I wanted to show a deeply engaging problem that doesn't take any technology to play with. But of course, we were using a high-tech way to gather together, and that changed the dynamic dramatically. Here are a few pros and cons:
- (+) About 18 people from far-flung parts of the world got to work/play together, without ever leaving our homes (though one person mentioned being at the library). I was in my pajamas.
- (-) I found it very frustrating to be the only one talking audibly. (For most of the session, everyone else used the 'chat' area to ask questions and make comments.)
- (+) Many participants used the whiteboard to draw their pancake slicing attempts. Even though it was probably new to most of them, it was pretty easy to figure out.
- (-) Participants couldn't collaborate with one another easily. It's so much more fun in person, with people showing a partner their ideas. When we're really physically together, our excitement is infectious.
- (-) I wanted to mention a bunch of books. In person, I would have picked each one up at an opportune moment, shown why it's cool in less than 15 seconds, handed it to a participant, and moved on. It took longer, and disrupted the flow more when I showed a book at the webinar, so I didn't mention all the books I thought might interest people. I made a list below.
- (-) I was distracted by the new environment; that will go away over time. I forgot to give my contact information out. If you have any questions or comments, you can leave them here, or email me at suevanhattum on hotmail. I also didn't judge time well, which I'll get better at with more experience.
Here are some of the questions and comments that came up:
- John Golden: When you pose a math circle problem do you try to avoid any further scaffolding? [I said that this past summer, when one student proposed a solution the other students didn't understand, I helped get the other students up to speed. But that wasn't an optimal solution for the problem of one person way beyond the rest. I didn't have a good answer in general to John's question. Anyone experienced with math circles care to reply to this?]
- Telannia: Do you find students getting frustrated and just zoning out? [I said sure, but maybe less than in a traditional lesson. Also, the Kaplan's tell a good story of a girl who seemed for many weeks not to be getting it, and who then proceeded to propose a vital step in the solution process, blowing the minds of her fellow students.]
- Telannia: Are the math circles you have done all on your blog? [I've mainly done this pancake problem, and Pythagorean triples. I'll write a post soon about some of my favorite topics in math circles I've attended. I've put a list of some good sites for these below.]
- John Golden: Do you care if people get to the answer or are you just looking at the process? [I find getting to the answer so exciting, but if it comes from the teacher instead of the students, we haven't really gotten there. I do leave people hanging sometimes.]
- Amanda Serenevy: We have been using Fermi Estimates in some of our Math Circles this summer. [The link is to a fabulous report her students have put together.]
- Maria: I want to ask people here - what are your favorite "low-tech" problem prompts? "Stuff" that you use?
Patterns - sniffing, breaking, and finding:
Sue: Avery put a list on his blog, of the habits of mind he’d like to encourage in his classes. I loved the pattern sniffing.
Sheng: Ben has that post about pattern breaking.
Maria: (a thought from a conference) Patterns do not lead to formulas at all.
Sue: With the pancake problem we're doing, we got a recursive formula, but not a functional description, of what's happening.
Online sources of math circle style problems:
- The Math Circle (the Kaplan's site)
- Riverbend Community Math Center
- National Association of Math Circles
- Colin recommends Project Euler, but warns that it's tech oriented.
- Tom Davis' Math Circle Topics
- Josh Zucker's problem page for the Julia Robinson Mathematics Festival
Books:
- Out of the Labyrinth: Setting Mathematics Free, by Bob and Ellen Kaplan, is a great introduction to math circles.
- Circle In a Box, by Sam VanDerVelde, gives some nuts and bolts for running math circles, and includes a great selection of problems.
- Problem-Solving Strategies: Crossing the River With Dogs, by Ken Johnson and Ted Herr, has some good problems, and good ideas about how to work on problem-solving in the classroom.
- The Art of Problem-Posing, by Stephen Brown and Marion Walter, has some good problems, and a good analysis of how to get students asking the questions.
- Mathematics: A Human Endeavor, by Harold Jacobs, is an alternative textbook - it's got great problems, they're engaging and get the student really thinking.
- The Art and Craft of Problem-Solving, by Paul Zeitz, has a cool analysis of the problem-solving process, and some interesting problems.
- How To Solve It, by George Polya, is a classic, and includes some intriguing problems.
- The Cat In Numberland, by Ivar Ekeland, introduces students (of almost any age) to deep concepts related to infinity, through an engaging story. Story-telling is a great low-tech way to make problems engaging.
- John Golden recommends Symmetry, Shape, and Space: An Introduction to Mathematics Through Geometry, by Christine Kinsey and Teresa Moore.
For my sabbatical report, I had to write up 3 new projects I'll be doing in my classes. Here's what I wrote about this topic:
Magic Pancake Problem
Introduction.I draw a circle on the board.
“Who likes pancakes?” [I usually get enthusiastic responses when I’ve done this in math circles.]
“What type of pancakes do you like?” [Collect a variety of responses.]
“This pancake is magic. When we cut it into pieces, each person’s piece will expand to be just the right size, when they're eating it and when it's in their belly. It will also become exactly the type of pancake you love best.]
“How many pieces before we've cut it?” [one…]
Data Gathering.
“If we cut it once, how many pieces will there be?” [two]
“Twice?” [there could be 2, 3, or 4… discuss goals… we want to feed us all]
“Hmm, should we keep track of what we’re finding out? What’s a good way to do that?” [Some of them will have used what they like to call a t-chart. That’s probably what I’ll use, unless something else intrigues the group.]
“Now, with your partner, do 4 cuts. (We’ll go on to 5, 6 and 7 cuts…)” [As they work, they’ll be checking against neighbors. When everyone has started thinking about 5 cuts, I’ll stop them to either say how many we all got for 4, or to get consensus.]
We continue, finishing 5, and starting 6.
Searching for Pattern.
“Hmm, does anyone see a pattern?” [At this point, the conversation can go in lots of different ways.]
We’ll discuss how we know our pattern stays true, an example of inductive versus deductive reasoning.
Once they do see a pattern, it’s still described in an inconvenient way (recursively), and we’ll search for a way of describing the pattern that will allow us to predict how many pieces come from 30 cuts. We will not find this in our first hour of exploration.
Graphing.
Perhaps on this first day, perhaps later, we will graph the number of cuts on the x-axis versus the number of pieces on the y-axis.
“If we connected these points, would we get a straight line?” [no…]
“What does this shape look like?” [a parabola, but they may not see this yet…]
This will be used during the introduction to the graphing section of the course, either by reviewing what we did before, or remembering our magic pancake, and doing this step for the first time.
Quadratic Relationships.
A few months have passed since we first looked at this problem, so we’ll do it over again. [Students will direct the steps, and they’ll do it quickly. This demonstrates for them how much they’ve learned.]
We now attempt to find a non-recursive relationship between number of cuts and number of pieces. The graph now may be recognized as a parabola, which tells us we have a quadratic relationship. Unlike most parabolas, we won’t see a vertex. [Hmm, real life problems are often more complex than textbook problems.]
The final answer involves a fraction, which adds another layer of complexity to the problem.
“In the problems we’ve done so far, we’ve been given an equation and we’ve graphed it. This time we have points and we want to find an equation that fits these points. Hmm…”
We will explore adding the numbers from 1 to n. We may explore triangular numbers also.
Closure?
It’s possible the group won’t figure out the relationship entirely. I will not show it to them. If this happens, I’ll discuss real mathematical research and the fact that real problems often take years to solve. I’ll also discuss with them the benefits of leaving an open question in their minds. [It keeps them thinking.] In our last session exploring this problem, I will review the concepts they’ve used in their (perhaps partial) solution of the problem.
That's it, folks. Please bring on your questions and comments.
Posts
Hi Sue
ReplyDeleteIt was interesting in some ways to observe what was happening in your session on Wednesday (or Thursday where I am!). You asked that those who knew the answer should take a back seat, I wonder if this happens in maths circles too - do students zone out because they can see what is going on?
No-one responded to my comment about noticing triangle numbers - the number of pieces from n cuts is basically the nth Triangle number plus 1. That may have been a way of turning a pattern into a formula... but I remain to be convinced that pattern-sniffing will always work or lead to an algebraic generalization (if that is the end-product you want).
The real challenge, as I am sure you know, is to help students begin the transition from concrete patterns towards the abstract formulations needed for higher levels of mathematics.
Hi Colin, Thanks for commenting. It was Saturday here. Does that mean it was Sunday where you are?
ReplyDeleteDid you zone out? I think someone who's done it before might. But someone who sees the relationship for the first time, I'd expect them to be excited enough to stay quite tuned in.
I remember now your comment about triangle numbers, but I didn't manage to hold it in my head long enough on Saturday to respond to it. (The electronic format of our gathering was very confusing for me.) I should have asked what you were seeing.
Maria said something along the lines of what you're saying - that patterns are concrete, formulas are more general, and getting from the one to the other is the hard part, and not just a matter of pattern-sniffing.
I wonder what you think of looking at first and second differences to see that you have a quadratic relationship.
Or, we could graph the points we'd get from setting x values equal to the number of cuts and y values to the number of pieces. We'd see a parabola, and could then work on finding the quadratic relationship.
I'll be coming back to this problem in my classes multiple times this semester, and I'll write about the progress my students make.
Hi Sue,
ReplyDeleteNo, still Saturday but evening not afternoon. I wasn't zoning out completely! I was actually thinking about the story (apocryphal or not) of Blaise Pascal in primary school, where the teacher set the students the task of finding out the sum of all the numbers from 1 to 200 - thinking it would keep the children quiet for half-an-hour or so. Pascal apparently produced the general formula for the nth triangle number and used that to calculate...
Of course, that level of insight is rare! I think that triangle numbers and square numbers will often provide inroads to quadratics because they are usually introduced fairly early on in concrete forms, even if you don't go on to make generalizations. Or visually representing n-squared as the nth and (n-1)th triangle numbers combined, etc.
If you plotted points, you would be able to see a curve forming, and using 1st and 2nd differences is a way to check or test for the presence of a quadratic/cubic/geometric relationship.
The more interesting questions, though, are why should the relationship be quadratic or why is it that the nth cut produces a maximum of n new regions... the inductive reasoning which underlies much of 'proper' mathematics.
Incidentally, the method apparently used by Pascal was to reverse the series, add to the original and then half it.
(1+2+3+.....+n)+ (n+(n-1)+(n-2)...+2+1) = 2 ((1+(n))+ 2+(n-1) +...)
It makes more sense if you set it out vertically and you can see that there are n instances of (n+1)...
A Christmas puzzle, related to sums/triangle numbers, is to ask how many gifts the True Love had sent all together by the end of the Twelfth Day of Christmas. ;-)
Is Pascal credited with that, too? I've always heard it as Gauss, and the teacher giving the numbers 1 to 100.
ReplyDeleteI knew about the Gauss story, which does seem to be wider-spread and better documented, but the Pascal story also seems plausible given his precocity.
ReplyDeleteAlso, that's why I said 'apocryphal', not being able to find online references to it... maybe some kind of French/German thing is going on... ;-)