Sixth Annual Math Circle Summer Teacher Training Institute

I've been to five out of the six, and this was the best one so far for me personally. One of the changes was to have the participants (who are learning how to run math circles) stick with a group of young people through the week. I facilitated the high school team. We had 8 adult participants, counting me, who ran math circles each afternoon for about 15 mostly high school age kids. Staying with one group of kids was great - we could really see the group of them coalescing over the course of the week.

One thing that was missing was the beauty and skill the Kaplans bring to this event each year. (There was a health problem, and rest needed to recuperate, so they couldn't make it.) I expect them to be back - as good as ever - next year, and hope to once again have the best time ever next July. 


Mornings 
We got to all participate in a big math circle together each morning.

On Monday, Sam Lichtenstein led us in an exploration of how much overhang you can get on a stack of dominoes. I had seen images of this sort of thing before, so I didn't start out hugely excited. But as we worked on it, my interest and excitement grew. I never felt like I had complete clarity on the physics involved, but we did come up with a mathematical solution that satisfied me.


On Tuesday, Nathan got us working on a puzzle that started out with a hydra on our lawn. You probably know that if you cut off a hydra's head, it will grow two heads in place of the one. Our lawn began at the bottom left corner of a chessboard with an infinite number of spots upward and to the right. The hydra was in that first spot, and when we chopped its head off, it grew two new ones - one to its right, and one just above it. We could chop off one of those and again get heads to the right and above that one. The other head was now blocked, though.

We used Othello boards to model the situation, and had lawns as shown. We got those two squares cleared and worked hard to clear the next diagonal. No one had yet done it when we gathered back together to discuss it. We went back and forth enthusiastically between trying to figure out how to do it, and trying to figure out a way to prove that we couldn't do it. Seems to me that's one sign of a good problem. A few of the high school age students attended morning sessions, and one of them wrote up a lovely solution to this problem - very clear and lively, with excellent illustrations.


On Wednesday, Amanda led us in exploring Rational Tangles. Many of us had played with this amazing puzzle/game before, but I think everyone loved having another chance to explore. You need four volunteers, two ropes, and a caller. The dancers start, each holding a rope end, with the ropes parallel. On "turn 'em round", the group rotates 90 degrees, so everyone is in the position where the person to their left had been standing. On "do si do", the person at the back left holds up their rope and comes forward, with the person at the front left going under and to the back. The ropes get all tangled up after a few random calls. And if you know the number secret, you can always get them untangled! The mathematical thought that goes with this is delightfully tangled.


On Thursday, Amanda led us through some interesting number puzzles. A number is onederful if it it composed of all ones in some (integer) base. 3 is onederful because it is represented as 11 in base two. Which numbers are onederful? Along with this exploration, we looked at divisibility rules: Why is a number divisible by 9 when its digits add up to a multiple of 9? What are the divisibility tests for 2, 3, 4, 5, and 6, and why do they work? 7 is the first number that has no really simple divisibility test. We came up with a great test together, by using modular arithmetic.

Suppose we have a number abc (where a, b, and c are the digits). Algebraically, our number = 100a+10b+c. Modular arithmetic is a way to pay attention only to the remainders. 100 is equivalent to 2 mod 7, since 100 = 7*14+2. (This is usually written using a three bar equal sign. I'll use a regular = with a 7 subscript for the rest of this post.) So 100a+10b+c =₇ 2a+3b+c. Let's look at just two digits. 10a+b =₇ 3a+b. If we multiply both sides by two, we change what we're looking at, but we don't change whether or not it's a multiple of 7. So now let's look at 6a+2b =₇ -a+2b = -1*(a-2b). Once again, we can ignore the multiplier if we're just interested in whether or not we have a multiple of 7. One more step: Imagine a number with many digits, where the last digit is b. Use an A to represent the rest, so Ab means 10A+b. The same logic we used above applies, and if A-2b is a multiple of 7, so is the original number. (And if one is not a multiple of 7 the other also is not.)

So our test has us take off the last digit, multiply it by 2, and subtract it from the number formed by the remaining digits. Is 123456789 a multiple of 7? 12345678-18 = 12345660, 1234566 -2*0 = 1234566, 123456-12 = 123444, 12344-8 = 12336, 1233-12 = 1221, 122-2=120, which is not divisible by 7. That means 123456789 is not a multiple of 7. Wild! (Did I do that right? Please correct me if I messed up.)

I'm going to end this too long post here. Maybe later I can describe Friday's circle, and some of the evening entertainments.

Good Problems: Geometric Construction

I enjoy the problems posted on the Five Triangles blog. It says they're appropriate for grades 6 to 8, but they are often hard enough to make me sweat a bit. This particular problem demanded to be solved, and made me sweat more than most.

When I finally got a chance to look it over carefully, my first response was bewilderment. Then I had a few ideas, got out paper and pen, and drew some lines in. I was stuck again, but when I stated my new simpler problem carefully, and drew a new picture for it, I got it. To check, I used geogebra to construct it with just points, lines, and circles.

Playing with these problems makes me want to teach a geometry course. 

Euclidean Construction: Science vs Magic Makes it a Game

I liked high school geometry, but I've never taught a geometry course. And I feel limited when I try to do geometric constructions. Algebraic reasoning and coordinate graphs feel very natural to me. But the idea of constructing a pentagon, for instance, is a bit bewildering. I'm a bit clumsy with a compass and straightedge - I often push the pencil into the device and have to fix it and try again. So a site that lets me play with geometric construction without the mechanical device sounded intriguing.

The first time I heard about it (don't remember who, and don't know how to search my feedly reader the way I could with google reader), I tried it out and just got confused. There are minimal instructions on the site, and I just wasn't getting how to get started. So I went on to something else.

A few days later, Brent at The Math Less Traveled mentioned it, and explained the interface when I asked. I went back and played. There are 40 challenges in groups of 4. I've gotten through 21 so far and am looking forward to getting further. Nico Disseldorp, who created this site, has been working with Javascript for less than 8 months, and set this site up just a few weeks ago. I'm glad he got it out during the summer so I have more time to play.

The screen starts out with just two dots. Click on one and drag. You'll see a circle. If you drag to the other point it becomes a line segment. If you drag in any other direction and stop when the circle intersects the other point, the circle is fixed. Once you've drawn two circles, their points of intersection offer you new points to work from. The number of moves you've used so far is displayed at the bottom of the challenges box. (You can extend a line you've made without it counting as a new move.)

The first 4 challenges involve constructing a triangle. If you make one at all, you've completed one challenge. If you make one inside the first circle you drew, you've completed another. And if you make one in 8 or fewer moves, you've completed yet another. The last of the set of 4 challenges is to make the triangle in 5 or fewer moves. So the 40 challenges involve constructing 10 figures.

This is my construction of the 'circle pack 7'. It took me 17 moves, and there's a way to do it in 14.

I think I'll return to this site often. Thanks, Nico!


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If you get excited about construction after playing here, and want a more flexible playground, you might like geometry toolbox.

Can You Do Math Without Understanding It? Sure.

I think I want to start a collection of articles like this one. The article explains a bit of what scientists have have done to show how plants do division to efficiently use their stored energy through the night.

I already knew about how dogs take the optimal path, running and swimming, to fetch a frisbee, a task we would use calculus for.

Of course, the plants and animals aren't doing what we do, but these stories get me shaking my head in wonder.

What other examples do you know of?






[Thanks, David Petersen, for tweeting this.]

Guest Post: Understanding is Misunderstood

Burt Furuta and I met long ago on one math education list or another. I always enjoy his perspective, and wanted to share it. Here’s Burt:

Sue often writes about math being challenging, engaging, fun. This post is about math in our schools, which is rarely engaging or fun for most kids. I want to explore why this is so. I think Sue was right when she said in this video that math in school is all about getting the right answer. Not that we don’t want our students to get the right answer. Of course we do. And we want them to understand the concepts we are trying to teach. Some people will say that we have no time in the school day for children to play with math ideas. If students understand the concepts and can solve the problems, then we've done our job. I don‘t think that way of thinking is the real problem - I’ll explain why later. I think the root problem is that we’re all graduates of the same system.

The vast majority of us, including those with the power to shape reform, believe that if we can compute the answer then we understand the concept; and if we can solve routine problems, then we have developed problem-solving skills. Being products of the system, we generally don’t have an appreciation of what conceptual understanding and problem solving really involve. We think we know what understanding is, but unless we've thought deeply about it, it is likely that we don’t. That could be because the word understanding has many levels of meaning - see this article by Skemp for more.

In this post, I’ll just focus on the fallacy that computing answers to problems means we understand the mathematical concepts related to the problem. At a conscious level, most of us get that computation and conceptual understanding are two related but different things. But at an unconscious level, we often treat them as the same thing. When students can compute the answer, we think they understand.

When we talk about conceptual understanding, what we often consciously or unconsciously mean is that students know how to use procedures to compute answers. This is evident in the majority of Liping Ma’s interviews with American teachers described in Knowing and Teaching Elementary Mathematics. The TIMSS video study reported in The Teaching Gap by Stigler and Hiebert describes how our math teaching is dominated by "memorizing definitions and practicing procedures." As Eric Mazur explained, this happens even with our best students at our most prestigious institutions—students compute answers to problems, we think they understand the concepts related to those problems, but in fact they don’t understand.

I am saying that just talking about conceptual understanding doesn't overcome the long history that we have in thinking that getting the right answer indicates understanding. We need to consciously focus on, and think deeply about, the difference between understanding procedural concepts that allow us to calculate answers to problems versus understanding the underlying math concepts. That understanding tells us: how elements of the problem are related to one another, why certain computations are appropriate, when they may not be appropriate, what the significance of that computed answer is, and to what purposes we may apply that information.

Conceptual understanding involves more than associating cues in a problem situation with a known computational procedure and correctly plugging in numbers to crank out the answer, which is how Mazur described his students' behavior in the above linked video. Conceptual understanding always involves a network of connected concepts that are not simply related to computation. When we understand concepts, we can recognize relationships in problem situations, think logically about how the elements of the problem are related, and see how one element may change as another changes.

Last summer I mentored three 7^th graders in math for several weeks. I say mentored instead of tutored because they were all straight A students from three of the best private schools in the state. I gave them a small set of scores and asked them to find the median and arithmetic mean, and to define the concepts median and mean. They had no trouble. They agreed on definitions that are typical: for a set of scores the median is the middle value (with as many scores above as below it) and the mean is the sum of scores divided by the number of scores.

Then I asked if the mean was also a kind of middle value. They didn't think so. I asked them to do some things that made them change their minds - the mean is also a middle value. We discussed how the median is a middle value with ordinal data, and the mean is a middle value with interval data. (If all you care about is which is bigger, you’re talking ordinal data. If you care how much bigger, that’s interval data.)  Concepts like the mean are often only taught as a formula or procedure. Is the purpose of the lesson limited to computing a value? If we teach the median as a middle score doesn't it make sense to also teach the mean as a kind of middle score? I wonder if there were a simple formula to determine the median, would we just give the formula and not mention that the median is a middle score?

In the first paragraph, I said the problem is not in thinking that the traditional teaching approach is more efficient in teaching math and that we don't have time to play with math ideas. Let me explain what I mean. Everyone wants our children to understand concepts. No one is saying that computational fluency is sufficient without conceptual understanding. People "know" that traditional methods are more efficient than playing with ideas - students get the right answer, therefore they “understand”. Arguing about teaching methods only brings heated words and hardened positions. The critical factor in reform is not teaching methods, but rejection of the belief that computing the right answer means understanding the concepts.

It is when people realize that students often don’t understand the concepts, despite being able to compute answers, that they will seek change. Eric Mazur's experience is a good example. His Harvard students did great with the computations, but they still didn't get Newton's laws, which form the conceptual foundation for all those computations. For example, his students did not really believe that the forces a light car and a heavy truck exert on one another in a crash are equal. He knew that a true understanding of Newtonian mechanics would make this a simple conclusion, and sought a better way to teach the concepts. Until then, he had thought he was doing a good job teaching. The realization that his students actually did not understand the concepts is what brought significant change to his teaching methods.

If the difference between computation and concept is not made clear, then to improve the system people will focus even more on getting students to compute the right answers. We are seeing that now, with the emphasis on improving standardized test scores. This is squeezing the life out of learning. We need to help people understand what conceptual understanding really is, and then they will see that real understanding can only come from engagement in activity, making guesses and mistakes, thinking hard, questioning and arguing with oneself or others, testing ideas—in short, some kind of purposeful "play" with math. And this is what is missing in school today. When we only memorize procedures to compute answers, math is boring. On the other hand, true understanding is inherently interesting. There’s decades of research on competence motivation, e.g. see this 10-minute video. Real, meaningful learning is fun.

Talking abstractly about concepts is not enough. So let’s think about this example, and what understanding concepts means:

Sarah was paid $10/hr for her summer job; while her sister was paid $12/hr at her job. To earn the same amount of money as her sister, Sarah work 60 hours more than her sister that summer. How much money did each girl earn? Before reading on, think about how you would solve it.

The typical response is to look for topical associations or simple relationships that cue the use of known procedures. Not finding any, an equation is set up and solved. This would be an easy algebra problem, solved almost mechanically. Those who don’t know algebra might try guess-and-check. Get the answer, then move on.

Let’s do more than get an answer. What are the relationships in the problem? Even if you use algebra to get an answer, look back on the problem. Using the relationships in the problem, could you solve it in other ways? What are some relevant math concepts here? How difficult is it to see the concepts and use the relationships to logically solve this problem without algebra?  How much time do we normally spend on analyzing relationships in problem situations, and relating what we find to prior discussions of those concepts?

One last question: For what grade level do you think this problem is appropriate? You might think it's too hard for most students, whatever their grade level. But third graders can solve problems as challenging as this when given a curriculum that helps them develop the necessary thinking skills. Jean Schmittau ran a research project in an elementary school in New York which used a curriculum originally developed by the Russian educator Vasily Davidov. She found that these third graders "were able to analyze and solve problems that are typically difficult for US high school students." Davydov's curriculum is impressive, and it shows what young children are capable of learning, but we don't need to clone it. What we do need to do is to get past the major obstacle in our own system, which is believing that computing answers to problems is all that is needed to understand concepts. When we teach for real understanding and real problem solving, we will put the joy back in learning and the meaning back in understanding.

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*Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy, European Journal of Psychology of Education. Vol. XIX.  I: 19-43

Links: How to Learn Math, WSJ Mentions Me

Jo Boaler wrote What's Math Got to Do With It, which is an excellent book (with an unhelpful title). This summer she will be conducting a free online course through Stanford Online.



Denise Gaskins has blogged about all the details.  I'll be taking this class. Will you join us?

 ~  *  ~  *  ~  *  ~  *  ~

I wrote a guest post for MathBabe, which is a modified (hopefully improved) version of my post last October on Proving the Pythagorean Theorem. The day after she posted it, the Wall Street Journal made it the headline piece for their Train Reading blog series. (I got almost double my usual traffic yesterday. Nothing like the spike I imagined.) What fun.

TwinDragon Fractal

Thank you Justin (at Math Munch), for pointing this out!

Writing, Vocabulary, and Teacher Inquiry

I was intrigued by this post on Anthony Cody's blog in so many ways. It's about science, not math, but I wonder if we can apply these ideas to math instruction?

Channon Jackson, a fourth grade teacher in Oakland, writes:

In my class [the students] do a science conclusion at the end of every investigation. It is a paragraph where they introduce the topic, give a definition of the key term, make a claim that goes with the focus question, give evidence and then give the scientific reasoning behind the claim. I ... found out that a lot of students were not writing.
...
I decided to have this new structure, where they have these free writes. They don't have to use complete sentences. They just can write whatever they want to write about the topic we were learning about. So if we were learning about rocks, they could write whatever they learned about rocks. If we were learning about magnets, it's just whatever you learned about magnets, but they still had to use vocabulary words. They would get five minutes free write, then sit with their table groups and share out to their table about what they learned.

Still, I had some kids that weren't writing.

Eventually, she allowed them to skip the vocabulary words in their writing.

And all of a sudden, ALL my students had something to say. They clearly understood the content. They now were competing with each other to see who could have the most things to share out at their table group.

She didn't want to give up on the vocabulary...

So the third thing I changed in my teaching was they would do the free-write, they would still do the table talk, we would have a class discussion, and then I would have them pull out their vocabulary words (we keep them on these cards.) I would tell them "Go back and look at the sentences you wrote, and see if there are any words that you could replace with one of your vocabulary words?" I found out that most of them could do that. Most of them could figure out "Oh, I used 'non-living' here, and I could say 'abiotic' instead." 

I love that she found ways to open up the process until it worked for her students. They knew so much more than she realized at first. And I believe the ones who weren't writing previously are learning so much more now that they're writing. I think they are internalizing the concepts more fully, and learning how to incorporate strange new words into their own personal vocabularies.

One of the things I loved about this post was that, without ever mentioning standardized tests, it gave a beautiful critique of a system built around testing. Standardized tests make no room for this organic process between a teacher and her students. They take time away that could be used in these delightful community-building ways, and they frame learning in such shallow ways.

The only example she gave of vocabulary - abiotic, which means non-living - confirms me in my bias against teaching vocabulary. The kids may need to know abiotic for the testing they must endure, but how does knowing a word like that help them in their understanding?

When I explain vocabulary in my math classes, I often tell students I would never test them on it, I just want us to be better able to communicate the ideas to each other. I think my position may be a bit extreme. I'm curious. Do you teach vocabulary, and can you help me understand how it helps students to learn math?

I know that certain concepts take a long time to internalize. We don't get very far along that path by teaching the definition of a term  explicitly, but it is still part of the process. Here's a bit from a great post about how definitions are handled in math:

Aspects of definitions that people new to abstract math don't always understand [include]: The definition gives a small amount of structural information and properties that are enough to determine the concept, [but] the information in the definition may not be the most important things to know about the concept. (from SixWingedSeraph, who blogs at Gyre and Gimble)

Definitions in Math
I can give an example of the necessity of precise understanding of vocabulary from my studies last night from W.W. Sawyer's book, A Concrete Approach to Abstract Algebra. I believe I took courses in both group theory and field theory about 25 years ago, so I know I've worked with these ideas before. But either the examples used were different, or I've completely forgotten the experience. I am loving my discoveries as I work through the exercises.

Sawyer definitely doesn't focus on vocabulary, and clearly avoids the typical definitions given at the beginning of most advanced math courses. Mostly, that's great. But then in an exercise, he asks whether two structures he has described are isomorphic. And, although I thought I knew what isomorphic means (having the same structure), I'm not sure. He's given me no definition of isomorphic to use, and I'm puzzled whether the difference I see makes the two non-isomorphic. Mathematics depends on careful definitions of words used, and I need a precise definition of isomorphic to work from to answer the delightful question he posed.

Here are the two structures he asks the reader to compare. (At this point, it may get a bit hairy if you're not used to thinking abstractly.)

One is the addition and multiplication tables for modulo 5 (also called mod 5) arithmetic, where we are only interested in the remainder after division by 5. This looks like

+	0	1	2	3	4			*	0	1	2	3	4
0	0	1	2	3	4			0	0	0	0	0	0
1	1	2	3	4	0			1	0	1	2	3	4
2	2	3	4	0	1			2	0	2	4	1	3
3	3	4	0	1	2			3	0	3	1	4	2
4	4	0	1	2	3			4	0	4	3	2	1

Can you see why? 2*4 = 8, which is 3 more than 5. In modulo 5 arithmetic, 8 is equivalent to 3, and we forget about the 8 and just write the 3. It turns out that this new arithmetic still has all the properties we would want: commutativity (a+b = b+a), an additive identity (0 works as it should, a+0 = a), a multiplicative identity (1 works as it should, a*1 = a), and so on.

Now we're going to compare this to another, slightly stranger structure. This time we'll use modulo 10 (so we get to just pay attention to the ones digit), and we'll only use the even numbers: 0, 2, 4, 6, and 8.

+	0	2	4	6	8			*	0	2	4	6	8
0	0	2	4	6	8			0	0	0	0	0	0
2	2	4	6	8	0			2	0	4	8	2	6
4	4	6	8	0	2			4	0	8	6	4	2
6	6	8	0	2	4			6	0	2	4	6	8
8	8	0	2	4	6			8	0	6	2	8	4

I had predicted that they would be isomorphic, and felt so sure that this was a simple problem that I almost skipped it. Imagine my surprise when I saw that the multiplicative identity wasn't in the position I had expected (the 2, since it's the first non-zero element) but was the number 6!

So here's my dilemma. The addition table is really the same, so if we were just considering addition, we would say that the two systems were isomorphic. And actually, the multiplication tables can be thought of as being the same in a strange way. We saw that 6 is "our new 1", since it's the multiplicative identity. And we could match up the other rows: in both tables 4 gives the numbers in reverse order (I think of it was being like -1), in both tables 2 goes through the numbers in the same order, and the 8 row in the mod 10 table seems to give the numbers in the same order as the 3 row in the mod 5 table. So considering multiplication alone, we could say the two systems were isomorphic. But. We need to consider both operations at once, I think. And then things are tangled up.

I just read the definitions over at wikipedia, and I know my next step, but it's pretty technical. (It took some vocabulary to help me wade through the articles and find the one I wanted. I had thought I was working with 'groups', but realized as I read wikipedia that having two operations makes these structures 'rings' or 'fields'.) After seeing how the numbers were tangled up in their multiplications, and not tangled in the same way in their additions, I wanted to believe the two structures were not isomorphic. But now I'm not so sure.

You're welcome to discuss this question in the comments too. (Are these two structures isomorphic?) I'll be working it out before I peek, and will enjoy any discussion of this issue. My main question in this post, though, is how vocabulary is related to the learning of deeper subject matter, whether it be science, math, or something else.

W.W. Sawyer - Math Hero

As I've come across more and more really delightful math books, I've often regretted the fact that there weren't books like this available when I was young. But I was wrong. It turns out one of the best writers of engaging math books was writing in the forties, fifties, and sixties. W.W. Sawyer wrote over a dozen books, and every one I've read so far is engaging, worth reading more than once, and full of ideas I can use to improve my teaching.

I started with Vision in Elementary Mathematics, then read Mathematician's Delight (mentioned in this post). At some point, I may have read Prelude to Mathematics. They're all great, but I read them when I was too busy to write up a detailed review. (I'm getting frustrated by the hodgepodge state of my bookshelves. I want to be able to find these three books, so I can say more about them.)

Now I've just finished What Is Calculus About? Also fabulous. I wondered what else the great W.W. Sawyer had written, and found this lovely archive.

I'm glad to see he agrees with my assessment of limits:  

The first chapter dealt with limits. No one sees any reason for thinking about limits before having some exposure to calculus, so I left chapter 1 for much later in the course. (from http://www.marco-learningsystems.com/pages/sawyer/illinois.htm)

I also liked Notes on the art of passing exams. 

So here was this great teacher, very aware of the issues we discuss on blogs - being less helpful, project-based learning, etc. - and yet I had never heard of him until recently. His books sold widely when they were first published, but now are treasures we uncover.

It's so sad to think of these wonderful books sitting on bookshelves, unknown. And it's exciting to know how much the internet helps us learn about good resources, however old they are.

I just got my copy of W.W. Sawyer's A Concrete Approach to Abstract Algebra in the mail. I would love to work through it together with a few friends. Anyone interested?

Math Videos: Vi is already an influence on others!

So there was this contest called Math-O-Vision put on by Dartmouth, that I never heard about until it was over. (I'm glad I heard about Mathagogy's initiative in time.) They invited high school students to create videos "at most four minutes in length telling original stories inspired by mathematics".



I learned of the contest just before the winners were announced. They were highlighting the finalists at that time, and I loved this video by Emily Griffith (which got a third place award). I wonder if she lost points for adopting Vi Hart's style? I may biased, having seen this video before the others, but I like it better than the top two - I think because both of those felt like commercials for math, and I'm more interested in the details of playing with math.

I had a bit of fun playing with listening to this and a Vi Hart video at the same time.





Check out all the winning videos at the Math-O-Vision site.

Sounds like they'll be doing the contest again next year. I will let the high school students in my classes* know.





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*Middle College High School is based within Contra Costa College, where I teach. I often have a number of high school students in my courses.