Shawn Cornally, at Think Thank Thunk, has written some great posts over the years about doing calculus classes in some fabulous ways. Last night he asked, "Perhaps the job really is impossible to do right. What then?"

Instead of thinking about doing it right, does it help to think about doing it well?

I've always felt that making kids go to school, and making students (even in college) take certain classes, are ways we disempower students, and lessen their curiosity and eagerness to learn. I feel like the system we're a part of does make it impossible to do it 'right'. But I love learning how to do it better.

Even though it's not math, I want to share a blog post about a 4th/5th grade class that's doing really cool things with geography. They're doing it very well!

## Friday, September 30, 2011

## Saturday, September 24, 2011

### Photos for Playing With Math

Do you have a good, crisp mathy photo that you took yourself? Would you like to see it (with credit to you, either by name or by reference to your mathy blog) in the upcoming book

If so, send it to mathanthologyeditor at gmail (that's me), along with this statement. "I took this photo, and I give permission to Sue VanHattum to publish it in

We'll do a random drawing from all the people who send photos we decide to use, and give a free copy of the book to one person from the list. (If there are more than 30 people on that list, we'll send out 2 copies of the book.)

If you're not one of the 35 authors, or 18 volunteer editors, this is your chance to join in on the creation of this fabulous book.

What do you think?

*Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet*?If so, send it to mathanthologyeditor at gmail (that's me), along with this statement. "I took this photo, and I give permission to Sue VanHattum to publish it in

*Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet."*Also include your preferred credit line. Your email should include the word 'photos' in the subject line.We'll do a random drawing from all the people who send photos we decide to use, and give a free copy of the book to one person from the list. (If there are more than 30 people on that list, we'll send out 2 copies of the book.)

If you're not one of the 35 authors, or 18 volunteer editors, this is your chance to join in on the creation of this fabulous book.

What do you think?

## Sunday, September 18, 2011

### Groups

I am not very good at implementing new procedures in my classes, and I haven't been reinforcing the roles students are supposed to take on in groups (according to the Complex Instruction philosophy). I've also been using groups sometimes for work on exercises (rather than deep, rich problems). Even so, I'm still finding the positive effects pretty substantial.

I've often had students work in ad hoc groups in the past, but there'd always be a few huge groups (where some of the students were just leaning on the industrious ones) and a few students who preferred to work alone. With the set groups of 4, there has been a different dynamic. They develop a concern for each other, and are working together so much more effectively. This new dynamic has improved student learning in all 3 of my classes, I think. Along with allowing retesting, I think groups have changed my classes so that students take lots more responsibility for their learning. My job continues to be to make it interesting...

On Wednesday evening I was worrying about my next day's lesson in trig intresting. I had spent much of our Wednesday hour on the proof of the Law of Cosines. It was the least interactive class we'd had, I think. I'd like to come up with a way to get them walking through a proof like this, with a little more action on their parts. Any ideas? (I'll be teaching this again next semester... And of course there are lots more proofs to come in this course.)

I didn't come up with anything brilliant, but had them work in groups on a lake problem I had given them earlier in the week - before they really had the tools to solve it. Their second task was to carefully draw a triangle, measure its sides, and solve for the angles. (Then pass it to another person in the group, and have them solve it too.)

I was enjoying going around and helping people find their mistakes. One person had just eyeballed his sides, and written 'measurements' from his rough estimation. His triangle didn't work out right somehow. I think my slight scorn may have gotten him to try again.

One student wanted me to check his work. I scanned it and said it seemed reasonable. He had gotten 37 degrees for one of the angles. I used my circle with angle lines to measure and it was right on. His response: "Wow, math is real!" That just about made my day. (Kind of shocking that he'd be surprised though...)

In Calc II, we're working on partial fractions, and I wanted to talk about the consequences of the Fundamental Theorem of Algebra - that any real polynomial can be factored into linear complex factors or into linear and quadratic real factors. I've never worked my way through the proof of that, and still feel a bit mystified that at the same time there is no formula possible to solve 5th degree polynomials. I'd never thought much about it, and the mystery of it (for me) makes me want to learn more about all this. (I may not ever get to it, though...) I went into class feeling high on math.

My evening class went pretty well too. I was showing my Intermediate Algebra students the process for solving 3 equations in 3 variables. I made up another silly coin problem, just so they could see the possibility of 3 equations in 3 variables being meaningful. (I have pennies, nickels, and dimes in my pocket. My 32 coins are worth $1.87, and I know I have twice as many nickels as pennies.) I also showed them a 3D coordinate system in the air, with one side of my desk being the x-axis, the front of it being the y-axis, and a line straight up from the corner being the z-axis. I place a few points in the air from that. I usually stop there. This time, I pointed to the corner of the room (floor), and talked about where negative values on each axis would put us (outside, in a closet, in the foundation), and then walked to (10,15,4), by pacing 10 feet along the wall, 15 feet into the room, and putting my finger 4 feet high. I then had two students do (10,20,3) and (12,20,3) at the same time.

When we did the system of equations, I didn't do anything new, but I felt like they were approaching it more sensibly than past classes. Most students want to write down a bunch of rules. I talk about figuring out which variable is easiest to get rid of by addition method, and then we get two new equations and solve the simpler problem. They seemed to be getting into it.

I felt so lucky that day to have work I love.

I've often had students work in ad hoc groups in the past, but there'd always be a few huge groups (where some of the students were just leaning on the industrious ones) and a few students who preferred to work alone. With the set groups of 4, there has been a different dynamic. They develop a concern for each other, and are working together so much more effectively. This new dynamic has improved student learning in all 3 of my classes, I think. Along with allowing retesting, I think groups have changed my classes so that students take lots more responsibility for their learning. My job continues to be to make it interesting...

**Trigonometry**On Wednesday evening I was worrying about my next day's lesson in trig intresting. I had spent much of our Wednesday hour on the proof of the Law of Cosines. It was the least interactive class we'd had, I think. I'd like to come up with a way to get them walking through a proof like this, with a little more action on their parts. Any ideas? (I'll be teaching this again next semester... And of course there are lots more proofs to come in this course.)

I didn't come up with anything brilliant, but had them work in groups on a lake problem I had given them earlier in the week - before they really had the tools to solve it. Their second task was to carefully draw a triangle, measure its sides, and solve for the angles. (Then pass it to another person in the group, and have them solve it too.)

I was enjoying going around and helping people find their mistakes. One person had just eyeballed his sides, and written 'measurements' from his rough estimation. His triangle didn't work out right somehow. I think my slight scorn may have gotten him to try again.

One student wanted me to check his work. I scanned it and said it seemed reasonable. He had gotten 37 degrees for one of the angles. I used my circle with angle lines to measure and it was right on. His response: "Wow, math is real!" That just about made my day. (Kind of shocking that he'd be surprised though...)

**Factoring Polynomials**In Calc II, we're working on partial fractions, and I wanted to talk about the consequences of the Fundamental Theorem of Algebra - that any real polynomial can be factored into linear complex factors or into linear and quadratic real factors. I've never worked my way through the proof of that, and still feel a bit mystified that at the same time there is no formula possible to solve 5th degree polynomials. I'd never thought much about it, and the mystery of it (for me) makes me want to learn more about all this. (I may not ever get to it, though...) I went into class feeling high on math.

**Systems of Equations**My evening class went pretty well too. I was showing my Intermediate Algebra students the process for solving 3 equations in 3 variables. I made up another silly coin problem, just so they could see the possibility of 3 equations in 3 variables being meaningful. (I have pennies, nickels, and dimes in my pocket. My 32 coins are worth $1.87, and I know I have twice as many nickels as pennies.) I also showed them a 3D coordinate system in the air, with one side of my desk being the x-axis, the front of it being the y-axis, and a line straight up from the corner being the z-axis. I place a few points in the air from that. I usually stop there. This time, I pointed to the corner of the room (floor), and talked about where negative values on each axis would put us (outside, in a closet, in the foundation), and then walked to (10,15,4), by pacing 10 feet along the wall, 15 feet into the room, and putting my finger 4 feet high. I then had two students do (10,20,3) and (12,20,3) at the same time.

When we did the system of equations, I didn't do anything new, but I felt like they were approaching it more sensibly than past classes. Most students want to write down a bunch of rules. I talk about figuring out which variable is easiest to get rid of by addition method, and then we get two new equations and solve the simpler problem. They seemed to be getting into it.

I felt so lucky that day to have work I love.

## Wednesday, September 14, 2011

### Please Donate: Maria Droujkova's Moebius Noodles Project

Maria Droujkova has done some wonderful work creating math projects for young children. I've seen early drafts of the Moebius Noodles book, and it's exciting.

Her project is featured for one week at tippingbucket.org. She has to raise $6200 in that time. (If she doesn't, the donations aren't taken. The project has been up for 5 hours so far, and over $1200 has already been donated.)

Disclaimer: Maria is starting a publishing company, Delta Stream Media, which will be publishing my book,

Her project is featured for one week at tippingbucket.org. She has to raise $6200 in that time. (If she doesn't, the donations aren't taken. The project has been up for 5 hours so far, and over $1200 has already been donated.)

Disclaimer: Maria is starting a publishing company, Delta Stream Media, which will be publishing my book,

*Playing With Math*. Our projects are somewhat intertwined.## Tuesday, September 13, 2011

### Richmond Math Salon this Saturday (2 to 5 pm) - Another Comic

I don't know why I'm working so hard on flyers - I never get around to putting them up. (But one student is taking them to the first grade she works in, so maybe it's good.) Anyway, I hated the bad typeface on the comic I made last week (See this post.) So I decided to play around with photo booth (on my mac) and Snagit (which I was introduced to at the Math Technology Bootcamp, maybe you can go next summer), and here's what I came up with. Not great, but I'm a rank beginner...

## Sunday, September 11, 2011

### How do you decide what to believe?

Many of my students would like to take notes on what I tell them, and memorize it all. I tell them that math isn't like that. The idea is to prove to yourself that things make sense.

Sometimes I talk about how growing up with a lawyer for a dad made arguing (usually amicable) one of our family sports, and how that helped me be good at math, because I question everything. Other times I suggest that it's a good idea to question authority. I tell them not to believe me, because I make mistakes. "How do you know that's true?" I ask.

Today I'm asking that about what happened on 9/11. I don't know what happened. I've watched the videos Nilo de Roock linked to a few days ago. I've followed some links he sent me, and read about the 3rd tower that's seldom mentioned and about Paul Wellstone's death.

On Friday a public health advocate was being interviewed on KPFA (podcast). She talked about how the government said the air nearby was fine just days later, when they couldn't yet know, and how people with cancers from the air pollutants are not getting help. She said she wasn't a 'conspiracy theorist', and that she's not the kind of person to suggest people lose all hope and feel powerless. I worry about the phrase 'conspiracy theorist'. Is anyone who doubts the official story a 'conspiracy theorist'? Why? I'm trying to be logical about this. I don't want to lose hope, but I sure want to try to figure out what's true.

Here's a television news archive, from a more mainstream source. The videos I watched from the first link above talked about how there was no reason for the buildings in the background to look so nondescript. It was a sunny day, and it makes no sense for there to be so little detail in the videos. (The hypothesis is that much of the news footage was faked.) Is that true? You may not see the details from the first videos in the official news archives, but you can make the comparisons they suggest, between 9/11 scenes of the Manhattan skyline and other videos of the same skyline.

Of course, those videos don't answer my dozens of questions. The biggest being, if the media was helping to show us a lie, that would put at least dozens of people (but more likely hundreds) in on some sort of conspiracy. How could that many secrets be kept?

What we do know is that about 3000 people died that day, and the world changed. It's a tragedy however it happened. And it's important to me to try to understand what really happened.

Sometimes I talk about how growing up with a lawyer for a dad made arguing (usually amicable) one of our family sports, and how that helped me be good at math, because I question everything. Other times I suggest that it's a good idea to question authority. I tell them not to believe me, because I make mistakes. "How do you know that's true?" I ask.

Today I'm asking that about what happened on 9/11. I don't know what happened. I've watched the videos Nilo de Roock linked to a few days ago. I've followed some links he sent me, and read about the 3rd tower that's seldom mentioned and about Paul Wellstone's death.

On Friday a public health advocate was being interviewed on KPFA (podcast). She talked about how the government said the air nearby was fine just days later, when they couldn't yet know, and how people with cancers from the air pollutants are not getting help. She said she wasn't a 'conspiracy theorist', and that she's not the kind of person to suggest people lose all hope and feel powerless. I worry about the phrase 'conspiracy theorist'. Is anyone who doubts the official story a 'conspiracy theorist'? Why? I'm trying to be logical about this. I don't want to lose hope, but I sure want to try to figure out what's true.

Here's a television news archive, from a more mainstream source. The videos I watched from the first link above talked about how there was no reason for the buildings in the background to look so nondescript. It was a sunny day, and it makes no sense for there to be so little detail in the videos. (The hypothesis is that much of the news footage was faked.) Is that true? You may not see the details from the first videos in the official news archives, but you can make the comparisons they suggest, between 9/11 scenes of the Manhattan skyline and other videos of the same skyline.

Of course, those videos don't answer my dozens of questions. The biggest being, if the media was helping to show us a lie, that would put at least dozens of people (but more likely hundreds) in on some sort of conspiracy. How could that many secrets be kept?

What we do know is that about 3000 people died that day, and the world changed. It's a tragedy however it happened. And it's important to me to try to understand what really happened.

## Thursday, September 8, 2011

### Richmond Math Salon - Saturday, September 17, 2 to 5pm

Yesterday I explored a bunch of comic making sites. This one was the easiest to use, but the writing looks pretty bad. pixton.com had nicer writing, but skinny characters. The others I saw also had character styles I didn't like. Anyone know of a free comic making site I might like better?

No theme this month, since it's my first time after about 3 months off. Just fun with math games and puzzles. Email me at suevanhattum on the hot mail system for more information. Or call me in the 510 area code at 236-80 four four.

Want to see what it's like? Check out this great video.

## Monday, September 5, 2011

### More Cool Math Videos: Erik Demaine

The Museum of Mathematics, which will open in NYC in 2012, has been hosting a series of talks titled Math Encounters. I'm watching Erik Demaine's talk on the Geometry of Origami, but it's so much more than that. It's also about the richness of his relationship with his father, and about the adventures they have. He shows a clip of his father blowing glass while blindfolded. (!) Why? Because it sounded interesting to focus on the tactile aspect of the process.

## Saturday, September 3, 2011

### Blanking on Tests

When a student tells me they blanked on the test, and couldn't do what they normally can, I'm never sure if it's really test anxiety (like they're describing) or if they just don't prepare well enough. I often talk to them about being super-well prepared, but I also point them to Math Relax, the guided visualization I made a few years ago.

If you have students who've built up lots of anxiety around math tests, you might want to recommend that they try this.

If you have students who've built up lots of anxiety around math tests, you might want to recommend that they try this.

## Friday, September 2, 2011

### Day 4 (posted 2 weeks later...)

In my

In

The 4th said she had a 'proof using words', but it was also just a description. I was amazed. They seem to have no conception of the meaning of proof. One student was making a very accurate diagram of a 3-4-5 triangle with the squares attached, and when I called it an example, he said he could change the sides to a, b, and c.

I asked the class, "How do we know this is true?" And they said we could measure it. I asked, "What if the third side is 4.9 inches, instead of 5 inches? Or 4.95?" They had trouble seeing why measurement wasn't enough. I'm going to learn so much from this class.

I showed them a visual proof and an algebraic proof, both starting with this tilted square inside a square. In the visual proof, you swing two of the triangles around, so the 4 triangles make two rectangular areas. What's left is one square area with side length a, and another with side length b. So elegant.

The algebraic proof describes the total area two ways:

(a+b)

A few simple algebra steps will do it.

It felt important to discuss the meaning of proof. I have no idea if it stuck, and I'd like to come back to this during this course.

Edit on 9-5: Unknown pointed me to another great Vi Hart video. She does pretty much the same proof I showed above, but using paper that she folds (and rips). It's a great demo.

**Pre-calc**class, I lectured. I thought I could pull them in at various steps along the way, but class felt a bit boring to me today. I wondered if two days in a row of facing front put them back into that passive student mode. As part of their homework I told them to email me with one thing they like about this class, one thing they don't, one thing they wish were different, and one question. 10-15 students wrote to me, and I started a list of pros and cons. I want to get their feedback so I know what the quieter students are thinking.from Wikipedia |

**Calc II**, I had asked them (as part of their homework) to either figure out a proof of the Pythagorean Theorem or look it up. Only 4 of them had done that (about what I expected), but 3 of the 4 gave this...The 4th said she had a 'proof using words', but it was also just a description. I was amazed. They seem to have no conception of the meaning of proof. One student was making a very accurate diagram of a 3-4-5 triangle with the squares attached, and when I called it an example, he said he could change the sides to a, b, and c.

I asked the class, "How do we know this is true?" And they said we could measure it. I asked, "What if the third side is 4.9 inches, instead of 5 inches? Or 4.95?" They had trouble seeing why measurement wasn't enough. I'm going to learn so much from this class.

I showed them a visual proof and an algebraic proof, both starting with this tilted square inside a square. In the visual proof, you swing two of the triangles around, so the 4 triangles make two rectangular areas. What's left is one square area with side length a, and another with side length b. So elegant.

The algebraic proof describes the total area two ways:

(a+b)

^{2}= 4*(1/2*a*b)+c^{2}A few simple algebra steps will do it.

It felt important to discuss the meaning of proof. I have no idea if it stuck, and I'd like to come back to this during this course.

Edit on 9-5: Unknown pointed me to another great Vi Hart video. She does pretty much the same proof I showed above, but using paper that she folds (and rips). It's a great demo.

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