I’ve been exploring a new problem with a couple of students recently that I find incredibly compelling, and I thought I’d mention it here. The main idea is looking at the behavior of functions of the form f(x) = ax + b in various mods. There’s actually a huge amount to explore here, from fixed points to invertibility to the length of loops when you repeatedly apply functions. But one apparent pattern in particular blew me away.
The question is this: in mod n, how many functions f(x)= ax +b are their own inverses?
For example, the function f(x) = 5x + 2, applied twice in mod 12, is equal to the identity. It’s direct to check: f(f(x)) = f(5x+2) = 5(5x+2) + 2 = 25 x + 10 + 2 = x (mod 12).
We proved that there are at least n+1 self-invertible functions of this form for odd n, and n+2 for even n… these are our minimal cases. But what about the non-minimal cases?
We worked out the cases up to 28 by hand, and bewilderingly, for every non-minimal n, the number of self-invertible functions was always a number that completely factored as products of 2s and 3s. For example, in mod 12 there are 24 = 2x2x2x3 linear functions that are their own inverses.
I can see no reason why this should be true, or what these things even have to do with each other. My hunch is that it may just be a coincidence. Still, if it is a coincidence, it’s a pretty marvelous one, working for n = 8, 12, 15, 16, 20, 21, 24, 28, which, by our reckoning, are the non-minimal cases. How long should we expect such a pattern to hold before we suspect that something is going on?
In any case, the behavior of these functions mod n has totally captured me for the time being. I can’t believe I’ve never thought about them with students before.
Math makes sense. Not only to mathematicians, it turns out. Math just makes sense. It’s internally coherent, and shows you so when probed. All the rules in math that seem like “just because”–you can think of them probably pretty quickly, like don’t ever divide by zero, or a number raised to the zeroth power equals one, or to divide fractions you multiply by the reciprocal–have very good, very sensible reasons behind them*. So sensible, in fact, that when you really see these reasons, the rigor and meaning of the mathematical framework strikes one as almost impossibly beautiful.
Math makes sense, but it doesn’t always make sense to us, as math doers, right away. However, the fact that it is internally coherent means there is a task for us, which is to uncover the purpose that underlies the definitions, rules, patterns, and behaviors we see. As mathematicians, when we sit down to work, we are in essence sitting down to a science of deep reason. Things have a place, a reason, a logical flow.
So there are two things happening here. One is the sense the subject itself makes–the internal consistencies that were built into the subject over the centuries as its practitioners added more ideas. The other is the sense we must make of it–the sifting and mining we do to uncover the reason of the mathematics. They are deeply related, both to each other, and to the practice of doing and thinking about mathematics. To miss the fact that math makes sense is to misunderstand one of the most essential ideas math was built around. Sometimes you have to work for it, though, patiently and persistently.
Many people go through the entirety of their math lives without ever understanding that math actually makes sense–indeed, is more firmly dedicated to sense than any other subject or area of study–because we aren’t teaching it in a way that illustrates this, and we aren’t teaching it in a way that encourages students to pursue the sense that’s there.
How do we teach math in a way that highlights its inherent sense, and pushes our students to dig into problems in an effort to make sense of them? Though there are many useful answers to this, here are a few:
In order to learn how to make sense of problems, students need to be given problems that resonate with deep, internal coherence.
They also need to be given problems that require their effort to uncover the reason behind the problem. These problems need to be hard enough to put students into productive struggle, but not so hard that they give up. Well-crafted math problems should be like adventures, with the opportunity for real diversions, points of interest along the way, and rewarding views after periods of sustained struggle.
Students’ questions should be used to find where the math isn’t making sense on a conceptual level.
Internally motivated math experiences can help students develop real persistence in sense-making. When students care about answering a question, and care about the answers they get, they work longer, harder, and also object when their work or the problem itself doesn’t seem to make sense.
Aesthetics help! Go for beauty!
As Common Core picks up steam around the country, there are going to be more conversations about making sense of math. Common Core includes a list of math habits of mind, or standards of practice, all students should be developing. Making sense of problems and persevering in solving them is the first of these practices, and for good reason. Because one of the most important things to know about math is that it makes sense, and that we can see it if we try.
*For example: a number raised to the zeroth power equals one. Why? Maybe I’ll let you think about why it makes sense, but here’s a framework to help—what happens as you go down this list of powers?
5^4=5 x 5 x 5 x 5 = 625
5^3 = 5 x 5 x 5 = 125
5^2 = 5 x 5 = 25
5^1 = 5
5^0 = ?
Do you see a pattern? To make that pattern hold, what is the “natural” value that 5^0 should have, that would make the most sense?
Our Teacher Circles have been seeing great results, and we’re gearing up for our spring session. Meanwhile, the Julia Robinson Festival is coming up on April 28, and we’ve already got 158 kids signed up, which is more than came last year. I’m hoping we’ll get to 300 this year!
Today we met with a Seattle principal to talk about working intensely at his school for at least a year. It’s an opportunity to take our intervention to a deeper level, and to move even more firmly from us affecting just the kids we teach to creating a positive impact on a school which will reverberate through the students who pass through it for years to come.
So far, this year has been amazing. And today… a good day.
There has been considerable backlash against processed food products in the last few years, and for good reason. A slew of health problems implicate what we eat, and processed food products are more product than they are food. As industry spread through the last century and we began to see its uses multiply, the convenience made possible by industrially processing food was seductive. We thought we could improve our lives with convenience, but really we just gave ourselves heart disease and diabetes. Humans should eat food. Real food. ‘Nuff said.
Likewise, in an effort to make math easier to teach, easier to grade, and easier to standardize, we have essentially run it through industrial-scale refining, removing much of what math really is, and leaving behind a quivering, viscous math product. Math, as it is currently taught in K-12 education, is like highly processed food. And much of the education industry is happily at work on the factory line, boxing the processed math product into consumable, digestible, tidy little packages.
When a mathematician tells people what she does, the most common response is ”I hate math” or “I suck at math.” People think they hate math, but what they don’t realize is they’ve had very little contact with Real Math. Rather, they’ve been on a carefully controlled diet of processed math, hardly even a shadow of its former self.
Processed math is contrived (i.e. tidy story problems that look nothing like their real-world counterparts). Processed math is meaningless (i.e. the reasons behind the rules and processes seem arbitrary and everything is a memorization game). Processed math is unmotivated (cue the classic student response to a problem: “What do they want me to do here?). Processed math is uninspiring. Seriously, what’s to love here? No wonder math has such a bad reputation.
Real math, on the other hand is juicy, messy, rich, and vibrant. It compels our creative attention. It allows for dynamic interplay between problem solvers and the problem itself. We can’t help but get sucked in. We can’t help but take ownership of the work. Math needs a revolution like food had a revolution, and every good revolution needs a catchy slogan (Michael Pollan’s quips come to mind). So here’s a start, in three words: Do Real Math. Or in five words: Real Math for Real People. Word.
There are two major thrusts to math education. One is to teach skills–how to combine numbers, for example, and the definitions and rules of things in the mathematical universe. The other is teaching how to think (as it pertains to the mathematical universe, though some would argue this qualification is unnecessary: how to think in math is, in general, helpful in general thinking). This second purpose is about developing mathematical habits of mind: how to approach solving a problem, how to be creative in problem solving, and other deeper thinking skills.
Math education has generally focused almost exclusively on the first of these two themes. I think in part this is because most people don’t know what mathematical thinking looks like, let alone how to teach it. Even many mathematicians would have to ponder a bit to be able to articulate how they think in their field. We are all much more accustomed to talking about math, rather than talking about how to think about math. It’s this second “about” that puts math habits of mind into a kind of ‘meta’ place. It’s a meta concept, and it takes some laborious self-observation to hammer out just what these ways of thinking are.
There is considerable debate over how to teach skills. There is debate over whether to teach skills. There is debate over which skills to teach. Do we teach the division algorithm or chuck it? Do we emphasize multiple techniques for multiplication or do we just give the time-honored standard multiplication algorithm? What are all these skills building towards, or are they building empty, skeletal towers students will abandon as soon as they are out of high school, a waste of considerable effort and energy? As teachers, educators, and specialists have struggled to sort all this out, the second major purpose of math education has been left to wilt.
There is something inherently challenging about teaching a person how to think, rather than just giving facts. When I was little I struggled to understand how light works: to know how it worked, I thought, we had to look at it, but looking at it means using the very thing we were trying to understand. This has the unsettling quality of an infinite loop, doesn’t it? Likewise, thinking about thinking can bring us into a kind of recursive trouble, almost of the Bertrand Russell paradox flavor. Teaching about thinking is so troublesome that in general we avoid doing it.
Sometimes the straight approach isn’t the most productive, it turns out. Few of us develop our habits of mind explicitly. Rather, they arise implicitly, subterraneally, through experiences that apply pressure on our thinking and shape it in near-geologic processes. A good math experience is one that offers this kind of pressure, that can gently sculpt our ways of thinking. While some skill-based work may be happening on the surface of a lesson, underneath we want the lesson to drag an iceberg through our mental habits, shaping them into math habits of mind. We’ll get sharper by going through this repeated scouring.
When we focus all the time on skill building, we lose the beating heart of teaching. You know what they say: give a person a fish, they eat for a day. Teach a person facts, they may or may not remember them and that will be that. Teach a person how to think, sharpen their mind, help them develop their insight and mental agility, and you’ve done everything we want good teaching to be.
I’d call this article, about how doubling the time students spent studying algebra led them to do better in math and also reading and writing(!) a case of burying the lead.
Why? Before anyone rushes to double the lengths of all algebra classes, make sure you read into the article, where Cortes, a researcher following the kids in the “doubled algebra” classes, considers why they work.
“And we said, why is this?” Cortes recalls. “So we looked at how the classes were being taught. The first algebra class is a typical lecture-style class, but the second class is designed to be more interactive. The teachers would break the students up into groups and have them discuss problems and write on the board. So they were learning math, but they were also learning how to read and write in the context of algebra.”
And there you have it. It wasn’t twice as much as they were getting before. It was something qualitatively different. What we’re seeing here is a description of a mixed approach to math, including lecture, group problem sessions, presentations, and class discussions. Any maybe that takes twice as long to do well, but it’s not much of a surprise that it will work better, make the math more meaningful, and have rippling positive effects.
An interesting question would be, what was the outcome for kids who took just the second session, and skipped the lecture?
1. Why on earth would you break the symmetry by using a parallelogram instead of two triangles in the biggest one? I’m convinced you must have run out of triangles.
2. How many ways can you make a 12-gon out of regular polygons?
3. What are the angles on the rhombuses?
4. What’s the interior angle of a dodecahedron?
5. If you try to make a tiling out of 12-gons, what shapes get left in the gaps?
6. Are the two smaller 12-gon the same size?
7. How much bigger is the bigger one?
8. Did you know that you can find the bottom left one in Lockhart’s book, Measurement?
From Sue VanHattum
9. In the biggest one, we’re depending on the edges of adjacent diamonds forming a straight line. Do we know that they do?
10. Are the two smaller ones the same size?
11. Is the bigger one an even multiple of them?
12. How can you figure out the area of these shapes?
13. If the two smaller ones are the same size, what (if anything) can you infer about the relative area of the square and the skinny diamond?
Thanks for these great questions! And also, for the demonstration that a well-developed curiosity can lead us from a rich environment in the pursuit of some pretty serious explorations.
Let’s consider question 11: Is the bigger one an even multiple of the two smaller 12-gons? (I’ll assume that the two small ones are the same size… though that’s worth thinking about too–see questions 10 & 6. Also see question 7.)
On the one hand, it seems like it would be pretty incredible if it were. That would mean you could take some number of the small 12-gons, break apart their pieces, and arrange them into the larger 12-gon. Is it possible?
Some bookkeeping should help us. Counting the pieces in the big 12-gon, and taking advantage of its 12-fold rotational symmetry, we see that it consists of:
24 tan rhombuses
13 blue rhombuses
We can see, and know from earlier play with pattern blocks, that 2 triangles = 1 blue rhombus, so we could rewrite this list as
24 tan rhombuses
13 blue rhombuses = 26 green triangles
24 tan rhombuses
22 +26 = 48 triangles
What about the small 12-gons?
Type 1 has 12 tan rhombuses and 12 triangles.
Type 2 has 1 hexagon, 6 triangles, and 6 squares, which is the same as 12 triangles and 6 squares.
Wait a minute! 6 squares plus 12 triangles is the same size as 12 tan rhombuses and 12 triangles! That must mean that 6 squares = 12 tan rhombuses… which must mean that 1 square = 2 tan rhombuses! Not obvious at all (and answers question 13, incidentally).
In any case, can we use 12-gons of type 1 and type 2 to actually build the big 12-gon?
The answer, amazingly, is yes! Two 12-gons of type 2 give you 24 triangles and 12 squares; Two 12-gons of type 1 give you 24 tan rhombuses and 24 triangles. Sum those pieces and you have precisely what you need to make the bigger 12-gon!
So, the big 12-gon is precisely four times the area of the small 12-gon. And you can literally break apart four of the small 12-gons to build the big one. Which raises another question for me:
14. How many of the little 12-gons do you need to build an even bigger 12-gon?
We seem to be experiencing a brief cultural moment, in cinema, at least, to look back at slavery. Lincoln and Django Unchained at least, take place in a six-year period where people enslaved each other with all the brutality that involved (Django) and orated on the moral imperative of doing so, or not doing so (Lincoln).
For me, of course, it is a pleasure to imagine, as writer Tony Kushner did, that a mathematical education had something to do with finding the moral clarity required to do the right thing, as in this pivotal scene in the movie.
For centuries, learning Euclid was considered essential for the proper development of the mind, his axiomatic take on geometry the cornerstone of logical thought, argument, and for some, even morality. (Plato famously wrote “Let none who have not studied mathematics enter here,” on his University, and advocated for the study of geometry as a pathway to understanding “the Good.”) And at its best, I do believe true mathematical experiences can lay a foundation that allows us to imagine what is possible beyond the ethical blindness of our own time.
How often it actually serves that purpose is another question. I don’t know if I’m ready to argue that mathematicians are more moral than anyone else.
Serendipitously, after I saw these two movies, I came across a powerful portrait of an escape from slavery and rise to becoming the towering figure he became in American history. I was reading Carl Sagan’s paean to science and skepticism, The Demon-Haunted World, which is quite good, if occasionally a bit longwinded; in chapter 21, he starts with the story of the slave boy Frederick Bailey struggling to learn the alphabet and eventually, with the help of the wife of the man who owned him, learning simple three- and four-letter words. When her husband found out, he chastised his wife, and—in Bailey’s presence!—said the following:
A n***** should know nothing but to obey his master—to do as he is told to do. Learning would spoil the best n***** in the world. Now if you teach that n***** how to read, there would be no keeping him. It would forever unfit him to be a slave.
[Since this is a family blog, I've blotted out the "n-word" in the passage above.]
And in that moment, Bailey saw the great secret: “I now understood… the white man’s power to enslave the black man. From that moment, I understood the pathway from slavery to freedom.” Bailey later escaped and changed his name to Frederick Douglass.
And here is the great truth of true education: it is the path to freedom. Learning unshackles the mind and opens the world. In the words of the slaveholder, learning “unfits” us to be slaves.
The hard question is, then, when do we see this kind of learning happen? Different lessons are taught in schools. For some, there is the elation and struggle of real learning; in others, the primary lesson is to obey, and schools shackle the mind rather than free it. Jean Anyon’s groundbreaking study of what is taught to kids of different social classes in schools should be a wake-up call to anyone paying attention. Ditto with Jonathan Kozol’s entire body of work, especially his bitter The Shame of a Nation (I haven’t read it, but I heard him speak about it when he was in Seattle some years ago), which underlines the shocking fact that schools are more segregated now than they were fifty years ago, when Kozol started writing.
There is no doubt in my mind about the power of education to lead us to freedom. It is the path to freedom. Paradoxically, we may sometimes resist this freedom, but real learning is too potent and too seductive not to win out in the end. As Douglass wrote about his experience of teaching other slaves, “Their minds have been starved… they had been shut up in mental darkness. I taught them, because it was the delight of my soul.”
The delight of my soul. That’s what teaching can feel like. That’s what learning does feel like.
(By the way, if you want to see a brilliant moment in acting and direction, take a look at Daniel Day-Lewis as Lincoln again, and note the pause around 20 seconds in… “In his book… mmmm”. What’s happening in the mind of the character? He’s connecting Euclid’s “self-evident” first axiom, that two things that are equal to the same thing are equal to each other, with the Declaration of Independence’s opening salvo, “We hold these truths to be self-evident, that all men are created equal.” A great moment.)
The title of this blog post is the last line to a beautiful, short film called GÖMBÖC. For a film where almost nothing happens, it’s compelling watching. And wonderfully, it captures the simple, profound, mathematical joy of thinking really, really hard about a beautiful problem.