Last summer, at a conference for the California Acceleration Project, Myra Snell used a cool way to set up random groups of participants/students. I wanted to use it in my classes, but just didn't get around to it. It seems, after almost 30 years of teaching, that it has become hard to change the way I run my classroom.

But I did change one thing this past semester. I noticed, while sitting in on a colleague's Calc III class, that I really appreciated the notices he wrote on the board at the beginning of each class. So I began to do it too. Maybe I could implement a few more good habits by watching other teachers during the second and third weeks of class.

Coming back to those random groups... I recently read research that found two effective strategies for getting students to participate more. One is visibly random groups. 'Visibly' means that they can't suspect the teacher of manipulating the group memberships. Myra's method is clearly random, looks easy to implement, and allows for up to four different groupings per class day. You have a slip for each student, with a number, a letter, an animal, and a food on it (for example). Those slips are set up so that no one is with any of the same other people more than once. I've asked Myra for her slips, but last night I was eager to think about it, and created my own. I don't know if this is the best way to do it, but I think it will work. Myra's slips had the 4 terms in a square and mine will be all in a row. I don't think that's a problem.

The second strategy which made a difference in student participation was student use of vertical whiteboards. The researcher(s?) compared paper and whiteboard, used vertically and horizontally. [Unfortunately, I can't find the research I originally read, which mentioned both the visibly random groups and the vertical whiteboards.] I'd like to try this out with the class I'll be teaching for the first time this fall, a compressed version of beginning algebra (first half of the semester) and intermediate algebra (second half of the semester). It's officially the same courses we've always taught, but I get to use a different curriculum, and will be using something project-based. I'm excited about implementing this.

## Wednesday, May 27, 2015

## Tuesday, May 26, 2015

### Machinery, Lines and Circles

On Facebook, someone posted an animation of how a sewing machine works. It wasn't enough to help me understand how the top thread manages to get around the bobbin mechanism. I searched on youtube, and nothing helped. This article on math and the sewing machine made me think for a moment that I was getting it, but I still am not. How is that bobbin mechanism held in place in a way that allows the thread to get around it? (Do you see how the top thread moves past the whole back of the bobbin? How is that possible?) They say that the bobbin is held snugly inside its case, but how is the out part attached?

I think I need a transparent sewing machine, so I can really see how this is working.

On thing leads to another (especially online!), and I ended up at this site from a museum for mathematics, called The Garden of Archimedes, in Florence, Italy, where I encountered this very simple statement about the difference between constructing a circle and a line - something I had never thought about before.

The simplest curves are doubtless the line and the circle. To draw circles, one uses a compass. It's sufficient to keep a constant distance between the tracing point and the centre, and one obtains a near-perfect circle, even with a primitive compass. At first sight, one would think that tracing a segment is also a very simple operation: you just need to use a ruler or pull a string taut. In fact, things don't work exactly like that. In order to draw a good straight line with a ruler, one needs the ruler itself to have a "straight" side, but the value of a ruled line depends on the ruler that was used to make it. So, who made the first ruler? To apply the same method to the circle would mean, for example, to take a coin and trace its edge - the circular profile would be "intrinsic" to the instrument itself.Inatead of using a ruler or straightedge, can't you use the "pull a string taut" method, with something a bit less flexible than string? Maybe something that freezes into position? Hmm... Apparently that's not the avenue that was followed. You can find out the fascinating history of the solutions people found for this problem by going to the Garden of Archimedes site.

It would be better to apply to the straight line the principle used to draw the circle, rather than vice versa.

## Tuesday, May 19, 2015

### Teaching My Son (Post One of Many?)

I started out really believing in unschooling. (Advice to self: Beliefs are dangerous.) My son has attended a free school, where he didn't have to attend classes (K-2), and then was homeschooled at the homes of friends (I'm a single parent), in groups of 2 to 8, with very little required of him. He has learned a lot over the years, but not in the conventional ways. If you're an advocate of unschooling, that may not sound like a problem at all. But for him it was. He thought he was 'behind' in reading, and felt bad about that. He totally avoided math because of how far behind he thought he was. He thinks he's dumb because he hasn't done the conventional academics.

Now he wants to go to a 'normal' school. So I signed him up for 8th grade at a charter school his friend goes to. (I've heard great things about it, and it is supposed to be project-based.*) Part of going to a regular school means catching up on all the 'regular subjects.' So I've begun requiring him to do 'academics' daily. (He asks if he has to. I say yes. He then shows subtle signs of relief. He really wants me to make him do this. This blows my mind.)

About a month ago, we started with 15 minutes of reading and 15 minutes of handwriting practice each day. I don't care about his handwriting. He does. He is so embarrassed about it that he resisted signing in for his trampoline class. A few weeks ago, I added spelling (his desire), geography (identifying the states), and math. This week we're adding science and an essay on the history of bikes. My opinion is that the only things he really needs to catch up on are math and writing (essays, stories, ...). It helps that we're doing this, because he also needs to become more aware of conventions - how to write dates, what schoolwork looks like.

For math, we're using

The first day that we did math, he was sitting next to me, saying his answers, waiting for me to confirm before he'd write them down. I did. (What he needs, as he takes on this huge emotional challenge, is support. Once he feels more secure, I'll be able to say things like "How can you decide whether that answer is right or not?")

On the second and third days, I noticed that his wrong answers were usually one off. To me, that meant he wasn't noticing things I notice about even and odd numbers. I printed out something from the nrich site that looked good. We haven't tried it yet.

It's very fun for me to be planning out his math curriculum. But this is very stressful for him, so our work time can be full of conflict. Once he buckles down and gets started, I get to quietly support him. Mostly I just confirm his answers. He is already seeing progress, and feeling good about it. I am trying to use Denise's technique of buddy math, offering to do every other problem myself, and then talking my way through it. He seems to prefer doing the problems himself most of the time, but let me do one problem last night.

The lessons he's working on are about finding perimeters. It has been a great way for him to work on adding numbers, with something extra thrown in. Most of the shapes have more than six sides. While we were working last night, I told him I noticed that he picked numbers that add to ten, which is a good strategy. I said that some people call those ten-bonds. He said he didn't know them all. I asked him for numbers that add to ten, and he got a bunch. The ones he hadn't mentioned, I asked him about: "Eight and ...?" He was surprised that there were only 5 pairs, I think. (At least two different stories in

I am exploring the balance between telling (ten bonds) and helping him to discover (hopefully we'll do that with odds and evens). I am so happy to be doing this, and marveling at how hard it was for me to see that he actually wanted me to make him do it.

_____

*Yes, I agree that charter schools are being used to mess up the regular public schools. Difficult situation all around.

Now he wants to go to a 'normal' school. So I signed him up for 8th grade at a charter school his friend goes to. (I've heard great things about it, and it is supposed to be project-based.*) Part of going to a regular school means catching up on all the 'regular subjects.' So I've begun requiring him to do 'academics' daily. (He asks if he has to. I say yes. He then shows subtle signs of relief. He really wants me to make him do this. This blows my mind.)

About a month ago, we started with 15 minutes of reading and 15 minutes of handwriting practice each day. I don't care about his handwriting. He does. He is so embarrassed about it that he resisted signing in for his trampoline class. A few weeks ago, I added spelling (his desire), geography (identifying the states), and math. This week we're adding science and an essay on the history of bikes. My opinion is that the only things he really needs to catch up on are math and writing (essays, stories, ...). It helps that we're doing this, because he also needs to become more aware of conventions - how to write dates, what schoolwork looks like.

For math, we're using

*Beast Academy*. We started with book 3A. Yes, the 3 means third grade. We don't mention it, but he knows this is "supposed to be" for younger kids.*Beast Academy*has challenging work, though, and if we make it though all eight of the levels (3A-D and 4A-D), I think he'll be pretty well-prepared to join a class of 8th graders. I will look over the 'standards' for 5th to 7th grade later this summer, and see what might be missing from what we're doing. I have made a math plan for the next 14 weeks, leaving out some of the topics in the*Beast Academy*books (perfect squares, variables, counting, logic, probability). I'm sure they are excellent, but my goal was to find a way to pare it down, so he gets as much as possible of the foundational skills he'll need, in the short time we have before he starts 8th grade.The first day that we did math, he was sitting next to me, saying his answers, waiting for me to confirm before he'd write them down. I did. (What he needs, as he takes on this huge emotional challenge, is support. Once he feels more secure, I'll be able to say things like "How can you decide whether that answer is right or not?")

On the second and third days, I noticed that his wrong answers were usually one off. To me, that meant he wasn't noticing things I notice about even and odd numbers. I printed out something from the nrich site that looked good. We haven't tried it yet.

It's very fun for me to be planning out his math curriculum. But this is very stressful for him, so our work time can be full of conflict. Once he buckles down and gets started, I get to quietly support him. Mostly I just confirm his answers. He is already seeing progress, and feeling good about it. I am trying to use Denise's technique of buddy math, offering to do every other problem myself, and then talking my way through it. He seems to prefer doing the problems himself most of the time, but let me do one problem last night.

The lessons he's working on are about finding perimeters. It has been a great way for him to work on adding numbers, with something extra thrown in. Most of the shapes have more than six sides. While we were working last night, I told him I noticed that he picked numbers that add to ten, which is a good strategy. I said that some people call those ten-bonds. He said he didn't know them all. I asked him for numbers that add to ten, and he got a bunch. The ones he hadn't mentioned, I asked him about: "Eight and ...?" He was surprised that there were only 5 pairs, I think. (At least two different stories in

*Playing with Math*address this issue - Prison Math Circle and The Math Haters Come Around.) When he was unsure of 5 plus 8, I told him that I sometimes forget that one myself, and one way to figure it out is to move 2 from the 5 to the 8, so you get 3 and 10.I am exploring the balance between telling (ten bonds) and helping him to discover (hopefully we'll do that with odds and evens). I am so happy to be doing this, and marveling at how hard it was for me to see that he actually wanted me to make him do it.

_____

*Yes, I agree that charter schools are being used to mess up the regular public schools. Difficult situation all around.

## Thursday, May 14, 2015

### Moebius Noodles is Delightful

*Moebius Noodles*is headed into its second printing soon. For the past few days I've been reading it over carefully to offer suggested edits. What a delightful task I gave myself! It has been so fun to remind myself of all the activities for young children Maria Droujkova and Yelena McManaman have put together.

Their suggestion for creating an iconic times table got me dreaming. How can I get my son (who "hates" math, unfortunately) inspired to take photos for a times table collection? I was dreaming of a website that would show the whole table on one page, with each photo pretty small. And when you hover over a photo, that one would show up big. I don't know how to do that, though...

Here's a photo (from lernertandsander.com/cubes) that feels like it belongs in the Grid section of

*Moebius Noodles*, except that there's no pattern to the pieces. Well, the rows and columns are a bit wonky too. Hmm...

Kate Nowak posted this on Facebook. The question that came to her mind (among other less mathy questions) was ... How do

*you*count these?

[Edited to add: In the comments, Joshua described a very cool pattern he saw, and suggested that it's like 9 plus 4 is 13, which looks like my diagram below.]

*Moebius Noodles*has four sections: Symmetry, Number, Function, and Grid.

The mirror book introduced in the symmetry section is so simple, and so cool to play with. Just get two small rectangular mirrors (at a dollar store), tape them together along one side, and use with photos or drawings, to see lots of symmetrical designs.

My favorite game in the function section is Silly Robot. The grownup plays the robot, and follows orders exactly (while always trying to find a way to mess up the intention of the orders).

If you know anyone with a child from one to eight who'd like to find ways to play around with mathematical ideas,

*Moebius Noodles*is a great resource.

And my book,

*Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers*, is delighting readers across the U.S. (and hopefully around the world). Here are a few photos of happy readers. Send me a photo of you with the book, and I'll add it to my collection (especially if you live far from me!).

This is shaping up to be a very fun summer...

## Tuesday, May 5, 2015

### Aunty Math

**Welcome to the world of**

**Math Challenges for K-5 Learners**

One of the reasons I put together a book was my fear that good online writing often just disappears. One of the sites I had really liked - and thought of including somehow in the book - was a site with stories from Aunty Math (Aunt Mathilda). It disappeared before I could contact the author. And for years, I thought it was just plain gone.

This evening I searched for Aunty Math, and found that someone had managed to get to this site through the Wayback Machine. It is now available as an archive. Check out all eleven past challenges. I think you'll enjoy them.

I would love to be in touch with the author, Angela G. Andrews. I googled her, but I don't see an email address. I'll just thank her here for her lovely stories. Thanks, Angela!

My book,

*Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers*, won't disappear. If you want a copy to appear in your mailbox, order one now.

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