Jan Nordgreen, Introducing Another Math Enthusiast

I've been enjoying Jan's blog for about a year now. Most of his posts come in pairs, one post with a photo of someone famous along with a good quote of theirs, the other post with a math problem/puzzle, usually embedded in a very silly dialogue. I enjoy his brevity, his sense of humor, and his choice in puzzles.

Just recently, I found out that he's written in much more depth, and has lots of intriguing essays gathered together here. My favorite essay so far includes thoughts on how to talk with your child's math teacher about math:

These are some of the questions I would ask [the teacher]. Is she interested in mathematics? Does she know the subject well? Does she believe the mind is a pot to be filled or a fire to be ignited? Does she like to solve problems? What kind of problems? Does she encourage the students to show perseverance when the riding gets rough, tolerance towards opinions held by others and pride in work well done? Does she teach the students general problem solving skills? Does she teach them guessing, pattern spotting and the difference between a guess and a proof? Why does she like to be around children? Why does she like to teach mathematics?  

 A few of Jan's puzzles will be featured in my soon-to-be-published book, Playing With Math: Stories From Math Circles, Homeschoolers, and the Internet. 

Enjoy!

These are a few of my favorite blogs...

I recently received my first blogging award (no badge included). It's the Stylish Blogger Award, bestowed on me by Laura Grace Weldon. Neither she nor I see ourselves as stylish, so I decided the award needed a little change for us (think of it as evolutionary pressure). The award is meant to be passed along, so I'm passing along the Blogger of Substance Award  to these blogs whose posts I've especially enjoyed chewing on:

Homeschooling & Parenting

• Tiff's FreePlayLife
• Laura Grace Weldon (now you get the new award, too, Laura)

Math, Math, Math

• Denise Gaskin's Let's Play Math!
• Dan Finkel's Math for Love
• Gary Ernest Davis' Republic of Mathematics
• John Golden's Math Hombre
• Pat's Blog
• Ben Blum-Smith's Research in Practice

Other Teacher Blogs

• Doug Noon's Borderland
• Michelle Martin's Rookery

Broadening My Horizons

• Chad Orzel's Uncertain Principles
• Barry Leiba's Staring at Empty Pages
• John Cook's Endeavor
• Civil Eats
• Mary O'Keeffe's Bed Buffaloes In Your Tax Code

In the Community Organizer category is Kate Nowak's f(t). And here's one last category, for my favorite non-blog sources: Julie Brennan's Living Math Forum, Maria Droujkova's Natural Math group, and James Tanton.

There are tons more blogs I love to read, but they may post less often, or may have hidden from my attention this morning for some other strange, esoteric reason. (To accept this award, please pass it forward to some of your favorite Bloggers of Substance, tweaking it in the process if you feel like it.)

The Stylish Blogger Award also required making a list of 7 random things about yourself. I think that needs to morph a bit, too. But to what? I'll list some things I could have blogged about but never have:

1. My first mathematical passion was codes and ciphers. Maybe reading all those books about the use of codes and ciphers in wars started me on my anti-war path.
2. I haven't posted my complete thoughts on the regions in a circle problem because I don't want to post the solution, but I'm planning to submit my 6-page paper on it to the Journal of Humanistic Mathematics.
3. In my twenties, I taught junior high math for a year and a half, and was fired because I was terrible at keeping the kids on task. I won a grievance, but figured I'd be happier teaching at college level.
4. I flunked Calc II in college. (It was an honors course, more at the analysis level.)
5. I never took trig in high school, because it sounded boring (too much memorization). I was never told that I'd need it for calculus (where I had to memorize lots of the trig identities to get by). When I finally really learned it (to teach it), I fell in love with it.
6. Online math communities have tripled the time I spend playing with math. At least. 
7. If you haven't noticed, I post about the rest of my life (once in a while) at my other blog. You can read there about my son, my chickens, my political views, kids' books, and more.

Let me know if you fall in love with a blog I've introduced you to.

Cracking the Lottery

Mohan Srivastava, a geological statistician living in Toronto, figured out a way to tell which scratch-off lottery tickets were winners, with 90% accuracy. He told the Ontario lottery authorities, and they took that particular game off the market. But most scratch-off games are flawed in similar ways. Here's a fascinating article from Wired on how he did it, and other flaws in the lottery systems.


[Thanks for the pointer, Jan.]

Math Teachers at Play #35

It's up at Let's Play Math. Lovely collection, along some some cute math poetry. Enjoy!

I'm in love with Vi Hart('s imagination)

Wind and Mr. Ug is going to keep me thinking mathematically, and metaphorically, for days.

Making Connections - Graphing Helps Me See Trig Identities

When you study trigonometry, there are dozens of identities to learn. I’ve taught the course 20 times, and with my bad memory, I still don’t have all the identities in my head. All the basics are there, but some of the identities I need for Calc II are not in my memory.

For example, it turns out that we need an identity that will change the form of sin²x, in order to evaluate the integral of  sin²x. While my students were taking a test on volumes of rotation a few weeks back, I was thinking about integration techniques, which I’d be teaching next. I wanted to solve this problem, and had no table of trig identities handy. So I thought I’d draw a graph to see what it might tell me.

The next day of class I showed it to my students, and today a student who was absent asked to see my 'derivation'. I was tickled. And then I thought maybe some of you might like to see it. This is how I can remember things - visually. Because then everything is connected.


I want to take each y-coordinate on this graph of y=sin x, and square it, to get y= sin²x. The points where y=0 stay put, and so does the point where y=1. The point at (3π/2,-1) moves to (3π/2,1). The portions of the graph very close to y=0 look almost like straight lines, so squaring those portions will get us a shape that’s close to a parabola near the y=0 points.

Now we have something that looks like the new graph I've added in here. Well, that looks like a trig function - what would it be? The frequency has doubled, so we use 2x. It looks like an upside down cosine, with horizontal midline at y= 1/2, and amplitude (distance from midline to top or bottom) = 1/2. So I get ...
y = 1/2 - 1/2 * cos(2x), which is usually written as ...
y = (1-cos 2x)/2. (Apologies for my limited html skills.)

Now that you've seen this, can you find the identity for cos²x?

Math + Hearts

Everyone else celebrates Valentine's Day today; my son and I celebrate Family Day. I adopted my son 8 years ago today; it was pretty exciting to finally become a mama (at 46) on Valentine's Day.


Check out these mathematical Valentines. I won't have time today to recreate the one I made years ago on an early HP computer with a plotter, but it used trig functions, and zigzagged up and down from one point to the next (odd-numbered points were something like 10% bigger than even-numbered points) for a somewhat lacy effect. You could do that in Scratch, I bet.

Maybe I'll get to edit this all day. Here's David Cox with Geogebra. And here's Vi Hart with Mobius Hearts.

Richmond Math Salon Meets Next Saturday - Shape Numbers

Here is one boy from last month, looking very proud of his big boat. It was fun working with the one family that came, but I need to figure out how to do the publicity thing better. I feel much more useful when I have 4 to 8 families here.

If you want to be notified of the salons, join the Yahoo group, here.

So this coming Saturday (February 19), from 2 to 5, we'll be playing with shape numbers: Triangle Numbers, Square Numbers, and Trapezoidal Numbers. We'll use pennies and other small objects to make these numbers.

I think it will be fun for young and old.

If you'd like to join us, please email me at suevanhattum on not-cold-mail. (My attempt at foiling the spambots.)

Tin Ceilings, Triangles, and Loving Math

James Tanton had a tin ceiling in his childhood bedroom with repeated squares on it. It got him wondering how many squares total, and how many ways to get from A to B, and ...

Now one of Dave Richeson's students has brought up another tin ceiling problem. Here's the pattern:

The question the student asked was "How many triangles (of any size) are there total?"

I avoided reading the solution post, so I could work on it when I had time, but I had a suspicion I might not manage to find the time. Then a conversation began on the Math 2.0 list (aka Math Future) about problem-solving, and how cool it would be to have a written description of different people's thinking processes as they work through a problem. (Mike South wrote: "Something that may be of use is first person narrative of an actual problem solution by various people.") I mentioned that I was thinking about trying to solve this problem, and two group members jumped on it, and contributed their solutions/ thoughts (which I haven't really looked at yet). Linda Fahlberg-S​tojanovska and Dani Novak each posted some cool stuff related to how they played around with it.

While I was on BART on the way to the Escape From the Textbook conference, I began work on the problem, and tried to write down everything I was thinking. It would be great to have this sort of thing from other people too. If you're interested, you may want to stop now, and go play. It was hard for me to remember to stop thinking about the problem and write down what I'd just been thinking, but I think I captured most of it. (I have probably conveniently forgotten to write down some false starts and dead ends...) I was so engrossed in the problem that I almost missed getting off at Civic Center. I jumped up just in time, ran over to the Muni, and got back to work.


My Process
Saw the problem at Dave Richeson's blog, skipped the solution post. Counted 4 little triangles per square. Saw at least one bigger triangle. Stopped there.

In bed, decided to organize my counting by size of triangle.

On BART, using my graph paper notebook, I draw a picture like that above, that shows a 2x2 grid of squares. (Each square uses 4 of the graph paper squares.) I count 4 little triangles in each square. 4 squares makes 16 of those. I draw a little picture of that size triangle and next to it I write '=16'.

Making a triangle from 2 littles gives 16 also. (There are 4 ways to get one of these triangles in each square.)

4 triangles... Wait, can't I use 3? Nah. [Later thoughts in brackets: While typing, I double-check this by looking at a representative size 2 triangle, and trying to attach each adjacent triangle to it. None work.]

Back to 4. Orienting the first one with its hypotenuse along the left edge of the figure, I see I can't move it up or down. There's another just like it with its hypotenuse along the vertical midline. That makes 16. [Typing time note: As I try to explain here what I saw, I write ... "No more that way (hypotenuse on left side). That's 2. Now use the bottom for the hypotenuse. 2 more." And I realize I'm only getting 8 this way. I search and search for more, and don't find them. I think I made a mistake on BART. I'll keep typing up my original work, though.]

So 4 triangles gives 16. That's it. 3x16=48, 3x2^4. This was 2x2 (of squares), so perhaps we'll have 3xn^4. Try a 3x3 (of squares) picture. I start drawing it. Uh oh. I see something bigger in the first one. Half the big square, duh. [Although I say 'duh' to myself, I don't feel bad. In fact I feel good. I know that it only feels dumb after I see something new.] 4 of those... Now I have 52. No sense trying to find a pattern yet.

I finish drawing 3x3. I'm looking for a way to classify the different triangles, so I'm redrawing one of each size, trying to see how they relate. I see that sometimes the hypotenuse is vertical or horizontal, and sometimes it's diagonal. Hmm, I also want to write each size as a multiplication, since I think that might help me see patterns. I'm calling the smallest triangles 2x1 (counting the graph paper squares as my units). I see that the smallest have a vertical or horizontal hypotenuse (I'll use VH for these). There are 4 in each square times 3^2 squares = 36. The ones that use 2 little triangles, 2x2 (on the graph paper), have a diagonal hypotenuse (D). There are again 4 in each square times 3^2 squares = 36.

Using 4 little triangles, 4x2,  back to VH. With hypotenuse on the left, I see one toward the top, and I can slide down a bit for a second one. I can slide the hypotenuse to the right, to the second or third vertical line. 4 (directions from VH hyp) times 2 times 3 = 24.

Using 8 little triangles, 4x4, back to D. Shorter side on left (top or slid down) or slid to the right one box (top or slid down), gives 4 times 4 directions (legs on left and bottom, bottom and right, right and top, top and left), or 4*2*2 = 16.

16 little triangles, 6x3, back to VH. Hypotenuse (or spine) at left gives 1 at left and one more (slide right). 4 directions. 4*2*1 = 8. 32 little triangles, 6x6, back to D. There are 4*1^2 = 4. The total for 3x3 (of squares) is 124 triangles.

I'm thinking that the way I label sizes will confuse anyone reading my account, so I decide to change to size labels that are based on the small squares being the unit. So the smallest triangle is 1x 1/2. I change all the sizes for the 3x3 square above. (Biggest becomes 3x3.)

Count for VH Diagonal spines, CVHD(a x a/2) = 4*(n-a+1)^2. Ahh, translating those sizes has made it easier for me to see patterns. These were the easier ones, now on to the Diagonal VH spines. (Oops! Don't miss your BART stop, Sue!)

 Diagonal VH doesn't go as smoothly: 4*3^3, 4*3*2, 4*2*1.

I guess it's time for 4x4. I draw this one smaller, to fit on what's left of the page. Counting is quicker now.

Oops. I left out the 4x4. There are 4*1^2 of them. The diagonal pattern has held, and I still see no clear pattern on the VH triangles. On to 5x5. I draw it, and I draw a representative triangle of each size. Ride over. Time for the conference...




I have typed up all my notes. Now I need to go back to drawing to finish thinking about the 5x5. I may not have time for that today, so I'm posting, and will edit later.

Escape From the Textbook Conference

Today was the conference. This morning I got on BART, and worked on a math problem the whole way in to SF. (I'll post about that tomorrow.) This post comes from the notes I took at the conference, and will be a bit choppy.

We were welcomed by Henri Picciotto. (Check out some of his cool stuff!) And then Jo Boaler, author of What's Math Got To Do With It?, spoke.

Jo Boaler's Talk
She points out 3 characteristics of good classrooms:

• They are mixed ability
• The students work on problem-solving
• The students work in groups

She and her graduate students worked with 6th and 7th grade students in a summer school program as part of their research. They developed an Exploratory Algebra course for the students that met for 5 weeks. The students came in ready to hate it (90% were 'made' to come by parents, teachers, or school), and left loving it. Boaler et al had 4 principles for their teaching. They wanted to:

• Engage the students as active and capable learners,
• Teach mathematical practices,
• Develop a collaborative mathematical community, and
• Help the students develop their own voices.

The students did much better in their fall math classes than students who had taken a regular summer school math class. (However, all gains were lost by the winter.) The two things that were most important to the kids were collaboration and agency (getting to choose how to proceed in a problem).

One student said: "After finding a pattern, you can stretch it in many ways, instead of just staring at it."

At first, when asked "How many squares (of any size) on a chessboard?", the students were willing to play around with the problem, but they were not willing to look systematically, or to record their thinking. The teachers encouraged them to try a smaller case, but the kids felt like that would be cheating. (!)

She showed some video clips of these classes, and asked us to discuss our reactions. That gave me just enough interaction to keep me focused. (Usually an hour-plus talk would be too long for me, and I'd be off thinking about something else. I actually managed to listen and think about it the whole time.)


Paul Zeitz, Puppies, and Kittens
We had a short break, and then came Paul Zeitz's talk. (Paul Zeitz is the author of The Art and Craft of Problem Solving.) Except that it was a math play session instead of a talk. Before we started playing, he said he thought math class should be more like:

• Field Biology
• Shop
• Sports 

Regarding the analogy with sports, he said, "I want to de-emphasize competition, but not to throw it out." He says math class should always be/involve:

• Hands-On
• Interactive
• Discovery
• Comradeship

"What you need to learn is how to investigate."

He gave us some handouts with good math games, and asked us to think about this kittens and puppies game.

The two players come in turns to the pet store, and each time have to buy at least one pet. The rules are that they can buy:

• As many puppies as they want, or
• As many kittens as they want, or
• An equal number of puppies and kittens.

The pet store starts out with 10 puppies and 7 kittens. The last person who can make a legal move wins. He played the game out once with us, and then had us play against our neighbor a few times. Then he showed us that if he could get the numbers down to 1 puppy and two kittens (or vice versa), the other player had no good move. (Buying one of the kittens leaves it equal, so he can take both the animals that are left. Buying both kittens leaves only the puppy, which he buys. Buying the puppy allows him to buy both kittens.) He asked for a participant who wouldn't mind losing, and very nonchalantly beat her. Paul called the 1 kitten, 2 puppies and 1 puppy, 2 kittens situations oases. We played against our neighbor, looking for other oases.

Then Paul showed us a way to graph the oases, which helps you find more of them. The patten is very interesting. (I won't wreck your fun by telling more.)

After Paul's session was lunch, where we got to chat with other teachers. The folks there were over 80% high school teachers, I'd guess. It was a lively crowd, and I had fun chatting with two people I'd just met and a colleague I get to see at every one of these conferences. After lunch we went to one of about 6 workshops. I went to Avery's and had a blast.

I liked that he focused on the idea of getting students to pose their own questions. After some discussion, he handed out unifix cubes, and asked us to make a patten that:

• Could be repeated indefinitely
• Can be counted

One person in our group hooked a unifix cube onto each face of a central cube. Our first stage was the central cube; our second stage was the object with 7 cubes total.  We talked about what our next stage would be and had (at least) two different notions of how to proceed. One person made the 6 'arms' 2 cubes long for the next stage; two others of us saw it as - put cubes on every exposed face of this object. That seemed to me to be a fractal (but maybe I'm wrong). I spent some time figuring out how many cubes would be used for each stage.

I loved each of the 3 sessions, and the energy of the teachers there. I hope there will be another conference like this in the fall.

School Starts, Blog Neglect Sets In

I see my last real post was two days before school started. That was a Saturday. On Sunday, I started having severe pain while I was trying to copy first day handouts. I've had this happen before, and have had all sorts of tests done. It could be gall bladder, ulcer, or about 3 other things. All tests come back negative (ie nothing seems to be wrong). It usually lasts about a week. This time it was 11 days. Luckily, teaching distracts me from that sort of pain, and so the best part of each day for me was while I was teaching. I'm fine now, and trying to eat better, in case it's gall bladder. It would be nice to know what's going on, but for now all is well. Meanwhile, my blog has suffered some serious neglect.

I have links I'd like to share. Like another cool math-related art site, called Algorithmic Worlds. (Thanks Dan, for your lovely post about this.) And this simple decimal game that John Golden put together.

But I'd also like to talk about how this semester is going. I knew it would be better than last semester, but I was still nervous, because of all the trouble I'd had with difficult students. It has been glorious. I am loving teaching Calc II. I am enjoying a huge Intermediate Algebra class. (I wanted to let in former students, so I had to let in the folks on the wait list too. I have over 50 students.) And I have a tiny Beginning Algebra class, where I'll have time to help each one of them. I'm doing so much I want to write about, but I haven't had time to write.

The SBG folks got me pushing myself to allow students to retake tests, and I made lots of versions of my mastery tests for Beginning Algebra last semester. With 3 different courses this semester, I wasn't sure I'd be able to do that for all 3. I promised the Intermediate Algebra students who'd had me last semester that I'd continue that, but I figured my Calculus students didn't need it so much. When more than half failed the Volume test I gave last week, I figured I'd need to let them do retakes too. Today I made two sheets that laid out what a student has to do to be allowed to retake their test (one for Intermediate Algebra and one for Calc). Here's the calc list:

Retaking the Volume Test

1. Re-do each problem. Find your mistakes and explain your old thinking and your new thinking. The hardest part is usually setting up the integral. Show a representative disk, washer, or ‘tube’, labeled.

2. Do a volume project. My first recommendation is to get a glass, measure it carefully (using a caliper), line its axis of symmetry up with the x-axis, and come up with an equation that represents the inside of the glass. Do a volume of revolution, and give the volume. Now check by measuring how much the glass holds. Come see my examples if this isn’t clear.

3. Make a sheet summarizing the differences between the two methods (disks and washers versus ‘shells’, ‘soup cans’ or ‘tubes’).

4. Do the 4 volume of revolution problems on the back of this sheet.

5. Come show me all this in my office, and I will make you a new version of the test.

When students know they can learn the material still, they're not so discouraged after doing badly on a test, and I can keep a lively atmosphere in our class.

I'm going to a conference in SF on Saturday called Escape From the Textbook (which you can watch through live stream). I'm excited about that.

I've got more to write about my classes, but my son is waiting for me to read to him.