For All/There Exists

Two different statements come up often in mathematical proofs: “for all” and “there exists.” “For all” makes a statement that is true in every case. For instance, for all numbers divisible by both 2 and 3, they are also divisible by 6. “There exists” makes a statement that is true in at least one case, but need not be true in every case. For instance, there exist whole numbers not divisible by any whole number except for 1 and itself.

I think a similar distinction is useful when talking about teaching. There are some things in the “for all” category — things that should always be true, that I should strive to do every class, every day. Academic safety falls into that category. Every student should feel like they can take risks, share ideas, and be wrong, all with unconditional support. I should strive never to compromise on academic safety. I fall short on this all the time, but I need to set “for all” as my goal and work to help every student feel safe every day.

There are other things that fall in the “there exists” category — things that students should experience but don’t need to happen every day. Learning through discovery falls here for me. I think every student should experience mathematical discovery. And it’s hard to get discovery right, so this can’t be a one-off every few months. At the same time, I don’t believe students need to discover something themselves to understand it, or need to experience mathematical discovery every class.

There are other examples. Practice is a “for all” — I value practice, and I want students to practice every mathematical concept they encounter. Representation is “there exists” — I can’t show every student a mathematician who looks like them every day, but I can strive to share mathematicians who represent each of my students several times over the course of our time together.

Discourse around teaching can get lost when we confuse “for all” with “there exists.” I need to hold myself to a high standard around academic safety, every day and for every student. But it would be easy to get defensive and say, “but this other student feels safe, so x student should feel safe too!” Teachers face a constant onslaught of decisions and information; I have to avoid cherry-picking examples to fit my narrative. And it’s easy to make the opposite mistake, to take something that should exist somewhere and assume it has to exist everywhere. The value I place on discovery doesn’t mean that every lesson has to be a discovery lesson, and doing so risks losing sight of my true goals. One of the biggest challenges of teaching is how many decisions I have to make a day, and how quickly I have to make them. I hope distinction can help me to better live out my values and avoid lazy shortcuts.

Time

Here’s a tension I think a lot about:

As a teacher I only have so much time. Time per class, classes per semester. I can’t easily fabricate more. I have to make a lot of decisions about time: what can I get done next class? Is there room for that in this unit? Do I go deeper with fewer ideas or try to touch on every little thing?

But constantly worrying about time can make teaching feel industrial. I can justify a lot of shitty teaching by saying “we just didn’t have time” or “its time to move on.”

I don’t have a great answer. It would be nice to ignore the limits on my time entirely, but that would come back to bite me worse later. I think my biggest takeaway is around how students feel in class. I don’t want students to feel rushed, or to feel like they need to get through something to get to the next thing, or that they are being shortchanged because of the limitations of the school schedule. If my students don’t feel like they’re cogs in a machine sprinting from one topic to the next, I’ll take it as a small victory.

Familiar Territory

After I graduated college I road-tripped across the country, stopping in different national parks. After my first year of teaching I backpacked the famous John Muir Trail in California. I had a long list of spectacular and well-known destinations I wanted to see, in the US and beyond. A year later I moved to Leadville, Colorado, excited to climb tall mountains and tick destinations off of my list.

Almost six years later I still love spending time outside, but I have different priorities. I don’t often visit nearby national parks — too crowded and too much time driving. I almost never climb the tallest peaks or ski the big resorts. I’d rather know my home better and better than do lightning tours of the “biggest and best” of the west. Leadville is a bit out of the way but still has great sights to see, and I enjoy exploring new corners of the abandoned mining district east of town, or getting to know a quiet new trail nearby, or finding a new perspective to appreciate what I see every day.

I try to see math the same way. There are spectacular puzzles and fields of math that I find exhilarating. There’s a time and place to share math’s great theorems. But there’s also magic in the hidden corners of everyday school curriculum, digging deeper in familiar territory. In the last weeks I’ve enjoyed finding patterns in equivalent forms of sine functions, exploring all the ways an arithmetic series can sum to 60 or 105, and finding clever new rational functions to graph. These aren’t the great theorems of mathematics and aren’t very interesting to professional mathematicians. But they’re fun for me. I enjoy seeing familiar terrain from a new perspective. And I think finding new puzzles in the everyday content I teach students is a great way to spark student curiosity. Math’s “greatest hits” are beautiful to me but not to everyone else, and especially not to students who have a hard time with math class.

Four Types of Problems

I’ve got an idea that I’m pretty sure is wrong, but is also maybe useful.

I love problems, and I love trying to better understand what types of problems help students learn math. Here’s a rough way to categorize problems:

Two axes. First, does the problem require a moment of insight, or is it something that one can solve by grinding, sticking with the problem, and trying different things? Second, does the problem have one strategy or several possible strategies?

Problems in the top left I would describe as puzzles. Catriona Agg’s geometry problems typically fall in this category. I learn a ton from seeing different ways to solve them, but if I don’t have that moment of insight I won’t figure it out. These problems can have value in the classroom, but only if students have support. If I leave someone without help as they struggle to find the right insight I’m setting them up for frustration.

Problems in the bottom left I would describe as riddles. I often think of the wolf, goat, and cabbage problem as the prototypical riddle, but plenty of everyday math problems turn into riddles when students don’t have the right tools. I’ve developed a pretty strong distaste for riddles. They tend to make people feel stupid, and they don’t lead to much interesting discussion because you either end up in the right place or flail, and there’s not much in between.

Problems in the bottom right I would describe as exercises. Exercises get a bad reputation, but I give my students exercises every day. They’re an important part of a balanced mathematical diet. They’re best in regular but relatively small doses.

Problems in the top right I would describe as discussion problems. These are everywhere, from a derivative problem where students can choose to use the product rule or simplify first, to everyday quadratic graphing problems that students can solve by factoring, making a table, or completing the square. When different students approach the same problem in different ways, and most of the class can work through the problem successfully, that problem lays the groundwork for some great learning. I had a misconception in the past that a problem needs to be rich and unique to be useful for discussion. But there are lots of problems I consider exercises until a student finds a new method, presenting an opportunity for a fun discussion.

One important caveat is that this distinction depends on students’ prior knowledge. A problem that requires insight for one student is accessible for another with different knowledge. An important but often neglected role of the teacher is to give students the knowledge they need to access problems, rather than hoping the problems will do the teaching on their own.

I want to say again that I’m pretty sure this is wrong. The world of problems is too complex to simplify into a neat little two-dimensional diagram. But I do think this can be a useful representation of the problems I give students. I think math class should involve all of the categories except for riddles. Giving students problems is always a delicate balance. I want problems to be accessible so students can feel successful and recognize their mathematical talents. But I also want to give problems that provoke curiosity and give students a glimpse of the beauty of mathematics. I think this way of conceptualizing problems can help me to manage that balance while giving students a rich experience in math class.

The Crest of the Peacock

I recently read The Crest of the Peacock: Non-European Roots of Mathematics. It’s a great read! I’d definitely recommend it. I’ve been trying to blog about it for a few weeks, but I’ve had trouble summarizing what I’ve learned into some neat and tidy lessons. Maybe that’s the point. The history of math is messy.

Two themes of the book are interesting to me. First, the history of math isn’t linear. It’s full of false starts and dead ends. One example is calculus; Chinese and Indian mathematicians anticipated calculus concepts centuries before Newton and Leibniz. That two European mathematicians get credit for inventing calculus is mostly an accident of history. There are tons of other examples of non-European cultures discovering ideas that Europeans later get credit for. We ignore them because they don’t fit into a simple trajectory.

Second, different cultures — including Europe — often tiptoed around ideas we now think of as obvious, seemingly almost there but not quite reaching a modern concept or notation. Place value, limits, functions, algebra, and more were all hinted at for centuries or millenia, across different cultures and contexts. Looking back now it sometimes feels cringeworthy. But everything is obvious in hindsight, and mathematicians were doing just fine with the concepts they had.

I see these themes as useful metaphors for how students learn math. Student learning is often less linear than teachers like to pretend. It happens in fits and starts. While we emphasize the ways mathematical ideas build on each other, that sequence is often arbitrary. Some students grasp “advanced” topics before “simpler” ones. Part of teaching is engaging with that complexity. Students also often seem like they’re close to understanding an idea, but take longer to get there than might be convenient. And that’s normal. Everything is obvious to a teacher, but some of the conceptual leaps that we take for granted are much larger than we realize.

The history of math is fascinating. It’s easy to tell a story about the Greeks, the Middle Ages, and modern European mathematics. It’s neat and tidy and confirms our preconceptions. But that history silences incredible mathematical learning in cultures around the world, learning that doesn’t fit into a little box. And that history also paints a portrait of math that sands away the rough edges and oversimplifies complexity. I think engaging with that complexity is a great way to see math with new eyes and new perspective.

Problems and Textbooks

If students only solve a narrow range of problems they will not be able to apply their knowledge in new contexts in the future. If my students solve a broad range of problems they are more likely to develop deep knowledge. All my clever demonstrations, cute explanations, and bad jokes matter a lot less than I like to think. What matters is students thinking about mathematical ideas, and the best way to get students thinking is to have them solve problems.

I’ve spent some time over the last few weeks trying to expand my knowledge of problems. Mostly, this has involved finding interesting problems in my textbook and working through them. I use the Larson, Hostetler & Edwards Precalculus textbook sporadically but most of my curriculum is a homebrew. While I enjoy the freedom to teach what I want to teach, I also run the risk of teaching topics in narrow ways based on my knowledge and biases.

There’s a lot of drudgery in textbooks, and some are better than others. I didn’t work through every problem. But flipping to the last page of each section led me to a surprising variety of interesting problems. I found plenty of challenges and new perspectives and saved lots of problems to use in my curriculum. I think textbooks can get an unfair reputation. For the most part they are resources of examples and problems. Examples and problems are the backbone of any math curriculum. Teaching straight from the textbook can be incredibly uninspiring if I just parrot the examples and assign 1-33 odd, but if I insist on inventing everything myself I’m missing the opportunity to save myself effort and expand my knowledge of many topics.

The largest danger of textbooks, in my opinion, is the structure of the text and not the problems themselves. There’s an implied pedagogy in typical textbook design. Start with examples, then assign students some repetitive practice. Fast students might get to some harder problems at the end, but probably not. I want to use some of the new and challenging problems I’m finding in my examples. Rather than only teaching the basics and hoping a few students can figure out hard problems, I can raise expectations by making challenging and unusual problems an explicit part of my teaching, providing students with support, leading discussions, sharing perspectives, and summarizing takeaways. The drudgery of textbooks to me is in the repetition of paint-by-numbers mathematics. I can enrich my curriculum by incorporating variety. It’s all there, it just needs to be structured in the right way.

Cognitive Biases as a Teacher

I love learning about cognitive science and psychology, and applying that learning to my classroom. But I’ve noticed that research often focuses more on understanding the minds of my students than understanding my own mind as a teacher. I’m not sure why this is, but there are plenty of things that I have learned about how my own mind works that influence my teaching. Here are three that I try to think about on a regular basis:

Confirmation bias. Humans don’t like to change our minds or be wrong. But more than that, our minds filter information around us in ways that confirm our previous beliefs. I think about this a lot on two levels. First, I have certain beliefs about what effective teaching looks like. I’m likely to focus on evidence that my pedagogy is working and ignore cases where it isn’t. Second, I make assumptions about my students, and I’m likely to reinforce those assumptions without effort. I need to consciously seek out evidence that disconfirms my beliefs — in this case, evidence that my teaching isn’t working as well as I want to believe, and evidence that my students are not who I assumed them to be.

The curse of knowledge. Humans tend to assume that others have the same knowledge we do, and struggle to recognize the ways our own knowledge allows us to do things. I like to think I know a fair amount about math. But that knowledge prevents me from understanding what it is like to be a student in a math class. This influences my ability to empathize with students, and also to break down content to help it make sense. Having proved lots of trigonometric identities in my life, I am fluent in lots of little skills that I forget I’ve even learned, and when I forget what those skills are I can’t set my students up for success. This means I need to take time to better understand what I already know, and stay open to learning familiar ideas from a new perspective.

Fundamental attribution error. Humans are biased to assume that the behavior of people in front of us represents who those people are. We assume that, if a person says something mean that they are a mean person, or if a person struggles to explain something that they are inarticulate. But the math classroom is only one context, and humans change between contexts. I need to resist the urge to categorize and judge my students, and give them the chance to break those boundaries. I’m often surprised by my students passions and interests outside of math. Many young people don’t like math very much, and for good reasons. I need to recognize when I am extrapolating based on a limited sample size, and seek out opportunities to broaden what I know about my students.

I’m far from perfect, and understanding some of my biases helps me to recognize when I make mistakes and correct them. These three psychological phenomena are always operating in the background of my mind. Better understanding each bias helps me to recognize their impact and correct for their negative consequences.

(Not) Learning From Problems

I love Catriona Agg’s geometry puzzles. I enjoyed playing with this one from two weeks ago:

I had a hard time solving it. I floundered for a while, then used a convoluted strategy involving four equations and four variables. The answer popped out, but when I looked in the replies I found many more concise ways to do it.

Truth is I’m not that great at solving geometry puzzles like this one. I often struggle with Catriona’s problems. I’m not great with common strategies like using similar triangles, or drawing a circle, or lots of other things. And here’s the tough part — as I’ve spent time exploring these puzzles over the last year, I haven’t gotten much better.

I think this is a useful example of the difference between solving a problem and learning math.

Catriona writes great problems, but the problems aren’t designed to teach me new things. They aren’t sequenced to build on each other, there aren’t opportunities to practice key ideas, there’s no summary or discussion of takeaways for a learner like me. It’s hard to learn solely from problems, especially on a topic I’m not already good at. I might come across an interesting principle in some problem, but I’m cognitively overloaded while trying to figure it out. By the time I figure out whatever the idea is I’m ready to move on to something else and forget what I might have learned. It’s not a recipe for durable learning.

If I want students to learn from problems I need to embed time to process and codify learning, practice opportunities, and opportunities to transfer the mathematical ideas to new contexts. The role of a teacher is to do all of that — to figure out what we want students to learn and get students thinking about those ideas. That’s different from having students solve a single problem and hope the learning sticks.

I don’t mean to be critical. I don’t think Catriona is trying to teach things on Twitter, only to have fun and share the beauty of math. But exploring her problems has been eye-opening for me as a novice. I love moments like this to better empathize with my students’ experiences in math class. Learning is hard, and it’s easy to forget that when I’m teaching the same topics and same problems I’ve used for years.

What I’ve Learned From #DisruptTexts

I’m hesitant to wade into the #DisruptTexts debate. I’m a math teacher, what do I know about choosing texts to read in English classes? But I’ve learned a lot observing the important work of questioning the traditional literary canon. I am sad to see bad-faith attacks that mischaracterize what #DisruptTexts is about, and I’d like to offer my perspective on the movement.

First, I want to engage with a legitimate argument for the canon. I read The Crucible in school. It explores a literal witch hunt, written during the era of McCarthyism in the United States. The play gives context to a historical era that I hope we can learn from. The play also gives context to the phrase “witch hunt.” I have a better understanding of that phrase and its implications because I’ve read The Crucible. One important aspect of this type of learning is that it’s often implicit. When I hear the phrase “witch hunt” I don’t immediately think of John Proctor, but my knowledge still helps me to better understand the world around me. This is only one example. There are other places in the traditional canon that build useful knowledge. And that’s the argument for the canon: these texts have stood the test of time, form a foundation for what it means to be educated, and provide access to cultural references.

There’s a caricature going around that #DisruptTexts is about banning books and throwing out the canon entirely. That’s not what I’ve observed. Instead, #DisruptTexts is about interrogating what students learn from the texts they read, and making informed decisions about what they should read and how they should read it. For instance, the website has a great article discussing The Crucible. Reading that article I learned a lot that I didn’t notice when I read the play in school. The play deals in stereotypes and elevates some perspectives at the expense of others. So while I learned useful lessons about witch hunts, the text also reinforced stereotypes and tired narratives about good intentions that also impact how I see the world today. A lot of that learning is implicit, but still shapes the way what young people learn in school. #DisruptTexts isn’t about banning or censoring. It’s about unpacking the lessons students learn from texts, teaching traditional texts with a critical eye toward those lessons, and replacing others with new, valuable perspectives.

The Crucible offered me one useful lesson, but those lessons aren’t unique to the traditional canon. In the last few years I have developed a deeper understanding of police violence by reading The Hate U Give, a deeper understanding of the complexities of immigration by reading Exit West, and a deeper understanding of prejudice by reading the Broken Earth trilogy. The canon doesn’t have a monopoly on important knowledge or important learning. That’s why I support the work of #DisruptTexts. I read far too many white authors and traditional narratives when I was in school. More diverse voices and perspectives would have enriched my education and broadened my world, and I’m still doing work to play catch-up.

So where does math come into this? I think that the #DisruptTexts folks are way ahead of any comparable efforts in the math community. The closest thing to a canon in math class is probably our race through algebra to calculus. There are lots of types of mathematical thinking, and we choose to value complicated symbol-pushing and abstraction as the end goal of high school math education. That’s a choice — there are lots of other directions we could head. We could choose statistics, probability, mathematical modeling, number theory, computer science, data science, and more.

Why do we teach algebra? There’s an argument for it, definitely. It’s the foundation of the math we use in disciplines like engineering. But there are also arguments against it. What I love about #DisruptTexts is the dialogue and community. They create space to have hard conversations around what texts to read, how best to read them, and what they want students to learn. Those are conversations I wish we had more in the math education community. Too often math educators see curriculum and standards as static, taking what we teach for granted and trying to figure out the best way to teach within those contraints.

To be fair, there are plenty of efforts heading in this direction. NCTM released Catalyzing Change, which addressed many of these themes, and plenty of schools are having similar conversations. But they haven’t percolated to the surface the same way #DisruptTexts has. And I admire the depth and sophistication of the conversations I see around the texts teachers use in English class. We need to ask some of those same questions. What do students actually learn from algebra – not what we wish they would learn, but what they actually learn? What knowledge do students use implicitly, without realizing they are using it? When do they use that knowledge? Which pieces are useful, and which are worth scrapping? What do we most want students to be able to do with math outside of the math classroom? To what extent do we teach students that they are bad at math? Would that change if we changed what we taught? What other implicit lessons do we teach without realizing it? What is the hidden curriculum of math class?

These questions and more are worth asking. And again, I know many teachers ask them every day. But #DisruptTexts provides a model for what it looks like to build a community around asking hard questions, and engaging in dialogue about those questions. I think we have a lot to learn, and I’m very grateful to the #DisruptTexts folks for offering a model that we can learn from.

A Small Change

When I teach rational functions, I always use this task after students have gained some fluency in graphing simple rational functions:

Last basketball season, [student name] made 21 of her first 30 free throws, and then went on a hot streak and made every single free throw after that. Write a function for her free throw shooting percentage as a function of shots taken (after the first 30).

What do the horizontal asymptote, y-intercept, x-intercept, and vertical asymptote represent in this situation? On what domain does this function make sense?

(credit to Rachel who I originally stole the problem from)

The problem has a few other fun extensions. I can give students new functions to interpret and describe what they say about a basketball player’s shooting skills, get into the weeds of domain and range, throw in a problem about field goal percentage and average points per shot, or more. The problems aren’t anything that special. I’m sure many teachers use similar problems in their classes. But I’ve found this sequence useful for starting interesting discussions, getting students to engage with applications of a topic that doesn’t have very many applications, and interpreting a complicated graph in context.

This year when I taught this problem I made one small change. I gave them the initial problem, to write free throw percentage as a function of shots taken. But I also offered a hint: shooting percentage = shots made / shots taken. In the past when I’ve used this problem, some groups figure out the function quickly and others get stuck because they don’t know where to start. And that might be fine if my goal was for students to learn about percentages and writing functions based on proportions. But the goal of this sequence of questions is to connect representations of rational functions. While I’d love students to be proficient at writing functions like this, it’s not a skill that comes up very often and it’s not my main focus. I’d rather help students write their function successfully, and have them spend more time trying to figure out what the different parts of the function represent in context.

It’s a small change, but I’ve found myself making changes like this more and more often. I wrote about this idea earlier this year under the title “More Explicit.” A lot of teachers pushed back. At times there’s an orthodoxy in math teaching that struggle is good and the teacher’s job is to be less helpful. I think this lesson is a good example of what I mean when I say that my teaching has become more explicit. I don’t mean that I’m doing everything for students. But struggle is best in small doses, under the right circumstances. Too much struggle leaves students feeling dumb. I’m making a lot of small changes like this one to focus struggle on the most important parts of a problem and to focus student thinking on the most important mathematics. There are a lot of fun math tasks in the world, but some take students on long and frustrating detours. That can be fine. But it’s also fine to be a little more explicit to help more students get where they’re going.