As a coach and former classroom teacher, I’ve learned that many students have experienced a mathematics education that has been hyperfocused on procedural fluency. When students have this experience, it can be easy for them to pick up the belief that math is about product, not process. When students aren’t the first ones to arrive at a potential solution, or they use a less efficient strategy, it can impact their perception of their ability to “do math.” That’s why it’s important to ensure that students are developing both conceptual understanding and procedural fluency as well as highlighting multiple ways of knowing. That’s why I utilize a group activity to show students that “doing math” isn’t simply answering questions; it’s exploring the diversity of processes and representations that can lead to an answer or inspire a deeper appreciation of mathematics.

Two fundamental understandings that guide all of my pedagogical decisions in mathematics education are (1) procedural fluency versus conceptual understanding and (2) multiple ways of knowing. Kilpatrick, Swafford and Findell define procedural fluency as “knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.”¹ Conversely, “conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which it is useful …. A significant indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes. To find one’s way around the mathematical terrain, it is important to see how the various representations connect with each other, how they are similar, and how they are different.”²

That’s why, rather than tackling traditional problems that prioritize procedural fluency, my first step is inviting students to construct mathematical arguments. The idea of mathematical arguments was introduced to me by Steve Leinwand. When I am trying to determine the task for students, I refer to the curriculum or standards. For example, one of the Grade 6 standards within the jurisdiction in which I am employed is to “extend understanding of fractions to improper fractions and mixed numbers.” One of the indicators embedded within this standard is to “explain, with the use of concrete or visual representations, how to express an improper fraction as a mixed number (and vice versa) and write the resulting equality in symbolic form.” I always ensure that what I am asking students to do is connected to the curriculum.

First draft of an argument. This argument was about demonstrating understanding between a set of equivalent fractions.

From there, I frame the task in language that is accessible to students. I might say, “Use pictures/representations, words and numbers to prove that an improper fraction and a mixed number represent the same value.” Students work together in groups of three to prove the argument by drawing representations, drafting explanations and finally by creating numerical examples. By deprioritizing the numerical aspect, I create space for students to express conceptual understanding through pictorial representations and words – and to notice that each form of their understanding (e.g. numbers, words, representations) has value.

Second draft of previous argument.

Next, the groups write an initial draft of their argument. As I support the groups in constructing their arguments, their understanding of the mathematical concepts deepens. There are so many “aha” moments during this time that the experience inspires groups to refine and clarify their understanding in a second draft. After the second drafts are complete, the class takes a gallery walk, and we discuss the strengths of each argument. This encourages students to notice the diversity of expressions, processes and connections their classmates have come up with to explore the same concept – and it underscores the idea that there’s no one single “right” way to do math.

Students participating in a gallery walk.

The next steps are trading peer feedback and revising arguments for a final time. I use a set of peer feedback questions to support groups in providing respectful and meaningful feedback to one another. I do NOT provide feedback at this time. It has been my experience that the students provide the same feedback as I do – but it is my belief that the groups are more receptive to feedback from one another rather than from the teacher. It also keeps groups engaged, accountable and invested. With feedback in hand, it is now time for groups to reflect on their own work and make final revisions.  

Example of student feedback.

Example of a final draft of an argument. This argument is connected to mixed numbers and improper fractions. Please note that the pictures used in this posting are from a Grade 5/6 classroom. As such, some of the arguments reflect the Grade 5 standard connected to equivalent fractions while others reflect the Grade 6 standard connected to mixed numbers and improper fractions.

Balancing conceptual understanding with procedural fluency through multiple ways of knowing as well as students’ shared sense of investment in the process is what makes this activity so empowering for students. Every time we take these steps together, students are reminded that they are powerful thinkers, capable learners – and growing mathematicians.

1. National Academies of Sciences, Engineering, and Medicine. (2001). Adding it up: Helping Children Learn Mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. (120.) Washington, DC: National Academy Press. https://doi.org/10.17226/9822

2. National Academies of Sciences, Engineering, and Medicine. (2001). Adding It Up: Helping Children Learn Mathematics. (118-119.)