreds, tens, and ones. The printed material *digit cards* focuses on tasks with digits from 1 to 9 each written on a card [26]. The students choose, for example, three digit cards and form all possible 3-digit numbers. Herewith, understanding the principle of position can be fostered. Further tasks focus on addition and *Fostering Basic Mathematical Competencies: Concepts and Materials for Teachers and Students… DOI: http://dx.doi.org/10.5772/intechopen.113257*

subtraction of 2- and 3-digit numbers, reaching either small or big sums or concrete target numbers.

The following exemplary tasks show how basic mathematical competencies for understanding place value, such as the principles of bundling and position, are fostered by the use of non-routine tasks and different representations. These examples are also used to show in how far the tasks and activities are suitable for inclusive mathematics settings. In general, all printed materials include tasks that can either be solved by students individually or by working in pairs or groups. Nevertheless, some tasks especially concentrate on students working in pairs or groups and encourage communication and cooperation.

One task of the material *patterns and structures* focuses on working with counters and a place value chart (see **Figure 4**). It can be discussed with students in how far a counter represents a different value when it is placed in different columns. The underlying idea of bundling and unbundling is also addressed in a further task where counters are shifted from one to another column. In addition, this task allows to focus on the transfer between a concrete/enactive and a symbolic representation of numbers also for vacant place values. For teaching mathematics in inclusive settings, it is important to use tasks that offer the opportunity for action-oriented learning. In this case, learning can be based on practical experiences, for example, with manipulatives, as well as on tasks that allow students to be actively involved in learning.

Two of the tasks in the material *connecting representations* explicitly address different ways of solution, which is also a central aspect in inclusive mathematics teaching. One task offers the opportunity to discuss about how a number can be represented cleverly with the hundred dots field (see **Figure 5**) and allows to reflect upon how to represent hundreds, tens, and ones. The transfer between an iconic/pictorial and symbolic representation might also be discussed regarding the correct order of digits and inversion errors.

Another task prompts students to represent the same number in different ways but with the same material (see **Figure 6**). This idea focuses especially on bundling and unbundling and offers students the opportunity to discover mathematical concepts and structures concerning the understanding of place value on their own. Again,

**Figure 4.**

*Example from the printed material "patterns and structures" with counters and a place value chart ([23], p. 22).*

**Figure 5.**

*Example from the printed material "connecting representations" with the hundreds dot field [25].*

**Figure 6.**

*Example from the printed material "connecting representations" with base ten blocks [25].*

students should use manipulatives and are confronted with non-routine forms of representation (e.g., 24 shown as 1 tens and 14 ones).

As one possibility to allow students working on their individual level, which is especially important in inclusive settings, different number sets are used in the material *digit cards*. For example, students should build two 2-digit numbers, add them, and try to reach a sum of 100 (or reach the sum of 1000 with two 3-digit numbers). The solutions, which involve the main idea of considering the carrying or regrouping of values with addition, are similar for the different sums but have a different level

### *Fostering Basic Mathematical Competencies: Concepts and Materials for Teachers and Students… DOI: http://dx.doi.org/10.5772/intechopen.113257*

of complexity for 2- and 3-digit (or even bigger) numbers. In addition, one task in the material (see **Figure 7**) allows students to either use the cards and place them on a desk in different order, like 174, 147, and 471, or find the various numbers just by changing the order of the chosen cards mentally. Some students might find only a few numbers; others might find all possible solutions. Because all numbers should also be ordered by size, students have to reflect on digits and their meaning depending on their position in a number as hundreds, tens, or ones (principle of position).

#### *4.2.2 Web applications*

Aside from printed materials, two web applications have also been developed in the project to support students' understanding of place value. These applications allow to implement tasks (and representations with virtual materials) that might not be used as a non-digital environment with manipulatives in the same way. However, two representations used in the printed materials are implemented: base ten blocks and the number line. All web applications will be available on the project platform (see https://maco.dzlm.de/stellenwertverstaendnis-bei-natuerlichen-zahlen).

The special features of the web application using *base ten blocks* refer to feedback possibilities. Since it can be challenging to provide individualized feedback to each student in the inclusive classroom, the web application has a benefit here. Students should represent a randomly given number with the base ten blocks (see **Figure 8**). After representing a number via drag-and-drop, the web application gives feedback whether the representation is correct or not. For the feedback, the web application also recognizes non-canonical representations as correct solutions, like representing 123 with 1 plate of hundred cubes and 23 single cubes. At the same time, the web application gives feedback and requests another form of representation with a complete bundling. This helps to foster students' understanding of base ten bundling.

The interface of the web application *number line* consists of a number line ranging from 0 to 10000, with thousands marked by longer dashes and their corresponding numbers (**Figure 9**). Shorter dashes between the thousands indicate hundreds. Clicking the "Start" button displays a prompt to mark a certain number on the number line, for example, 1634.

Clicking on any 1000 interval zooms into this section by displaying a new number line with a finer scale. If, for example, the interval from 1000 to 2000 is selected, a number line from 1000 to 2000 appears, in which the hundreds are marked by longer dashes (**Figure 10**). A color marking in the first number line indicates which range with a finer scale is now displayed enlarged in the second number line.


**Figure 7.** *Example from the printed material "digit cards" ([26], p. 2).*

#### **Figure 8.**

*Interface of the web application "base ten blocks".*

#### **Figure 9.**

In the end, the idea of locating adequate intervals and zooming in leads to a number line where 1634 can be marked exactly (**Figure 11**).

By enlarging intervals step-by-step, the focus is automatically set on the digit values of the given number from big to small. In addition, orientation on the number line is necessary, since only successive zooming into the correct sections in each case allows to finally mark the number accurately.

In both web applications, particular attention was paid to an intuitive usage with an easy interface. Tasks and feedbacks can be read aloud. One possibility for differentiation is also given: Teachers can set the size of the randomly generated numbers in both web applications. Herewith, some students might represent numbers up to 1000 or 10000 (or another individual chosen number) with base ten blocks.

#### *4.2.3 Explanatory videos*

While the printed materials were developed with a focus on using it in classroom or in learning settings between student and teacher, the web applications and especially the explanatory videos allow students also to work on their own. More specifically, four explanatory videos focusing on different relevant aspects for supporting

*Fostering Basic Mathematical Competencies: Concepts and Materials for Teachers and Students… DOI: http://dx.doi.org/10.5772/intechopen.113257*

#### **Figure 10.**

*Interface of the web application "number line"—First and second number line.*

#### **Figure 11.**

students' understanding of place value were developed: *bundling and unbundling*, *stepwise calculation (addition by place value)*, *principle of position*, and *connecting representations*. All videos introduce Jonas and Merve, two primary students, and their teacher as comic figures. The teacher gives tasks to the students, and a fictive interaction between the two students starts while solving the tasks. In between, the teacher gives some explanations as well. Most of the time, a small image of the person who is talking in that moment can be seen in one corner of the video. The "action" happens in the middle of the picture (see **Figure 12**). In the following, some examples for the content of each video concerning the support of students' place value understanding are given. All explanatory videos will be available on the project platform (see https:// maco.dzlm.de/stellenwertverstaendnis-bei-natuerlichen-zahlen).

The first video focuses on the idea of *bundling and unbundling*. A certain number of cubes (base ten blocks) lies on a table. Jonas and Merve have to count them (**Figure 12**, left). By putting cubes together in groups of ten, they discover the idea of bundling for smart counting (**Figure 12**, right). As the video goes on, the other parts of the base ten blocks are introduced, along with how to represent the number of cubes using a place value chart. Later on, the students are confronted with two alternative solutions, one with a canonical (2 hundreds, 3 tens, 5 ones) and one with

*Interface of the web application "number line"—Further number lines.*

**Figure 12.** *Example pictures of the video "bundling and unbundling".*

a non-canonical (1 hundred, 13 tens, 5 ones) representation of the number 235 in a place value chart. In the end, 78 cubes (represented by 7 bars of ten and 8 single cubes) must be fairly distributed between Jonas and Merve. For this distribution, the students get used to the idea of unbundling (here one bar of ten into ten single cubes).

The second video focuses on *stepwise calculation (addition by place value)*. Jonas has 43 cubes, and Merve has 25 cubes (each represented by base ten blocks with bars of ten and single cubes). They have to find out how many cubes they have together. By adding 43 and 20 and afterward 63 and 5, the stepwise addition is first visualized with the base ten blocks and then explained with a written calculation. Next, the children analyze if adding 43 and 5 and afterward 48 and 20 leads to the same result. In the end, further tasks like 425 + 213 are explained in similar ways.

The third video focuses on the *principle of position*. First, Jonas and Merve decompose the number 4547 into 4·1000 5·100 4·10 7 + ++ to reflect upon the same digit (4) having different meanings (either 4000 or 40) depending on their position in the number. Afterward, the children must compare two numbers and decide which one is bigger (2497 and 5812, 6859 and 6374, 9768 and 32145, 453872 and 453782). This, for example, helps them to also pay attention to the quantity of digits and the sequence of digits, and not solely to compare the leading digit to decide which number is bigger. In the end, Jonas and Merve choose three digit cards (see also Section 4.2.1) and should form out of them the three biggest 3-digit numbers.

The fourth video focuses on *connecting representations*. Jonas and Merve must represent the number 47 with various representations: Base ten blocks, place value chart, dot field, bead frame, short chains, number line, hundred board, as well as money. Herewith, different representations of the same number are contrasted. For example, it is compared which parallels can be drawn between the hundreds dot field and the hundred board or the number line and the short chains concerning hundreds, tens, and ones. Specific attention is paid to horizontal or vertical structures and representations of the tens and the ones.

The last picture in each video gives the student a new but similar task to the one the children worked on before. Herewith, the students should apply what they have learned.

### **5. Conclusion and perspectives**

In this chapter, the meaning of place value for primary mathematics and central ideas for teaching this topic have been brought out. Starting from the idea to foster basic mathematical competencies—also of major importance with regard to inclusive

#### *Fostering Basic Mathematical Competencies: Concepts and Materials for Teachers and Students… DOI: http://dx.doi.org/10.5772/intechopen.113257*

mathematic classes—various materials were created within the project MaCo. The materials and tasks for the topic of place value can be used in different learning arrangements like individual work, small group work, or whole classroom activities, which are especially important also in inclusive settings. Moreover, the materials and tasks can also be used more or less independently from the classroom as self-study materials. In addition, the materials and tasks can be used for different purposes as they can function as activities for revising a topic like place value, or for deepening. There are already some first experiences of trying-out the materials at school: the tasks using counters and the place value chart (see Section 4.2.1), for example, really challenged primary students to think about bundling and unbundling, including a transfer between different forms of representations (concrete/enactive and symbolic). Further tests and trials with accompanying research are planned.

To ensure that the important topic of place value is taught in an adequate way, the professionalization of teachers as well as facilitators is of major importance. There was positive feedback from participants on both, the MOOCs for teachers and the workshop for facilitators. Besides the good structure following the three steps of identifying, diagnosing, and supporting, the teachers appreciated, for instance, the design of the MOOC (e.g., meaningful use of the online format, alternation of input and activities) and the discussion of practical implementations (e.g., concrete examples for classroom practice, concrete hints for supporting students). Following teachers' feedback after the first MOOC, further background information on the typical difficulty of writing numbers inverted (right to left instead of left to right; see Section 2), and examples of appropriate classroom activities have also been integrated in following MOOCs as well as in the material for facilitators. Beyond that, the facilitators valued to have a well-structured template for their own courses and the collegial exchange on equal footing.

By offering these different elements, MaCo tries to realize a coherent concept in order to assist teachers as well as facilitators in enabling students to catch up in mathematics. The development of modules and materials for central basic competencies for different grades and school levels provides a rich source for designing inclusive settings.
