**4. Enactive, iconic, and symbolic representations**

Bruner's modes of representations begin with the enactive, which includes manipulatives, or concrete, physical objects. The second representation is iconic, which represents any visual representations like diagrams, number lines, bar models, and graphs. The third representation is symbolic, which are abstract symbols like equations and algorithms. According to Bruner [2], students access their background knowledge of the representations to help make connections when the abstract symbols are isolated from other contexts. Concrete materials provide an opportunity for students to build background knowledge with iconic images depicting the meaning of the abstract symbols. When new abstract symbols are introduced, students can use their visual background knowledge as a retrieval mechanism to help remind them of the relevant concepts.

Instructional tasks heavily focused on abstract symbols tend to draw out the use of rote, memorized skill practice, which has been shown to compete with the development of spatial reasoning skills [57]. One way to help students make connections between numbers and symbols is to incorporate concrete materials

for students to manipulate during their practice and application [58]. Including concrete manipulatives for mathematical tasks has been shown to improve student understanding and retention of the practiced concept [59]. The use of concrete materials in isolation does not always guarantee that students will flexibly transfer the concrete representation to the symbolic representations [60]. Solving problems strictly in symbolic form leads to inefficient solution strategies, entrenchment of operational procedures, and inconsistent errors [42, 54, 57]. As a whole, mathematics instruction that isolates the symbolic representations leads students to manipulate symbols without conceptual understanding and a weakened ability to solve problems outside of their procedural understandings [61]. Alternatively, instruction designed to include a progression of representations beginning with an enactive or physical model to then an iconic or visual representation to a symbolic form can support a deep understanding of the mathematics [2, 62, 63].

Many students have difficulty in understanding concepts without being able to first visualize an idea in their mind [3]. Visualization helps students use figures or shapes in their mind to recall, understand, make connections, clarify, and remember new information [64]. Mathematics curricula loaded with only symbolic representations require students to memorize procedures, denying the student an opportunity to utilize their visual thinking in the process of building conceptual understanding. However, including visual representations into daily mathematics lessons can support the learning process and increase conceptual understandings [65].

Strong visualization and spatial reasoning skills contribute greatly to students' ability to organize the structure of equations and mathematics [66]. Mathematical models can be a way to connect one's visualization to their understandings of the problem [67]. The model connects the visualization into the spatial layout of an equation so students can devise a solution to solve the problem [68]. As students visualize the problem, they flexibly decode the context into the spatial layout of an equation [69].

When given the opportunity, students can develop the necessary spatial skills to visualize mathematics. Gesturing assists students to communicate their thinking. Mental rotation and spatial visualization can strengthen students' ability to solve nontraditional equations. Therefore, promoting spatial reasoning and modeling (EIS) early on in students' learning can promote mathematical competency and algebraic thinking.

#### **5. Developing mathematical thinking**

Curriculum should include ways to promote spatial reasoning through mathematical modeling to develop students' conceptual understandings [47, 70]. Mathematical tasks should include both traditional and nontraditional equations [44, 46]. The use of mathematical modeling should connect through a progression of enactive models, iconic models, and formal, symbolic models. Iconic models are one way to introduce spatial reasoning tasks and can be integrated throughout the instructional year to increase students' flexibility with the structure of equations and mathematical competency [8, 34].

The Developing Mathematical Thinking Institute (DMTI) offers a comprehensive curriculum designed to encompass all of these components for students to develop procedural and conceptual understandings. The DMTI curriculum is an alternative to the typical curriculum for teaching mathematics to help teachers develop a different approach to how mathematics is taught [71].

The DMT framework consists of five key elements for teachers to reflect upon as they plan, prepare, and instruct mathematics lessons: taking student's ideas

**119**

**Figure 1.**

*The Importance of Spatial Reasoning in Early Childhood Mathematics*

attention to the structural components in mathematics.

sample solution for the students to use as a model.

equivalence through the use of mathematical modeling.

*Sample solution for making 10 in Module 3 of the DMTI curriculum.*

seriously, encouraging multiple solution strategies and models, pressing students conceptually, addressing misconceptions, and maintaining a focus on the structure of the mathematics [72, 73]. Using students' informal strategies values their thinking and gives the teacher insight as to the level of understanding each student has. Teachers use the five elements of the DMT to develop more efficient strategies and multiple models for solutions to mathematical problems. Students are encouraged to talk with others about their thinking, compare solutions, and make corrections to their errors. One of the most critical components of the framework is to draw

One of the ways the DMTI curriculum builds student thinking is through the inclusion of Bruner's [2] enactive, iconic, and symbolic models. Each module is comprised of lessons with tasks centered on the EIS framework to develop a strong foundation for the development of conceptual understanding and for solving problems [72]. For example, students in first grade are given a contextual problem about 10 children playing in sandbox, where they need to determine whether six of the children are boys, and then how many children are girls? Students first demonstrate their thinking using unifix cubes, followed by drawing an iconic bar model to match their unifix cubes model. The symbolic representation of the numbers is then attached with labels. For example, to highlight the variety of ways to represent the number 10, students are asked to demonstrate the other possible representations for making 10 following the EIS progression. Modeling all of the possible combinations for 10 emphasizes the idea of equivalence, and using the EIS progression helps all students to visualize how the numbers relate to one another. **Figure 1** provides a

As students become fluent with facts within 10, they are introduced to the variety of ways to compose the teen numbers using units of tens and ones. For example, one task is to represent each teen number using units of one. Eventually, students begin to recognize the inefficiency of counting each unit of one. At that point, the teacher introduces a more efficient way of building the teen numbers by using a unit of 10. Over time, students independently build efficient models for larger numbers based on their previous experiences building with units of one. Once again, tasks such as these expose students to relational thinking and highlight the structure of

The DMTI curriculum encourages students to represent solutions to contextual problems, explain their solutions, and then generalize their understandings to other concepts (see **Figure 2**). An example of this is with contextual compare problems presented in Module 3 where students represent the number of blocks used to build two different towers. The task states that one tower is eight blocks tall, and another

*DOI: http://dx.doi.org/10.5772/intechopen.81564*

#### *The Importance of Spatial Reasoning in Early Childhood Mathematics DOI: http://dx.doi.org/10.5772/intechopen.81564*

*Early Childhood Education*

for students to manipulate during their practice and application [58]. Including concrete manipulatives for mathematical tasks has been shown to improve student understanding and retention of the practiced concept [59]. The use of concrete materials in isolation does not always guarantee that students will flexibly transfer the concrete representation to the symbolic representations [60]. Solving problems strictly in symbolic form leads to inefficient solution strategies, entrenchment of operational procedures, and inconsistent errors [42, 54, 57]. As a whole, mathematics instruction that isolates the symbolic representations leads students to manipulate symbols without conceptual understanding and a weakened ability to solve problems outside of their procedural understandings [61]. Alternatively, instruction designed to include a progression of representations beginning with an enactive or physical model to then an iconic or visual representation to a symbolic

form can support a deep understanding of the mathematics [2, 62, 63].

port the learning process and increase conceptual understandings [65].

Many students have difficulty in understanding concepts without being able to first visualize an idea in their mind [3]. Visualization helps students use figures or shapes in their mind to recall, understand, make connections, clarify, and remember new information [64]. Mathematics curricula loaded with only symbolic representations require students to memorize procedures, denying the student an opportunity to utilize their visual thinking in the process of building conceptual understanding. However, including visual representations into daily mathematics lessons can sup-

Strong visualization and spatial reasoning skills contribute greatly to students' ability to organize the structure of equations and mathematics [66]. Mathematical models can be a way to connect one's visualization to their understandings of the problem [67]. The model connects the visualization into the spatial layout of an equation so students can devise a solution to solve the problem [68]. As students visualize the problem, they flexibly decode the context into the spatial layout of an

When given the opportunity, students can develop the necessary spatial skills to visualize mathematics. Gesturing assists students to communicate their thinking. Mental rotation and spatial visualization can strengthen students' ability to solve nontraditional equations. Therefore, promoting spatial reasoning and modeling (EIS) early on in students' learning can promote mathematical competency and

Curriculum should include ways to promote spatial reasoning through math-

The Developing Mathematical Thinking Institute (DMTI) offers a comprehensive curriculum designed to encompass all of these components for students to develop procedural and conceptual understandings. The DMTI curriculum is an alternative to the typical curriculum for teaching mathematics to help teachers

The DMT framework consists of five key elements for teachers to reflect upon as they plan, prepare, and instruct mathematics lessons: taking student's ideas

develop a different approach to how mathematics is taught [71].

ematical modeling to develop students' conceptual understandings [47, 70]. Mathematical tasks should include both traditional and nontraditional equations [44, 46]. The use of mathematical modeling should connect through a progression of enactive models, iconic models, and formal, symbolic models. Iconic models are one way to introduce spatial reasoning tasks and can be integrated throughout the instructional year to increase students' flexibility with the structure of equations

**118**

equation [69].

algebraic thinking.

**5. Developing mathematical thinking**

and mathematical competency [8, 34].

seriously, encouraging multiple solution strategies and models, pressing students conceptually, addressing misconceptions, and maintaining a focus on the structure of the mathematics [72, 73]. Using students' informal strategies values their thinking and gives the teacher insight as to the level of understanding each student has. Teachers use the five elements of the DMT to develop more efficient strategies and multiple models for solutions to mathematical problems. Students are encouraged to talk with others about their thinking, compare solutions, and make corrections to their errors. One of the most critical components of the framework is to draw attention to the structural components in mathematics.

One of the ways the DMTI curriculum builds student thinking is through the inclusion of Bruner's [2] enactive, iconic, and symbolic models. Each module is comprised of lessons with tasks centered on the EIS framework to develop a strong foundation for the development of conceptual understanding and for solving problems [72]. For example, students in first grade are given a contextual problem about 10 children playing in sandbox, where they need to determine whether six of the children are boys, and then how many children are girls? Students first demonstrate their thinking using unifix cubes, followed by drawing an iconic bar model to match their unifix cubes model. The symbolic representation of the numbers is then attached with labels. For example, to highlight the variety of ways to represent the number 10, students are asked to demonstrate the other possible representations for making 10 following the EIS progression. Modeling all of the possible combinations for 10 emphasizes the idea of equivalence, and using the EIS progression helps all students to visualize how the numbers relate to one another. **Figure 1** provides a sample solution for the students to use as a model.

As students become fluent with facts within 10, they are introduced to the variety of ways to compose the teen numbers using units of tens and ones. For example, one task is to represent each teen number using units of one. Eventually, students begin to recognize the inefficiency of counting each unit of one. At that point, the teacher introduces a more efficient way of building the teen numbers by using a unit of 10. Over time, students independently build efficient models for larger numbers based on their previous experiences building with units of one. Once again, tasks such as these expose students to relational thinking and highlight the structure of equivalence through the use of mathematical modeling.

The DMTI curriculum encourages students to represent solutions to contextual problems, explain their solutions, and then generalize their understandings to other concepts (see **Figure 2**). An example of this is with contextual compare problems presented in Module 3 where students represent the number of blocks used to build two different towers. The task states that one tower is eight blocks tall, and another

#### **Figure 1.**

*Sample solution for making 10 in Module 3 of the DMTI curriculum.*

#### **Figure 2.**

*Example of student work mat from Module 3 of the DMTI curriculum.*

tower is six blocks tall. Students are asked to represent both towers using unifix cubes and determine whose tower is tallest and by how much. Next, students draw an iconic representation of the towers, paying attention to the spatial relationship between the number seven and four. The drawing should depict that one tower is taller than the other, and the enactive model is used to determine the difference between the numbers seven and four. Last, students connect their understandings of the relationship between the two towers back to the symbolic representation by notating 8–6 = 2. As students fluently build models to represent the context, they are then asked to look at a given set of numbers, build the models with unifix cubes to match, draw an iconic representation of the models, and create their own story to match their model. Students work in partners to listen to the story, but then also explain the relationships between the two towers. With this activity, students often times gesture with their hands to explain how many more blocks are in one tower than the other tower.

As suggested by NCTM [21], the DMTI curriculum intentionally focuses on building students' conceptual understandings of mathematical concepts through spatial reasoning tasks. Each task presents students with meaningful problemsolving situations where they are encouraged to begin to represent their thinking through enactive mathematical modeling, followed by an iconic representation depicting their thinking, and lastly with a connection to the symbolic representation of the problem. Students are encouraged to communicate their thinking with partners to check for understanding or assessing any misconceptions that may arise. The structural components are intentionally highlighted within each lesson to foster deep conceptual understanding and help students generalize their knowledge to other tasks throughout the year. Overall, the DMT framework delivers a comprehensive curriculum designed to increase students' mathematical understanding and improve spatial reasoning.

#### **6. Summary**

Most elementary students begin to develop their awareness of the equal sign's functionality at an operational level, where the equal sign acts as a symbol to perform a calculation or action [42]. When the bulk of instruction is focused on procedures and computing facts, many elementary students develop a shallow understanding of the equal sign and consider it an operational symbol [45, 46]. Mathematics instruction for early elementary classrooms should foster relational

**121**

*The Importance of Spatial Reasoning in Early Childhood Mathematics*

thinking by including tasks designed to draw attention to how numbers relate to one another and develop the flexibility to think of numbers in a variety of ways to establish the idea of equivalence [8, 48]. Mathematical tasks should include both

As educators become more aware of the need for relational thinking tasks, it is important to recognize the critical role spatial reasoning and mathematical modeling play in the overall development of algebraic thinking and the equal sign. The National Research Council report [1] and the National Council of Teachers of Mathematics [21] suggest more spatial reasoning be integrated into the elementary mathematics curriculum to promote relational thinking skills. Spatial visualization, gesturing, and mental rotation have been shown to increase student performance in

Mathematical modeling gives students a visual representation to explain their mathematical thinking [74]. The use of mathematical modeling should connect through a progression of concrete representations, visual or iconic representations to more formal, and abstract representations [62]. We will examine whether curriculum that supports students' conceptual understandings through the integration of relational thinking, spatial reasoning, and mathematical models by incorporating Bruner's EIS framework improves students' spatial reasoning and relational thinking.

This study was conducted to investigate whether there was a significant difference in first grade students' performance in spatial reasoning when they learn to construct and compare numbers using iconic modeling. The study examined spatial reasoning for first grade students whose teachers either received a curriculum built on the use of enactive, iconic, and symbolic representations (EIS group) and an adopted traditional curriculum (traditional group). Students in both groups were tested using the Primary Mathematics Assessment Screener [75] in September, prior to the mathematics instruction, and again mid-May after the mathematics instruction. Student performance was compared across time. Thus, this study used a 2 (EIS group versus comparison group) × 2 (pretest versus posttest) design. The dependent variable was the students' knowledge of spatial reasoning measured with the PMA-S. The goal of this study was to determine whether student achievement on the PMA-S differed between the EIS and traditional groups and whether achievement differed across time. The following research question was investigated: What is the effect of integrating iconic representations through student drawings in conjunction with the enactive, iconic, and symbolic teaching methodology into mathematics instruction on first

The study consisted of first grade classrooms from five school districts. Two of the school districts serve between 15,650 and 26,240 students, and three of the districts serve between 600 and 1725 students. There were over 2600 students with Limited English Proficiency (LEP) comprising approximately 8% of the total districts. In these districts, the student demographics were 79.3% white, 10.3% Hispanic/Latino, 5.9% Asian, 3.3% black, 0.9% Native American, and 0.8% Pacific Islander. First, grade classrooms were chosen on the basis of similarly matched demographics related to students who received free and reduced lunch assistance. There were 10 teachers in the EIS treatment group and 12 teachers in the traditional comparison group. The treatment group used the DMTI curriculum [76], and the comparison group used

grade students' spatial reasoning and relational thinking performance?

Bridges in Mathematics [77], and Math in Focus, Singapore Math [78].

The Primary Mathematics Assessment [75] is a formative assessment that includes a screener and six diagnostic measures. The PMA-Screener (PMA-S) builds

*DOI: http://dx.doi.org/10.5772/intechopen.81564*

mathematics [8].

**7. Overview of the study**

traditional and nontraditional equations [44, 46].

*The Importance of Spatial Reasoning in Early Childhood Mathematics DOI: http://dx.doi.org/10.5772/intechopen.81564*

thinking by including tasks designed to draw attention to how numbers relate to one another and develop the flexibility to think of numbers in a variety of ways to establish the idea of equivalence [8, 48]. Mathematical tasks should include both traditional and nontraditional equations [44, 46].

As educators become more aware of the need for relational thinking tasks, it is important to recognize the critical role spatial reasoning and mathematical modeling play in the overall development of algebraic thinking and the equal sign. The National Research Council report [1] and the National Council of Teachers of Mathematics [21] suggest more spatial reasoning be integrated into the elementary mathematics curriculum to promote relational thinking skills. Spatial visualization, gesturing, and mental rotation have been shown to increase student performance in mathematics [8].

Mathematical modeling gives students a visual representation to explain their mathematical thinking [74]. The use of mathematical modeling should connect through a progression of concrete representations, visual or iconic representations to more formal, and abstract representations [62]. We will examine whether curriculum that supports students' conceptual understandings through the integration of relational thinking, spatial reasoning, and mathematical models by incorporating Bruner's EIS framework improves students' spatial reasoning and relational thinking.

### **7. Overview of the study**

*Early Childhood Education*

than the other tower.

**Figure 2.**

improve spatial reasoning.

**6. Summary**

tower is six blocks tall. Students are asked to represent both towers using unifix cubes and determine whose tower is tallest and by how much. Next, students draw an iconic representation of the towers, paying attention to the spatial relationship between the number seven and four. The drawing should depict that one tower is taller than the other, and the enactive model is used to determine the difference between the numbers seven and four. Last, students connect their understandings of the relationship between the two towers back to the symbolic representation by notating 8–6 = 2. As students fluently build models to represent the context, they are then asked to look at a given set of numbers, build the models with unifix cubes to match, draw an iconic representation of the models, and create their own story to match their model. Students work in partners to listen to the story, but then also explain the relationships between the two towers. With this activity, students often times gesture with their hands to explain how many more blocks are in one tower

*Example of student work mat from Module 3 of the DMTI curriculum.*

As suggested by NCTM [21], the DMTI curriculum intentionally focuses on building students' conceptual understandings of mathematical concepts through spatial reasoning tasks. Each task presents students with meaningful problemsolving situations where they are encouraged to begin to represent their thinking through enactive mathematical modeling, followed by an iconic representation depicting their thinking, and lastly with a connection to the symbolic representation of the problem. Students are encouraged to communicate their thinking with partners to check for understanding or assessing any misconceptions that may arise. The structural components are intentionally highlighted within each lesson to foster deep conceptual understanding and help students generalize their knowledge to other tasks throughout the year. Overall, the DMT framework delivers a comprehensive curriculum designed to increase students' mathematical understanding and

Most elementary students begin to develop their awareness of the equal sign's functionality at an operational level, where the equal sign acts as a symbol to perform a calculation or action [42]. When the bulk of instruction is focused on procedures and computing facts, many elementary students develop a shallow understanding of the equal sign and consider it an operational symbol [45, 46]. Mathematics instruction for early elementary classrooms should foster relational

**120**

This study was conducted to investigate whether there was a significant difference in first grade students' performance in spatial reasoning when they learn to construct and compare numbers using iconic modeling. The study examined spatial reasoning for first grade students whose teachers either received a curriculum built on the use of enactive, iconic, and symbolic representations (EIS group) and an adopted traditional curriculum (traditional group). Students in both groups were tested using the Primary Mathematics Assessment Screener [75] in September, prior to the mathematics instruction, and again mid-May after the mathematics instruction. Student performance was compared across time. Thus, this study used a 2 (EIS group versus comparison group) × 2 (pretest versus posttest) design. The dependent variable was the students' knowledge of spatial reasoning measured with the PMA-S. The goal of this study was to determine whether student achievement on the PMA-S differed between the EIS and traditional groups and whether achievement differed across time. The following research question was investigated: What is the effect of integrating iconic representations through student drawings in conjunction with the enactive, iconic, and symbolic teaching methodology into mathematics instruction on first grade students' spatial reasoning and relational thinking performance?

The study consisted of first grade classrooms from five school districts. Two of the school districts serve between 15,650 and 26,240 students, and three of the districts serve between 600 and 1725 students. There were over 2600 students with Limited English Proficiency (LEP) comprising approximately 8% of the total districts. In these districts, the student demographics were 79.3% white, 10.3% Hispanic/Latino, 5.9% Asian, 3.3% black, 0.9% Native American, and 0.8% Pacific Islander. First, grade classrooms were chosen on the basis of similarly matched demographics related to students who received free and reduced lunch assistance. There were 10 teachers in the EIS treatment group and 12 teachers in the traditional comparison group. The treatment group used the DMTI curriculum [76], and the comparison group used Bridges in Mathematics [77], and Math in Focus, Singapore Math [78].

The Primary Mathematics Assessment [75] is a formative assessment that includes a screener and six diagnostic measures. The PMA-Screener (PMA-S) builds a profile of students' strengths and weaknesses for six dimensions: number sense and sequencing, number facts, contextual problems, relational thinking, measurement, and spatial reasoning.

One of the diagnostics includes a series of questions for shape composition. There are three subsections which include—shape composition without the need to rotate, composing a figure requiring overlapping of pieces during translations, and composing a figure by filling in a missing space.

A two-way design was used to explore the main effects on the different treatments, EIS instruction and Traditional instruction and their interactions under different conditions, pretest and posttest. The research question was analyzed using a 2 × 2 analysis of variance (ANOVA) to explore whether scores on the pre and posttest was dependent upon the type of instruction. Repeated measure analysis of variance (ANOVA) allows a look at change over time using the PMA-S given two times over 9 months of instruction with different conditions (EIS and traditional instruction). Main effects and interactions were analyzed on the independent variables (EIS and traditional instruction and time) from the dependent variable PMA-S scores.

## **8. Findings**

A two-way repeated measure ANOVA was conducted to determine whether there was a significant difference in growth between the EIS group and the traditional group for relational thinking and spatial reasoning. The PMA-S screened four other subset dimensions, facts, context, sequence, and measurement, which were not included in the design of the study.

For the relational thinking subtest, there was a main effect for TIME with a statistically significant difference for both groups (EIS and traditional)—scores increase from pretest to posttest, F(1, 449) = 105.2, MSe = 0.9, p < 0.001. There is also a main effect for groups with a statistically significant difference between EIS and traditional, F(1, 449) = 5.6, MSe = 1.2, p = 0.019.

There was a statistically significant interaction between both groups and time on relational thinking, F(1, 449) = 13.2, MSe = 0.9, p < 0.001, η<sup>2</sup> = 0.03. This indicates that the difference between the change in students' knowledge of relational thinking in the EIS and traditional groups was dependent upon the type of mathematical instruction. Based on the profile plots of estimated marginal means of relational thinking in **Figure 3**, EIS (group 1) and traditional (group 2), EIS and traditional groups' trajectories indicate different patterns of mean scores over time. The p-value for the two-way interaction effect is <0.001, indicating mean relational thinking changed differently over time depending on whether students were in EIS or traditional.

To better understand the interaction, tests of simple effects were conducted. These results showed for the EIS group, scores on the relational thinking scale increased significantly from pretest to posttest, t(242) = 10.2, p < 0.001. For the traditional group, scores on the relational thinking scale also increased significantly from pretest to posttest, t(242) = 4.6, p < 0.001. Thus, for both groups, scores increased from pretest to posttest. The EIS and traditional groups were also compared separately on the pretest and then on the posttest. These results showed that for the pretest, the groups differed significantly, t(449) = 4.5, p < 0.001. For the posttest, the groups were not significantly different, t(449) = 0.53, p = 0.6. For the pretest, scores were greater for the traditional group than for the EIS group.

Taken all together, the results of these analyses show that scores on the relational thinking subtest scores did not differ across groups. However, significant

**123**

**Table 1.**

**Figure 3.**

*The Importance of Spatial Reasoning in Early Childhood Mathematics*

interaction suggests that the change from pretest to posttest was not the same for the two groups. As seen in **Table 1**, the change was greater for the EIS group than for the traditional group. The EIS group began the study with significantly lower scores on the relational thinking subtests. The EIS group shows statistically higher

For the spatial reasoning diagnostic, there was a main effect for TIME with a statistically significant difference for both groups (EIS and traditional)—scores increased from pretest to posttest, F(1, 449) = 85.2, MSe = 0.6, p < 0.001. There was also a main effect for groups with a statistically significant difference between EIS

There was a marginal significant interaction between both groups and time on

that the difference between the change in students' knowledge of spatial reasoning in the EIS and traditional groups was dependent upon the type of mathematical instruction. Based on the profile plots of estimated marginal means of spatial reasoning (**Figure 4**), EIS and traditional groups' trajectories indicate slightly different

To better understand the interaction, tests of simple effects were conducted. These results showed for the EIS group, scores on the spatial reasoning scale increased significantly from pretest to posttest, *t*(207) = 7.4, p < 0.001. For

**Group Mean SD Mean SD** EIS 0.74 0.77 1.61 1.2 Traditional 1.14 1.1 1.55 1.1

= 0.01. This indicates

**Pretest Posttest**

gains than the traditional, thus confirming EIS has an effect.

and traditional, F(1, 449) = 3.9, MSe = 0.9, p = 0.05.

patterns of mean scores over time.

**Relational thinking**

*Relational thinking descriptive statistics.*

*Estimated marginal means of relational thinking.*

spatial reasoning, F(1, 449) = 3.3, MSe = 0.6, p < 0.071, η<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.81564*

*The Importance of Spatial Reasoning in Early Childhood Mathematics DOI: http://dx.doi.org/10.5772/intechopen.81564*

*Early Childhood Education*

ment, and spatial reasoning.

PMA-S scores.

**8. Findings**

or traditional.

composing a figure by filling in a missing space.

not included in the design of the study.

and traditional, F(1, 449) = 5.6, MSe = 1.2, p = 0.019.

relational thinking, F(1, 449) = 13.2, MSe = 0.9, p < 0.001, η<sup>2</sup>

a profile of students' strengths and weaknesses for six dimensions: number sense and sequencing, number facts, contextual problems, relational thinking, measure-

One of the diagnostics includes a series of questions for shape composition. There are three subsections which include—shape composition without the need to rotate, composing a figure requiring overlapping of pieces during translations, and

A two-way design was used to explore the main effects on the different treatments, EIS instruction and Traditional instruction and their interactions under different conditions, pretest and posttest. The research question was analyzed using a 2 × 2 analysis of variance (ANOVA) to explore whether scores on the pre and posttest was dependent upon the type of instruction. Repeated measure analysis of variance (ANOVA) allows a look at change over time using the PMA-S given two times over 9 months of instruction with different conditions (EIS and traditional instruction). Main effects and interactions were analyzed on the independent variables (EIS and traditional instruction and time) from the dependent variable

A two-way repeated measure ANOVA was conducted to determine whether there was a significant difference in growth between the EIS group and the traditional group for relational thinking and spatial reasoning. The PMA-S screened four other subset dimensions, facts, context, sequence, and measurement, which were

For the relational thinking subtest, there was a main effect for TIME with a statistically significant difference for both groups (EIS and traditional)—scores increase from pretest to posttest, F(1, 449) = 105.2, MSe = 0.9, p < 0.001. There is also a main effect for groups with a statistically significant difference between EIS

There was a statistically significant interaction between both groups and time on

that the difference between the change in students' knowledge of relational thinking in the EIS and traditional groups was dependent upon the type of mathematical instruction. Based on the profile plots of estimated marginal means of relational thinking in **Figure 3**, EIS (group 1) and traditional (group 2), EIS and traditional groups' trajectories indicate different patterns of mean scores over time. The p-value for the two-way interaction effect is <0.001, indicating mean relational thinking changed differently over time depending on whether students were in EIS

To better understand the interaction, tests of simple effects were conducted. These results showed for the EIS group, scores on the relational thinking scale increased significantly from pretest to posttest, t(242) = 10.2, p < 0.001. For the traditional group, scores on the relational thinking scale also increased significantly from pretest to posttest, t(242) = 4.6, p < 0.001. Thus, for both groups, scores increased from pretest to posttest. The EIS and traditional groups were also compared separately on the pretest and then on the posttest. These results showed that for the pretest, the groups differed significantly, t(449) = 4.5, p < 0.001. For the posttest, the groups were not significantly different, t(449) = 0.53, p = 0.6. For the pretest, scores were greater for the traditional group than for the EIS group. Taken all together, the results of these analyses show that scores on the relational thinking subtest scores did not differ across groups. However, significant

= 0.03. This indicates

**122**

**Figure 3.** *Estimated marginal means of relational thinking.*

interaction suggests that the change from pretest to posttest was not the same for the two groups. As seen in **Table 1**, the change was greater for the EIS group than for the traditional group. The EIS group began the study with significantly lower scores on the relational thinking subtests. The EIS group shows statistically higher gains than the traditional, thus confirming EIS has an effect.

For the spatial reasoning diagnostic, there was a main effect for TIME with a statistically significant difference for both groups (EIS and traditional)—scores increased from pretest to posttest, F(1, 449) = 85.2, MSe = 0.6, p < 0.001. There was also a main effect for groups with a statistically significant difference between EIS and traditional, F(1, 449) = 3.9, MSe = 0.9, p = 0.05.

There was a marginal significant interaction between both groups and time on spatial reasoning, F(1, 449) = 3.3, MSe = 0.6, p < 0.071, η<sup>2</sup> = 0.01. This indicates that the difference between the change in students' knowledge of spatial reasoning in the EIS and traditional groups was dependent upon the type of mathematical instruction. Based on the profile plots of estimated marginal means of spatial reasoning (**Figure 4**), EIS and traditional groups' trajectories indicate slightly different patterns of mean scores over time.

To better understand the interaction, tests of simple effects were conducted. These results showed for the EIS group, scores on the spatial reasoning scale increased significantly from pretest to posttest, *t*(207) = 7.4, p < 0.001. For


**Table 1.** *Relational thinking descriptive statistics.*

**Figure 4.** *Estimated marginal means of spatial reasoning.*

the traditional group, scores on the spatial reasoning scale also increased significantly from pretest to posttest, *t*(242) = 5.5, p < 0.001. Thus, for both groups, scores increased from pretest to posttest. The EIS and traditional groups were also compared separately on the pretest and then on the posttest. These results showed that for the pretest, the groups differed significantly, *t*(449) = 2.8, p < 0.01. For the posttest, the groups were not significantly different, *t*(449) = 0.36, p = 0.72. For the pretest, scores were greater for the traditional than for the EIS group, and on the posttest, scores were the same across both groups.

Taken together, the results of these analyses show that scores on the spatial reasoning subtest were equal on the posttest across both groups. However, the marginally significant interaction suggests that the change from pretest to posttest was not the same for the two groups. As seen in **Table 2**, the change was greater for the EIS group than for the traditional group. The EIS group began the study with significantly lower scores on the spatial reasoning subtests. The EIS group shows statistically higher gains than the traditional, thus confirming EIS has an effect.

In summary, the instructional method (EIS vs. traditional) did have a significant effect on first grade students' spatial reasoning. The study demonstrated statistical significance between the treatment groups who implemented the EIS instruction and comparison group who used traditional mathematics instruction. The next


**125**

*The Importance of Spatial Reasoning in Early Childhood Mathematics*

for educators, and recommendations for further study.

section will provide details of the interpretation of findings, practical implications

The primary focus of the study was to look at the effects on students' conceptual understandings of relational thinking and spatial reasoning when integrating the EIS representations into first grade mathematics lessons. As Cheng and Mix [8] revealed through their research, the need to integrate spatial reasoning tasks is critical for the development of students' conceptual knowledge. Similar claims can

The EIS group performed statistically higher in relational thinking than the traditional group, doubling mean scores from pretest (0.74) to posttest (1.27). Previous work has shown students who are instructed to solve equations strictly in symbolic form struggle with algebraic thinking [79]. Integrating EIS representation into first grade mathematics lessons with a balanced set of equations has shown to be effective at developing students' relational thinking and spatial

As Cheng and Mix [8] revealed through their research, the need to integrate spatial reasoning tasks is critical for the development of students' conceptual knowledge. Similar claims can be made based on the results from this study. We conclude that the integration of spatial reasoning had positive effects on first grade students' spatial reasoning skills, relational thinking, the development of concep-

The findings support the notion that the integration of EIS representation into mathematics lessons offers students sufficient conceptual knowledge to develop number operations and mathematical competency [48]. Gain scores in facts and context are found to be consistent with earlier works from Carbonneau and colleagues [61], who suggests mathematics instruction should refrain from isolated skill and procedural practice in lieu of the development of conceptual understanding. Curriculum designed to include a progression of enactive, iconic, and symbolic models supports students' conceptual understanding [2, 62, 63]. Students in the EIS group were instructed to enactively build and iconically represent their math facts simultaneously. In doing so, they increased their conceptual understanding of the mathematics. K-12 reform has included an integration of meaningful lessons designed to enhance algebraic thinking across all mathematical domains, and altering the curriculum to include spatial reasoning tasks has shown to improve mathematical performance [54]. Our investigation has demonstrated a positive effect on students' spatial reasoning, relational thinking, and overall mathematical competency when first grade mathematics

*DOI: http://dx.doi.org/10.5772/intechopen.81564*

be made based on the results of this study.

tual understanding, and mathematical competency.

lessons integrate EIS representations.

**9. Interpretation of findings**

reasoning.

**Table 2.** *Spatial reasoning descriptive statistics.* section will provide details of the interpretation of findings, practical implications for educators, and recommendations for further study.
