Category Archives: Arithmetic

Great Thinking!

light bulbOne of the bonuses of working with students one-on-one is that I can get a window into their mathematical thinking by asking questions and having them “narrate” their work as they proceed.  Several of my recent conversations demonstrate unique and flexible thinking that helped my students work through computations effectively.

How might we enable this type of visible thinking in our classrooms?  I tried these prompts:

  • Tell me what you were thinking.
  • What would you do to compute this if you didn’t have a calculator?
  • What math operation did you do first?
  • Can you explain the steps you did in your mind?

What follows are some observations of “great thinking” by my students, who used flexible number sense for success.

A. Using number sense for easier multiplying or dividing

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B. Fantastic Fraction Flexibility

Here, noticing the multiplier is easier than solving the proportion directly:IMG_7207

In this question about slope, the student noticed that the first denominator must be 2 if the fractions are to be equal.IMG_7210

C. Simplifying with Radicals

It is hard to find perfect square factors of large numbers; this strategy employs factors and the student never found the original product.IMG_7208

Here, the student used an alternative to the traditional process of rationalizing the denominator.IMG_7213

D. Factors and Multiples

Rather than set up proportions, this student noticed multiplesIMG_7205

I’ve avoided the term “cancel” for some time, so here we talked about “dividing away common factors”. IMG_7206

E. Other interesting choices for computation and order of operations

The formula for area of a trapezoid can be thought of as “height * average of the bases”.  This student also applied binomial multiplication to compute without a calculator.IMG_7215

Here the typical order of operations would do the distributive property first.  Instead, the student saved a step.IMG_7211


Also on my mind as I write this are three things I have read this week.  First, James Tanton addresses a multiplication mistake made by an education official in this blog post, forgetting what 7 x 8 is.  Tanton suggests that the “best answer” in this case would have been “This is a tricky one. But I do have in my head that 7×7 is 49, so 7×8 is seven more than this: 56” because it highlights the thought process and downplays the memorization aspect of number facts.  His discussion is thought provoking (and he applies the same idea to trig identities too!).

The second item I was contemplating was in Sarah Carter’s Monday Must Reads post; she and I both liked David Sladkey’s “No eraser allowed” technique of insisting students leave their mathematical thinking available for the teacher to see.

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The third concept informing my thinking is the book study I’ve been doing with colleagues from Teachers Teaching with Technology (#T3Learns) on “Visible Learning for Mathematics” (2017) by Hattie et al.  Hattie describes a spiral relationship between surface learning, deep learning, and transfer learning that enables students to achieve.  He notes that surface learning is NOT shallow learning (p 29) but is instead “made up of both conceptual exploration and learning vocabulary and procedural skills that give structure to ideas” (p 104) that “sets the necessary foundation for the deepening knowledge” (P 131) on the path to understanding. Techniques such as number talks, guided questioning, worked examples (including accurate work and work with mistakes), and highlighting metacognitive strategies all enhance the process of surface learning for students, according to Hattie.

I constantly remind students to “Show Your Mathematical Thinking”—which is the updated version of “Show Your Work.” By asking them to “narrate” their thinking, I am focusing on my students’ surface learning to build their foundation of skills and tools on their learning journey.