“Thinking Mathematically”: A Tour

Mark Chubb writes the “Thinking Mathematically” education blog. He has taught grades 5-8 and serves as an instructional coach for the DSB of Niagara in Ontario, Canada. He’s also an Additional Qualifications instructor. Here are a few highlights from the blog.

How Not to Start Math Class in the Fall – 2020″

At the beginning of the post, Chubb links Tracy Zager’s 2016 post on the same topic for her “Becoming the Math Teacher You Wish You’d Had” blog. Zager wrote about her daughters’ first days of second and fourth grade math classes, which both began with math tests.

She discusses how teachers can balance the need to assess the knowledge of new students so they can plan their instruction accordingly against the need to “set a tone and climate for mathematics…[by] build[ing] community and trust and relationships and an atmosphere conducive to collaboration and risk taking and inquiry and learning.”

In Chubb’s post, he notes three common points teachers have discussed with him about distance learning this school year that will impact classes next fall:

  • Learning over the past few months has not been ideal for many students;
  • Learning about our students’ thinking has been difficult, at best, for us, making it difficult to sequence learning, consolidate big ideas, and use various students’ thinking to drive conversations; and
  • There will be a huge discrepancy between how much / what students have learned over the past few months

“What first moves we make when school returns matters more this year than ever. This leads me to wonder, will our decisions be driven by thoughts of how to fill gaps or how to build a community of learners?” he wrote.

He then discusses issues that can occur with the “gaps driven” approach and suggests other ways of “thinking about how to start all new learning with experiences that will help bridge current understandings with what your students will be learning, [which] will need to be a focus.”

“Taking an Equity Stance in Math Class”

Should all students learn the same things? Should they learn different things based on their abilities and readiness? This post explores these questions and more.

“Instead of seeing the issue as simply whether or not we want a classroom of students to be doing the same things or if we should be providing some students with different things, we should also consider what is actually being learned by the students,” Chubb writes. He presents a matrix with four options about student learning: everyone is doing the same tasks and learning the same things, everyone is doing the same tasks but learning different things, everyone is doing different tasks and learning different things and everyone is doing different tasks but learning the same things.

After going into further detail on each of those points, he discusses broader ideas about what it means to take an equity stance in mathematics (“we both believe that every student can achieve, and understand that every student might need different things from us”), and how we can aim for equity by expanding “who is considered a math student” and “what is accepted as mathematics.”

“Exit Cards – What do yours look like?”

In this post, Chubbs describes using exit cards to determine how individual students are learning and thinking. He discusses four purposes of exit cards and offers sample exit card prompts that could be used to fulfill those purposes.

For instance, “Write a question you’d like to ask or something you’d like to know more about” is a prompt designed for meta-cognitive reflection/connection. “Create 2 addition questions, one that is easy to solve mentally and one that is harder. Use a number line to explain how to answer both. What makes one of the questions harder?” is a question targeted towards concepts. “How many ways can you solve 68 + 18?  Explain each way. Which was the most efficient for you?” is a prompt targeted towards procedures and “Phillip explained that 100cm2 is the same as 1m2.  Explain why he is correct/incorrect” is a prompt focusing on clarifying misconceptions.

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