**Abstract**

It is important to recognize the critical role spatial reasoning, relational thinking, and mathematical modeling play in the overall development of students' central understanding of mathematics. Spatial reasoning predicts students' later success in higher levels of mathematics, such as proportional thinking and algebraic reasoning. The National Research Council report implores educators to recognize the importance of developing spatial reasoning skills with students across all areas of mathematics. This chapter describes a study that used the Primary Math Assessment—Screener and Diagnostic to assess students' spatial reasoning and relational thinking. The results highlighted curricular resources to improve students' understanding of mathematics. Students' mathematical spatial reasoning improved significantly.

**Keywords:** spatial reasoning, relational thinking, early childhood, mathematics, achievement, DMTI

#### **1. Introduction**

It is important for educators to recognize the critical role spatial reasoning along with mathematical modeling plays in the overall development of mathematical skills and understanding. It is a fundamental bridge to algebraic thinking and conceptual understanding. The National Research Council report [1] urges educators to recognize the importance of developing these skills with students across all areas of mathematics.

Bruner's [2] modes of representation describe the process of enriching students' understanding by working through enactive, iconic, and symbolic (EIS) models. The enactive (physical) and iconic (visual) models are critical to help students develop connections to a task and allows for better recall of mathematical ideas. It is critical for teachers to expose students to different methods of modeling relationships with multiple representations. Students will have a better opportunity to generalize and build on existing foundational knowledge of equivalence throughout their mathematical careers.

Many students have difficulty in understanding concepts without being able to first observe a pictorial image of an idea in their mind [3]. Mathematics curricula loaded with symbolic representation require students to memorize procedures, denying the student an opportunity to utilize their visual thinking modality in the process of building conceptual understanding. On the other hand, curricula that embed more iconic models may allow for students to deepen their understanding of the mathematics and improve their skill levels [4]. Thus, we wanted to investigate whether there was a significant difference in first grade students' performance in spatial reasoning after being introduced to mathematics that included a plethora of iconic modeling.

## **2. Spatial reasoning**

Spatial reasoning is strongly correlated with achievement in mathematics [5–7]. Students who perform better on spatial tasks also perform better on tests of mathematical ability [8–10]. Spatial reasoning involves (a) composing and decomposing shapes and figures, (b) visualization, or the ability to mentally manipulate, rotate, twist, or invert pictures or objects, (c) spatial orientation, or the ability to recognize an object even when the object's orientation changes, and (d) spatial relations, or the ability to recognize spatial patterns, to understand spatial hierarchies, and to imagine maps from verbal descriptions [10, 11]. Recent evidence indicates that spatial reasoning training can have transfer effects on mathematics achievement, particularly on missing term problems (e.g., 7 + \_\_ = 15), which are important in developing algebraic understanding [8].

In addition, spatial reasoning skills and mathematical competency are directly related to each other [12–15]. Learning with specific spatial reasoning tasks improves students' abilities in the Science, Technology, Engineering, and Mathematics (STEM) fields [16, 17]. And there is a strong link between spatial reasoning ability and geometry where strong visuospatial skills predict how well students will complete 3D geometry tasks [18–20]. As educators become more aware of the need for spatial reasoning tasks, it is important to recognize the critical role mathematical modeling plays in the overall development of mathematical thinking.

The National Research Council report [1] urges educators to recognize the importance of developing spatial reasoning skills with students across all areas of mathematics. And the National Council of Teachers of Mathematics [21] suggests more spatial reasoning be integrated into the elementary mathematics curriculum to promote relational thinking skills. Mathematical modeling may be a key component to help students explain their thinking when representing algebraic concepts.

Mix and Cheng [22] found that students with strong spatial reasoning skills do well in mathematics [23]. Spatial reasoning is a critical element for developing ways students think about equations. Given the opportunity, students' spatial reasoning skills can increase when practice is integrated and supported throughout mathematics instruction [24]. By the time students reach kindergarten, their spatial reasoning skills predict their overall mathematical success [25]. Therefore, students' educational experience in elementary school should have an intentional focus on improving spatial reasoning skills.

The focus of the next section is to highlight the connection between spatial reasoning and spatial orientation on a number line, gesture, visualization, and mental rotation. For instance, a crucial component to understanding ordinality (the position of a number in relation to its location on a number line) and magnitude (the size of a number) is the development of a spatial representation of numbers in connection to the symbolic representations [26]. The number line has been shown in cognitive studies to be important for the development of numerical knowledge [27–29]. Ramani and Siegler [30] report that students who play board games such as Chutes and Ladders increase rote counting skills, number identification, and the

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[37, 38].

**3. Relational thinking**

*The Importance of Spatial Reasoning in Early Childhood Mathematics*

was creating and conveying mathematical thinking.

increase student performance in mathematics [8].

within equations for simpler computation [8].

conceptual understanding of numerical magnitude. Additionally, activities which include puzzles, video games, and blocks with significant connections to spatial reasoning skills and mathematical competency improve accuracy of symbolically

Problem-solving tasks regarding orientation, transformations, and movement of shapes create an opportunity among students and the teacher to engage in rich, mathematical discourse. As students discuss their thinking, they will use their hands to gesture while attempting to convey their thoughts surrounding the task. Gesturing allows students to explain the visual imagery taking place inside their head as they work on problem-solving specific tasks [32]. Students' gestures represent the movement of the transformation and create an avenue for their thinking to emerge through the discussion. Alibali and Nathan [33] found gestures to be an excellent tool for teaching students how to solve spatial transformation tasks by placing an emphasis on the importance of moving the pieces without the actual physical movement. In essence, they used their hands to gesture what their mind

The ability to gesture what the mind is thinking is dependent upon students'

Students who are allotted time to practice mental rotation have demonstrated the ability to solve a series of multi-step word problems [36]. Mental rotation consists of the ability to look at an object or picture of an object and visualize what it might look like when rotated in 2D or 3D space. The most recent study of spatial training with mental rotation was conducted with young students developing number sense, counting sequence, fact fluency, and missing term problems [8, 22]. Although the other areas showed improvement with the spatial training, missing term problems such as 2 + \_\_ = 6 indicated the most significant effect size. Much like the relational skills needed to find the most efficient way to solve missing term problems, the completion of mental rotation tasks during spatial training helped to strengthen students' ability to visualize the necessary transformations of numbers

It is important to note that mental rotation and spatial visualization are both subsets to spatial reasoning and much of their characteristics overlap [34]. Developing both skills is a powerful way to connect back to the bigger idea of conceptual understanding for relational thinking, spatial reasoning, and equivalence

In addition to spatial reasoning, relational thinking or early algebraic reasoning is critical for long-term success in mathematics. Students need time to develop relational thinking, with practice designed to explicitly examine the way in which numbers relate, and ways that those relations can generalize to other

ability to visualize mathematical transformations [34]. The ability to think relationally requires students to visualize how numbers can be manipulated and rearranged in an equation [35]. Therefore, visualization is a key component across mathematical topics [34]. Spatial visualization tasks require students to create an image in their mind, hold the image, and then mentally transform or manipulate that image to be different. Some examples of these types of tasks include composing and decomposing pattern blocks to determine a new composed image, imagining transformations and perspectives of a three-dimensional cube, or activities that involve mentally folding a two-dimensional shape to form a new three-dimensional shape. In addition to spatial visualization, mental rotation has also been shown to

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

representing a number line [31].

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

*Early Childhood Education*

iconic modeling.

thinking.

**2. Spatial reasoning**

developing algebraic understanding [8].

improving spatial reasoning skills.

process of building conceptual understanding. On the other hand, curricula that embed more iconic models may allow for students to deepen their understanding of the mathematics and improve their skill levels [4]. Thus, we wanted to investigate whether there was a significant difference in first grade students' performance in spatial reasoning after being introduced to mathematics that included a plethora of

Spatial reasoning is strongly correlated with achievement in mathematics [5–7]. Students who perform better on spatial tasks also perform better on tests of mathematical ability [8–10]. Spatial reasoning involves (a) composing and decomposing shapes and figures, (b) visualization, or the ability to mentally manipulate, rotate, twist, or invert pictures or objects, (c) spatial orientation, or the ability to recognize an object even when the object's orientation changes, and (d) spatial relations, or the ability to recognize spatial patterns, to understand spatial hierarchies, and to imagine maps from verbal descriptions [10, 11]. Recent evidence indicates that spatial reasoning training can have transfer effects on mathematics achievement, particularly on missing term problems (e.g., 7 + \_\_ = 15), which are important in

In addition, spatial reasoning skills and mathematical competency are directly related to each other [12–15]. Learning with specific spatial reasoning tasks improves students' abilities in the Science, Technology, Engineering, and Mathematics (STEM) fields [16, 17]. And there is a strong link between spatial reasoning ability and geometry where strong visuospatial skills predict how well students will complete 3D geometry tasks [18–20]. As educators become more aware of the need for spatial reasoning tasks, it is important to recognize the critical role mathematical modeling plays in the overall development of mathematical

The National Research Council report [1] urges educators to recognize the importance of developing spatial reasoning skills with students across all areas of mathematics. And the National Council of Teachers of Mathematics [21] suggests more spatial reasoning be integrated into the elementary mathematics curriculum to promote relational thinking skills. Mathematical modeling may be a key component to help students explain their thinking when representing algebraic concepts. Mix and Cheng [22] found that students with strong spatial reasoning skills do well in mathematics [23]. Spatial reasoning is a critical element for developing ways students think about equations. Given the opportunity, students' spatial reasoning skills can increase when practice is integrated and supported throughout mathematics instruction [24]. By the time students reach kindergarten, their spatial reasoning skills predict their overall mathematical success [25]. Therefore, students' educational experience in elementary school should have an intentional focus on

The focus of the next section is to highlight the connection between spatial reasoning and spatial orientation on a number line, gesture, visualization, and mental rotation. For instance, a crucial component to understanding ordinality (the position of a number in relation to its location on a number line) and magnitude (the size of a number) is the development of a spatial representation of numbers in connection to the symbolic representations [26]. The number line has been shown in cognitive studies to be important for the development of numerical knowledge [27–29]. Ramani and Siegler [30] report that students who play board games such as Chutes and Ladders increase rote counting skills, number identification, and the

**114**

conceptual understanding of numerical magnitude. Additionally, activities which include puzzles, video games, and blocks with significant connections to spatial reasoning skills and mathematical competency improve accuracy of symbolically representing a number line [31].

Problem-solving tasks regarding orientation, transformations, and movement of shapes create an opportunity among students and the teacher to engage in rich, mathematical discourse. As students discuss their thinking, they will use their hands to gesture while attempting to convey their thoughts surrounding the task. Gesturing allows students to explain the visual imagery taking place inside their head as they work on problem-solving specific tasks [32]. Students' gestures represent the movement of the transformation and create an avenue for their thinking to emerge through the discussion. Alibali and Nathan [33] found gestures to be an excellent tool for teaching students how to solve spatial transformation tasks by placing an emphasis on the importance of moving the pieces without the actual physical movement. In essence, they used their hands to gesture what their mind was creating and conveying mathematical thinking.

The ability to gesture what the mind is thinking is dependent upon students' ability to visualize mathematical transformations [34]. The ability to think relationally requires students to visualize how numbers can be manipulated and rearranged in an equation [35]. Therefore, visualization is a key component across mathematical topics [34]. Spatial visualization tasks require students to create an image in their mind, hold the image, and then mentally transform or manipulate that image to be different. Some examples of these types of tasks include composing and decomposing pattern blocks to determine a new composed image, imagining transformations and perspectives of a three-dimensional cube, or activities that involve mentally folding a two-dimensional shape to form a new three-dimensional shape. In addition to spatial visualization, mental rotation has also been shown to increase student performance in mathematics [8].

Students who are allotted time to practice mental rotation have demonstrated the ability to solve a series of multi-step word problems [36]. Mental rotation consists of the ability to look at an object or picture of an object and visualize what it might look like when rotated in 2D or 3D space. The most recent study of spatial training with mental rotation was conducted with young students developing number sense, counting sequence, fact fluency, and missing term problems [8, 22]. Although the other areas showed improvement with the spatial training, missing term problems such as 2 + \_\_ = 6 indicated the most significant effect size. Much like the relational skills needed to find the most efficient way to solve missing term problems, the completion of mental rotation tasks during spatial training helped to strengthen students' ability to visualize the necessary transformations of numbers within equations for simpler computation [8].

It is important to note that mental rotation and spatial visualization are both subsets to spatial reasoning and much of their characteristics overlap [34]. Developing both skills is a powerful way to connect back to the bigger idea of conceptual understanding for relational thinking, spatial reasoning, and equivalence [37, 38].
