**3. Relational thinking**

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

areas of mathematics [39–42]. One way to improve conceptual understanding is to increase the exposure of problem-solving tasks involving nontraditional equations. It has been shown that students as young as kindergarten and first grade have informal knowledge of number relations; however, the mathematics presented in traditional textbooks do not explicitly draw out these relations, allow time for the relations to organically emerge, or instruct students to determine how the ideas can be generalized (Blanton and Kaput [40]). Consequently, there is a need for mathematics instruction to incorporate more than just the traditional format of equations into daily lessons and include ways to represent relational equivalence [43, 44].

One aspect of relational thinking is equal sign. 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]. For instance, students with an operational view of the equal sign will reject any equations presented outside of the traditional format, a + b = c, and will define the purpose of the equal sign as a cue to perform the calculations on the left side of the equal sign to get an answer [47]. However, given more exposure to a variety of equations, students can become more flexible with their thinking and progress to different levels of understanding [40]. Mathematics instruction for early elementary classrooms should foster relational 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].

Matthews et al. [49] developed a construct map based on the research of Carpenter [47] and Hunter [50] to explain the continuum of relational thinking for students' thinking. The first level of student understanding is called rigid operational. Students at this level are calculating traditional or missing term equations. Traditional equations, written a + b = c, place the equal sign as a function for solving the addition problem a + b to produce an answer. This traditional format instills an operational view of the equal sign [51]. With exposure to nontraditional equations, such as a = b + c, students become more flexible in their determination of a correctly written equation. However, their view of the equal sign still remains as a cue for calculation. As students move into the basic relational stage, their flexibility to solve equations written with operations on both sides of the equals sign increases. However, it is not until the final stage, comparative relational when students consider the number relations on each side of the equal sign to determine equivalency and their need to calculate diminishes. This level of relational thinking demonstrates students' knowledge about how the equal sign relates to the entire equation, where they are looking for relatable numbers in the equation prior to solving the problem [52]. Identifying these relationships in equations and their connections with the numbers is a critical component of mathematical understanding. Developing and applying the knowledge of relational thinking to solve mathematical equivalence problems will increase early algebraic understanding [41, 44, 46, 53]. Students who think at the comparative relational level have a strong understanding of the equal sign and a deeper connection to algebraic reasoning [47, 50].

The natural tendency for students as young as kindergarten is to demonstrate an operational view of the equal sign; however, they do have the capabilities to think relationally if given the opportunity [45]. Therefore, relational thinking skills should be explicitly taught at an early age to avoid a deep-rooted set of operational skills [54]. Relational thinking involves flexible thinking to determine how numbers can be manipulated before answering a problem. Using relational thinking to

**117**

*The Importance of Spatial Reasoning in Early Childhood Mathematics*

number is manipulated the equation remains equivalent.

solve an algebraic equation requires the conceptual understanding that each time a

Providing students with a progression of nontraditional number sentences focused on numerical relationships and patterns will develop relational thinking. As a starting point for young students reversing the order of the number sentence to begin with the answer such as 3 = 2 + 1 presses students to accept that the answer does not always need to be after the operation [49, 55]. Next, students develop their understanding of the term equal as they begin to recognize that both sides of the equation compute to the same quantity through exposure to nontraditional equations written with the operations on both sides of the equal sign [47]. Students who possess the conceptual knowledge of equivalence recognize transformations can occur by adding the same number to both sides of the equal sign without changing the structure of the equation. For example, when asked whether the equation 18 + 3 = 16 + 5 is true or false, students who are taught to think about the relationship between 18 and 16, notice that 18 is 2 more than 16, and reason that it must be true because 5 is 2 more than 3. Unfortunately, if students are not taught to look at equations relationally, then the transformations between 18 and 16 simply become proceduralized and learned as memorized rules [52]. This strategy shows a level of relational thinking in which students use number relations to make the problem more manageable. Thinking relationally, therefore, is different from applying a collection of memorized mathematical rules and procedures [56]. Students who think relationally identify number relations and reason about which transformations

Providing students with true or false equations can be another way to press students to think about number relationships. Equations such as 14 + 18 = 13 + 17 are more compatible with instructing students to see number relationships because a numerical answer is not required. Engaging students in a discussion of how the numbers relate to each other to determine whether the equation is true or false strengthens their conceptual understandings of equivalence (Carpenter [47]). Students with sufficient conceptual knowledge of how these number properties are applied have the understanding to transfer their procedural knowledge of mathematical equations to algebraic thinking [48]. Meaningful discussions about number relationships and the transferability of those ideas helps students make

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

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

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

make sense in a particular problem [42].

more mathematical generalizations [39].

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

mechanism to help remind them of the relevant concepts.

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

*Early Childhood Education*

equivalence [43, 44].

establish the idea of equivalence [8, 48].

areas of mathematics [39–42]. One way to improve conceptual understanding is to increase the exposure of problem-solving tasks involving nontraditional equations. It has been shown that students as young as kindergarten and first grade have informal knowledge of number relations; however, the mathematics presented in traditional textbooks do not explicitly draw out these relations, allow time for the relations to organically emerge, or instruct students to determine how the ideas can be generalized (Blanton and Kaput [40]). Consequently, there is a need for mathematics instruction to incorporate more than just the traditional format of equations into daily lessons and include ways to represent relational

One aspect of relational thinking is equal sign. 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]. For instance, students with an operational view of the equal sign will reject any equations presented outside of the traditional format, a + b = c, and will define the purpose of the equal sign as a cue to perform the calculations on the left side of the equal sign to get an answer [47]. However, given more exposure to a variety of equations, students can become more flexible with their thinking and progress to different levels of understanding [40]. Mathematics instruction for early elementary classrooms should foster relational 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

Matthews et al. [49] developed a construct map based on the research of Carpenter [47] and Hunter [50] to explain the continuum of relational thinking for students' thinking. The first level of student understanding is called rigid operational. Students at this level are calculating traditional or missing term equations. Traditional equations, written a + b = c, place the equal sign as a function for solving the addition problem a + b to produce an answer. This traditional format instills an operational view of the equal sign [51]. With exposure to nontraditional equations, such as a = b + c, students become more flexible in their determination of a correctly written equation. However, their view of the equal sign still remains as a cue for calculation. As students move into the basic relational stage, their flexibility to solve equations written with operations on both sides of the equals sign increases. However, it is not until the final stage, comparative relational when students consider the number relations on each side of the equal sign to determine equivalency and their need to calculate diminishes. This level of relational thinking demonstrates students' knowledge about how the equal sign relates to the entire equation, where they are looking for relatable numbers in the equation prior to solving the problem [52]. Identifying these relationships in equations and their connections with the numbers is a critical component of mathematical understanding. Developing and applying the knowledge of relational thinking to solve mathematical equivalence problems will increase early algebraic understanding [41, 44, 46, 53]. Students who think at the comparative relational level have a strong understanding

of the equal sign and a deeper connection to algebraic reasoning [47, 50].

The natural tendency for students as young as kindergarten is to demonstrate an operational view of the equal sign; however, they do have the capabilities to think relationally if given the opportunity [45]. Therefore, relational thinking skills should be explicitly taught at an early age to avoid a deep-rooted set of operational skills [54]. Relational thinking involves flexible thinking to determine how numbers can be manipulated before answering a problem. Using relational thinking to

**116**

solve an algebraic equation requires the conceptual understanding that each time a number is manipulated the equation remains equivalent.

Providing students with a progression of nontraditional number sentences focused on numerical relationships and patterns will develop relational thinking. As a starting point for young students reversing the order of the number sentence to begin with the answer such as 3 = 2 + 1 presses students to accept that the answer does not always need to be after the operation [49, 55]. Next, students develop their understanding of the term equal as they begin to recognize that both sides of the equation compute to the same quantity through exposure to nontraditional equations written with the operations on both sides of the equal sign [47]. Students who possess the conceptual knowledge of equivalence recognize transformations can occur by adding the same number to both sides of the equal sign without changing the structure of the equation. For example, when asked whether the equation 18 + 3 = 16 + 5 is true or false, students who are taught to think about the relationship between 18 and 16, notice that 18 is 2 more than 16, and reason that it must be true because 5 is 2 more than 3. Unfortunately, if students are not taught to look at equations relationally, then the transformations between 18 and 16 simply become proceduralized and learned as memorized rules [52]. This strategy shows a level of relational thinking in which students use number relations to make the problem more manageable. Thinking relationally, therefore, is different from applying a collection of memorized mathematical rules and procedures [56]. Students who think relationally identify number relations and reason about which transformations make sense in a particular problem [42].

Providing students with true or false equations can be another way to press students to think about number relationships. Equations such as 14 + 18 = 13 + 17 are more compatible with instructing students to see number relationships because a numerical answer is not required. Engaging students in a discussion of how the numbers relate to each other to determine whether the equation is true or false strengthens their conceptual understandings of equivalence (Carpenter [47]). Students with sufficient conceptual knowledge of how these number properties are applied have the understanding to transfer their procedural knowledge of mathematical equations to algebraic thinking [48]. Meaningful discussions about number relationships and the transferability of those ideas helps students make more mathematical generalizations [39].
