Saturday, April 30, 2011

Richmond Math Salon - Last Session of the Season!

photo by Roy Robles
If you live in the California Bay Area,  you may want to join me next Saturday from 2 to 5pm for my monthly math salon.

I have games like Rush Hour, Quarto, Set, Blink, and Blokus; puzzles like tangrams and pentominoes; math toys like Polydrons and Perplexus; and resources that parents like to look at. Typically, kids and parents play with a game or puzzle until everyone has arrived, and then we come together to try out the activity of the month.

This month I'll be telling a story (a take-off on the Greek story of Icarus) to go with a game that I've used with kids before. Kids love playing with it, and get lots of arithmetic practice thrown in for good measure. This game is part of an open problem in mathematics.  I'll also be bringing in a few other puzzles related to unsolved problems in math.

Email me at suevanhattum on the warmer mail system* if you're interested.

*My attempt at foiling the spambots.

Lecture versus Students Figuring Things Out

I've kept this page up in a tab on my browser for a week. Each time I come back to it, I'm not sure what to do with it. EdNext* is describing a study done by researchers from Harvard. From the intro:
A new study finds that 8th grade students in the U.S. score higher on standardized tests in math and science when their teachers allocate greater amounts of class time to lecture-style presentations than to group problem-solving activities.  For both math and science, the study finds that a shift of 10 percentage points of time from problem solving to lecture-style presentations (for example, increasing the share of time spent lecturing from 60 to 70 percent) is associated with a rise in student test scores of 4 percent of a standard deviation for the students who had the exact same peers in both their math and science classes ...
It seems clear to me that we do not learn as well when we're passive as when we're active. This study seems to argue the opposite. The article points to another article that goes deeper into the research; at the end of that one, they do say:
Newer teaching methods might be beneficial for student achievement if implemented in the proper way, but our findings imply that simply inducing teachers to shift time in class from lecture-style presentations to problem solving without ensuring effective implementation is unlikely to raise overall student achievement in math and science. On the contrary, our results indicate that there might even be an adverse impact on student learning.
My take on any sociological research, which includes education research, is that there are too many variables involved to do really good research. What happens is that many researchers frame a question in a way that is bound to bias the results toward a conclusion they already favor.

But still, I'd like to be able to explain a result like this one.

I know from experience that teaching from one of the 'reform' (back when reform meant something else) calculus texts would have been very hard without getting trained. I was trying to find ways to change my teaching, and would have worked from the project-based 'reform' texts, but couldn't see how to do it. It became clear to me that you couldn't change the way math is taught by providing a good textbook. The way classrooms 'should' work has been imprinted on us through 16+ years of sitting in classrooms as students. It's pretty hard to do something different, and then to do that different thing well. So I do agree with their conclusion that newer teaching methods need to be implemented in 'the proper way', which involves lots of work retraining ourselves.

Jo Boaler has done lots of research on math education. I'd like to know what her take on this study is.

What do you think?

*EdNext is part of the Hoover Institution, which is a conservative think tank.

Wednesday, April 27, 2011

Math Pickle!

A while back John Golden wrote about a game he made up called Decimal Point Pickle. Something about pickles seems to attract math folks. Gordon Hamilton has created a web site called Math Pickle, on which he's posted a slew of wonderful videos of his work with kids, playing with math.

He was the guest host tonight of the Math 2.0 Webinar (a series Maria Droujkova founded and organizes), and showed us half a dozen delightful puzzles and games. For the first half hour, he showed us open mathematical problems that he uses with kids. The first one was called No-3-In-a-Line. Can you put 8 skyscrapers on a 4x4 grid so no 3 are in a line? (The open mathematical problem is finding the largest nxn grid that can have 2n cells filled with no 3 in a line.)

The second open problem he showed us was first posed in 1916 by Issai Schur. Here's his video of kids working on it.

Next he showed us an elegant game he created, called Mimizu (Japanese for earthworm). (Watch this video, at 1:15 in.)

I mentioned my collection of easy to understand open problems, and how much kids like the problem that is alternatively called Hailstone numbers or the Collatz Conjecture. He's worked with that, and jazzed it up with the Deadalus and Icarus story. Lovely additions! (And there's a pdf on the site to help you use it with kids.)

Thank you, Gord!

There was lots more - check out the recording. And check out his site!


I'll be using lots of these fun activities in my next math salon. (Coming up soon. 10 days until Saturday, May 7. Interested? Local to Richmond, California? Email me at suevanhattum on the warmer mail system.)

Tuesday, April 19, 2011

More Number Theory

This week is my spring break. Luckily my son's school is taking spring break the same week (for the first time ever), so we flew to Florida to visit my parents. I brought The Art and Craft of Problem Solving, by Paul Zeitz, along for fun. Here's my version of a problem I've been playing with. (Modified from 2.2.19 on page 38.)

My favorite candy come in packages of 5 or 6. What's the largest number for which I cannot buy exactly that many candies? Can this be generalized?
I got a result yesterday. I liked its symmetry but couldn't figure out why it worked. I think I have the why down now, but I'm not sure I could explain it well yet. I thought you and your kids might enjoy playing with it.

The beach is fun, too.  :^)

[Edited on April 24: The generalization discussed in the comments involves using two numbers that are relatively prime, ie GCF(A,B)=1. JD blogged 5 years ago about "The McNuggets Puzzle", using 3 numbers, which share some common factors. Cool extension!]

Friday, April 15, 2011

Math Teachers at Play #37

This edition starts with a puzzle (thank you, Caroline!): Use five 5's to make 37. I don't see a way yet. Do you?

I just had a lovely time following up the links. One of my favorites was...


Thursday, April 14, 2011

Computer Science: Sorting

Long ago, my first community college teaching gig was a computer science course. I think we taught a few sorting algorithms. This video, linked at Flowing Data, was a delightful reminder of one of them, the 'bubble sort'. (And it makes me want to get to a contra dance.)

Wednesday, April 13, 2011

Math Circle Institute

Each summer Bob and Ellen Kaplan get together with Amanda Serenevy and a few dozen other kindred souls, and play with math. This year will be the 4th annual Institute.

It's held at Notre Dame where the food is pretty amazing. (Well, it's not up to local, sustainable, organic standards, but it's quite yummy.) If you go, your days there will be full of math and camaraderie.

Here's their blurb on their website: (edited a teeny bit)

Math Circle Teacher Training Institute

We will hold our fourth Math Circle Summer Teacher Training Institute on the Campus of Notre Dame, in South Bend Indiana, from July 10th to 16th, 2011.

Demonstrations of our approach, practice sessions in running Math Circles, discussions of theory and practice, and conversations about selected math topics will be hosted by Bob and Ellen Kaplan, Leo Goldmakher, and Amanda Serenevy. Participants will work with children in 1st through 12th grades each afternoon to try out their own Math Circle ideas.

Tuition is $800 for the week, room and board included.
Inquiries can be made by e-mail to

I went for 3 years in a row. I'd go again, but there's a technology workshop I've just gotta check out this summer. Maybe I can get back next summer. Another mom and I dubbed it Math Spa. (We swam every day the first year.)

The participants range from folks whose math skills had me intimidated (I am so over that now!) to people who are just beginning to discover the joys of math. We all worked together, and no one expected you to know things you didn't.

Go! You'll be happy you did.

Sunday, April 10, 2011

Links: Symmetry, Education, Codes

I'm now following another education blog, Re-Educate Seattle, by Steve Miranda. Good starting points - his take on redefining  education intrigued me, and his story about how focused students were,  with no teacher in sight. This blog often brings a smile to my face. (I do wish schools like this were available as part of the public schools.)

My first math-related love was codes and ciphers. Recently two code-related links have looked interesting:

    Sunday, April 3, 2011

    Book Review: Rediscovering Mathematics, by Shai Simonson

    Rediscovering Mathematics: You Do the Math, by Shai Simonson is a great book. It starts with his piece on How to Read Mathematics, which I love, and wends its way through lots of mathematical problems, and lots of ideas about how to teach and learn math.

    When my review copy arrived yesterday I was disappointed to see the hard cover. I suspected that would mean a higher price than most people would find comfortable. The damage is even worse than I expected - $65 (or $53 for MAA members). I think it's a shame to limit the distribution of such good material with a cover price like that. [I had considered approaching the MAA about publishing my book, but perhaps they don't know how to produce books at affordable prices.] Note added five years later: Now it's available used for just over $30. Much better!

    The only other objection I have to this lovely book is that solutions immediately follow the problems posed, so it's hard to resist the temptation to peek. I just now managed to resist temptation, and sat with the problem below (from page 175) until I got it. I don't teach geometry, so my geometric intuitions aren't well-honed. It took me a while, and I reverted to algebraic reasoning for parts of it.
    Given the square ABCD, with side length 1 and circular arcs centered at each vertex, find the area of the region at the center - without using calculus.

    I broke it down into shapes I could handle, and got an answer something like his but not quite. I haven't yet found the discrepancy. (My process was very different from his.)

    Simonson includes a number of problems I haven't seen before, which is quite a feat after all the grazing I've done online in the past few years. And the problems are at lots of levels, so there is much to chew on whatever your mathematical sophistication.

    I like his perspective on how math should be taught and learned. Here he is on memorization (page 169):
    Every mathematical idea has a story. To remember the idea, just recall the story. In mathematics, the stories are proofs and the endings are theorems. The more you turn a proof into a story, the easier it is to remember the ending. Can you tell me what you did last summer? Of course you can. Did you memorize that? Surely not; there is a context and one thought leads to another. Of course it can get a little tedious recalling a story a hundred times just to get to the ending, so sooner or later one just knows the ending. This is the kind of memorizing that a student should do with mathematics.
    An example of this comes earlier. On page 131 he writes, "In the 1960's, it was popular in the U.S. for the middle school math curriculum to include a square root algorithm. The logic of the algorithm ... is clever but cumbersome, unintuitive, and inaccessible to students and most teachers." He compares this method to a Babylonian method: To find the square root of x, guess (r1), divide x by your guess (x/r1), for a better estimate average your guess with this quotient (r2=(r1+x/r1)/2), repeat. Simple, elegant, transparent.

    I didn't learn an algorithm for finding the square root in my U.S. math classes, but I did learn one during my junior year, while attending high school in Brazil (as an exchange student). I've wished I could remember how it worked, but never could dredge it up. I don't think it was anything this straightforward.
    Most students will not forget the Babylonian method because it makes sense. It is a story. It can be remembered because it can be reconstructed. There is context, purpose, and structure. (page 132)

    I'll be sharing quite a few nuggets from this book with my students. If you can afford it, I highly recommend getting this book.

    Friday, April 1, 2011

    April Fool's Day

    My son is 8 and is excited about April Fool's Day. The only mathy joke I know that he'd like is...

    Q: Why was 6 afraid of 7?
    A: Because 7 ate 9!

    But that's not really an April Fool's joke. Anyone have any good ones I can 'share' with him?

    [Pat writes about a good mathy April Fool's joke (at 1975), but it's not my son's speed.]
    Math Blog Directory