The Importance of ‘Understanding’ in Mathematics

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Wendy Ward Hoffer

Wendy Ward Hoffer

Every now and then, I think back to my high school calculus courses. I can still hear the rattling sound of our teacher energetically chalking symbols at the front of the room. He would spend the majority of fifty minutes at the board working conic sections and integrals with aplomb. We copied down everything he wrote. Once in a while, he would holler at someone to tell him what to do next. I always sat in the same seat, where I sometimes ducked behind the big boys who sat up front and liked to be called on. Somehow I would get hollered at anyhow and have to cough up a derivative or domain—either to be scolded or praised. I was often unsure which to expect. We learned math by copying.

When the AP exam presented us with big blank sheets of paper on which to scrawl solutions all our own, I remember desperately wishing I understood more about what all the symbols represented and which I was meant to put where. Despite my long years as a devoted mathematical spectator, I had yet to grasp the underlying point. During the exam, I wrestled to combat an existential crisis: Why were we doing this?

Why Math Workshops?
The purpose of teaching and learning mathematics is understanding. When we understand, we can remember, transfer knowledge to new contexts, apply concepts to novel situations, look at problems from varied perspectives, and explain in ways that make sense to others. Though this was not my initial experience as a calculus learner, that course did propel me to recognize the importance of understanding, the need that I—and all students—have to develop strategies that will help us to make meaning of mathematics for ourselves.

To that end, a math workshop is an ideal forum for learners to construct their own mathematical understanding through rich interactions with both content and peers, in line with Lev Vygotsky’s theory of social constructivism (1978). The big idea behind a math workshop is that whoever is doing the majority of the speaking, solving, justifying, and explaining is doing the learning—since our purpose is to teach students math, our workshops need to be about students doing the work of mathematicians. In a workshop, learners are actors, not audience members, and teachers are coaches, not sages. A math workshop is a structure that turns over the work of learning math—and the responsibility for doing so—to students.

Students working togetherLearners thrive in a math workshop when they are apprenticed as capable, independent problem solvers. Workshop model instruction affords learners the time to experience, not just to observe, all eight Mathematical Practices detailed by the Common Core. In order to, first, “Make sense of problems and persevere in solving them,” students need opportunities to face challenges and experience the productive struggle called for by NCTM’s Mathematical Teaching Practices. Workshop is the cauldron of mathematical grappling—we set learners up with challenges, support their progress, and together look back on all they achieved and came to understand.

Learners can tackle more challenging problems collaboratively than they could independently; their comprehension is catalyzed by hearing the thinking of their peers. For this reason, discourse—engaged, accountable conversations about mathematical content—plays a key role in a math workshop. Discourse affords learners with opportunities to reason, argue, and critique the thinking of others. Students need training and skilled facilitation to develop their capacity for generative conversations, and we know from international comparative studies of classroom practice that discussing and defending mathematical ideas promotes students’ mathematical understanding.

 

This entry is adapted from Wendy Ward Hoffer’s recently published book, Developing Literate Mathematicians: A Guide for Integrating Language and Literacy Instruction into Secondary Mathematics.

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Thinking Strategies Institute (Winter 2023)

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