I’ve been collaborating with the English-language newspaper in India called The Hindu for the past few months, producing puzzles for a column called A Mathematician at Play.
It’s a great collaboration. I produce the puzzles, and they make them look beautiful.
I’m going to start sharing these puzzles on the Math for Love blog. Some of these puzzles are classics, others are original. All of them involve some kind of thinking or insight that strikes me as pretty, or surprising, or delightful.
Puzzles will go out Mondays for the next few months. Answers will go out Fridays.
The first puzzle, below, is a series of grid puzzles. I hope you have fun trying these out! Share them with your kids or students, or work on them with your friends, and let me know what you think in the comments!
I’ve just witnessed one of the best back-and-forths ever on Twitter, and I had to share it. It’s a phenomenal example of the pileup that allows great teachers to push each other and discover more was possible in their lesson ideas and plans than they had thought.
This is why you should join the Math Twitter Blog o Sphere (#MTBoS, #iteachmath)
The promise of mathematics is that it will teach students to think.
Sadly, there’s been no solid evidence that math class actually succeeds, statistically speaking, in achieving this end. A pessimist would conclude it’s a hopeless project. But far more likely is that math classes, statistically speaking, haven’t been spending much time doing tasks that lead students to think, or learn to think.
The researcher Peter Cathcart Wason came up with a series of games and tests in the sixties to explore common failures in human reasoning. The first of these was the 2, 4, 6 Puzzle.
Happily, the 2, 4, 6 Puzzle is a delightful and devious way to play with kids (and adults!) in your life, be they students or children. Our writeup of the lesson is here.
The brilliance of the game is that it demonstrates to the player how staying in their comfort zone, avoiding “errors,” and sticking with safe guesses prevents them from solving the puzzle. It teaches inductive reasoning, and why avoiding confirmation bias is important.
In other words, it teachers students how to think.
One encourages readers to commit poems to memory, and recite them to each other; the other encourages us to learn the names of the trees that surround us, and start paying attention to their active, wild lives.
Two things jump out at me as I read these articles. The first is that memorization, so often maligned in educational circles, has a place in learning. It serves a function: to focus attention, to help us notice the invisible. There is no question that the role memorization plays in education can be overstated and overplayed. In math education especially, entire subjects are reduced to incoherent laundry lists of isolated, irrelevant facts that we’re made to memorize for some reason. Still, I worry that saying memorization serves no role at all is to remove a meaningful learning tool.
The second thing I notice is my own state in reading these articles. My mind quiets and fills as I think of the poetry I know, and want to know. I want to organize a party where everyone shows up and recites a poem to the group. Even though my own knowledge of botany is laughably poor, the invitation to spend time with a tree is welcome, and while I may not seek out the opportunity to learn the names of trees, I’d certainly be happy to if the opportunity arose.
Memorization is one of the great bugaboos of mathematics learning. As someone who usually didn’t have a great memory—acquiring foreign languages, remembering biological and chemical structures, and so on was always tricky for me—math was always appealing because it didn’t require much memorization. Or if it did, it was a kind of memorization that fit my love of stories. I can remember lines in a play if I understand what the characters want, and what motivates them; in this case, their words make sense. Math is like this for me too: the unexplained pattern cries out for explanation, and the facts and formulas are short hands for the arguments that make the structure sensible. On the other hand, memorizing all the rivers in Europe was an empty exercise when I had to do that in seventh grade. Those river names are all gone from memory now. I bet if we had learned the stories of history at the same time, I would have been more interested, though; I still remember the Arno, in Italy, because in Arieti’s historical novel The Parnas the characters have to decide whether or not to flee to the river, which marks the dividing line between the Allied and Axis forces in Pisa. I care, so I remember.
The point of all this is that context matters. Unmotivated memorization is one of the great ways to kill interest. But even in an age where information is everywhere, I maintain that memorization serves a function in learning, as long as it acts as an invitation to see something new.
For example, I find being able to know and be fluent with some basic arithmetic helps you see new ideas and patterns in mathematics, just like knowing the names of trees helps you see the natural world alive around you. I once led a class where we interested in the areas of tilted squares on a grid.
What is the area of this square?
There’s no simple formula to calculate the area of this square (well, there is, in a way, but my students were young enough not to know it). And memorizing formulas to calculate the areas of geometric shapes is usually a waste of time and energy. “Memorize arguments, not formulas,” as a professor of mine used to say. Which is to underline the point that most – probably 95% – of the memorization we ask students to do in math class is a waste of time, and counterproductive.
But there still is that 5% that’s worth it. And in this case, after we’d spent several days finding the areas of squares and organizing our lists in various ways to make sure we had them all, we had discovered this list of numbers, representing the areas of tilted squares:
What in the world is going on here? This is the kind of pattern that can leave us bewildered unless we actually have enough earlier patterns memorized to see structure in the noise. And this is precisely the kind of pattern that emerges in math sometimes, where there’s a deep, strange connection that reaches in from a completely different section of mathematics.
On the one hand, who cares. But on the other hand, exploring the interplay of patterns in the sequence above leads to a truly tremendous breakthrough known as quadraticreciprocity, which in its own right inspires vast new explorations in mathematical thought. But unless you have some previous patterns and arithmetic in your working memory, finding this kind of pattern is pretty much out of reach. This may seem like a peculiar example, but I think this is common in math: instant recall of certain objects, patterns, and algorithms helps us to see what can be invisible otherwise.
Forcing students to memorize something is like making them put something in their backpack before you hike up a mountain. It works best if they want to climb the mountain, and believe this tool in their backpack is going to be useful. And some tools can be picked up on the way! However, if they have to stop and go back to grab a missing tool every time they take a step, they never experience the beauty of the hike. That, to me, is the experience of the student who has to pull out a calculator when they are subtracting one and two digit numbers; they are constantly distracted from the beautiful experience of finding meaningful patterns.
So memorization has a place, but we have to be very selective. The moment the preparation goes too long, the experience becomes about being loaded up with a heavier and heavier backpack, filled with junk we don’t need, to climb a mountain we don’t want to go up anyway. Confusing memorization with learning is a huge error. Recognizing that memorizing a small body of selective knowledge at the appropriate time in order to aid learning and reveal beauty in the mathematical (or real) world is something else entirely.
Confusing memorization with learning is a huge error.
Still, I get a little nervous defending memorization at all. I recently had a back and forth on twitter with Eugenia Cheng, discussing if there’s any merit to memorizing in math at all. The pendulum has historically been so far in the direction of memorizing that the impulse to push it back is strong. If we had to memorize poems we didn’t care about (or that were in different languages we didn’t understand) for 12 years, we’d quickly come to say that memorizing poetry was pointless too. And yet, I can’t shake the sense that we are in danger of going too far; that teachers see the value in the selective use of memorization for students, and are afraid that it’s so strongly counter-indicated by experts that they should never do it.
So even as I emphasize the journey up the mountain, and all the reasons for taking it, I’ll keep reassuring teachers that they are allowed to help students memorize a select body of terms, ideas, and algorithms that will make the journey richer for them.
But keeping your eyes on the real point of learning – that’s wanting to take the trip – is always primary. As a sign off on that note, here’s a quote from Paul Lockhart’s new book Arithmetic: “There are a lot of people who hate arithmetic (far too many to count!), and it makes me sad. Usually it’s because they were made to do something they weren’t interested in doing. Let’s not have that be you.”
One thing I realized in looking this post over is that I didn’t really define what memorization means to me. I tend to find context-less memorization (mnemonic devices and the like) useless or worse than useless. For me, useful memorization is the act of going over facts and ideas that you understand to keep the details fresh, and which might slip away from you otherwise. It’s memorizing lines in a play where you have a sense of the motivations of the characters, as opposed to the useless memorizing lines in a language you don’t understand. Sometimes definitions and vocabulary need to be memorized, but I usually like those to come from a genuine need to name something as well. So I guess there’s much more I need to say on this topic to articulate my own thinking.
Ben Orlin has some nice articles on the evils and value of memorizing in learning. (Interestingly, I don’t agree entirely with his “memorize on Monday” idea, but both articles are still definitely worth reading.)
In the run up to the upcoming year of Math Teacher Circles, we decided to offer a kickoff workshop for 4th – 8th grade teachers to drum up some excitement for the upcoming year. (The “we” in this case is myself, from Math for Love, and Jayadev Athreya of the Washington Experimental Math Lab, run through the UW.) This partnership (M4L + WXML) led a series of eight Math Teacher Circles last year, for K – 8th grade teachers, and they were great. The model is one of my favorites: teachers opt in to meet once a month to study topics in math and pedagogy, then try things out in their classroom between sessions. The circles are a fantastic way to meet and collaborate with colleagues from other schools, focus on fascinating and critical issues in math teaching, and energize your practice. They’re also free! Sign up now for elementary or middle school math teacher circles.
This year, we’re hoping to put together resources that will allow others to take this model and adapt it around the state/country/world. To that end, I wanted to summarize the last week’s kickoff workshop, especially because it was a remarkably fun workshop to be a part of.
Part 1: Introduction and Motivating Examples
We had about 35 teachers in attendance, teaching math for 4th – 8th graders, plus a few mathematicians. The workshop began with conversations among groups about people’s math autobiographies. What were the most memorable experiences in people’s mathematical histories, positive and negative?
Some of the big ideas that came from these discussions:
Math is emotionally charged. We feel a lot – pride, shame, satisfaction, etc. – in the course of doing math.
One teacher can have a huge impact on how someone views themselves mathematically.
Those moments where the math is meaningful creates a sense of empowerment; alternatively, those moments when math is empty and we’re working without understanding undermines that sense of personal power.
After the discussion we showed Robert Kaplinsky’s mind-blowing How Old Is the Shepherd video. This video always baffles me, especially because I suspect that younger students wouldn’t respond the way these 8th graders (they’re eighth graders!) do.
There’s an idea that certain institutions create a mirror image of the very outcomes they’re trying to prevent: there is a certain kind of sickness that exists only in hospitals, and a certain kind of ignorance created only in schools. We need to take this seriously. If we’re not careful, we estrange students from their own sense of knowledge, and they end up not being able to make the obvious observation: that something doesn’t make sense.
Part 2: Conjectures and Counterexamples
I’ve been developing a structure to help students to help invite students into a genuine mathematical process, starting with their own understanding. We call it Making and Breaking Conjectures. In the next section, we talked about this structure.
First, what are conjectures? What are counterexamples?
Conjecture. A mathematical hypothesis. A guess of the underlying structure or pattern based on what we know so far.
Counterexample. An example that proves a conjecture false.
Mathematics as a field progresses by way to conjectures and counterexamples. The good news is, we can use them even with very young kids.
We played the game Counterexamples to get a sense of how this works. (Lesson plan PDF here. Online here.) The game is super-simple: the teacher makes a false conjecture, and the students prove it false with a counterexample.
Teacher: All pets have four legs.
Students: No! Because birds have two legs!
Teacher: Okay – refined conjecture: all pets have two or four legs.
Students: What about a snake?
Teacher: That’s a pet with no legs. So I’ll refine my conjecture again. All pets have at most four legs.
Students: What about a spider?
And so on. So compelling is the game that we almost got off track when we considered conjectures about area and perimeter. But there was work on conjectures to be done first.
To warm up our observing/noticing/wondering/conjecturing muscles, we started with a paperclip.
This is a classic exercise, and Kindergartners tend to beat the pants off of adults. After spending ten minutes or so, groups had come up with everywhere from 20 to 65 uses for a paperclip. That was just the warmup, though, so we didn’t go too deep into what the uses actually were; the main event was still on its way.
Considering this question was trickier, and required a little more discussion to draw out observations and questions. By the time the conversation was done, however, we had a tidy collection of questions to consider.
How many rectangles can be formed by connecting dots with straight lines? (Refinement: diagonal lines vs only horizontal and vertical)
How can you connect all the dots with connected straight lines drawn in a continuous path? How many lines does it take?
How many sides could a polygon have on the grid? Could it be all 16 dots?
Can you make a square with any number of dots on its perimeter?
What grid came before? What grid comes after?
Maze: how many different paths are there from the bottom left to top right? Refinement: direct route or can you loop back?
If you can define “1” differently (length or area), what it the area of the whole grid?
How many different angles can you form if you connect three points?
How many different areas can you get if you form a triangle with three points on the grid?
How many lines of symmetry can you make? What if the lines don’t have to be on the dots?
How many symmetric shapes can you make with corners on the dots?
How many lines would there be if you connected every dot to every other dot?
Can you find two triangles with the same area that are not similar?
How many different lengths could you find by connecting two dots?
Can you make a triangle with area 1/3? (If the whole is 1)
How many fractions could you represent on the grid?
Take that list in for a minute. There are weeks of beautiful, high level problems to explore here. For each, you can start by casting around. Soon, you’ll find you have conjectures. Once you have a conjecture, you can try to break it right away by looking for a counterexample. Refine and repeat until you end up with something that seems to be true, and then you can put together an argument, with luck and a little insight, into why it actually might be true. We’re really doing mathematics!
With all these questions to consider, we wanted to provide some guidelines. So we posed a choice: teachers could choose the problem that inspired them most and work on that one. Or they could work on the question of how many different areas a triangle could have if its corners are on the grid. (Having a default question is something we’ve found can help prevent groups from becoming aimless. With an actual classroom, you might want to skip some of these steps, and start from a more tightly focused, teacher-chosen task in the first place. Still, opening up the entire thinking process can definitely be worth it, if you’re ready to take the step.)
And with that, we sent the groups out to work on their own. However, we had one more idea ready at hand to ensure things went as well as they could go.
Aside: Thinking Classrooms
Peter Liljedahl has been developing a series of concrete steps to change the classroom to support genuine thinking in mathematics. The first, central step is to use tasks, that is, make real, meaningful mathematical experiences the heart of what you ask students to do. But you can help students actually transition, and the steps Liljedahl has determined to encourage this transition are bizarrely simple to employ. Here’s how you begin, according to Liljedahl:
Use vertical, erasable surfaces (i.e., whiteboards)
Assign groups in a visibly random fashion
It seems almost too easy, but the room was electric with thinking energy. Here’s what it looked like.
It’s hard to overstate how getting everyone standing and working together impacts the quality or thinking and engagement in the room. One of Liljedahl’s recommendations is to use just one marker per group. You can see the value of that suggestion in the video below, as the carrier of the marker is drawn back up to the board to write in details of someone else’s computation.
Don’t have enough whiteboards? You can write on windows too!
With so many teachers considering such a range of problems, we decided to let groups pair up to share what they had discovered with each other.
All in all, we were totally energized by the workshop, and I think the teachers were too. They’ve got a whole bunch of great questions to pose about grids, and the motivation to use task, whiteboards, and visibly random groups in their own classrooms.
And we’ve got a goal for this year’s math teacher circles, especially for the middle school level: create a series of PD sessions that will help everyone turn their classroom into a Thinking Classroom, where students use Conjectures and Counterexamples to power genuine mathematical experiences.
You can sign up for our 2017 – 2018 Math Teacher Circles now:
There was one more takehome point for the teachers: October will be the launch of the Global Mathematics Project! We shared another grid-related resource from the GMP.
Not only is this kind of problem a super-fun way to launch an exploration with students, it’s also grid-related! And it’s also just a taste of the kind of math that’s going to be highlighted during the Global Math Project in October, 2017. We’re looking to give 1,000,000 a genuine math experience the week of October 10. Interested? You can learn more and sign up here.
So, a long post, but hopefully just a beginning of what will be a framework to help others lead their own Math Teacher Circles, and help making authentic and beautiful mathematical experiences the center of math class.
I’m hosting a math night for parents and students tonight at a local elementary school tonight. These are always fun events, because
I get to preach how games and puzzles are some of the best ways to support math understanding at home
We get to play said games and puzzles and have a great time.
See the bottom of this post, where you can see my handout of puzzles and games to play at home, and other resources to check out.
In addition to my other favorites, I think I’m going to share the “Polka Loop Puzzle.” This is a classic puzzle we included as one of the puzzles to play in Tiny Polka Dot.
Last weekend I took this puzzle, among others, to the National Math Festival in DC. It was, not surprisingly, a blast. My only sadness is I was so busy sharing the puzzles and games I brought that I didn’t get to see all the other spectacular presentations.
I got to spend a lot of time with the folks from ThinkFun, from the Julia Robinson Festival nationwide, from Gathering for Gardner, and others, since we were all in the same section of the event. One big theme: the values of games in creating a math-positive culture at home. Everyone basically had their own variation on the sentiment that “What books are to reading, games are to math.” (One variant I was struck with: “What books are to writing, games are to math.”
Students would start by solving the puzzle from 1 to 5, and then add on a 6, then a 7, until they had it solved up to 10. One student really showed me something new when she concocted, and solved, a totally new variation on the puzzle I’d never seen before. Instead of putting one card on the bottom of the deck, you put one card on the bottom for each letter in the number that’s coming next.
Want more math games to play at home? Check out the handout I’m passing out at the Math Games Night tonight. And let me know what great games and puzzles I should add!