Nuggets III: Problem Design

I have to make this one quick because I have a cold and am trying to pack up my life and move uptown. But I wanted to finish my little “Nuggets” series with a thought inspired by Catherine Twomey-Fosnot and Maarten Dolk’s awesome Young Mathematicians at Work books. Twomey-Fosnot runs Math in the City, a math education think tank and professional development center run out of City College. She is writing a new book about algebra with the research mathematician Bill Jacob. I’m excited.

Anyway, people often talk like it’s a choice between developing students’ understanding of concepts and developing their technical ability. My experience is that everybody agrees that the two goals support each other, but there are major differences when push comes to shove in terms of where people believe the emphasis belongs. Maybe you all saw everything clearly from the jump, but I feel I spent a lot of years locked in this dichotomous framework. (Partly because of some experience with curricula like IMP that do a very good job with one of the goals and not the other.) What Young Mathematicians at Work did for me was to abolish the framework.

Nugget: You can develop conceptual understanding and technical ability (for example, computational ability) with the exact same lesson. The secret is to embed the technical instruction in the design of the problems you assign.

It’s necessary to take great care in designing the problems so that they support the development of skills over time. According to Fosnot (and I’ll take her word for it), very few American curricula have given adequate care to the sequence of problems and how it supports this development. My own consciousness was definitely raised by reading Young Mathematicians at Work about the extent to which (for example) the choice of the numbers matters.

For example, consider the following pair of questions:

At Sweet Virgo Desserts, a small chocolate cake costs $7.00. An apple dumpling costs $3.50.

1) How many chocolate cakes can you get for $49?

2) How many apple dumplings can you get for $49?

In 2007, a few months after reading Young Mathematicians at Work, I gave this pair of problems to a classroom of very-weak-skilled 6th graders, who would have balked at #2 (“you want me to divide by a decimal without a calculator?!”) if it had come first. They answered it easily and without any help after being asked and answering #1 first.

The two problems are formally identical. The only difference is the numbers. The important thing is not that #2 is harder; it’s that the way the numbers are chosen makes #1 a hint for #2. It’s also a hint with an applicability far beyond this problem: if n is hard to divide by, would 2n be easier? Pretty soon, the same class was using the technique to solve straight division problems accurately in their heads. (I’ve unfortunately lost the followup worksheet so I can’t tell you what problems; but they were things like 60 / 7.5 and 15 / 1.25.) This is a piece of computational technique; and teaching it this way supported the development of the students’ conceptual understanding of division at the same time that their proficiency with certain division computations was improving. The goals don’t have to be addressed separately.

Maybe you all think this is obvious. But I’m still constantly hearing folks (most recently, a college professor, a former high school principal, and the parent of a mathematically precocious 7-year-old) say things like “but at some point, they just have to memorize those times tables.” Meaning, “all this talk about understanding is really wonderful but you have to admit that there are some things you just have to bang into your head.” I used to be plagued by doubts of this form. Now I’m not. Yeah, you have to learn the times tables, but there’s never a reason to bang something into your head. Can’t remember 6×7? Great, do you know 6×6? How are they related? You go thru that a few times and not only will you remember 6×7 but you’ll be building the groundwork so that later it’ll seem intuitive that 6(x+1) = 6x+6.

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Nuggets II addendum: more problems…

This is a followup to my last post. I promised some more problems in which there is an initial pattern that’s wrong. Here is one more. It is not nearly as good as the points-on-a-circle problem I discussed before, for reasons I’ll say below. But I’m brainstorming here, and hope you’ll join me, so anything’s better than nothing. (And actually I think it’s a cool problem in its own way.) Thanks Kate, jd2718, and Gilbert for contributing ideas so far.

The problem involves seeking a simple formula that produces prime numbers only. As you probably know, in spite of centuries of research no such formula is known to this day. There is some fun history around this. For example, Fermat believed that 2^k + 1 was prime whenever k was a power of 2. It is prime for k=1, 2, 4, 8 and 16. However for k=32, the number is 4,294,967,297 which was found by Euler to be equal to 641 * 6,700,417. Now, even in the age of computers, no other prime of the form 2^k + 1 has yet been found. Of course, I’d avoid putting kids in the position of having to calculate 4,294,967,297 or to show that it’s not prime.

Anyway, the idea is to get a class engaged in a search for such a formula. My idea for how the lesson goes would be to try out some examples with them to show them what is being sought. Like, maybe 4n + 3 which equals 3, 7, 11 for n = 0, 1, 2 but then fails for n = 3, or, starting with p_1 = 2 and then recursively doubling and adding 1 which gives you 2, 5, 11, 23, 47 and then the next one fails. (A closed form for this last one would be
3*2^n – 1 for n = 0, 1, 2, 3, 4.) This second formula is a good replacement for Fermat’s conjecture, because it gives you 5 primes before it fails, just like Fermat’s conjecture, but the primes are a reasonable size and the one that fails (95) is obviously not prime. Anyway, once they understand what’s being sought, the problem is to find such a formula. They will totally fail and they have no tools that will help them, so don’t let them stew too long. Then, show them a very pretty creation of Euler’s:
n^2 – n + 41. This quadratic polynomial is (amazingly) prime for n=0, 1, 2, …, 40. So there’ll be some initial excitement as this one seems to answer the question. But actually, it can’t possibly answer it. And the class may be able to see that the n=41 case will fail without actually doing the calculation. Even if the calculation is needed, it can still lead to a cool conversation.

Now this isn’t as rich as the points-on-a-circle problem because the inordinate primality of n^2-n+41 is sort of a mathematical accident; there isn’t a rich story behind it (at least not one I’ve ever heard), so once the pattern is noticed and then broken there’s nowhere to go. But it does at least give students some experience of the fact that if a rule holds for small cases, it doesn’t mean it always holds. And the breakdown at n=41 is accessible to reasoning alone, without calculating. So it’s a win for the power of mathematical reasoning over raw pattern-noticing.

Other ideas in this vein? (Problems where there is a “obvious” or “apparent” pattern or conclusion that is actually wrong?)

Nuggets II: Proof

Paul Lockhart has a lot to say. He was a research mathematician, and now he teaches kids. His essay A Mathematician’s Lament is a passionate screed against stultifying math education. (Which Lockhart sees as typical math education.) I am not alone among lovers of mathematics in saying that when I read it I experienced many, many moments of “Yesss!!! EXACTLY.” (Well, actually, “worrrd!” I was a teenager in the nineties; don’t laugh at me.) Another math lover with the same reaction was Keith Devlin, NPR’s Math Guy, who devoted his column in the MAA’s website to it in March 2008. This year, the essay was expanded into a very short book and published by Bellevue Literary Press (with a foreword by Devlin). Keith Devlin thinks everyone in math education should read it and I think I do too.

(This is not a 100% endorsement of Lockhart’s whole point of view. There were several moments in the book I found dismissive of the hard work that thousands of teachers are putting in across the country to try to teach math in a passionate way inside the constraints of traditional schooling. But the man is angry, and with good reason, so there you go.)

Anyway, amid all the things Lockhart talks about, one jumped out at me and clicked into place in my mind.

Nugget: The impulse toward rigorous proof comes about when your intuition fails you. If your intuition is never given a chance to fail you, it’s hard to see the point of proof.

From A Mathematician’s Lament (p. 72 of the book version):

“Rigorous formal proof only becomes important when there is a crisis – when you discover that your imaginary objects behave in a counterintuitive way; when there is a paradox of some kind.”

THIS IS SO TRUE. And it’s IMPORTANT. I think the issue wouldn’t be so pressing if mathematical argumentation were more intrinsically part of every math class, at every level, going back to little kids, as it should be. But in the absence of this, what happens is that when kids reach a point in their mathematical education where they are asked to prove things, they find
a) that they have no idea how to accomplish what is being asked of them, and
b) that they don’t really get why they’re being asked to do it in the first place.
The way out of this is to give them a crisis. We need to give them problems where the obvious pattern is not the real pattern. What you see is not the whole story! Then, there is a reason to prove something.

The way it typically goes is that all of a sudden in high school geometry, you’re being asked to prove something that’s just as visually obvious as the given information. Clearly it’s a pointless question. Meanwhile, you can’t do it, because everything true about the diagram seems equally true to you. The art of proof is about taking things that you’re sure of and touching them, prodding them, turning them around, pushing them against each other, until you become sure of more things, and continuing like so until you become convinced of the thing to be proved. To do it you need to feel it. When you do it well, your gut is giving you a reading on what you’re sure of and what you’re not sure of. (With students and teachers I work with, I’ve taken to calling this gut readout “your internal compass for rigor.”) But in a typical geometry proof, the thing to be proved feels just as true as the givens. You’re being told you know one and not the other, but this is not a difference you can feel. There is no internal compass for rigor to guide your path.

This is a situation exacerbated by everything that’s happened before geometry class. Every time you noticed a pattern, it was the right pattern, unless it was demonstrably wrong. For example, maybe you got a chance to experience the following awesomeness:
1+3 = 4
1+3+5 = 9
1+3+5+7 = 16
1+3+5+7+9 = 25
BLAM! The sums of consecutive odds are squares!
Now this observation is very cool when you first notice it. But where is this going? A lot of the time, it stops at the observation. The teacher gives the class a chance to see a pattern, they see it, that’s cool for 1.5 seconds, and then that’s it. Everybody moves on. The kids in such a class are being trained not to understand the need for proof. Even the teacher is acting like seeing is believing, so why, when you later get to geometry, are you suddenly being asked to “prove” things you can obviously see?

It’s much better if after the initial “aha” moment, there’s some sort of quest for an explanation. Lockhart describes such a quest, for an explanation of this exact pattern, in the book version of his essay, on pp. 106-117. But something is still missing if the only kinds of experience the students have fit this (see a pattern) – (explain it) – (see another pattern) – (explain it) cycle. The students are never getting a chance to see the wrong pattern.

I visited Paul Lockhart’s class at St. Ann’s School, where he teaches, a few weeks ago. At one point he said to his class (I’m paraphrasing because I don’t remember exactly) – “One thing that will happen this year is that your intuition will suggest something is true, and then you’ll look for a way to establish its truth, and find it, and refine it into a solid argument. That’ll be a good exercize. But far better for your mathematical development will be when your intuition will suggest something to you, and it’s wrong. You’re dead wrong. And then you see that you need a richer understanding of what’s going on.” So, a propos of this, I’m making a case that we give our kids lots of chances to have their intuition be wrong. The earlier the better. Nothing will develop the internal compass for rigor more powerfully.

Sensing a danger of being misunderstood, let me get concrete. I’m not talking about doing anything to undermine students’ trust in their reasoning. Cultivating a student’s trust in her own reasoning is what I believe math education is most centrally about. This is not about telling students they’re wrong, it’s about giving them a crisis. I’m talking about giving them problems that suggest one pattern on the surface when really something else is going on. In this way, students’ own reasoning is what puts their intuition in check.

I’m about to tell you the best problem I know like this. Tomorrow I’ll put up some more (though they’re not as good.) I’m hoping that some of you will add to the list of problems. They’re of vital importance and, though they’re easy to come by at the level of active mathematical research, I’ve encountered very few at the K-12 level. We need a repository!

The best one I know I learned from Bob and Ellen Kaplan’s book Out of the Labyrinth, which I wrote about last week.

Take a circle. Put 2 points on the circumference and connect them with a line. Into how many regions is the circle divided? Two.

Now add a 3rd point on the circumference and connect it with lines to the other two. How many regions now? Four: the points make a triangle so the interior of the triangle is one region and another one between the circle and each edge.

Add a 4th point and connect it with the other three. How many regions? Eight. Count ’em.

A 5th point? Sixteen. If you’ve never seen this problem, you should be drawing right now because you don’t want to miss the full glory of this.

Alright, I see where you’re going, you say. Does it fit the case with only one point? Oh yeah, that’s just one region. Pretty neat, but what’s the big deal?

Draw the 6th point. Connect it to the others and count the regions. How many? Thirty-tw… Thirty-ONE? What? Did I count wrong?

No, you didn’t. That power of two thing you saw, a bulls-eye for the first five cases, is a miss on the sixth. (And a tantalizingly near miss, at that.) What’s really going on here?

Now if you want to look it up, it’s discussed in Out of the Labyrinth, pp. 71-74. But I recommend, if you’ve never seen this problem before, that you try it out yourself. What’s the maximum number of regions you will get with 7 points? With n points? And why?

The beauty of this problem is that the wrongness of the initial “obvious” pattern gives the search for the truth much more urgency. And, more importantly for the present conversation, it gives the student a reason to care about proof. I can stand up here and say “you’ve given me the first five cases, but you haven’t proved it” till I turn blue, but if you’ve never seen something work five times and then fail later, there’s some level on which you don’t believe me.

So this is what I’m advocating: let’s give students problems where there’s a superficial pattern that’s not the real deal. The need for mathematical argumentation is going to spring from these problems like corn from the Iowa soil. (Forgive the corny metaphor; I’m just excited.)

And folks: what other problems like this do you know?

Nuggets I

The following is the first of what I hope will be three posts about thought-provoking books about math teaching, each accompanied by a treasured insight I got out of reading it. All three books (Bob and Ellen Kaplan’s Out of the Labyrinth; Paul Lockhart’s A Mathematician’s Lament; Catherine Twomey Fosnot and Maarten Dolk’s Young Mathematicians at Work) have a lot going on. If you read (or have read) any of them, I’m confident you’ll have (or have had) all kinds of thoughts totally unrelated to the ones I’m about to share. But the thoughts I am highlighting have all felt very exciting to me and that’s why I’m sharing them. It was like something clicked into place and answered a question that had been loitering inchoately in my mind.

Out of the Labyrinth: Setting Mathematics Free
by Robert and Ellen Kaplan

I’ve written about my appreciation for this book elsewhere. So without further ado –

Nugget: Mathematics is a vital interplay between the general/abstract and the specific/concrete. Without generality and abstraction, mathematics lacks power and grandeur. But without specifics, mathematics lacks life.

From Out of the Labyrinth:
“The spirits of Hilbert and Ramanujan lean over our efforts: the one ever lifting us up toward the form of the whole, the other dipping down again and again to catch at the invigorating singular…. This stirred soup is a spiral nebula, exceptional in each of its stars.” (p.157)

(To explicate the cultural reference: Ramanujan and Hilbert were both early twentieth century mathematicians. They represent opposite poles of the spectrum from general to particular. Hilbert was the grand theorist, among other projects attempting to find a formal system that could unify the entire edifice of mathematics. Ramanujan was a delighter in the details. In a famous anecdote, when his friend G. H. Hardy remarked that he had arrived in a taxi whose number was 1729 and that this seemed a dull number, Ramanujan exclaimed, “No Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”)

Everybody talks about the need to make math concrete for students, but what clicked into place for me when I read Out of the Labyrinth is that mathematicians at every level have literally the same need. What counts as concrete is different at different levels but this is the only difference. A problem is not compelling if you can’t feel it, if you can’t turn it and prod it in your mind. It’s not that at some level of mathematical development you cease to need the problems to be real. It’s that as you deal with and get to know a mathematical object, at any level of abstraction, through its interactions with things that are already real for you, it becomes real. Then questions about it can become interesting.

Students learning algebra for the first time often experience equations containing variables as a sort of hazy, insubstantial thing. The methods for manipulating the equations seem arbitrary, and there is nothing satisfying about solving them because they barely seem to exist in the first place. I had the good fortune to encounter variables for the very first time in a context that made them extremely real, so I never had this experience. But I remember that haziness, that alienating lack of substance, from my first encounter with the definition of a group. “What is this pointless object that kicks all the substance out of the rich terrain of Number and leaves a hollow shell?” I remember thinking.

This is funny to me now since my subsequent mathematical education has made groups just as concrete as numbers. I now experience the symmetric groups on 3 and 4 elements as familiar little toys. The point is that this was a process. Groups did not begin to feel real to me until I saw how they explained and unified things I could already touch: numbers, the symmetry of shapes, motions in space. A long, slow process of solving problems, cultivating techniques, and most importantly investigating interactions and connections between things I already understood, made concrete objects out of what had first to me literally felt like thin air.

The philosophical lesson I draw from this is that a powerful way to look at teaching mathematics is that it is a process of cultivating this sense of concreteness in more and more abstract objects. The process begins in the concrete details of a problem – that’s what gives it life. Then it pulls upward, beginning to perceive the ethereal layer above the details that tie them together with other details. Then you try to hold both layers at the same time, and perceive their relation, until the concreteness of the details seep into the ethereal layer and it gains fleshy substance. Then you can move higher still. (First number, then variable, then function, then operator, for example, over the course of a decade.)

More concretely, a math lesson has to start from a problem that feels real and tangible for the students; but this doesn’t have to mean “real-life”. It just means that they need to be able to hold the problem, touch it, manipulate it in the mind. This depends on their mathematical development. If it’s a calculus class and the precalculus class did its job, the graph of y=sin x is a real, tangible thing. So a provocative question about the graph of y = sin x is more compelling than a routine question about a car’s speed.

Conversely, a problem that does not feel concrete lacks life, and concreteness grows slowly. We should beware of problems which require us to teach something before the problem can be posed. The lesson must begin with an interesting question about a tangible reality. The reality can be pysical, social, or purely mathematical. It just has to be real.

Required Reading for Math Teachers II

I’m excited and grateful about the positive response to Required Reading I. Therefore I’m a tiny bit trepidatious about my followup since it’s probably going to be a little more controversial. Be that as it may, I think it’s really, vitally important, so here goes:

Praise for Intelligence

Carol Dweck is a developmental psychologist who has made a career of studying how people’s beliefs about their traits influence their performance. She’s written a lot of good stuff but I want to call your attention to this article published in American Educator in 1999. The article summarizes research conducted by Dweck and others, most critically a study she and Claudia Mueller published in 1998 in the Journal of Personality and Social Psychology, 75 33-52 entitled “Intelligence praise can undermine motivation and performance.” I could not find the full text of the study online but the American Educator article summarizes the methodology and results.

Take-home lesson: Praise kids for what they have control over. Do not praise them for what they do not have control over. In particular, do not communicate to them that they’re smart, gifted, talented, intelligent, or the like, when they do something easily.

Telling kids they’re smart or gifted when they do something easily communicates to them that people will stop thinking they’re smart if they ever break a sweat. They become fundamentally afraid of struggle. This inhibits them from growing.

In the research Dweck presents, she and her colleagues took fifth graders and divided them into three experimental groups. All the students were given a set of puzzles that was designed to be “challenging but easy enough for all of them to do quite well.” Afterward, children in the three groups were told the following things:
Group 1: “Wow, you got x number correct. That’s a really good score. You must be smart at this.”
Group 2: “Wow, you got x number correct. That’s a really good score. You must have worked hard.”
Group 3: “Wow, you got x number correct. That’s a really good score.”

Afterward, the students were asked questions: how did they like the task? Would they like to take the problems home to practice? How smart did they feel? The three groups responded similarly to each other.

Next, all three groups were given a harder set of problems, on which they didn’t do as well. Then they were asked again how they liked it, would they like to take home the problems, and how smart did they feel? Lo and behold, the students who had been told “you must be smart at this” now did not feel smart at all, did not enjoy the task, and did not want to take the problems home. The students who had been told “you must have worked hard,” in contrast, enjoyed the task as much or more than the easier one. The third group had results between the other two.

Finally, the three groups were given a third set of problems similar in difficulty to the original set. The “you must be smart” group did the worst, and significantly worse than they had done on the original set. The “you must have worked hard” group did the best, and significantly better than they had done on the original set.

I think these results speak for themselves. If you think you are smart because you succeed with ease, you have a devastating and totally unavoidable conclusion to draw the minute you do not succeed with ease: you are not actually smart. There is nothing left to do but to desperately hide your struggle and hope no one finds you out. This is rough on your performance as well as your psyche. Meanwhile, if you find that your effort, your diligence, better yet your perseverance in the face of setbacks are what get the teacher excited about you, then a real challenge (meaning something you’re doubtful you can do) is actually an opportunity to enact your value. Dweck and Mueller also did some more experiments, fleshing out these details, that I won’t go into – the American Educator article (linked to above) is worth your read. But I have one thing to add. (Actually I have an unbelievably huge amount to add, but I had to limit it somehow.)

I take this research result to have very strong implications about how we should talk with kids about their performance. However, I take it to have no implications at all about what kinds of work they should be doing. For example, I think it challenges us to be very careful around the idea of “giftedness.” I would not mind if the whole idea of “mathematical giftedness” was given a rest for a good long while. But that doesn’t mean I don’t think children who love math shouldn’t get the opportunity to explore this love.

For example, if you are the parent or teacher of a kid who loves math and is breezing through her schoolwork, please please please give her opportunities to study mathematics that she finds exciting and challenging. But don’t tell her this is because she’s “gifted.” That puts her in Dweck’s experimental group 1 (“you must be smart at this”) and so sets her up to become freaked out and alienated from mathematics when the going gets rough. Tell her it’s because she’s excited by it. Praise her not when she flies through something easily but when she sticks with a problem past the point where she was ready to give up. Highlight her own sense of accomplishment when she does something that was really hard for her – “I was so proud when you did that because your patience paid off. I bet you’re proud too, huh?” Acknowledge resourcefulness – “I like how you looked for a new way to see it when your old way wasn’t working.” And show her that you value her enjoyment of the experience of doing math. When she first sees a pattern, she will be excited about it. When she figures out how to solve a new kind of problem, she will feel powerful. When she first sees a connection between disparate-looking objects, she will be in awe. These are the experiences that motivate a lifelong relationship to mathematics, so when she has them, let her know you value that. “Tell me what it was like when you saw that pattern.” If her enjoyment of math is bound up with being “gifted” it is fragile; so train her to enjoy math for its own sake.