Sunday, February 24, 2013

What would Suzuki Method for Math Look Like?

Grant Wiggins' blog features a post by Susan Fine on the Suzuki method. This image delighted me:
"Review is a significant feature of the method, emphasized in daily practice as well as in annual festivals with “Playdowns,” in which the program begins with an advanced piece, and the students play down to the Twinkles: five versions of Twinkle, Twinkle Little Star, each with a distinct rhythm and learned at the start of Book I. Students join in the playing when the program comes to their most advanced piece."
This is the opposite of a contest model, in which everyone starts, and contestants drop out until only one is left.

Later she points out:
I often reflect on how the combination of inspired pedagogy and content-rich curriculum creates an ideal learning environment. This point seems rather obvious, yet I am hard pressed to come up with other examples of the two being so beautifully wedded to one another on behalf of learning...

What would math instruction look like if it followed a similar model?

Friday, February 22, 2013

Carnival of Mathematics: Coming Soon

I'll be hosting the 96th Carnival of Mathematics here. Do you have a post that should be included? If you do, submit it soon. The deadline is March 1st (next Friday).

To give you a taste of what the Carnival offers, here's a link to the 94th Carnival. (The 95th was in Spanish, and the English translation leaves a bit to be desired.)

Tuesday, February 19, 2013

The Collatz Conjecture

If your number is even, cut it in half.
Write down your new number, and repeat.

Do you always end up at 1, 4, 2, 1, ... ?

Collatz's Conjecture was that you do. No one has found a counter-example in the century or so since he proposed his conjecture. No one has a proof that it will always happen either.

Kids like playing around with this. So I like it. But I hadn't played around much with it until now.

Discrete Mathematics With Ducks mentions this problem / puzzle / conjecture in Chapter 5, on algorithms. So I made a spreadsheet, and am looking at the lengths of the series for each number up to 100. I noticed two things:
1. Quite a few consecutive pairs of numbers have the same length series. (I'm counting how long until the number 1 shows up.) It seems to always be an even number and then the number after it.
2. I haven't counted them all yet (I'm doing it by hand), but there seems to be a big gap. Most sequences have lengths under 25, and a few have lengths over 100. So far, I haven't found any sequence lengths in between 25 and 100.
I made the spreadsheet by putting the numbers 1 through 100 in the first column. Every other cell gets this (with the cell name changed automatically):    =IF(INT(B1/2)=B1/2,B1/2,3*B1+1)

What do you notice about the Collatz series? What do you wonder?

Friday, February 15, 2013

Math-Storytelling: What math stories do you like?

I woke up early, thinking about how much I like storytelling as a way to ground math lessons.

• I'd like to get my son working on the Life of Fred books. I thought I bought the first one. But where is it? I have so many math books, it might take a detective to find it. (Fred goes from basic arithmetic past calculus.)
• I was also thinking that I need to get A's address, so I can send her Math Girls I and II. (The math here is pretty high-level.)
• I got up, and look what I found in Google Reader! The textbook had a typo, and the class ran with it. Then they got Grant Wiggins in on the act, and he gave them some more food for thought - hay, actually. The class that's having all this math fun is a middle school algebra class.
• I've mentioned before how differently I've organized my calculus course. (Why, oh why, do most textbooks start with a whole chapter on limits?!) So of course, I'm silly enough to think maybe I should write a calculus textbook. If I do, it's gotta have some characters who are wandering through the wilds of calculus, or something like that...

Other math storytelling:

• The Cat in Numberland, which will be available soon, is good for almost any age. The math is simple enough for a child to think about, and yet deep enough to get even most adults scratching their heads.
• The Number Devil explores lots of topics that can be enjoyed by middle schoolers on up.
• The Man Who Counted can be enjoyed by upper elementary students on up, and the story is delightfully intricate.
• Surreal Numbers was perhaps the first math story I encountered, many years ago. Proof-oriented and dense, with just enough storyline to add a little spice.
•  This list and this overlapping one include a few other math stories.
• Art of Problem Solving is starting a series for kids called Beast Academy. So far they have 3 of the 4 books for 3rd grade. They eventually intend to provide beasts for 2nd through 5th grade. (Too bad they use grade designations - I think older kids would like these, but my son would probably be annoyed at doing something that's for 3rd grade.)

If you like to tell stories, it might be even more fun to make up your own. Maria Droujkova created Math Storytelling Day on her (and my!) birthday to celebrate making our own stories. Here's one a friend made up.

What have I left out?

Sunday, February 10, 2013

Derivatives of Sine and Cosine

Next week I'll be working with my students to figure out these two derivatives. I don't want them taking notes on all the steps, so I made a handout that reflects what I think we'll do in class. I'm embarrassed to admit how many hours I put into this. The good thing is that I know the details of these proofs a lot better after having typed this out.

I used Word with MathType, which has usually looked fine to me. I'm starting to wish I knew LaTex. (It's definitely on the list of technology I want to learn soon.) I used Geogebra for diagrams, took snapshots using Jing, and pulled the .png files into my Word doc. There's probably a better way to do that too.

First is an estimation activity:
Estimating the Derivative From Graphs of Sine and Cosine by Sue VanHattum*

Then here is the complete proof I typed up:
Derivatives for Sine and Cosine by Sue VanHattum

What would you do differently?

[Note added on 2-21-14: I just taught this for the third time from my handouts. I loved how it went yesterday. When I came back to this post, I re-read the comment from Alemi at The Virtuosi, and studied the blog post referenced. It is exactly what I was hoping might exist: a nice short argument with the same result. Check it out!]

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*This is my first time using scribd in a very long time. I hope these are useful to someone. How do I get the documents to show up in my blog post?

Thursday, February 7, 2013

Discrete Math: Energy Level in the Classroom

My discrete math course is small - about 14 students. It's easier for me to build the enthusiasm up with a bigger class, of say 20 to 30 students. But for the first 3 weeks it seemed like things were going well with this small group. I love the topics we've addressing. (I'm following the textbook Josh Zucker recommended, Discrete Mathematics with Ducks, by sarah-marie belcastro.) And the students seemed engaged.

This Monday, there was no energy in the room. Students were less engaged and responsive, and one was even sleeping. (After class, someone suggested that the Super Bowl might have been part of our problem. I never would have thought of that...) I left class depressed. It doesn't matter how much I like this material, if I can't get the students interested.

On Wednesday, I came in early and asked questions of the students who were there. "Do you like this class?", "What's working for you?", "What isn't?" We usually sit in a circle, but I left them as they were so I could keep asking my questions. As more students arrived, they joined in the conversation. And the desks stayed in rows.

A number of them said they'd like more lecture, so I jumped into the problem we'd been considering on Monday. R's conjecture was that if the are more odd vertices than even in a graph it will be impossible to trace a path over every edge just once, and if there are more even vertices than odd, this will be possible. M had an alternate conjecture, that if there are more than 2 odd vertices, it will be impossible, and with 2 or less it will be possible.

We had an example with 4 odd vertices, so I asked if we could get an example with 3 odd vertices. That turned out to be the perfect question (for that group at that moment). They worked alone, with a partner, or with a small group. They came to the board to show their examples. I said, "This is how mathematicians work. We look for examples, and after enough failure, we try to use our failure to prove it's impossible. If we have trouble with that, we might go back, with new thoughts from the proof attempt, to looking for an example."

So we tried to prove it was impossible. Our attempts gave us plenty of chances to use the language of graphs (edges, vertices, degree), that they are just learning. One student said it might be easier to show that it's impossible to have just 1 odd vertex. I liked that. I said, "One good strategy is to start with a simpler case. Another is to generalize. Maybe we could try to prove that any odd number of odd vertices is impossible."

We weren't getting there quickly, and some students were still looking for an example. S came to the board to show his example. It turned out that he'd counted the edges wrong at some of the vertices. He said sorry, and I talked about how we want to celebrate our mistakes. He came up to the board later, so at least he's willing to take the risk, although he might not be celebrating yet.

M made up a new notation for keeping track of how many even and odd vertices we had. I loved that.

V said something like "We have to notice what doesn't matter." (I think his wording was better than this, but I don't have it now...) I loved what he said, wrote it on the board, and asked people to put it in their notes. If they're going to take notes, I want their notes to be on problem-solving strategies as much as on the content.

Another student, maybe R, suggested adding up the vertex degrees. There was some discussion, and it seemed like they'd pulled together everything they needed. So I said I thought it might be time to write down what they'd come up with. I did that, with lots of questions to them, and lots of unsolicited suggestions from them. We struggled with defining degree of a vertex.

We had an example on the board with a loop, so defining degree as 'the number of edges attached to that vertex' seemed wrong. (The lower vertex in this drawing has degree 3, but only has two edges attached to it.) I meant to check our text, but didn't manage that during class. Her definition is no better. We settled on: The degree of a vertex is the number of edge endings at that vertex. I loved that we were dealing with the issue of why definitions need to be precise.

I then wrote what I thought I had heard from a student: The total degree is the sum of all the vertex degrees. But I see now that I should have gotten this step from the students. I think a bunch of them suddenly saw it after I wrote that. It would have been better if I'd asked the student who had offered it to help me with the wording.

This whole process took over an hour. If I had done more of the 'work', it would have used up 5-10 minutes of class time, and the students would have believed they understood it. It would have seemed pretty obvious to most of them. And yet, they wouldn't have learned nearly as much.

Here's our proof:

I thought after class about Bob Kaplan's list of ways he attempts to 'disappear'. Yesterday I felt like I achieved that to some degree. I mentioned that after class to a student, and he said students would have talked longer if I hadn't stood up. Yep, I still have a ways to go.

And yet, it was standing up at the beginning of class (instead of sitting in the circle with them) that got the energy level up. Interesting...

Critique of Common Core, Focused on Math

I read Diane Ravitch (skimming past a majority of the numerous posts) so I can stay aware of what's happening to public schools across the nation. This may be the first time I've seen math discussed in depth on her blog. Gary Rubenstein wrote a great post on Common Core Standards, and how they affect math class. He used the Pythagorean Theorem as his example, describing some of what he'd do with 8th grade students. (My college students would have fun with this, too, and would learn some valuable math from a lesson like this.)

His take on how to assess was refreshing. See if you can explain the group of pictures below after reading the article.

The idea is to shade the bottom two pictures with the blue, yellow, and brown from the top pictures to show what's the same. I got something algebraically, but what I did wasn't nearly as elegant as the shading he showed.

Friday, February 1, 2013

Resource: Math Relax!

At least a quarter of my students would say they have test anxiety, and blank on tests. I think many of them haven't figured out yet how to study effectively, and don't really know as much as they had hoped when they walk into a test. But there are likely still quite a few who really do know the material but get overcome by their anxiety.

About 15 years ago, I decided to create a resource for these students. I had read a great book, by Margo Adair, on creating guided visualizations, and was ready to apply the ideas to this common problem. I used music that I may or may not have had a right to use. When I came to my current college, they were concerned about that. So I re-recorded my guided visualization with beautiful flute music that my colleague, Wayne Organ, composed just for this, and then played. He also provided the studio and all the technical expertise for recording.

Back then I was burning copies of this on cd's for my students. (It was and is too big to send as an email attachment.) One day a few years back I heard two students talking in the math lab. One was telling the other about this cd she had called Math Relax, and how much it had helped her. I felt like I was in a dream - it sounded like a commercial for what I'd made. I stuck my head out the door and said I had created that, but I guess she was embarrassed - she didn't react much.

Eventually, I put it on my blog. I'd like to have it out in a more public spot, and YouTube seems like the place. But that needs visuals. Last week a student of mine offered to help me with some of my projects, as a volunteer. Wow!  He has put together some photos (creative commons) and made my sound track into a video.

So ... Math Relax is now available on YouTube!

And here it is ...

Credits
Voice: Sue VanHattum
Flute music: Wayne Organ