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### By Kristin Klingensmith and Joe Dostilio

People of all ages and in all spaces use relational thinking on a regular basis.

• A person makes sense of what is happening in a baseball game because they are familiar with softball.
• Someone else identifies the bad guy in a book or movie because they made connections to another book they have read or movies they have seen.
• The child who wants to maintain a \$7 allowance reasons that if the value of each chore is doubled, they only have to complete half the number of them.
• The hiker who reasons that if they go three times faster, they can go 3 times the distance in the same amount of time.

The truth is that relational thinking is fairly common, and people, children and adults alike, naturally use relational thinking as they move through life.

You might be asking yourself what this has to do with mathematics, and if you are, great question! For the past couple of years, the IFL math team has been exploring relational thinking and its role in learning mathematics with understanding (rather than through memorization or rote procedures). One of the defining characteristics of learning with understanding is that knowledge is connected (Greeno, Collins, & Resnick 1996, Hiebert & Carpenter 1992, Kilpatrick et al. 2001), and relational thinking is one way of connecting knowledge. We know from research and our work in schools that too often in mathematics classrooms, learning is fragmented into a series of discreet skills, ideas, and procedures. There are few opportunities for students to step back and look at how the skills, ideas, and procedures are related and make connections to the structures of mathematics.

In this article, the first in a series, we look at relational thinking through the lens of numeric and algebraic reasoning. Our goal for all the articles in the series is to highlight ways in which relational thinking may appear and be supported in mathematics classrooms to enhance the learning opportunities afforded students.

##### Relational Thinking and Numeric and Algebraic Reasoning

In mathematics, a common definition for relational thinking refers to the recognition and use of relations between elements in arithmetic or algebraic expression and the properties of number and operations (Carpenter, T.P.; Franke, M.L.; Levi, L., 2004; Molina, M. 2009; Stephens, M.; Ribeiro, A., 2012).  Inherent to this definition of relational thinking is an understanding and use of mathematical structures, which is directly connected to Mathematical Practice 7 from the Common Core State Standards—Look for and make use of structure.

One of the mathematical structures central to numeric and algebraic reasoning using relational thinking is the property of equality, an understanding of the equal symbol as a balance and as a representation of equivalence rather than a signal to calculate an answer.

We have provided some examples to illustrate what relational thinking can look and sound like when engaging in numeric and algebraic reasoning. The examples have been chosen intentionally to distinguish reasoning rooted in relational thinking from reasoning based on calculation because relational thinking moves beyond following a prescribed sequence of procedures or solely using calculation as reasoning.

###### Example 1: Is 56 + 39 = 55 + 40 a true statement? How do you know?

Here’s a numeric reasoning example from an elementary classroom of how relational thinking may look different than thinking based on calculation. In this example, the students have been asked to decide if the numeric statement 56 + 39 = 55 + 40 is true and share how they knew.

The two work samples indicate that the statement is true but illustrate different ways of coming to this conclusion. Which do you think shows relational thinking?

If you said Work Sample A shows relational thinking, you probably noticed that the students used reasoning related to number and equivalence. In Work Sample A the student focuses on determining how the expression to the left of the equal symbol can be transformed into the expression to the right side of the equal symbol. The student indicates that moving 1 from 56 to 39 transforms the expression into 55 + 40. In this sample, relational thinking was used to look globally at the statement 56 + 39 = 55 + 40 and use mathematical structures related to number and meaning of the equal symbol to reason about the statement’s truth.

In Work Sample B the student, who also proves the statement is true, does so by calculating the sums of 56 + 39 and 55 + 40 to show that both expressions equal 95. While there is nothing inaccurate about this student’s approach, there is no evidence that the student is thinking about the relation between the two numerical expressions. Therefore, it would be worth asking this student to consider how their peer’s work shows that 56 + 39 is related to 55 + 40.

###### Example 2: What is the unknown in 3 x 8 = 6 x ____? How do you know?

Here’s an algebraic example of relational thinking.  In this example, students were asked to determine the unknown in the statement 3 x 8 = 6 x ____ and be ready to show how they knew.

Some students determined (or knew) that 3 x 8 = 24 and then figured out the unknown was 4 by using either multiplication with an unknown factor or dividing 24 by 6. These approaches to finding the unknown are mathematically accurate, but these approaches are not examples of relational thinking.

Students who use relational thinking may reason about the structural meaning of the factors, the number of groups and amount in each group. Take a look at the representations below. Structurally speaking, how does the transformation from the first diagram to the second diagram illustrate the unknown in 3 x 8 = 6 x ____? Did you see where the unknown, 4, appears? The first diagram shows 3 groups with 8 items in each, and the second diagram shows 6 groups with only 4 items in each. To maintain the balance signified by the equal symbol in 3 x 8 = 6 x ___, when the number of groups doubles from 3 to 6, the number of items in each group is halved from 8 to 4. Structurally this occurs because there is a multiplicative relationship between the number of groups and the size of each group, so when the number of groups changes, the size of each group changes by the multiplicative inverse. It would be worth asking students who used multiplication and/or division approaches to consider how the diagram shows how the expression 3 x 8 is related to the expression 6 x 4.

##### Orchestrating Opportunities for Relational Thinking through Numeric and Algebraic Reasoning

When we regularly and intentionally look for opportunities to surface and discuss relational thinking, we provide learning experiences that move beyond calculating to produce an answer. Allowing opportunities for and discussions around students’ manipulation of numbers in ways that speak to structure and properties benefits every student. Different students see different relationships and patterns. Having multiple students sharing their relational thinking not only validates their voice, it also messages that math is not about one way to “get it right.”

Let’s look at one more example to consider how relational thinking opportunities can be orchestrated in a classroom by posing purposeful questions to discuss students’ solution paths from a high-level task.

###### Example 3: Treat Bags Task

In this example students engaged with the high-level task shown below.

• Mary is making 6 treat bags for a party. Each treat bag has 5 pieces of candy and 3 toys.
• Joseph is also having a party. He makes 12 treat bags. He puts 5 pieces of candy and 3 toys in each bag.
1. How many pieces of candy does Mary need? How many toys does she need?
2. How many pieces of candy does Joseph need? How many toys does he need?
3. Who needs the most treats, Mary or Joseph? Explain how you know who needs the most treats.

Most of the small groups used a combination of addition and multiplication to determine that Joseph needed more treats since he needed 96 and Mary only needed 48 treats.

Some of the groups figured out how many pieces of candy Mary and Joseph needed by multiplying 6 x 5 and 12 x 5, respectively. Then they figured out the number of toys each needed. After which they added the pieces of candy to the number of toys to find the total number of treats each needed. The image to the right shows one group’s work using the “multiply then add” approach.

A few of those small groups added first, 5 candies + 3 toys, and determined that there were 8 treats in each bag. Then they multiplied 6 x 8 for Mary and 12 x 8 for Joseph to figure out the total number of treats that each of them needed.

These pathways, regardless of the order multiplication and addition were used, are mathematically accurate and logical and are based on computation. The teacher anticipated that students would primarily use calculations and decided to plan questions in advance that would create opportunities for students to engage in relational thinking.

Some of those questions are provided below. As you read through them, consider the relational thinking that may be used to answer them.

• Can we write 6 x 8 = (6 x 5) + (6 x 3)? Why or why not?
• Is 12 x (5 + 3) = 12 x 8 a true statement? How do you know?
• One student said they knew that Joseph needed more treats since he needed twice as many treats as Mary. What about their treat bags would allow us to make the claim that Joseph needed twice as many treats as Mary?
• How can we justify 12 x 8 > 6 x 8?

You probably noticed that the first two questions ask students to reason about equivalency and are rooted in the distributive property of multiplication over addition. The last two questions ask the students to reason about the structure of multiplication, specifically the meaning of the factors, using an understanding of inequality.

These questions were designed to be posed during the whole class discussion after students had a chance to share their groups’ work with their peers. Since students would have already calculated 48 and 96, the teacher can use these questions to press students for reasoning that moves beyond calculations. When posing such questions, it is important to allow several students to respond to each question. Doing so helps to ensure that different perspectives and reasoning are shared, and that more than one way of reasoning is valued.

##### Final Thoughts (at Least For Now…)

This article is not meant to minimize the importance of computational work or the reasoning related to computation, but rather to create space and expectations to move beyond just calculations.

The examples of relational thinking illustrated in this article are just some of many that span kindergarten through high school. These examples represent a type of thinking that can come from anyone, not just from those thought of as being the “most capable.” We hope they offered some insight into how relational thinking may look and sound when applied to numeric and algebraic reasoning and posed some things to consider when orchestrating opportunities for students to engage in relational thinking.