## Mathematical Representations: A Window into Student Thinking

### By Amaru Pareja

##### Syracuse City School District

**How do we prepare thoughtful questions and make connections between representations?**

When students share their brilliance with diagrams, manipulatives, words, or numbers, they share an abundance of information that can be woven into a storyline of reasoning and logic. It takes practice to listen carefully for that brilliance and make connections among those pieces of our puzzle that lead to the deep understanding of mathematics we want for every student, but with time, our pedagogical moves at leading these discussions get better and better. The most important aspect of this skill, however, is that we are using student representations to do so. Otherwise, we are the ones doing the heavy lifting. By using students’ representations, their mathematical voice is front and center and they see themselves as doers of mathematics!

When solving a problem, it is common to inherently relate it to others we have previously solved. We begin with proven strategies of success and do our best to find connections between our past understandings and the mathematical challenges ahead. This connection, or bridge, is how we grow our brains and our base knowledge about a topic.

Creating connections among multiple representations, explanations, and expressions or equations is not simple and requires thoughtful questions to consider. The work of Peg Smith and Mary K. Stein in “5 Practices for Orchestrating Productive Mathematics Discussions” illuminates the importance of anticipating how students may reason about a given problem to better prepare thoughtful questioning that challenge our students’ thinking. The more prepared we are as educators, the more opportunities we can create for our students to consider the multiple representations produced in each classroom and recognize their similar structures.

**Here are some questions to try in your classroom to connect representations (adapt based on student-produced representations and the math being taught).**

- How are the two representations similar? How are they different?
- How do both equations represent the context of the problem?
- How does the graph show the same information as the table?
- Where do we see the unit rate in the table and in the equation?
- Where is the solution to the problem in each of the drawings/diagrams?

**What does it look like and sound like to connect representations?**

A sixth-grade class in October is beginning to study ratios by using the *Buying Eggs* task. (The task and lesson guide are available here.) The mathematical learning goal for the task is that students will explain and demonstrate with visual and numeric representations that a ratio is a multiplicative relationship between two quantities.

**Buying Eggs Task**

**The market sells small cartons of eggs containing 6 eggs and medium cartons of eggs containing 12 eggs.**

- Compare the number of eggs in the medium carton to the number of eggs in the small carton. Write an equation that shows how you made your comparison.
- How many small cartons of eggs must you buy in order to buy the same number of eggs as is contained in 1 medium carton? 2 medium cartons? 5 medium cartons? 50 medium cartons? Describe in words how you made your decisions.
- The market also sells large cartons of eggs containing 30 eggs. Describe in words how you can determine how many small cartons of eggs you must buy in order to buy the same number of eggs as is contained in 4 large cartons.

Extension: Describe in words how you can determine how many small cartons of eggs you must buy in order to buy the same number of eggs as is contained in any number of large cartons.

The teacher opts to use Desmos.com because it offers a teacher dashboard that permits monitoring student work during the lesson, allowing the teacher to select which student work to use in a whole classroom discussion.

During the independent work time, some students draw containers of eggs, others use expressions, brief written explanations of their thinking, and some students even create a table. Insights about the multiplicative relationships are certainly visible, but not to every student. So how does the teacher support students to synthesize their individual thinking into a cohesive, shared understanding?

**The market sells small cartons of eggs containing 6 eggs and medium cartons of eggs containing 12 eggs.**

- Compare the number of eggs in the medium carton to the number of eggs in the small carton. Write an equation that shows how you made your comparison.

From the student work, the teacher selects three different ways to make the comparison and asks, “How does each response say the same thing?” The question is designed to have students make connections between a visual and numeric representation as they include yet another representation of the mathematics, verbal expression.

Students respond to the question, “What patterns do you notice?” After discussing the context of the responses, many agree that you always need to buy double the number of small containers as medium containers to match the same number of eggs.

At this point, students are still thinking additively but the goal is to highlight the multiplicative nature of ratios. So, two new pieces of student work are added to the discussion—a representation of the eggs followed by numbers to represent the eggs in cartons and a table. Students are asked what relationships are exhibited in each piece of work. The orange circles and arrows are added as multiple students state that you need to have two cartons of small eggs for every one medium carton of eggs. When any student calls out the relationship, they are asked to tether their statement to both the number pattern in the diagram on the left, and the table on the right.

Students are using repeated reasoning over several small-to-medium carton relationships, as well as across representations, and begin to identify the structure or pattern behind this multiplicative relationship.

As students continue to work on the task, one student provides his thinking visually as he draws eggs in a carton, but then upgrades his model to include numbers to represent how many eggs in each carton as opposed to drawing an individual dot for each egg. What a time saver and brilliant mathematical move! Could others perhaps be willing to share this strategy in future lessons? It is through seeing and understanding others’ representations that students begin to figure out the options they have for showing how they solve a problem, and only in knowing all of the options can they choose the one(s) that work best for them! Students drawing diagrams with dozens of dots can now consider using numbers to save them time and energy. Students using numbers can consider using expressions or equations and, if enough sense was made, everyone can consider why a table is such a useful method for organizing numerical information to look for patterns.

**How do we overcome challenges of access to representations in a virtual teaching space?**

Poor internet connection and getting used to a whole new structure of school are only some of the hurdles our students face when trying to communicate with peers and teachers. However, being able to communicate is the first step in orchestrating any rich math discussions. Without access to tools that help us first to explore math concepts deeply, then communicate it clearly with our peers, we remove opportunities for equity and access to all students.

It is even more imperative now that we work extra hard to provide our students with many options to share what they know as possible. We must find platforms/websites that work within our local contexts to provide open-ended tasks that allow students to use multiple representations to share their thinking. We must listen even more closely now to what they understand and connect that meaning to the thinking of others to build a stronger, shared knowledge.

Helping students make important mathematical connections is to shift the ownership of the mathematics to the students—it honors their thinking, their knowledge, their ideas, and it empowers them to identify themselves as a “math-doer” worthy of making important discoveries and solving challenging problems.

**Virtual manipulative websites to try:**

*NCTM’s Illuminations**National Library of Virtual Manipulatives**Mathigon**Math Learning Center**Geogebra**Knowledgehook**WhiteBoard**PowerPoint DIY manipulatives**Desmos (including the “Magic Link” to snap photos of student work on paper or whiteboard)*

Amaru is a proud teacher in the Syracuse City School District who specializes in intervention for grades 3-8.

When he is not playing with a new math problem, he’s eating something delicious with people he loves.

You can follow Amaru on Twitter @mramarupareja.

**Tell Us About Your Use of Math Representations in a Virtual Classroom**

- In what ways are your students representing their mathematical thinking?
- How are you leveraging representations to help students gain a deep understanding of the math?

*Tell us here. We would love to feature your story or maybe even ask you to write something for Bridges!*