Mistake-making to sense-making

Here are my slides and my notes from my five minute Pecha Kucha-style presentation at Educon. The focus of my presentation was on my journey as someone who started his teaching as viewing students as mistake makers to being a teacher who views students as sense-makers.

  1. I’m going to talk today about my journey from a teacher who tried to correct students’ mistakes to someone who paid attention to student thinking and participated in mathematical reasoning with my students.

  2. When I first started teaching, I monitored students’ behaviour, carefully recording dots when they failed to hand in their homework, dots when they were late, dots when they did not participate, dots when they were absent. I held those dots over my students’ heads like the Sword of Damocles. I was the dot master!

  3. I noticed that my students made predictable mistakes and I modified my lessons to address those mistakes. “Don’t forget to change the sign of the second number.” “x times x is x-squared, not 2x.” “Don’t cancel the x’s!”

  4. However, I was often confused by what they were doing. I became curious about where student mistakes come from. Why is this student writing a +1 there? What does this mean to him? Why are students doing these crazy things??

  5. I began to realize that what students said and what they did was the result of something they were doing that I could not directly observe. I formed a hypothesis: students think. In fact, I realized that students think quite a lot.

  6. I thought my job was to intervene on how they were thinking rather the product of that thinking. I realized that if what students do is a product of their thinking then I need to know what they are thinking not just what they write. I needed to be able to read minds.

  7. Unfortunately, I still thought my job was to fix their thinking as if it were something that had been broken by their experiences. I thought that my students were just thinking wrong things, and therefore all I had to do was correct their thinking.

  8. “A teacher is a mechanic for the mind”, I said to myself, “And in order to repair it, I just need to know how it works.” I wanted to reach into my students’ minds and fix them. I viewed my students as broken and my job was to make them whole again.

  9. I listened to what my students said. I carefully watched what my students did to help me prognosticate their actions. “If I just know enough about how they think,” I thought, “I can help them think better.”

  10. I was judge, I was jury, and I executed based on my understanding of student thinking. I tried students for the crime of thinking differently than I and sentenced them to more explanations of the only truth that mattered, my truth.

  11. This approach has flaws. It has limitations. 10 thoughts per kid. 20 kids a class. Six classes a day. Five days in a week. 34 school weeks in a year. That’s 204 thousand thoughts a year to pay attention to. It was overwhelming.

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  12. And you know what? One day I realized that the way the kids thought yesterday wasn’t the same they thought today. What I learned about student thinking was out of date by the time I wanted to use it because, just like me, kids don’t think the same way everyday.

  13. I needed to be able to responded to thinking live in the moment, rather than teaching while blindfolded. I started having mathematical conversations in the moment with my students and listening to them have conversations in order to uncover their thinking and respond now.

  14. I anticipated student thinking instead of student mistakes. Here’s 11 different ways I solved a problem. Guess how many of these ways were actually used by my students? None of them. Human cognition is incredibly complex.

  15. I questioned my beliefs about mathematics and why we teach it. What do I want my students to get out of mathematics class? More importantly, what do my students want to get out of mathematics class?

  16. We are conditioned to think of ideas like this as being right or wrong, correct or incorrect, true or false. Instead, let’s figure out how what this kid has done make sense to her. Let’s remember that that students’ mistakes are the result of thinking.

  17. There are other benefits to making student thinking visible in a classroom. An ongoing mathematically-rich conversation engages all of my students in thinking about mathematics in ways which give them agency and authority.

  18. How do we design mathematics classes where students don’t end up thinking they’ve spent 13 years memorizing arcane rituals? Let’s make mathematics class about learning about thinking rather than about trying to avoid mistakes.

  19. Children are not broken! It is not our job to treat them as things to fix. Children are sense-makers! Our job as educators is to provide experiences so students develop models for understanding the world.

  20. Here are my sons. Let’s work together to build a world that treats them and all other children as sense-makers within it.



  • I made a mistake in one of the fields above, wrote my comment and was then told about the mistake. My thoughts were deleted.

    This may also happen in the classroom!

  • Everywhere I look I see “check your work”, and sometines “compare with your estimate”, as the only advice (instruction) to the students. This is very inadequate, as they will check their work using the same misconceptions as they used in the first place. It is only any use for picking up 3 + 4 = 9 or losing an exponent.

    Problems should always have semi independent checking paths. Solve an equation – put the value back and do the arithmetic. Find factors – if you know that factors and roots have alot in common then same as above. Algebraic manipulation (simplify, take out common factor) – put some not too simple numbers in at the start and at the end. Finding indefinite integrals – differentiate your answer.

    This forces the student to think, when the independent check fails. And it can be done at any level.

    Then their questions are going to be along the lines of “HELP! I cannot see where I went wrong”

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