Thursday, July 30, 2009
When I'm teaching I often think out loud. Talking about powers, I often count on my fingers: 2 (first finger) times 2 (second finger) is 4, times 2 (third finger) is 8, (fourth finger) 16, (fifth finger) 32. I know my students haven't memorized powers of 2, so the quickest way to figure them is to multiply repeatedly by 2 and keep track on our fingers.
Of course, students who haven't memorized their times tables, but who add well, can do the same. Fives are the easiest: 7 times 5 is... 5, 10, 15, 20, 25, 30, 35 (and we stop when our 7th finger pops up).
So counting on our fingers is useful any time we're trying to figure something that needs a repeated step. Perhaps the thing I want to figure can be memorized. But if I haven't memorized it yet myself, the most efficient way to figure it will likely involve fingers.
Learning math is a process. Young children can add 2 blocks and 3 blocks by touching each one and finding the total. At some point, they learn that they can imagine the blocks and count them by touching a different finger for each one. And those fingers are one of the first steps towards the power math has to generalize.
What happens when a person becomes embarrassed about counting on their fingers? If they still want to think, they'll hide it. That's the better option. A worse option happens to way too many students: This may be the point where discomfort with not living up to someone else's expectations makes them give up on math, and then they just guess.
What do you think?
Saturday, July 25, 2009
I think my favorite bit is Mark Dominus' piece on error-correcting codes at the doctor's office. It's in the advanced math section, but it's mostly straightforward.
Friday, July 24, 2009
Let the myth-busting begin! I've gathered together 4 different perspectives on the issue. Mostly we're looking at nature versus nurture here, but there are lots of angles to this.
We all know what drives this myth. There are lots more male mathematicians (PhD level, anyway), engineers, and statisticians. And at least in the U.S., there have been big differences in math comfort level and skill level between women and men. But there is some great research on how these differences are connected to the broader culture (thereby having nothing to do with innate ability).
International comparisons show the cultural variability
The World Economic Forum has created a measure of the gender gap in each country. It turns out that, in countries with a smaller gender gap overall, there is usually also a smaller gender gap in math performance, or even a reversal - in Iceland, girls do significantly better than boys in math. Unfortunately for those of us who live here, the U.S. is not even near the top of the list for gender equality.
Of course, there are naysayers who doubt that this is the whole story. Larry Summers, then president of Harvard, made the claim that perhaps the average ability wasn't much different between men and women, but men, so he claimed, have more variability, and so there are both more dunces and more geniuses among men. (Slate wrote a good article on the Summers fiasco.) Interesting hypothesis, but wrong. Janet Mertz gave a talk at a conference I attended in April (see my original post here), on a paper she co-authored, which looks at that very top level of performance. In countries with greater gender equality, there are more women in these top ranks, and the link to culture is clear.
So, in general, the more equality overall, the better women do in math. But how does this operate on an individual level? Here's some psychology research on that. The term 'stereotype threat' refers to the fear that people have that their behavior will confirm a stereotype. That fear interferes measurably with performance on tests, when gender or race is brought to attention before the test. Actually, the reverse can happen too: Men, reminded that men generally do better in math than women, will do better than they otherwise would on a math test.
I learned about this at that April conference, where Fred Smyth, of the Full Potential Initiative, gave a great talk about it. Here's another good summary of the research, from NYU.
Not different abilities, but perhaps different learning styles
Of course, that's just one piece of the puzzle. Here's another. There are differences overall in learning styles among girls and boys, according to Jo Boaler. In her book What's Math Got to Do with It? she discusses girls' stronger need to understand why. In classrooms where that need isn't honored, girls are turned off to math, more than the boys are. Boaler also describes how classes that focus on the 'why' are better for both boys and girls. She quotes a great book, In a Different Voice, by Carol Gilligan. Gilligan claims that women are more likely to be 'connected' thinkers and men are more likely to be 'separate' thinkers. (But we may want to ask whether that difference comes from nature or nurture.)
More Myths to Bust
About a decade ago, when I read Women in Mathematics: The Addition of Difference, by Claudia Henrion, I was so relieved. It was what I'd been looking for for years. All the other books I'd read on women in math were either collections of biographies or cheerleading (we-can-do-it, rah rah). Finally, in 1997, Henrion brought us some explanations. She described the myths about what it means to be a mathematician, and then gave examples of women mathematicians whose lives disproved those myths. I found a great review of this book at Thus Spake Zuska. She listed the myths this way:
- Mathematicians work in complete isolation
- Women and mathematics don't mix
- Mathematicians do their best work in their youth
- Mathematics and politics don't mix
- Only white males do mathematics
- Mathematics is a realm of complete objectivity
Plain old sexism
"Maybe you're just not cut out for this." I heard that from a math prof when I was an undergrad at the University of Michigan. I never knew whether it was sexism or not. Maybe he said it to struggling male students too. But plenty of women have shared stories with me of blatant sexism directed at them in math class or from a math teacher. Yes, it's still happening. And it's still having a destructive effect. But perhaps this is an example of exponential decay - we'll always see a bit of it, but maybe it's heading closer and closer to zero.
Myth #1 has had more effect on my life than any of the others, but it's not the biggest problem in most people's math lives. More myth-busting coming soon!
Thursday, July 23, 2009
1. It's all about arithmetic. (Elementary school math, at least.)
and its corollary,
2. Gotta memorize those time tables.
Why do I call these myths?
1. Math is so much more than arithmetic. It's shapes, logic, problem-solving, and lots more. Arithmetic is one piece of a huge puzzle that can keep us engaged all our lives if we don't get discouraged or bored by narrowing it down.
2. Sure, they'll need to know their times tables, for all sorts of reasons. But if someone doesn't memorize easily, give them something more intriguing to think about, where they get slowed down, but not stopped, by not knowing their times tables. The skill will develop in this need-to-know context.
So I've been curious what all the math myths are that are floating around out there. Here's a list I found online, that originally comes from Mind Over Math, by Kogelman and Warren, a great book for overcoming math anxiety:
Twelve Math Myths
1. Men are better at math than women.
2. Math requires logic, not intuition.
3. Math is not creative.
4. You must always know how you got the answer.
5. There is a best way to do math problems.
6. It's always important to get the answer exactly right.
7. It's bad to count on your fingers.
8. Mathematicians do problems quickly, in their heads.
9. Math requires a good memory.
10. Math is done by working intensely until the problem is solved.
11. Some people have a 'math mind' and some don't.
12. There is a magic key to doing math.
The word myth doesn't have to mean untrue, but in this context I think it does mean that - inaccurate things lots of folks believe. And yet, there is often some way in which the belief is true, which is why it gets its power.
I am planning to do a series of posts on these myths (and any others my readers bring up), in which I explore them one by one. What's true? What's not? Why do people believe these things?
What math myths have you run into that aren't on this list?
Monday, July 20, 2009
Background from #36 (part 1):
Imagine arranging the positive integers in a spiral pattern.
The numbers from 1 to 16 look like this in the spiral pattern.
10 9 8 7
11 2 1 6
12 3 4 5
13 14 15 16
The location of each number corresponds to an X,Y Cartesian coordinate where the number 1 is at the origin: (0,0). 2 is at (-1,0). 3 is at (-1,-1). 4 is at (0,-1). 5 is at (1,-1). 6 is at (1,0). 7 is at (1,1) and so on.Here's this week's contest problem:
- Come up with an algorithm that tells what number is at an arbitrary X, Y coordinate.
- Come up with an algorithm that tells the X, Y coordinates for an arbitrary positive integer.
Thursday, July 9, 2009
There's a whole bunch here. Get comfy, settle in, and enjoy!
What's Special About This Number? says the Maoris used base 11, 112 = 30 + 31 + 32 + 33 + 34, 113 = 32 + 192 + 312, and 11 x 11 = 65 + 56 (palindromic equality). And of course Wolfram Alpha (WA) has something to say about 11, too.
For this issue of Math Teachers at Play, we have games, geometry, arithmetic, logic, and some test prep.
And here's one from Watch Math on Slopes of Perpendicular Lines.
John Cook, from The Endeavor, brings you Three rules of thumb. I've converted feet to miles, but never seconds to nanocenturies!
Tony, at This Young Economist, brings us 30-20-10 Pricing. How much do you really get off when you get a 30% discount, and another 20% discount, and another 10% discount? Not as much as you might expect...
Denise, at Let's play math!, brings us Solving Complex Story Problems II. She says, "Diagrams make it easy to model a multi-step word problem. When I was in school, we wouldn't have seen this sort of problem until algebra, but with these models, a 5th-grader can solve it." I've been hearing a lot about Singapore's bar diagrams. I'm looking forward to trying them out with some kids.
Misty, at Homeschool Bytes, says, "Make math interesting by mixing regular household items with math concepts, like a handful of candy, two circles, and the concept of venn diagrams." See her post on Venn Diagrams.
jd, over at JD2718, presents 5 logic puzzles. I think logic puzzles are my favorite mathy pastime. Thanks for some doozies!
And I thought I'd throw in one last bit. A few weeks ago I wrote a post about math and mountain climbing, and just now found out about this lovely post about math being like trees.
[Next time I host, I'll try to add a bit more spice. I've put this together in the middle of a marvelous week-long institute on running math circles, put on at Notre Dame by the folks from the Boston area Math Circle. That concludes this edition. Submit your blog article to the next edition of math teachers at play using the carnival submission form. Past posts and future hosts can be found on our blog carnival index page. Technorati tags: math teachers at play, blog carnival.]