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Mathematical Magic

(via Robert Kaplinsky)

I notice that I’ve seen this gif on social media a few times recently.

I notice that the exterior angles seem to add up to 360 degrees.

I notice that I’m drawing on a lot of prior knowledge to make that conclusion. There are lots of angles one could highlight in the original polygons; I understand which ones are exterior. I understand the idea of angles greater than 180 degrees, so that a full rotation makes sense as 360 degrees. There are probably other things I don’t realize I know because I’ve spent lots of time solving angle problems with polygons.

I wonder if this would make any sense to a typical 8th grader.

I wonder when in an instructional sequence this gif would make the most sense.

I wonder if this gif makes a substantive mathematical idea seem like “magic.” I wonder what message that sends about the nature of mathematics.

I wonder how often I explain an idea in a way that connects with students who already understand an idea, but not with students who feel confused.

Prerequisite Knowledge

I had an odd extra day in one of my precalc classes last week and picked up my college combinatorics and graph theory textbook for some math to explore. We spent the period playing with graph theory. This fun problem from Play With Your Math was an awesome place to start:

We played with this for a bit, and then we looked at colorings of graphs. How many colors do you need to color a graph of the United States? What about Canada? (Hint: they’re different!) Then we played with the Bridges of K√∂nigsberg. The map below is of the Prussian city of K√∂nigsberg, with bridges highlighted in green. Leonhard Euler wanted to go for a walk crossing each bridge exactly once, without repeating any bridges. This led to explorations of Euler and Hamiltonian paths.

It was a pretty relaxed class. I’m not sure how much students learned, but it was a fun one-day excursion into some neat math. The best thing about the class was that it relied on so little prerequisite knowledge; students didn’t need to know about fractions or factoring or functions, but could play with new and challenging math on their own terms. I’ve taught Algebra II and Precalculus this year, and I hear from so many disaffected students that geometry is the only math class they ever enjoyed because they didn’t feel behind from the beginning and the math made sense to them. But geometry is an outlier. Most of our math curriculum is designed sequentially in the race to calculus, building on ever more complicated layers of algebraic manipulation.

Is mathematics fundamentally sequential, or do we just choose to make it so? I wonder what a school math curriculum would look like if it were designed to minimize the impact of prerequisite knowledge, to help every concept feel accessible to every student. Which topics would we eliminate? Which topics would we add? Which topics would we teach differently? My current teaching load feels like it is designed to do the opposite; there are so many places a student is likely to feel confused because of something they missed or forgot from a year or several years before. How does that make students feel? Which students who aren’t invited into the math conversation now might be if we approached math in a new way?

When to Nix the Tricks?

I love Nix the Tricks. I try to teach in ways that encourage students to make sense of mathematics, to believe that mathematics makes sense, and to understand what they’re doing rather than following a recipe provided for them.

But today, I told a student to flip and multiply when dividing fractions.

I’m teaching trig identities with my precalc students. I love this part of the unit; it’s abstract and challenging at first, but over time students start to see problems as little puzzles to figure out. I don’t mess with the product-to-sum and other more obscure identities, focusing more on reciprocal, quotient, and Pythagorean identities. I love questions like this one:

It seems so counter-intuitive that these expressions could be equal, yet they are. Students need to know their basic identities, need to be comfortable with algebraic manipulation, and need to remember some things about fraction operations. I like this unit because I find students can transition from seeing these problems as inscrutable and pointless symbol-mashing to seeing them as satisfying and logical puzzles.

But this isn’t a unit on fraction operations. Students were working on this problem:

Those divisions are hard! How do they work again? And we’re early in the unit, so there are some feelings of frustration that what we’re doing doesn’t make sense and students feel dumb. Frustration with fraction division is layered on top of all that. I could dive into an explanation of why dividing by a fraction is the same as multiplying by the reciprocal, but what I really want right now is for my students to feel successful working with trig identities and recognize the ways that they already know most of the math they need to solve these problems. Digging into fraction division feels like a distraction from the key ideas of the lesson, when students are happy to be quickly reminded of a procedure that will help them solve the problem in front of them.

Does this make me a bad teacher? Maybe I missed an opportunity to anticipate the difficulty with fractions and preteach some of those ideas before the unit. Maybe slowing down to spend time on the conceptual basis of fraction division is the right move. Maybe I should revisit fractions tomorrow. But these moments come up all the time, especially with precalc students. Are these moments distractions from the heart of the math we’re working on, or opportunities to circle back to math students have seen before?

Conceptual Complexity and Computational Complexity

I taught expected value a few months ago. This is one task I used, from Illustrative Mathematics, that represents what I thought of as an important learning outcome for my students:

Here are a pair of graphics from a neat article on shooting in the NBA, via FiveThirtyEight:

There’s a ton of math here! But I’d argue that the math is different than the math in the first task. It could still be used to teach expected value, but this second task values a different type of thinking. I’d call the second type of thinking “conceptually complex.” It takes some background knowledge and reasoning to parse, but it doesn’t involve extensive calculation, just some multiplying by two and three. It does lead to some great thinking about decision-making and takes the abstraction of expected value and maps it into a context where the math matters.

I’d call the first type of thinking “computationally complex.” The first question is unlikely to matter to students; I don’t think they care very much about Bob’s bagel shop. It gets at some useful mathematical ideas, but in Bob’s bagel shop, calculation is an obstacle between students and the math I want them to learn. They need to parse probabilities written as decimals rather than percentages, pay careful attention to which values to use as the price, and multiply and add decimals.

I often find myself valuing computational complexity in math. When students struggle calculating complicated things, I often feel a need to support them at managing that type of complexity, and prioritize practice and explicit instruction that supports their computational reasoning. It’s how I was taught math. It’s what I was good at in school math. I’m probably a math teacher in part because of the messages sent to me that being fast meant being good. But I wonder what math class would look like if conceptual complexity was valued equally as computational complexity. What would class look like? Which students would feel smart? Who would pursue math beyond high school?

A Trigonometry Menu

We’re back from break, and I wanted to start with some review of sine and cosine functions to refresh students’ memories. I was inspired by this recent post by Nat Banting to try a “trigonometry menu.” It looked like this:

It was fun! Here’s what I like about it:

  • It requires students to think in the opposite direction they usually do. Nat Banting calls this “upstream thinking.” Rather than being given a model and reasoning about it or solving something with it, students are building models to certain specifications.
  • It has a low floor and a high ceiling; students can access questions they are more comfortable with first, but solving the task with only three or four functions requires some pretty sophisticated thinking.
  • It elicits thinking about relationships. Graphing functions or building functions to match a graph or data can feel formulaic, and encourage students to follow a set procedure without much thinking: find the midline, amplitude, phase shift, and period, set up the graph or equation, rinse, repeat. With the trig menu, I heard students talking about the relationship between the midline and the amplitude, visualizing what different constraints might look like together, and reasoning about which constraints are mutually exclusive. I think this type of flexible thinking is a really valuable opportunity for students to apply their knowledge in a new way.

At the same time, it was tough to get all students to move past writing individual functions for each constraint and think about which constraints can be combined and which cannot. I want to try this type of task again in the future, but I also think it needs to fit in a particular place in the curriculum. Review after a break, when some students lacked the confidence and fluency to work flexibly, probably wasn’t the best place for it. I think this task can be an important stepping stone between typical practice and more sophisticated reasoning, but I think it functions most equitably when students have a solid foundation of fluency with the basic components of trig functions, rather than pausing to review how to find the period from an equation halfway through. This task provides some useful opportunities for thinking, but I want to use it for more sophisticated reasons than just because it feels fun and different.

Mistakes

Math teachers disagree about mistakes. Just check out the comments on this blog post from Dan Meyer. Some folks argue that students don’t make mistakes, they just sometimes answer a different question than the question the teacher was asking. Others argue that mistakes entrench unhelpful ideas in students’ memories, and that mistakes are the enemy of learning.

Here’s a bit of thinking I’ve seen a few times in my precalc class recently, in a unit on graphing sine and cosines functions. We’re early in the unit, and a student is writing an equation for a sine function, maybe like this one:

She writes the period as pi over 2, reasoning that the period is how long a function takes to repeat, and the y value is the same at both 0 and pi over 2.

One path forward is to label this thinking a mistake and explain the “correct” way of finding the period of a sinusoidal function. But that path serves to invalidate the student’s knowledge, and pass up on an opportunity to help them connect what they already know to what I would like them to know. A second path is to help the student see the valuable thinking they did, offer another example, and link their thinking to the missing pieces. For instance, using the same reasoning but focusing on either the maxima or minima of the function, one gets a different answer. In this situation, if I have an instinct to label student thinking a mistake, I miss an opportunity to build off of what they know and help them see themselves as effective mathematicians. If I choose to see their strengths and find the question they are actually answering, I have an opportunity to validate their mathematical thinking and expand what they know, rather than trying to replace something I’m labeling as a misconception with the “correct” way of doing it.

But there are actually two different places I see this mistake. One is early in the unit, when students are first trying problems and applying their knowledge. The other is a few days later, practicing the concept to consolidate their understanding and solidifying their fluency. And I think those two places require different approaches. When I’m introducing a topic, I want to find every way I can to build off of students’ prior knowledge, to help students feel successful with a new concept, and to give students a sense of agency in their ability to make sense of mathematical ideas. These are important opportunities to look at mistakes as opportunities for learning and draw out the valuable ideas even when student thinking falls short of where I want them to be.

During practice, I fall on the other side. Practice is where students make associations and consolidate their understanding. Practicing the wrong way just leads to confusion and frustration. This doesn’t mean I want to shame students making mistakes during practice; I still want to connect what they know to where I want them to get. But I’ve spent my entire life confusing affect and effect; I’ve used them wrong as often as I’ve used them right, and even when I look up the distinction I forget it in a morass of incorrect usage. This confusion be really frustrating for students; I don’t want them to feel like I let them practice something the wrong way and confused them when I had the chance to correct them earlier. Even where the “mistake” is a matter of arbitrary convention, when students feel wrong they often get frustrated and disengage with math class. I’d love to celebrate mistakes and help students see mistakes as an essential step on the path to understanding, but I also want to make sure that those mistakes are authentic opportunities to build understanding and not undermining future success for students who already struggle to feel successful in math class.

There are two psychological phenomena relevant to mistakes. The first is the generation effect. Having someone guess an answer before learning it improves the quality of learning. Even if the learner generates an incorrect answer, as long as they get quick feedback on that answer they are likely to learn more than if they were just told the information to begin with. From this perspective, asking students to try to figure things out and then giving corrective feedback as necessary is an effective teaching strategy. Those mistakes are valuable; they’re both authentic to the practice of mathematics and improve the quality of learning with immediate feedback.

Second, the new theory of disuse suggests that we don’t ever really forget things, they just become less and less accessible. Even if I can’t remember my friend’s phone number from twenty years ago, if I am reminded of it I will retain it much better than if I had never learned it. Anything students learn incorrectly can never be replaced, we can just try to make the new learning more readily accessible. Mistakes from this perspective are permanently damaging. I think of my struggle to remember the difference between affect and effect. I have used them incorrectly so many times that I struggle to remember their correct usages; the wrong usage is embedded just as well as the correct one. Making a mistake once will be much less accessible once students have had a chance to practice the concept, but making a mistake over and over again can start a cycle of confusion and frustration.

I think I have a responsibility to help students see the ways they can be successful in math class, and to see mistakes as opportunities for learning. But all that nice talk about how much we can learn from mistakes feels limited to the first case, where it’s early in the learning process and there’s lots of time to practice and consolidate ideas. When students are practicing concepts incorrectly, I still want to help them feel like their thinking is valuable, but I also need to be proactive in catching and changing their practice before they start a cycle of negative feelings and confusion. And in lots of situations, this is a blurry line! There’s not always an easy answer.

Moral of the story: I think mistakes are great, but there are important boundary conditions that help me understand when mistakes are helpful and when they might be counterproductive.

Popsicle Sticks and Accountability

When I started teaching, I was told that cold calling was important because taking hands selects only confident students, and I might think a class understands a concept when the silent majority are all confused.

I think there are lots of other ways to avoid selection bias. I can give exit tickets or mid-class hinge questions, circulate to look at student work or listen in on student conversations, use whiteboards to quickly see answers to a particular question, and more broadly cultivate an environment where students check their own understanding and feel empowered to speak up and ask questions. Cold calling is a bit of a blunt instrument; it samples randomly, but only one student at a time, and can still misrepresent where the class is.

But my core issue with cold calling was the shame certain students — and it was always the same students — felt when they were called on and didn’t know what to say. Some students bring negative experiences to math class, and putting them on the spot is likely to entrench negative feelings toward math and threaten the social safety of mathematical risk-taking and idea-sharing.

At the same time, I want to create a sense of accountability in my class. I don’t mean accountability in the sense of punishing students when they don’t participate. For me, accountability is creating an environment where students know I care about their learning, I will make sure that every student is set up to succeed, and I follow through to see if they’ve learned and do something about it if they haven’t.

In Ilana Horn’s book Motivated, she describes three norms of participation that can help to create a sense of accountability:

  • Everyone participates
  • Listening matters
  • The focus is on mathematical ideas

I’ve started to use popsicle sticks to cold call students again, but rather than trying to see which students know or don’t know a certain answer, my focus is on creating a sense of accountability and reinforcing these norms. I like popsicle sticks because they are visibly random — students don’t feel like I’m picking on them or trying to catch them not paying attention. And I use them only in a few specific places where there is no right or wrong answer, but instead ask students to share a mathematical idea with the class:

  • After students attempt a problem in groups, or reflect on an idea and share with partners, I call on students asking, “How did your group approach the problem?” or “What is something useful that you or your partner shared?”
  • After looking at a mathematical prompt, for instance a Connecting Representations routine, but with pencils down and before solving, I ask students, “What did you notice that might be mathematically important?”
  • After reflecting on a situation and making estimates of answers to a problem, I ask students for their estimates.

In each of these cases, I’m not looking for a right or wrong answer, but for students to share ideas and approaches. Every student is expected to participate. And in particular when I ask students what they talked about with their partner or group, the message is that they are expected to listen to each other’s ideas.

More broadly, while popsicle sticks are something my students groan about at times, I’ve found that using them in a few specific places helps to create an environment where students know they are expected to participate and that I care what every student thinks, without shaming students for not knowing answers.