Tracy Zager gave my favorite talk at NCTM, and it is the talk that has kept me thinking the most. Tracy spoke at Shadow Con with the title “Breaking the Cycle”. Tracy works with pre-service elementary teachers, and when interviewed 63% of them revealed negative associations with math. When she took their descriptions of their mathematical experience and put them into a wordle, it looked like this:
Those are from teachers. Teachers who teach math! Does that scare you? It scares me. And it scare me more that I know there are students who feel that way in my classroom every day.
Tracy contrasted that wordle with this one, of the words that mathematician use to describe their experiences doing math:
A little different, right? This is a great summary of my goals for my classroom. To move students away from a fixed, negative mindset, of math as producing answers to questions that they don’t care about, or failing to do so and feeling frustrated and ashamed. And toward a mindset of playing with math, exploring numbers and structure, and wondering about the world around them
Every speaker at Shadow Con ended their talk with a call to action — a concrete way for attendees to follow up on the experience and bring them to their schools and classrooms. Tracy’s call to action is simple, and takes her ideas a step further. There were probably 350 people in the room at Shadow Con. She challenged each of us to take the words that mathematician use to describe their experience with math, and to share it with our colleagues — including everyone, not just the teachers who will be most amenable to it. Then, take a favorite word from the wordle, and make that word central to one lesson that we plan in the coming weeks. After we teach the lesson, share with our colleagues what that lesson was like, and where we can go next.
I think her challenge is incredibly important, and captures the spirit of Shadow Con — transcending the typical constraints of a talk at a conference that reaches the teachers in the room but no further. As I thought about it, I have one reservation that I want to be careful about as I move forward.
This is an article by writer and cognitive scientist Daniel Willingham called “Inflexible Knowledge: The First Step to Expertise”. He makes several points that I think are critical to implementing Tracy’s call to action effectively.
1. Inflexible knowledge — knowledge connected to concrete examples, that students struggle to apply in new contexts — is the natural first step of learning. It is different than rote knowledge, which is memorized but meaningless, but less applicable than the deep knowledge we want our students to gain. Inflexible knowledge is not bad — it is the natural foundation for deeper learning.
2. When students only have inflexible knowledge, they struggle with abstraction. Students fail to apply the knowledge they have in new situations, or situations that are different on the surface than the examples they have seen, even if they have the same underlying structure. This is normal. However, all of the things we are asking students to do — be creative, take leaps, wonder, play, innovate — they are unlikely to be possible if students only have inflexible knowledge of the content.
3. Willingham distinguishes between novices and experts in that novices are not the same as experts just a bit worse. Novices think about a topic completely differently than experts do — they lack the depth of knowledge and connections between ideas in the discipline that experts have. They think in concrete terms, noticing surface features and using a great deal of working memory to solve problems. On the other hand, experts rely on a great deal of deep knowledge to analyze problems quickly, make connections between other situations they have seen, and identify relevant similarities and differences. This is the heart of the creative, elegant work that we want of our young mathematicians.
Willingham’s research intimidates me, and on the surface it seems to make Tracy’s call to action impossible. How can we expect to treat students like real mathematicians if they think in a fundamentally different way about mathematics? I have three takeaways from synthesizing Tracy’s talk and Willingham’s ideas.
First, students can be introduced to mathematical concepts in engaging, concrete terms that allow them to play with mathematics in contexts that make sense to them. Looking at this image
and noticing and wondering about it doesn’t require mathematical knowledge — students can be curious about what happened with their knowledge of the world around them, where they are the experts. This is the first place where students can act like mathematicians, where it motivates the mathematics that we want them to learn.
Second, there is not a dichotomy between novice mathematicians and expert mathematicians. Instead, students work toward expertise in each of the domains they study. If we only give students a brief, surface-level tour of mathematical topics, they will never develop the depth of knowledge required to truly think mathematically. If, instead, we follow the spirit of the Common Core — to dive deeply into worthwhile mathematics, build students’ knowledge with a wide variety of rich problems, and ask them challenging abstract question as they gain expertise, they will both understand math more deeply and be able to practice math like mathematician.
Third, complex mathematical reasoning can happen with content that is much simpler than the students’ grade level. Play With Your Math is a great example — if my priority is for students to play with and explore mathematics, and we are only three days into a unit on exponential functions, I will likely only frustrate my students, or further divide them between students who “get math” and students who don’t. But this image
has a great deal of mathematical reasoning, has a low floor of entry, provide opportunities for justification and argument, has incredible depth as students play, and allows students to experience the practice of mathematics without the roadblocks of difficult content.
I know I fall into traps in my classroom because of my love of mathematics — giving students a difficult problem too early, encouraging them to be problem solvers, make connections, and persevere if they get stuck — and they get frustrated because they are not ready. I think Tracy’s call to action is absolutely critical to building in students the mathematical knowledge we all want for them — but it is most important for the 20% of my students who struggle with math the most. Many of my students play naturally with mathematics — because they are the sense-makers and the perpetual experts. It is the students who don’t have that knowledge that I most need to reach with Tracy’s call, and I need to do that by providing them with tasks that allow them to truly have the curious, imaginative, adventurous and joyful experiences that all math students and teachers should have. And I need to provide these tasks at times when they have the knowledge to access them, to wonder about them, and to answer the questions they ask — to create a loop of positive feedback that leads to more positive experiences with mathematics.