## Saturday, February 27, 2010

### Sue's Top Ten Issues in Math Education

In my last post, I mentioned my top ten list of 'problems'. I decided to call it my top ten 'issues' because I wanted to frame them in the positive - not what's wrong, but how to do it right. However, this is still a list of the top ten problems, because most of this is done wrong in most classrooms.

And here they are:

1. If you're going to teach math, you need to enjoy it.*
The best way to help kids learn to read is to read to them, lots of wonderful stories, so you can hook them on it. The best way to help kids learn math is to make it a game (see #3), or to make dozens of games out of it. Accessible mysteries. Number stories. Hook them on thinking. Get them so intrigued, they'll be willing to really sweat.

2. If you’re going to teach math, you need to know it deeply, and you need to keep learning.
Read Liping Ma. Arithmetic is deeper than you knew (see #6). Every mathematical subject you might teach is connected to many, many others. Heck, I'm still learning about multiplication myself. Over at Axioms to Teach By, I said (back in September), "You don't want the product to be 'the same kind of thing'.  ...   5 students per row times 8 rows is 40 students. So I have students/row * rows = students." Owen disagreed with me, and Burt's comment on my last post got me re-reading that discussion. I think Owen and I may both be right, but I have no idea how to use a compass and straightedge to multiply. I'm looking forward to playing with that. I think it will give me new insight.

3. Games are to math as books are to reading. Let the kids play games (or make up their own games) instead of "doing math", and they might learn more math.
Denise's game that's worth 1000 worksheets (addition war and its variations) is one place to start. And Pam Sorooshian has this to say about dice.  Learn to play games: Set, Blink, Quarto, Blokus, Chess, Nim, Connect Four.

4. Students are willing to do the deep work necessary to learn math if and only if they’re enjoying it.
Which means that grades and coercion are really destructive. Maybe more so than in any other subject. People need to feel safe to take the risks that really learning math requires. Read Joe at For the Love of Learning. (Maybe you'll get to read him here soon.) I'm not sure if this is true in other cultures. Students in Japan seem to be very stressed from many accounts I read; they also do some great problem-solving lessons.

5. Math is not facts (times tables) and procedures (long division), although those are a part of it; more deeply, math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways.
My favorite math ed quote of all time comes from Marilyn Burns: "The secret key to mathematics is pattern."

U.S. classrooms are way too focused on procedure in math. It's hard for any one teacher to break away from that, because the students come to expect it, and are likely to rebel if asked to really think. (See The Teaching Gap, by James Stigler.)

See George Hart for the artform. I don't know who to recommend for the language angle. Any recommendations?

6. Math is not arithmetic, although arithmetic is a part of it. (And even arithmetic has its deep side.)
Little kids can learn about infinity, geometry, probability, patterns, symmetry, tiling, map colorings, tangrams, ... And they can do arithmetic in another base to play games with the meaning of place value. (I wrote about base eight here, and base three here.)

7. Math itself is the authority - not the curriculum, not the teacher, not the standards committee.
Read Math Mojo – you can’t want kids to do it the way you do. You have to be fearless, and you need to see the connections.

8. Real mathematicians ask why.
If you’re trying to memorize it, you’re probably being pushed to learn something that hasn’t built up meaning for you. See Julie Brenna's article on Memorizing Math Facts. Yes, eventually you want to have the times tables memorized, just like you want to know words by sight. But the path there can be full of delicious entertainment. Learn your multiplications as a meditation, as part of the games you play, ...

Just like little kids, who ask why a thousand times a day, mathematicians ask why. Why are there only 5 Platonic (regular) solids? Why does a quadratic (y=x2), which gives a U-shaped parabola as its graph, have the same sort of U-shaped graph after you add a straight line equation (y=2x+1) to it? (A question asked and answered by James Tanton in this video.) Why does the anti-derivative give you area? Why does dividing by a fraction make something bigger? Why is the parallel postulate so much more complicated than the 4 postulates before it?

9. Earlier is not better.
The schools are pushing academics earlier and earlier. That's not a good idea. If young people learn to read when they're ready for it, they enjoy reading. They read more and more; they get better and better at it; reading serves them well. (See Peter Gray's recent post on this.) The same can happen with math. Daniel Greenberg, working at a Sudbury school (democratic schools, where kids do not have enforced lessons) taught  a group of 9 to 12 year olds all of arithmetic in 20 hours. They were ready and eager, and that's all it took.

In 1929, L.P. Benezet, superintendent of schools in Manchester, New Hampshire, believed that waiting until later would help children learn math more effectively. The experiment he conducted, waiting until 5th or 6th grade to offer formal arithmetic lessons, was very successful. (His report was published in the Journal of the NEA. Although some people disagree about the success of this experiment, there is nothing published which contradicts his evidence. I'd like to find more information about how this project ended.)

10. Textbooks are trouble. Corollary: The one doing the work is the one doing the learning. (Is it the text and the teacher, or is it the student?)
Hmm, this shouldn't be last, but when I look at the list, they all seem important. I guess this isn't a well-ordered domain.  ;^) Textbook Free: Kicking the Habit is an article by Chris Shore on getting away from using a textbook. (After clicking the link, click on 'Articles'.) I've been duly inspired, and will report in the fall about how it goes for me in my classes to teach without a textbook. See dy/dan on being less helpful (so the students will learn more).

31. Multiplication is not (just) repeated addition, it’s much richer than that.
Wait. I said that already... (I warned you, it's just not in my top ten.)

What do you see as the biggest issues or problems in math education?

____
*I know, top ten lists are supposed to start at number ten to keep the suspense up. But the suspense is gone - I already told you my top two in my last post. And I can't help it, I just have to start at the top.

## Friday, February 26, 2010

### What is Multiplication?

"Oh, come on, Sue! What kind of a question is that?"

I know, but think about your answer before you read on, OK? What do you think multiplication is? I ask because I want you to have your own sense of it in your mind before you read the folks I'm going to quote.

***     ***     ***     ***     ***     ***     ***     ***     ***     ***     ***     ***

It seems like a simple question. But when you get involved in discussions about how to teach, it isn't. Many elementary teachers present multiplication as repeated addition, and really, it's much more than that - areas, combinations, stretching, and more. (Here's a cool poster, created by Maria Droujkova.) Many math education experts think that calling it repeated addition is a big problem. [Keith Devlin's articles started this discussion. Jason Dyer just posted on this issue from a computer science perspective.]

I personally think that this is one tiny facet of the real problem, and in a comment at Rational Mathematics Education, I said, "I see plenty of problems in the way math gets taught, but this would not be in my top ten list." (After saying that, I decided to figure out what my top ten list would be - I'll post on that soon.) The top two problems, in my opinion, are that so many elementary teachers don't like math, and that they don't have a deep understanding of the math they teach.* We have a vicious circle going, where those who dislike math teach the young to dislike it - and that's a hard thing to change.

Back to the question at hand. Devlin says 'multiplication is not repeated addition'. I agree that it's not just that, but he and others say it's not that at all, and that saying it is messes kids up. I think it's more a case of the translation between English and math-language being rough sometimes. I trust that if we haven't gotten a student to give up thinking, they'll eventually construct their own definition of multiplication, as it becomes clear to them through what they do with it.

Here's a scenario: My son (7 years old) wants to know how much 5 dimes are worth. He says 10, 20, 30, 40, 50 while holding up fingers. The process he's using to figure it is skip counting, which feels like repeated addition to me. But what he's thinking about is 5 dimes x 10 cents for each dime, which is multiplicative reasoning. So the repeated addition is the process he uses to solve his multiplication problem.

It's important to note here that I didn't suggest this 'problem' to him. It was something he wanted to know. I didn't tell him he was doing multiplication, and I don't plan to 'extend his learning' with other problems that involve multiplicative reasoning. I expect his natural curiosity will lead him to explore lots of situations where he'll reason in whatever way helps him figure out what he wants to know. It's important to me not to push. I've noticed plenty of kids of mathematicians who don't like math, and I really want to give him the space to develop his own relationship with the beauty in math.

Most people who write about this are imagining a conventional classroom, where all the students are supposed to be 'learning' the same thing at the same time. (An impossibility, no?) When I imagine that classroom, I see a little child coming up to the teacher after class, worried about this multiplication thing, and the well-meaning teacher trying to be reassuring, and saying, "Don't worry, it's just repeated addition." What the teacher is doing is connecting the new material with something old and familiar.  This is how our brains work; through connections. The teacher is also recognizing the child's concern about what they will have to do, and since math lessons are so procedurally-based in this country (on my top ten problems list), she's telling the child s/he can do the multiplication problems by repeated addition. So there are positive aspects to this sort of response, but there are also ways in which it's somewhat problematic.

As we learn new concepts, we go through a phase where we feel confused. Recognizing that, and even celebrating it, is important. (Thanks, Maria, for that insight. It's hard to celebrate our confusion, though, when we're worried about grades.) I'm trying to think of an example that most kids would feel at home with... It's not confusion exactly, but when you learn to ride a bike, it feels all wrong, until suddenly, it feels right. Learning something new can be like that.

Maybe that's a part of what I'd tell that worried child. I might also refer to the repeated addition metaphor to help them feel calm, since I know plenty of people shut down when confronted by the mysteries of math. But I'd also give them an easy area model to think with, so they'd see a basic 'real' multiplication problem. Repeated addition can get at it, and yet it's really something new - a shape made with 3 rows of 5 squares is also 5 rows of 3 squares. But (and here's my problem with imagining that 'conventional' sort of classroom) I think it's better to be playing with areas enough that the kids will tell me, "Oh wow, look at this! 5 threes is the same as 3 fives!"

Devlin also says exponentiation is not repeated multiplication, and functions are not processes. He says you're starting with a lie if you explain these concepts using these metaphors. I disagree. We start out thinking of exponents as meaning repeated multiplication, and then we expand and extend that, to see exponential growth in a more continuous sense. (A 4% annual growth rate can be helpfully seen as multiplying by 1.04 each year, but the growth doesn't happen at one point in the year - it's smooth.)

Here's Devlin on functions: (Dec 08)
...a significant proportion of university mathematics students do not have the correct concept of a function.  Do you? Here is a simple test. ... Consider the "doubling function" y = 2x (or, if you prefer more sophisticated notation, f(x) = 2x.) Question: When you start with a number, what does this function do to it?
If you answered, "It doubles it," you are wrong. No, no going back now and saying "Well what I really meant was ..." That original answer was wrong, and shows that, even if you "know" the correct definition, your underlying concept of a function is wrong. Functions, as defined and used all the time in mathematics, don't do anything to anything. They are not processes. They relate things.
I think seeing functions as processes is a fine perspective to start from - and very few students will go far enough in math to need another point of view. I also think Devlin's insistence is likely to make people think math is stranger and harder and less knowable than it really is. If our elementary teachers were well-educated mathematically, they could weigh in with their own opinions on this subject. I'm concerned that Devlin's tone sets up the notion that there is one right answer to this. (And his question was quite a setup, wasn't it? "What do functions do?" "Gotcha! They don't do anything.") Real mathematicians ask why, which is what I'm doing, along with some of the people  I respectfully disagree with. But others are focusing more on the 'right answer' to this pedagogical question than on the reasons, which encourages the wrong approach to math and its pedagogy.

Here's one more part I'd like to think about (Keith Devlin, July 08):
Part of the problem, I suspect, is that many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength.
...
Where does the "abstracted from everyday experience and developed by iterated metaphors" mathematics end and the "rule-based mathematics that has to be bootstrapped" begin?
What if the mathematics that has to be bootstrapped in order to be properly mastered includes the real numbers? What if it includes the negative integers? What if it includes the concept of multiplication (a topic of three of my more recent columns)? What if teaching multiplication as repeated addition (see those previous columns) or introducing negative numbers using an everyday (explicit) metaphor (such as owing money) results in an incorrect concept that leads to increased difficulty later when the child needs to move on in math?
I don't believe that explicit metaphors like these get in the way, unless just one metaphor is used all the time. I agree with Devlin's claim that the strength of mathematics lies in its ability to use abstraction, but I disagree that starting from the concrete is dangerous or even problematic. I'll address that issue in a future post.

The real questions for me are broader: Are students getting a chance to explore lots of different multiplicative relationships? Are they maintaining their curiosity and rage to learn? Is math presented as a tool they can develop to help them think? I want schools in which: teachers are respected for the hard work they do,  they're given time daily in which to have professional discussions with their peers about what they are trying to help students learn, and they come in ready to approach math with comfort and joy.

[Edited on 3-1 to add: Surprisingly to me, the discussions on this topic have often become hostile. It's important to me that people treat each other decently here at my bloghome, and I turned comment moderation on when I first posted this, to enforce that. I am rejecting any comments that don't meet this standard. Here's what you see when you post a comment:
I would like this blog to be a safe place for people to disagree. Please do not attack the integrity of the person you disagree with. (Any comments which do so will not be accepted. If I can email you with my concern, I will. My email is suevanhattum on the hotmail system.)

Comments with links unrelated to the topic at hand will not be accepted.
Perhaps I should have said it more thoroughly. I will ask you to rewrite if you treat another person badly, or if you malign the intelligence of people on 'the other side'. etc. One comment has been rejected so far.]

_______
*Note to any elementary teachers reading this: I think a good K12 teacher is a saint. You work harder than I do, you have less autonomy, and you get paid less. If you can really reach kids, you also make a bigger impact than I do. I'm guessing the fact that you're here means you either like math or want to do better with it. I'm grateful for all you do. Say hi in the comments, email me, point me to things I should know.

## Monday, February 22, 2010

### Project Euler

My colleague from the computer science department at Contra Costa College, Tom Murphy, thought I might enjoy some more math problems, and pointed me to Project Euler. Here's their intro:
Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.
My programming skills are a bit rusty, so I'm starting with purely mathematical problems. The first one I did is:
If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
[I asked for the problems to be listed in ascending order of difficulty, and this one came up first.]

I figured it out, and used my calculator to do a simple multiplication and addition problem at the end. The 'forum' discussing this problem had lots and lots of programs listed. I was shocked people would ask a computer to solve this one.

I decided to do one a day. This morning I checked the next two and figured I'd at least want to use Excel on them, although I bet there's an elegant way to solve each of them without it. The fourth problem was once again simple enough to solve in a minute, with my trusty TI-83 doing my multiplication.

It looks like a nice set of problems to challenge students with. Thanks, Tom.

## Saturday, February 20, 2010

### Richmond Math Salon - Symmetry

Lots of people showed up, right at 2. The adults played with the math games even more than the kids this time. The kids love the trampoline, but they still came back in to play with polydrons, mirror books, Blink!, Set, and tangrams. And to read You Can Count on Monsters. I'll tell more later. I want to get this post up so Madeline, and others, can comment here. (Hey Madeline, you're awesome too!) [Now edited - photos and activity details added.]

Polydrons
I had just gotten my big box of polydrons yesterday, and was excited to share them. We had oodles of fun with them. The first thing I built when I opened the box on Friday was a tetrahedron. Each face is an equilateral triangle. There are 4 faces total - a pyramid on a triangular base. The picture you see here is not a tetrahedron, because the base is a square.

Then I built an octahedron, using 8 of the same equilateral triangles. I don't have photos yet, so here's a generic octahedron. My son built a cube. So we had 3 of the 5 Platonic solids. On a Platonic solid, every face is exactly the same, and each face is a regular polygon. (Regular polygons have all sides the same length, all angles equal. What we call a side on a 2 dimensional polygon is an edge on the 3 dimensional solid.) There's one more criteria: At each vertex (think corner), the same number of faces have to meet. To me, it feels wild that there are only 5 possible ways to do this.

As we sat down to build, I said I wanted to build an icosahedron (20 faces) and a dodecahedron (12 faces). D built an icosahedron pretty quickly. One of the parents was trying to build one, and thought he'd used the wrong triangles. (He hadn't.) I was excited - I got to explain one reason we're limited to just 5 Platonic solids. I showed him (and a few others) how you can see that the angles in an equilateral triangle are 60 degrees, and 6 of them make a whole circle. So putting 6 together, they'll lie flat. (It looked like the picture here, on the top, but not on the sides.) We need to put less of them together at the vertex if we want it to poke up. As we talked about changing it, he mentioned having lived in a yurt. So we kept his yurt, and started over for the icosahedron, which has 5 triangular faces at each vertex. I had a lot of trouble getting the last face snapped onto mine, and D helped me.

So 3 triangular faces at each vertex makes the tetrahedron, 4 at each vertex makes the octahedron, 5 at each vertex makes the icosahedron (20 faces total). You can't use more because 6 faces at each vertex would lie flat, and you can't use less because 2 faces at a vertex won't have space in between. The cube has square faces, 3 to a vertex. We can't make something with 4 squares meeting a a vertex; same problem, it would lie flat. Next shape is a pentagon, with 5 sides. If you put three of those together at each vertex you'd end up with 12 sides. D wanted to do that, but we didn't have enough pentagons. I'll have to order those separately.

I got these polydrons very cheaply through Educator's Outlet. (When you go to the page I've linked to, you'll see the prices slashed, but then when you check out, they're cut even more. I paid about $30 for what would normally cost a few hundred, I think.) Mirror Books I got this idea from Maria Droujkova (who posts at Natural Math). Tape two little mirrors together. Draw something and set the mirrors in a V just behind. You get a kaleidoscope. Draw a straight line, and set the mirrors on it. As you adjust the angle between them, you get all sorts of polygons. We also played with writing upside down, writing in cursive and then drawing the mirror reversal of the writing, making a simple path and trying to follow it with the pen while looking in the mirror (way hard!), and looking at our faces in the mirrors (use a 90 degree angle to see your face as others see it, which doesn't happen in a regular mirror). These were all suggestions from participating parents, yeay! More Blink! and Set are the games I put out every month, so people can sit right down to something as others are arriving. (I see that Out of the Box sold the rights to Blink! to Mattel in 2007.) The tangrams also come out pretty often. The book, You Can Count on Monsters, by Richard Schwartz, is about prime and composite numbers, but can actually be enjoyed by very young kids, because it has a page for each number, 1 to 100, and each one has a different monster on it. It's a delightful concept. I had a few other activities planned, but never got around to them. I wanted to do the rep-tiles activity explained here. And I wanted to do an activity where you find a line of symmetry in an odd-shaped figure, that I found In James Tanton's book, Math Without Words. [Note to participants who post comments: This is a public space, so just use initials if you mention other kids.] ## Thursday, February 18, 2010 ### Math Teachers at Play #23 (and #22) When I link to the MTaP, I like to point out one or two of my favorite items. That got me in trouble last month. I never finished reading all of the goodies at MTap #22, and so neglected to post a link to a fabulous blog carnival. So this month I'm posting a link right away. When I get time to read it all, I'll add a comment, or another post, or something. Here's MTaP #23, thanks Dan! Perhaps on Sunday I'll take my laptop to Catahoula Coffee and relax with a latte while I read my MTaP. ## Tuesday, February 16, 2010 ### Tutoring and Conics My tutoring sessions with Artemis* continue to go well. I never prep, except for fleeting thoughts about topics and problems he might enjoy. Today, I showed him the books that came a few days ago from James Tanton. (I will blog about them soon.) We started to look at Thinking Mathematics, Volume 3: Lines, Circles, Trigonometry and Conics, and before we'd looked at two pages, I asked if he'd like to figure out the equation for a circle. Sure. I drew a circle, and asked him for the definition. He said, "All the points are the same distance from the center." I said that what we were going to do is called analytic geometry - the marriage of geometry and algebra that Rene Descartes helped found. I drew x and y-axes and asked him what we should call the center. He said (x,y). I felt bad that I'd asked the question, since I didn't want to use (x,y) for the center. So I talked about how we have a tradition of using x and y for the points that would move around (said while tracing over the circle), and so the center would traditionally get another name. One tradition would be just to use (a,b), another would be to use (h,k). I have no idea why we use h and k... He chose a and b. I pointed to his definition and asked how we'd think about the distance. He said, "Use the Pythagorean Theorem." My algebra students tend to think the 'distance formula' is something separate, so of course it tickles me that he thinks of it in a way that feels more basic to me. I drew a triangle with a bad circle around it... ...and had him tell me what to do next. He worked it out to $r=\sqrt{\left(x-a\right)^2+\left(y-b\right)^2}$ and I told him the tradition is to write it as $\left(x-a\right)^2+\left(y-b\right)^2=r^2$. Then we looked at $x^2+y^2+4x-10y=7$ and completed the squares together. He told me the center and the radius. We decided to move on to other conics, and started with the definition which states that a parabola is all the points equal distance from a focus and a directrix (a line). But we also were talking about how you cut the cone to get the conics, and I said I had never figured out how we know that a plane cutting the cone parallel to the side gives us the same thing as this definition that uses focus and directrix. We worked it out for an example, using x,y,z coordinates. After a bit of fiddling around, we called our cone $z=2\sqrt{x^2+y^2}$ and our plane $z=2\left(x-1\right)$. We used a 3D graphing calculator to check whether it was right. A bit of algebra gives us $y^2=1-2x$, which is a parabola opening to the left. That was nice, but has nothing to say about focus and directrix, and of course it's not a proof. We decided to look it up. The explanation I found is for an ellipse, not a parabola, but we decided to work our way through it. It's titled Dandelin's Spheres, after the French/Belgian mathematician Germinal Dandelin (1794 – 1847) who came up with this proof - and it's dazzling! I've never seen this before, and I want to know why - it's so elegant, and pretty simple. At the end, it says the hyperbola and parabola can be thought through following almost the same steps. I'm going to do it! This was the first time I did some stretching mathematically while tutoring him. It's going to happen more and more. I'm very curious when he'll "outgrow" me. ______ *Artemis (not his real name) is 8, and is very smart. Note: The equations were done at CodeCogs. I had to redo the drawing because the website I used to put it up is gone now. ## Saturday, February 13, 2010 ### Math Poetry Wiki! My poetry challenge got more comments than any previous post on this blog. And the poems people wrote are quite delightful - much too delightful to be buried in the comments. So I set up a wiki. I've gotten permission to repost their work from a few of the authors, but not all. Please let me know if you'd rather not have your work posted there, and I can just put a link instead. I'd love to have one poem show up on the home page, in some rotation. Does anyone know of a way to make that happen? It took me until today to respond to my own challenge. Here's what I came up with: The Pleasure of Struggling I can’t get this. It doesn’t make sense. What are they talking about? I will never get this. It’s crazy. Wait… Hmm… Ah! If I put this with this… No… Hmm… Oh my! Look at that! How cool! May I have another, please? On re-reading this, I see it doesn't have to be about math. But it is. ;^) Your turn! Write a poem, and add it to the Math Poetry Wiki. ## Tuesday, February 9, 2010 ### One More Reason Why High-Stakes Tests Are Not a Good Idea Have you heard of Campbell's Law? On Bridging Differences, an educational policy blog, Diane Ravitch writes: Numbers don't lie, do they? Well, yes, they do. A major front-page story in The New York Times on February 6 described a major study conducted by criminologists who found that the numbers do lie. More than 100 retired, high-ranking police officers in New York City told them that intense pressure to produce improved crime statistics had led to manipulation of the data. For the past 15 years or so, the city boasted that its data system, known as CompStat, had brought about a major reduction in crime. But the survey said that the data system had encouraged supervisors and precinct commanders to relabel crimes to less serious offenses. The data mattered more than truth. Some, for example, would scout eBay and other Web sites to find values for stolen items that would reduce the complaint from a grand larceny (over$1,000 in value) to a misdemeanor. There were reports of officers who persuaded crime victims not to file a complaint or to change their accounts so that a crime's seriousness could be downgraded.

This is not only a major scandal, it is a validation once again of Campbell's Law, which holds that: "The more any quantitative social indicator is used for social decisionmaking, the more subject it will be to corruption pressures and the more apt it will be to distort and corrupt the social processes it is intended to monitor."

Anyone who wants to learn more about Campbell's law and how it applies to education should read Richard Rothstein's Grading Education and Daniel Koretz's Measuring Up. Or Google Rothstein's "Holding Accountability to Account," if you want to see what happens when data becomes our most important goal.

## Monday, February 8, 2010

### A Link and a Problem

Steven Strogatz is back, with his second math article in the weekly NY Times series which will go for 15 weeks. This one is called Rock Groups. He mentioned a puzzle series by John Tierney, and the one posted today reminds me of a problem I've posed in my Math for Elementary Teachers course - but this one's got a better storyline.

The Problem
Sol, at Wild About Math, asked for help solving a problem his brother heard on the radio:

Bob and Alice are both millionaires. They’re both curious to know who is richer but they don’t want to tell the other one how much money they have. Without engaging a trusted third party, how can they both know who is richer?

I have played with a similar problem that I think goes like this:
10 mathematicians are out to dinner, and want to know their average salary. Without anyone finding out anyone else’s salary, how can they do this?

I remember that I saw the solution and liked it. (I may have solved it myself, even, but I'm stumped again now - the delights of a bad memory...)

Sol wants the answer. I'd prefer hints, myself.

## Wednesday, February 3, 2010

### In Honor of Black History

On my other blog, I'm doing a series on African-American picture books. On this blog, I'd like to tell you the story of a Black mathematician.

Years ago, I wrote an essay on Vivienne Malone Mayes, for a course in African American history. As I looked around online today, I came upon a sad facet of her story, which will remind us that racism is hardly overcome. The Black Women in Mathematics page on her describes her experience at the University of Texas, in her PhD program:
In graduate school she was very much alone... In her first class, she was the only Black, the only woman. Her classmates ignored her completely, even terminating conversations if she came within earshot. She was denied a teaching assistantship, although she was an experienced ... and excellent teacher. She wrote: "I could not join my advisor and other classmates to discuss mathematics over coffee at Hilsberg's cafe.... Hilsberg's would not serve Blacks. Occasionally, I could get snatches of their conversation as they crossed our picket line outside the cafe." She could not enroll in professor R.L. Moore's class as he explicitly stated that he did not teach Blacks.
Part of the significance of this is R.L. Moore's fame as a math educator. The 'Moore Method', whereby his students did not use textbooks and provided all the proofs in class, is famous, in part because more mathematicians came out of his program than typically come through any one teacher. So his personal racism is all the more abhorrent.

My essay retold what I learned from Women In Mathematics: The Addition of Difference, by Claudia Henrion. This book contains interviews with both Vivienne Malone Mayes and Fern Hunt, both Black mathematicians. Both interviews point out the advantages of Black colleges, either studying at a Black college, as Malone-Mayes did, or working at one, as Hunt did.

Vivienne Malone Mayes went to Fisk University in Tennessee, and earned both her B.A. and M.A. there. She returned to Waco, and ended up teaching at Bishop College, a small Black college nearby. For years she had encouraged her better students to go on to get doctorates, so that they could come back and teach in the Black colleges, which would help the colleges to become accredited. Her students finally persuaded her to follow her own advice. There were no Black colleges in Texas that offered Ph.D’s, so she applied to Baylor University in Waco. In 1961, she was denied admission there because they did not admit Blacks. She then applied to the University of Texas at Austin, was admitted, and earned her Ph.D. in 1966. Five years after refusing her admittance as a student, Baylor University offered her a position as a professor, which she accepted.

When Vivienne started college at Fisk, her major was chemistry. But two of her teachers there inspired her love of mathematics, and so she switched her major, did graduate work, and became a college teacher herself. This switch was seen by her family as quite impractical, but Fisk had already been influencing her thought in other ways, so that she would say “we were DuBoisites”. (A huge influence in her life, her father’s history, views, and advice reflected the philosophy of Booker T. Washington - get the training that will get you work.)

The two teachers who encouraged her to enter the study of mathematics were Evelyn Boyd Granville and Lee Lorch. “Evelyn Boyd Granville was one of the first Black women to receive a Ph.D. in mathematics in the United States.” (p.200) Seeing another Black woman doing mathematics was important in giving Malone Mayes the confidence that she, too, could do this. Lee Lorch was a white teacher who (in Malone Mayes’ words) “believed that the students could understand the material, not just learn to do it”. (Lorch was white, but his commitment to interracial equality was clear. He was subpoenaed by the House Committee on Un-American Activities for his actions in support of integration, and subsequently lost his position at Fisk because of this.)

Malone Maye’s goals as a teacher were first, to support and respect her students so they would begin to have a sense of self-worth, second, to give them the tools of self-empowerment, and third, to create a path of opportunity - in math, the answers are right or wrong, when you know what you’re talking about, it’s clear, so she felt that her students would face less discrimination in this field.

Malone Mayes was paid substantially less than her similarly qualified colleagues at Baylor. She sued, received a $5000 raise, and was still$7000 below her colleagues. Her health deteriorated over the years, partly due to the stresses of her work at Baylor, and she died in 1995, at the age of 63.

## Monday, February 1, 2010

### Math As Story

Once upon a time, math came packaged in textbooks, and most people had never (ever) even seen a math story. Unless you count things like: "The westbound train leaves at 1pm going 50 miles per hour and the eastbound train leaves at 2pm going 60 miles per hour. If they're 300 miles apart, when will they crash?" (Except the textbook would have said 'meet', leaving out the tiny bit of drama I couldn't help adding.)

I love stories, and I love how much they can add to the appeal of math. History is one storytelling genre that adds a lot of drama to math. Why do we need calculus? Seems to me every calc course should start with enough history for students to see the story unfolding. Where did that crazy number i come from? What's up with geometry proofs? (I still don't know enough to tell these stories properly. It's one of the things I'm doing during my sabbatical year.) Biographies of mathematicians are also a great way to ground all the headiness of math in the details of a life. (My review of The Man Who Knew Infinity, about Ramanujan, is here.)

If you follow this blog, you've seen some of my storytelling attempts (Eight Fingers, Crash and Count). They pale in comparison to some of the literary delights below. Today was a blockbuster day for storytime in the math blogosphere.

Glenn is posting an ongoing adventure story about a place called Verdania, at his blog, Off the Hypotenuse. Each chapter ends with a math puzzle. In the current chapter (Chapter 10), the main characters, who were shipwrecked, are leaving the children's village, and heading to the adults' village. At the end, we're asked to figure out the lengths of 3 paths the characters could travel to get from Sentry Point 1 to Adult Village. If you want to start at the beginning, he's made a new blog with just this story.

But don't leave Off the Hypotenuse behind; there's lots of other great posts there. I am delighted over and over as I try to catch up on all his older posts. The strange thing is, I can't figure out who to thank for pointing me there. I just could not retrace my steps successfully the day I found it.

Glenn linked today to another new delight: Number Gossip, hosted by Tanya Khovanova. It's not exactly stories, but it's in the same spirit, and fun.

Then Jason Dyer, at Number Warrior, pointed to a gory delight over at Emily Short's blog. Word problems with a decidedly 'unfortunate events' twist.

Then there's the pirate story, over at The Math Factor.

Dave Richeson, at Division by Zero, pointed to a great series starting up in the NY Times, written by Steven Strogatz, on:
... the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it’s so enthralling to those who get it.
It starts with a story about penguins in a hotel, ordering fish, fish, fish ... fish. Check it out!

It made me chuckle to see Richeson mentioning Strogatz. I just finished reading Steven Strogatz's lovely new book, The Calculus of Friendship, after hearing him give a talk about it at the Joint Mathematics Meeting a few weeks ago, and I've just started reading Dave Richeson's exciting new book, Euler's Gem.

One last link, not quite a story: Mike Croucher, at Walking Randomly, posted this great piece on the Math Carnivals. (Sorry I neglected to link to the last Math Teachers at Play, over at Math Hombre. It's lovely, and I'm still working on getting through all the links.)

The math blogosphere is exploding with stories today. Yeay!