By Laurie Speranzo and Beatriz T. Font Strawhun
In the last math article, we explored relational thinking with numeric and algebraic reasoning. In this article we unpack ways in which students use relational thinking through connectionmaking when problem solving.
In English Language Arts, we commonly hear about making connections that are texttotext, texttoself, and texttoworld. Can these ways of thinking relationally between text, self, and world have implications for math? If so, what do each of them mean? What can they do for students?
In classrooms where students are truly authoring their understanding of the mathematics, the central “text” is written by the students. It is their solution path or their representation, and we can think of the relational connections too!
The student’s work in relation to

 other students’ work through comparing and contrasting student solution paths.
 themselves by asking where the student’s prior knowledge, current thinking, and lived experience can be seen in
their work.  the context or real world by examining how the student’s work reflects the context of the task OR how their thinking
can be applied to real world contexts.
Classroom Example 1: Students Relating Their Work to Other Students’ Work
SCENARIO
As we enter the classroom, we see a word problem projected on the board:
In our class, one out of every six students has a pet.
If there are 24 students, how many students have pets.
The students have been working independently and are ready to engage in a wholeclass discussion. The teacher starts by saying, “I saw a lot of drawings and of course, not all of you had the same, which is good.” One student, Eid, is invited to draw their representation on the board and explain their thinking.
The teacher says, “Raise your hand if you got something like this. If you didn’t, it means you had something different. So, Eid, explain your thinking. Students who have something different, listen so you can compare your thinking or model to theirs.”
After a second and a third student shares their diagrams, the teacher asks, “Looking at these diagrams, do we get the same answer? How is it that all three are different but can be correct?”
BENEFIT TO STUDENTS
When we think about what relational texttotext connections might mean in the math classroom, we consider opportunities in which students are working to understand how their strategy or representation relates to those others used or produced in their class. This might mean that students are looking at and interpreting their partner’s work or a representation shared at the front of the room in relation to their own work.
Creating this opportunity allows students to grow their “math toolbox.” It requires that they defend their own strategy and in doing so may solidify or revise their thinking. It may also lead to students trying out a new strategy the next time they are solving a similar problem because they found one shared by another student to be more efficient.
THINGS TO TRY IN THE CLASSROOM

 Anticipate ways students may solve tasks and intentionally sequence the order in which you will ask for solutions to be shared.
 Plan questions to encourage students to compare their thinking with that of their classmates’.
 Pair students in ways that allows both of them to gain a deeper understanding of the math through making connections between their thinking and representations.
Classroom Example 2: Students Relating Their Work to the Real World
SCENARIO
Students are working in pairs on a task. Two students seated together have each written some of their independent thinking on their papers but have now turned to talk to each other. They speak in Spanish as they take turns explaining what they did and why.
The teacher approaches the partnership and asks them in English, “How did you solve the problem?”
The students explain in English what they did and what they understand so far.
The teacher asks, “What would it look like if you graphed the relationship that you just described to me? Try it.”
As the teacher walks away, one of the students says, “Excuse me, miss. ¿Nos ayudaria una calculadora?”
The teacher responds, “Si, you can use a calculator to help you with determining what to graph.”
BENEFITS TO STUDENTS
In this classroom, student thinking was prioritized over monolingual communication. The students were free to discuss the math in their language of comfort. They were able to fluidly express what was in their heads without searching for the words in their second (or third) language. The students were able to see themselves as doers of math whose ideas were worthy of sharing, with each other and with their teacher.
By being able to be their authentic selves in their partnership, they knew they had a place in the classroom and could ask their teacher questions in either language. They were likely better prepared to share their thinking in English with their teacher when the time came. This classroom exchange shows one facet of more equitable education for students. When students are valued as mathematicians, the math thinking is at the forefront and the mode(s) for communicating understanding can be flexible and expanded over time.
THINGS TO TRY IN THE CLASSROOM

 Ensure that the students see themselves as authors of the math. Ask individuals “How are YOU thinking about this?”
 Honor the lived experiences students bring with them into the classroom. Accept alternative algorithms or processes that students may have learned/developed before they came to your classroom.
Classroom Example 3: Students Relating Their Work to Themselves
SCENARIO
The students are working on a task that involves walking 6.9 miles each day for 6 days. The teacher asks, “Is there a visual in this problem?” When the students respond, “No,” the teacher challenges the students, “Can we create one?” All the students respond in the affirmative and get to work.
As the teacher circulates, they ask individual students:

 “Why do you think a number line is a good representation of the problem?”
 “What from the context made you think that multiplication would help you?”
 “What was the distance traveled each day? Show me where you have that in your diagram.”
 “You have an expression on your paper. Describe the picture in your mind that goes with this problem?”
 “When you are hiking how do you keep track of the distance you travel?”
BENEFITS TO STUDENTS
With each of these questions, the teacher is asking the student to connect to the realworld context—either through their mathematical representation or, in the last one, a personal connection.
The teacher does not imply that there is just one way of visually showing this context. Number lines were used by many students but so were equal group models and expressions/equations. And in both cases, the teacher asks students to connect the math with the realworld context.
By asking students to be explicit about the ways they represented the context, they can continue to build their schema for seeing the world mathematically.
THINGS TO TRY IN THE CLASSROOM

 Have students explain how their solution/representation connects to the realworld context. You may specify, “Where do you see the number of days in your number line?” or ask why their choice of model best represents the context.
 Change names in contexts to match those of your students. Adjust context to reflect the students’ community.
 Pose the question, “What if you were the character in this situation?”
 Ask, “What do you know from other situations that help you think about and solve this one?”
Providing opportunities for students to leverage relational thinking around their work—to others’ work, to themselves, to real world contexts—allows students to be honored as mathematicians and to grow as doers of mathematics. Thinking relationally about your own work allows for new lenses, for solidification that your strategy or pathway works, and for opportunities to shift your thinking. It means the students are doing the heavy lifting: analyzing, evaluating, modifying, and solidifying understanding of the mathematics.
We would like to thank the following teachers whose classroom practice inspired the scenarios in this article:

 Maria Apodaca, Fabens Middle School in Fabens Independent School District (TX)
 Monica Palafox, Tornillo Junior High in Tornillo Independent School District (TX)
 Karina Trevino, Canutillo Middle School in Canutillo Independent School District (TX)
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