Mathematical Representations: A Window into Student Thinking

By Amaru Pareja

Syracuse City School District
A representation created by a student is a window into their mind. For a teacher, a student’s representation is an invitation to learn about how that student is reasoning about a problem they are working to solve. If a teacher values students’ thinking, they need to consider how to make it possible for all students to represent that thinking. If a teacher’s goal is to grow their thinking, they must create opportunities for students to produce it in a way that provides insight into their reasoning. This act of sharing one’s thinking has become challenging lately, but it is as critical online as it is in person for students to share their thinking via representations. Research shows (Lesh, Post, and Behr, 1987) that there are five representations of the math; however, being able to connect the representations to each other is the hallmark of a strong mathematician.

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.

  1. 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.
  2. 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.
  3. 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 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.

  1. 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 agree that the repeated addition expression 6 + 6 is an equivalent to the multiplication expression 6 x 2, but the teacher presses the students to share where they see each expression in the visual diagram (the students identify 2 groups of 6 eggs each).  In the next few pieces of shared student work, students again created equations and diagrams: 6 + 6 = 12, 12 + 12 = 24, and a drawing of two medium cartons of eggs, each split down the middle with six eggs on either side.

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!

Tagged with: Accountable Talk® Discussions, Agency and Voice, Equitable Instruction, Formative Assessment, High-Leverage Teaching Practices, Online Instruction, Partner Spotlight

Bridging to Research: Numeracy and Math Sense in Young Children, An Interview with Dr. Melissa Libertus

By Lindsay Clare Matsumura and Kristin Klingensmith

Institute for Learning

Big Ideas from the Interview with Dr. Melissa Libertus

  • The seeds of mathematical reasoning begin in infancy, starting with the ability to discriminate between quantities.
  • Young children’s mathematical reasoning skills develop during everyday activities, such as grocery shopping, cooking, and even riding in a car. When young children engage in intentional talk about quantities that they see in everyday life, their reasoning deepens.
  • Adults can support children’s mathematical skills in activities such as arts and crafts by engaging in discussions about mathematical observations like the number of crayons in a box, which picture has the most flowers, patterns in beaded bracelets being made, how shapes fit together, etc.
What got you interested in studying young children’s math development?

I loved math as a child, but I saw that lots of other kids were struggling to learn it. I started tutoring math early in life and that got me to think about how best to explain concepts and solve problems in ways not regularly taught in class. One of my biggest joys was to see how concepts clicked for students. As an undergraduate, I majored in cognitive science and worked with professors studying math education and how to best teach kids algebra. I realized that by the time kids came to algebra, a lot has already happened that determines success, and I wanted to start younger. So, I worked with infants for my dissertation to understand the foundations of mathematical thinking—what happens before kids get to school and how that lays the foundations for math.

What does mathematical thinking and reasoning look like in infants? What are those early foundations? 

When it comes to understanding the basic foundations of mathematical thinking present in infancy, there are two main concepts that we believe are present. The first is that infants have an understanding of small quantities and their exact numbers. Essentially, they can tell that one ball is a different number of objects than two balls, and that’s different from three. But the exact understanding of being able to tell these different quantities apart is pretty much limited to those small sizes.

Secondly, while having a precise understanding of small quantities, they also have an appreciation for large quantities, but it’s imprecise. A lot of research shows that infants can discriminate between different quantities of objects, but it needs to be a certain kind of difference between the quantities for them to be able to do so. For example, they might be able to tell that a set of 24 balls is more than a set of 8, but they wouldn’t be able to tell the difference between a set of 24 and a set of 18 because those quantities are too similar.

How do thinking and reasoning change after infancy?

Over the first few years of life, approximate quantity representations become more precise. In addition, infants are able to use their understanding of quantities to learn about basic arithmetic operations. For example, if I show an infant about 10 objects and add another set of objects to this, the rough outcome might be something like 15 or so. Infants know that there are no longer 10 objects, and they know there are more than 10. And if I take something away, there are less than 10 objects. Thus, infants have a rough idea of where the results might end up, but it’s all just an approximate understanding of these quantities, not necessarily precise. Precision grows later on when kids learn to count and do arithmetic with quantities and symbols.

Could you tell us about your research looking at the development of children’s mathematical thinking and reasoning?

One of the most surprising findings of my early research was that we see individual differences in infants’ rudimentary understanding of quantities from a very young age. By the time they are six months old, we already see some infants who can better discriminate between quantities. Even though everybody has access to this knowledge, there are some who can tell that quantities that are close in size are actually different, and others who need quantities that vary significantly to be able to tell them apart.

One of my longitudinal studies showed that this difference in infancy predicts kids’ math abilities three years later, which was important. We tested infants and then retested them when they were in pre-school. We saw that those children who as infants, had an easier time discriminating between quantities were the ones who later scored higher on a standardized math assessment. So, we’re curious where these differences come from at this very young age. What happens in infancy that puts some kids on a trajectory to developing a more solid representation of these quantities and others to have more imprecision? And if that matters for math, what is it that we might be able to do from a very young age to help those kids who, for whatever reason, might be behind?

How might parents/families/caregivers influence the development of children’s mathematical thinking and reasoning? 

This is an interesting question, and we don’t know the full story yet. However, so far, we have explored a few different origins for variations in basic number concepts present from birth, and at least part of it seems to be running in families. So, we have some studies where we’ve found that parents who have more precise representations of approximate quantities tend to have kids with better and more precise approximate number representations. But then there seem to be other factors too. We know from a lot of other research that how parents talk about math with their kids, the activities they do, even in the context of everyday play and day-to-day activities, matters for the kids’ mathematical thinking. There may be a link between parents’ own math ability and what they do with their kids, how often they point out quantities in their environment, how often they talk about numbers and math, what kinds of activities they do with the kids that could foster an understanding of math, and parents’ attitudes about math.

There seems to be some parallelism between what you were saying about what possibly could be a connection with parents’ attitudes about math and the experiences that are afforded their children. We know that in reading, as well, there may be a nature component, but there is likely also a nurture component.    

Absolutely. I think what you’re saying is absolutely true. Yet, it’s important to know also that reading and math both need to be emphasized. We often see that children’s math abilities do not just rely on general cognitive abilities and appropriate cognitive stimulation in general, but that there’s also something specific to math. Talking to children or reading to them by itself does not foster math abilities. Instead, the interactions need to specifically targeted mathematical concepts. There are many ways to incorporate math concepts into daily activities with young kids that are age appropriate and fun, such as counting steps when walking up the stairs, comparing prices of produce when shopping, or finding as many numbers as possible when driving in the car.

Do you have to have a lot of formal math learning to provide these experiences, or can the average parent or family member engage young children in meaningful ways?

I think, especially when we’re talking about young children, it is not necessary to have a formal instruction in math through college. You don’t need to have taken calculus to be able to provide the kind of mathematical interactions that are appropriate for young kids. People who use math on a regular basis [i.e., carpenters, contractors, etc.] and see how math is really used in everyday life may have great ideas to offer beneficial experiences to young kids. Going back to what we were talking about earlier, there’s also something to be said about the importance of attitudes towards math that people bring to the table. If you approach the activities with which you engage your child with a sense of enjoyment, this could translate into really important lessons that children learn.

What are some things that adults can do to support children’s mathematical development?

There are many opportunities that naturally arise for parents, caregivers, and teachers to talk about math with young kids. Parents shopping with their kids, for example, can talk about the prices of things, compare what costs more, what costs less, what is heavier or what’s lighter, compare different sizes of produce, or count how many bananas are in a bunch. Cooking also provides great opportunities to talk about measurement and different units of measurement. Talking about concepts like time also helps kids develop their mathematical understanding. For example, you can ask kids ‘How long does it take us to get from A to B?’ or “How fast can we drive? “And then point out the speed limit on signs as you pass them. This helps kids learn about why numbers and measuring things are important and connect this understanding to the symbolic language that we use to express these kinds of concepts.

Activities that don’t look like math on the surface can also help kids develop their math skills. For example, making a necklace with a pattern in the beads is a great activity that kids enjoy that people would not necessarily identify as a math activity. Finding patterns and recognizing those and finding shapes and how you can make patterns out of different types of shapes are great activities for building mathematical skills that can be included in arts and crafts projects.

What are some things that adults can do to support very young children?

There are lots of math-related concepts you can talk about even with one- and two-year-olds. They might not be able to fully get a concept yet but pointing out things like “this is a big apple, and this is a small apple” is something that even a really young child can understand, and you can build on that from there.

I have to say though, that while I think that starting early and getting into the habit of seeing these opportunities probably would be helpful for very young kids, there hasn’t been much research about this. In my lab, we’re beginning some studies now looking at what parents of two-year-olds are doing, and then following these kids over time to understand what kinds of activities and what kind of input are appropriate and most beneficial for kids down the road. We really need solid evidence and more data to be able to give guidance and advice about what kinds of adult-child interactions in very early childhood best support kids’ math development.

For more information about Dr. Libertus’s research on numeracy and math sense in young children, check out her other publications.


Tagged with: Early Childhood, Math

Rosita’s Reads March 2

Spatial thinking is crucial to student success in science, technology, engineering, and mathematics (STEM). The question is “Can spatial thinking actually be improved?” In this article, Newcombe examines how spatial cognition is indeed malleable and discusses techniques that educators already have for developing spatial thinking, through practice, language, gesture, maps, diagrams, sketching, and analogy. The author also explores the benefits of building these techniques into curricula. 

Tagged with: Rosita's Reads

Rosita’s Reads February 23

Getting Clearer: Schooling Loss, Not Learning Loss

Getting Clearer
January 13, 2021

The authors argue that students are not experiencing learning loss during the pandemic but rather school and peer interaction loss. They contend that students are not failing more; the system is. While students are not necessarily learning what is typically measured in school, students are learning—under constraints, about the human condition, in communities, and in the digital world. From a perspective that schools and districts will never get back to where they were, the authors offer next-steps for renewing learning communities and learning outcomes with families and community partners, using the pandemic as an opportunity, not a setback.


Tagged with: Rosita's Reads

IFL Partner Captures Connecticut Superintendent of the Year Award

By Michael Telek

Institute for Learning

Dr. Paul Freeman has come up through the classroom in Connecticut. The Superintendent of Guilford Public Schools gained much of his education there, honed his craft in Connecticut classrooms, and has given countless hours to Connecticut organizations aimed at improving educational opportunities for students.

Late last year, Dr. Freeman’s passion and dedication to education was recognized as he was named Connecticut’s Superintendent of the Year. During video remarks following the announcement, he shared his thoughts about the work of now and how the work of now is situated at the center of a three-part Venn diagram.

“The superintendents that I know, now more than ever, are working very hard to make sure that students are safe and healthy and continue learning,” said Dr. Freeman. “To be asked to represent this group of people, particularly this year, it’s just really humbling.”

With a light-hearted laugh and beaming smile, Dr. Freeman admits he has seen a lot in what is now approaching a thirty-year educational career; however, he has never seen anything like these last 11 months. COVID-19 has delivered a devastating blow to educators across the world, forcing teachers and administrators to push the limits of their imagination to ensure students are becoming effective, enthusiastic, and independent learners.

“Our veteran teachers are approaching this year as if they are first year teachers, they’re redesigning every lesson, they’re rethinking every objective and every goal for their students, and they’re asking, ‘How do we do what we have always done in these completely new conditions?’,” said Dr. Freeman.

This school year has been tough on everyone, including students and parents. Dr. Freeman says he has been inspired by the resiliency of the school community and their ability to rise to the challenge presented by the virus.

“Even when things go back to normal, there are things that we are discovering instructionally that are bending the structures that we thought were unbendable in the past.”
~ Dr. Paul Freeman

“Even when things go back to normal, there are things that we are discovering instructionally that are bending the structures that we thought were unbendable in the past,” said Dr. Freeman. “We are learning things about flipping the classroom, and remote learning, and archiving direct teaching opportunities for kids to re-watch multiple times.”

As part of the search for an educator on whom to bestow their highest honor, the Connecticut Association of Public School Superintendents consider four standards that are exhibited in exemplar educators: communication, community involvement, leadership for learning, and professionalism. You would be hard pressed to find someone who could check those boxes off better than Dr. Freeman.

In education, leaders must meet the needs of multiple, distinct constituencies. Elementary students are certainly different from high schoolers, and the parents of those students will have different viewpoints. That does not even include the needs of teachers and members of the community. That is why communication is key, according to Dr. Freeman. “I think that the ability to communicate in both directions is just a foundational skill to be the leader of a public-school system. You’ve got to be able to speak clearly and articulate what we’re trying to achieve here in the system, but you also need to listen really carefully,” said Dr. Freeman.

This foundational skill was on display as Dr. Freeman heard the concerns over the use of “Indians” as the school mascot by members of multiple Tribal Nations, a local human rights commission, diverse community members, and his own students. Dr. Freeman recommended to discontinue the moniker in Guilford. Shortly after the decision, he told local media “this is just one step in ongoing work.” The district has since initiated a curricular audit around race and equity and launched professional learning opportunities in culturally responsive and sustaining instruction.

Dr. Freeman is living proof that an investment in knowledge is in the best interest of students. Carrying four degrees from three different schools (two in Connecticut), he looks to pay it forward, even if that is to students outside his own district. He serves as an adjunct professor at the University of Connecticut, a National Advisory Board Member of the National Center for School Safety in Michigan, and once COVID hit, Dr. Freeman answered the call by serving as co-chair of the Governor’s COVID19 Learn from Home Task Force. Under his watch, the task force got 60,000 laptops out in quick order in communities of need.

The superintendent of the year award does not go to someone satisfied with the status quo, instead it is earned by an educator unafraid of change and driven to do whatever it takes to grow their pedagogical prowess. Dr. Freeman says the Institute for Learning has been foundational towards the way that the district approaches work in the classrooms.

“The IFL’s been enormously important to our success and to our ability to continue to move forward. The fellows that we have worked with in district have become part of our team, and we enormously appreciate them,” said Dr. Freeman. “The annual learning opportunities that we’ve been able to be part of have more often than not sparked continued future work that we’ve brought into and done in our district.”

The IFL has been partnering with Guilford Public Schools for nearly a decade, empowering teachers, and students to create the conditions and capacity to support high quality classroom practice. Dr. Freeman specifically pointed to improving quality classroom conversation through Accountable Talk® as a gamechanger for Guilford.

“We in fact, as part of our hiring process, will often show prospective candidates a student discourse occurring in the classroom and using that as a conversation piece to assess where those educators are in their understanding of student discourse and their ability to see things that are positive in it to push us to continue to get better,” said Dr. Freeman.

® Accountable Talk is a registered trademark of the University of Pittsburgh.

Tagged with: Instructional Leadership, Partner Spotlight