Mimicking vs. Thinking

This post is a reflection on the past school year with my plans for the upcoming year. It is heavily influenced by Building Thinking Classrooms in Mathematics by Peter Liljedahl, which I can’t recommend enough. Also check out Robert’s reflection post for a good summary of it. I will not be going into that level of detail, but focused on the big picture philosophy of it.

Reflecting on my 6th grade math class last year, I diagnosed two problems. First of all, my instruction wasn’t sticking. Many students memorized facts and procedures in the short term for a test but two weeks couldn’t remember how to solve a problem. Two months later students acted like they had never seen the math before.

There was little retention of content or skills so did students ever really learn it in the first place???

My initial solution to this lack of retention is to spiral my curriculum so that I am reviewing concepts throughout the year. This way students will consistently experience the algorithms for different types of problems multiple times.

The second issue is that many students lacked number sense. My instruction was not building a deep comprehension of things such as place value and why algorithms such as carrying, borrowing, or long division work. This comes back to the first problem, students were just memorizing a technique for a quiz or test with no understanding of WHY they were doing it. Therefore, they could not apply the math to a new situation that was different in any way from the examples that I had shown.

The solution to this problem is more complicated than just spiraling content. Reviewing procedures does not build comprehension. I need to involve more number talks, modeling of teacher thought, and asking students to explain their thinking, not just give answers. Then I discovered Building Thinking Classrooms in Mathematics. Peter Liljedahl’s introduction chapter revealed that my analysis was all wrong.

Lack of retention and poor number sense were not the underlying causes in my instruction, but signs and symptoms of the real problem: students were mimicking me instead of thinking.

Consider the analogy of someone with shortness of breath and numbness in their left arm. We might think the solution is a regular exercise routine to build endurance along side some physical therapy for the arm. But that would only be addressing the signs and symptoms of the real problem: a heart attack. Not only is this a misdiagnosis, but the physical exercise could potentially make the situation worse sending the patient into cardiac arrest. Spiraling my content and using number talks while helpful, is treating the symptoms while ignoring the underlying problem.

Last year I tried a couple of PBL projects and some 3 Acts Tasks. It did not go as well as I had hoped. Students needed their hands held the whole time. They were unable to problem-solve independently but always needed me to demonstrate how to do the work. Why? Because that’s what they always do in math class: mimic–I do; we do; you do. We have trained students to expect the teacher to show them exactly how to do a problem before they “practice” the same thing on their own. Before attempting a PBL project, my students needed scaffolding (which I neglected to provide) on how to think mathematically on their own.

I find it interesting that mimicking is the preferred approach of both high and low achieving students. Strong academic math students are often good at memorizing and manipulating algorithms but lack a deep understanding of what they are doing and why it works. I know that is true of my own mathematical journey as a student. It wasn’t until college that I began to unpack the why behind even simple math algorithms. Low achieving students often struggle to remember which procedure to use and when.

So high achievers are good mimickers and low achievers are poor mimickers, but virtually no one understands the actual math!

Peter Liljedahl explains what is missing: thinking. He makes a strong case that mimicking needs to be abolished from our math classes in order to shock students into thinking mode. His entire approach of 14 practices to teach thinking is based off from his observational research and conversations with students and teachers. I won’t get into all of the details here (buy the book!) but I plan to implement most of his ideas this year.

Peter’s approach is compatible with my student-centered PBL philosophy with students working on open ended problems in random groups of 3 at vertical whiteboards. The direct instruction comes at the end of class as the teacher guides students to consolidate their learning after they have experienced the math.

The thing that has me most excited about this year is students spending more time collaboratively doing math, rather than listening to me explain math procedures. I look forward to watching my students grow their thinking and problem solving skills.

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