Scott Kim, the author of the excellent Inversions, is launching a crusade to bring more puzzles to math education. He gave a talk on this subject at the Gathering 4 Gardner. (See the slides.) Martin Gardner, of course, is the author of the long-running Mathematical Games column in Scientific American. He inspired who knows how many zillions of people to become math teachers. I've attended many math conferences over the years -- without question, by far the longest line I've seen at a math conference was to get books signed by Martin Gardner.
As readers of this blog and users of my Web site may know, bringing puzzles into math education has been a good part of my life's work. One way I've done this is literally, bringing topics from recreational math directly into the classroom and building lessons around them. For example, I have brought pentominoes into more classrooms than anyone, with a book of Pentomino Activities, Lessons, and Puzzles, and a separate box and cards of pentomino puzzles in print continuously since 1984. A new version of the book is coming soon from Didax. At least, I hope it's soon because it is going out of print at McGraw-Hill. (A few copies are left here!) See the geometric puzzles page on my site for more along those lines. My Geometry Labs book (free download!) is infused with puzzles and a puzzle mindset throughout.
But also, my approach to teaching math in general has been puzzle-informed. I've developed much curriculum with a puzzler's ethic. My algebra book Algebra: Themes, Tools, Concepts came under phenomenal attack precisely because of that approach. (For one example, the critic felt that the McNuggets problem should be reserved for graduate school!)
Or see this example of my approach to the use of technology. Not the mind-numbing "graph this, graph that, what do you notice?", but "what functions would yield this interesting design?" Or the construction unit I teach in my geometry classes. More examples can be found throughout my Web site.
Bottom line: the kind of energy the right puzzle brings into the classroom is phenomenal, with all types of kids. To get teachers to buy into this, the challenge is making the connection between puzzles and the core curriculum, rather than promote puzzles as something that just happens on the side. I have done my bit in this direction, and I wish Scott much success with his campaign.
--Henri
Saturday, May 12, 2012
Monday, April 23, 2012
Presentation at SF State, May 1
I was invited by the San Francisco Teachers' Math Circle to present some problems. Here's the basic info.
--Henri
When: Tuesday, May 1 at 6:00 PM
Where: Trailer P (behind Thorton Hall) at San Francisco State University
3699 19th Avenue, San Francisco, CA
As far as I know, the event is free, and dinner is included!
I will present a pattern blocks-based problem which will be familiar to people who have taken my Visual Algebra workshop. (But it's a problem one can revisit, as there are lots of solutions!)
If there's time, we'll follow up with additional problems drawn from Algebra: Themes, Tools, Concepts, and Geometry Labs.
I have chosen problems that are accessible to students, at one level, but where there's more to find out at a teacher level. If you haven't heard me talk before, this would be a good opportunity to get acquainted. I hope to see some of you there!
Friday, April 13, 2012
Early Bird Discount
About three months ago, I mentioned my summer workshops for teachers and summarized how they will differ from their previous incarnations.
Brief recap:
- Hands-On Geometry, June 18-21 in San Francisco
- Visual Algebra, August 13-15 in New York City
- Re-imagining High School Math, August 16-17 in New York City
The workshops are sponsored by the Center for Innovative Teaching, which I direct, and will be held respectively at the Urban School of San Francisco and at Chapin School in New York. I hope to see some of you there!
There is more info on my Web site, and on the CIT site.
--Henri
[I updated this post to reflect the changed deadline, and to add the link to the CIT site]
Brief recap:
- Hands-On Geometry, June 18-21 in San Francisco
- Visual Algebra, August 13-15 in New York City
- Re-imagining High School Math, August 16-17 in New York City
The workshops are sponsored by the Center for Innovative Teaching, which I direct, and will be held respectively at the Urban School of San Francisco and at Chapin School in New York. I hope to see some of you there!
Breaking news! The deadline for the Early Bird discount rate for my Hands-On Geometry workshop (15% off) has been pushed back to April 30!
There is more info on my Web site, and on the CIT site.
--Henri
[I updated this post to reflect the changed deadline, and to add the link to the CIT site]
Saturday, March 31, 2012
The Quadratic Formula
In the period following the publication of Algebra: Themes, Tools, Concepts, we started asking our Math 1 students to write and illustrate a short report or poster, tying in four representations of a trinomial in the form x^2+bx+c. (Math 1 is Urban School's Math Department's version of Algebra 1, sort of.) I just posted a version of this assignment on my Web site.
One year, while grading this assignment, it occurred to me that it should be possible to use the "constant sums, constant products" setup as a path towards the quadratic formula. And sure enough, it is. I wrote an article about this for the "Delving Deeper" department of The Mathematics Teacher. ("A New Path to the Quadratic Formula", February 2008.) The proof involves neither completing the square, nor parabolas!
A worksheet guiding you through the proof has been on my site for a long time, but I just added a presentation of it on slides. (No audio, but the steps are spelled out sufficiently, I hope.) Let me know if it doesn't work on your computer.
You can download the presentation (3.2MB, in .mov —QuickTime— format) if you want to have it in your own computer. In fact, this version has helpful animations which do not work in the online version.
Finally, as long as I was adding these things, I thought I might as well organize the Parabolas and Quadratics page better, dividing the many links into three approximate groups: Algebra 1, Algebra 2, teachers' mathematics.
I still have a little more to add to the quadratic part of the site, but it will have to wait.
--Henri
One year, while grading this assignment, it occurred to me that it should be possible to use the "constant sums, constant products" setup as a path towards the quadratic formula. And sure enough, it is. I wrote an article about this for the "Delving Deeper" department of The Mathematics Teacher. ("A New Path to the Quadratic Formula", February 2008.) The proof involves neither completing the square, nor parabolas!
A worksheet guiding you through the proof has been on my site for a long time, but I just added a presentation of it on slides. (No audio, but the steps are spelled out sufficiently, I hope.) Let me know if it doesn't work on your computer.
You can download the presentation (3.2MB, in .mov —QuickTime— format) if you want to have it in your own computer. In fact, this version has helpful animations which do not work in the online version.
Finally, as long as I was adding these things, I thought I might as well organize the Parabolas and Quadratics page better, dividing the many links into three approximate groups: Algebra 1, Algebra 2, teachers' mathematics.
I still have a little more to add to the quadratic part of the site, but it will have to wait.
--Henri
Thursday, March 29, 2012
Playing Games
I attended an Escape from the Textbook! meeting last weekend. The first part of the meeting focused on the mathematics of the game of Set, and the second part launched a conversation about assessment, which will continue in future meetings. Avery Pickford took notes, and posted them on his excellent blog, "Without Geometry, Life Is Pointless". (Great name for a blog, and the converse is true: without life, geometry is pointless!)
One issue that came up at the meeting was: what could we learn about grouping in the mathematics classroom, from our experience playing the game of Set in groups of four? If you don't know the game, it shouldn't be too hard to learn about it on the Web, but you do need to know one thing about Set before you read on: this is a game where participants call out as soon as they see a "set" (as defined by the rules), so there is a lot of pressure to think fast.
One participant, who did not know the game, was in a group with three much more experienced players. This was, as you might imagine, quite overwhelming, and she commented that this episode confirmed her belief that it was best to organize students in homogeneous groups. The fact that her group-mates were thoughtful and considerate (probably much more so than many students in middle or high school) really didn't help. She would have been much happier in a group of all beginners.
While it's easy to determine who has different levels of experience in a given game, it is not so easy to make homogeneous groups. Among beginners, some may be fast learners. Among experienced players, some may be experts. And homogeneity becomes all the more difficult to achieve in a more general situation, as almost any group-worthy mathematical task involves a range of different skills and understandings, and rewards a range of different approaches. Thus my preference for random groups, changing every couple of weeks. (I've written more about this for my department, and shared it on this page.)
In my group we addressed the heterogeneity by taking turns instead of just calling out "set!" This helped calm things down, and made it much easier for me to think. (Like many students, I have a hard time thinking under time pressure.) We were informal about this, so that if someone was taking a very long time to find a set, others gently notified us they had found one, and they were able to take their turn early. In fact, the generally less frenetic atmosphere made it possible for our resident expert to explain how she thought about the game, and how she helped students think about it.
As a rule, I'm not a big fan of games in the classroom, precisely because they emphasize and reinforce status inequalities among students. I much prefer puzzles, which on the one hand more closely mirror the process of doing math, and on the other hand are easier to set up in non-competitive or less-competitive formats.
But back to Set. The strongest connection with the standard curriculum is with counting, as encountered at the beginning of an introduction to probability. In particular, our group found the question "how many cards can you have and still not have a set?" quite interesting, but we ran out of time before we answered it. (It turned out to not be as hard as I thought, and I was able to figure it out later that afternoon.)
--Henri
PS: the next Escape from the Textbook! meeting will be on June 2 in San Francisco. More info will be posted in the Escape! community on edWeb.net, which you can join here.
One issue that came up at the meeting was: what could we learn about grouping in the mathematics classroom, from our experience playing the game of Set in groups of four? If you don't know the game, it shouldn't be too hard to learn about it on the Web, but you do need to know one thing about Set before you read on: this is a game where participants call out as soon as they see a "set" (as defined by the rules), so there is a lot of pressure to think fast.
One participant, who did not know the game, was in a group with three much more experienced players. This was, as you might imagine, quite overwhelming, and she commented that this episode confirmed her belief that it was best to organize students in homogeneous groups. The fact that her group-mates were thoughtful and considerate (probably much more so than many students in middle or high school) really didn't help. She would have been much happier in a group of all beginners.
While it's easy to determine who has different levels of experience in a given game, it is not so easy to make homogeneous groups. Among beginners, some may be fast learners. Among experienced players, some may be experts. And homogeneity becomes all the more difficult to achieve in a more general situation, as almost any group-worthy mathematical task involves a range of different skills and understandings, and rewards a range of different approaches. Thus my preference for random groups, changing every couple of weeks. (I've written more about this for my department, and shared it on this page.)
In my group we addressed the heterogeneity by taking turns instead of just calling out "set!" This helped calm things down, and made it much easier for me to think. (Like many students, I have a hard time thinking under time pressure.) We were informal about this, so that if someone was taking a very long time to find a set, others gently notified us they had found one, and they were able to take their turn early. In fact, the generally less frenetic atmosphere made it possible for our resident expert to explain how she thought about the game, and how she helped students think about it.
As a rule, I'm not a big fan of games in the classroom, precisely because they emphasize and reinforce status inequalities among students. I much prefer puzzles, which on the one hand more closely mirror the process of doing math, and on the other hand are easier to set up in non-competitive or less-competitive formats.
But back to Set. The strongest connection with the standard curriculum is with counting, as encountered at the beginning of an introduction to probability. In particular, our group found the question "how many cards can you have and still not have a set?" quite interesting, but we ran out of time before we answered it. (It turned out to not be as hard as I thought, and I was able to figure it out later that afternoon.)
--Henri
PS: the next Escape from the Textbook! meeting will be on June 2 in San Francisco. More info will be posted in the Escape! community on edWeb.net, which you can join here.
Wednesday, March 28, 2012
Additional notes on ATTC
I just added one final page about Algebra: Themes, Tools, Concepts. It consists of errata and other notes I took in the first few years after the book was published, in anticipation of a second printing or second edition that never happened. It may be a useful complement to the abundant notes in the Teachers' Edition.
--Henri
--Henri
Thursday, March 8, 2012
My other blog
Check out my other blog: Word Salad, not related to math education. With Joshua Kosman, I co-construct the cryptic crossword for The Nation magazine. The blog is about that.
--Henri
--Henri
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