If you are looking for something fun and educational to do with yourself this summer, you should consider the Math Circle Teacher Training Summer Institute. It is run by Bob and Ellen Kaplan (founders of the Boston area math circles, and authors of Out of the Labyrinth: Setting Mathematics Free, along with Amanda Serenevy (founder of Riverbend Community Math Center.)

You will learn about how children can learn math through exploring open-ended problems, playing, and asking questions. There are basically two components. You get to participate in a math circle, working on some challenging, fun, unfamiliar math, and you also get to formulate a math circle and try it out with real-live kids. During down time you can tour the impressive campus, walk/swim, or if you are not mathed out, there are books and puzzles and origami stuff galore JUST LYING AROUND. People also tend to get together in the evenings (and long into the night) to keep working on the unresolved math from the day's activities. Notre Dame's cafeteria feels like Hogwarts and the food is actually quite good. And VERY cool and smart people attend this thing. This is where I met some of my favorite math-enthusiast buddies: Ben, Alex, Jesse, and Sue. This is also where I got the idea for the Delving Deeper article I wrote for Mathematics Teacher that was published in December.

It takes place at the University of Notre Dame and is only one week long : arrive Sunday, July 8 and depart Saturday, July 14. It is a good deal at $850, which covers both tuition and room and board. Learning about the Math Circle approach will make you a better teacher, mathematician, parent, and human being. But if you don't believe me and want to find out if it aligns with your goals, your best bet is to read Out of the Labyrinth and see what it is all about. I hope you decide to go, and you love it as much as I did.

## Saturday, April 28, 2012

## Wednesday, April 25, 2012

### Squareness, continued

Yesterday there was lots of grappling with what we might mean by the "squareness" of a rectangle, and how one might come up with a way to express squareness with a single number.

Today, I was a little bit grumpy with my first class right off the bat, because first thing this morning a Resource teacher came to tell me a girl complained about her group yesterday not working and talking about weed and making her uncomfortable. Does anyone know how to do group activities and guarantee everyone is on task at all times? Because I don't. (And people who claim they do are lying. Let's be real.) I didn't know how to respond to her. "Oh ok I'll seat them in rows and demand silence every day" is the normal level of sarcasm of my internal monologue, but I'm a professional, so I just said "Okay, I'm sorry, thanks for letting me know."

Anyway. I wanted students to evaluate the validity of some conjectures, so that they will have a vocabulary to use when we start proving conjectures about similar figures. And hopefully learn and/or recall some geometry along the way.

I also didn't want to just drop the idea and move on to something else, even something else related, because I want them to know that what I ask them to do in class has value and meaning, so that they trust that what they are being asked to do is for a reason, even if the reason isn't immediately clear. (I have paid closer attention to this principle this year, and it seems to have paid off...some days more than others. Keep reading.)

I gave them the following on a 1/2 sheet of paper, a piece of graph paper, and access to any geometry tools they wanted to use.

I found today more frustrating than yesterday. I had to be a bit of an ogre about "I'm going to collect this and grade it" because otherwise, I was sensing kids giving up in response to a little frustration, instead of working through the frustration to understanding. I hate being like that because it sucks all the fun out of the room. But honestly I don't know what else to do. I get that there are more fun things to do with similar figures (go outside and measure shadows and find the height of the flagpole! etc), but honestly I find those kinds of activities a bit juvenile for tenth grade - middle school-ish, if you will. Not that they are bad, but it depends on your goals. We are going for more formality in their Geometric arguments at this level, and I don't know of any of those more-fun activities that stand up to the rigor required to work through what we did today. And honestly, once they engaged and understood the assignment, they were good to go. It just took some prodding to get them to that point.

Today, I was a little bit grumpy with my first class right off the bat, because first thing this morning a Resource teacher came to tell me a girl complained about her group yesterday not working and talking about weed and making her uncomfortable. Does anyone know how to do group activities and guarantee everyone is on task at all times? Because I don't. (And people who claim they do are lying. Let's be real.) I didn't know how to respond to her. "Oh ok I'll seat them in rows and demand silence every day" is the normal level of sarcasm of my internal monologue, but I'm a professional, so I just said "Okay, I'm sorry, thanks for letting me know."

Anyway. I wanted students to evaluate the validity of some conjectures, so that they will have a vocabulary to use when we start proving conjectures about similar figures. And hopefully learn and/or recall some geometry along the way.

I also didn't want to just drop the idea and move on to something else, even something else related, because I want them to know that what I ask them to do in class has value and meaning, so that they trust that what they are being asked to do is for a reason, even if the reason isn't immediately clear. (I have paid closer attention to this principle this year, and it seems to have paid off...some days more than others. Keep reading.)

I gave them the following on a 1/2 sheet of paper, a piece of graph paper, and access to any geometry tools they wanted to use.

1. Consider the following rectangles with given side lengths:

(a) 5 by 2

(b) 12 by 9

(c) 15 by 6

(d) 8 by 8

(e) 16 by 4

2. Draw them on graph paper. Come up with what you think the ranking should be, from most square to least square, just by visually looking at them.

3. Here are some methods for ranking squareness that your classmates conjectured yesterday. Some of them are valid, and some of them are not. Use each of them on all the rectangles above. List the ranking from most to least square produced by each method. Which methods are valid and why? Please respond to this in writing, right on the graph paper. Any tools you need are available on the table.

(f) The absolute value in the difference in length and width. closer to zero, the more square it is.

(g) Longer side divided by shorter sides; closest to one is square

(h) Find the areas of the shapes and compare how close it is to a square number.

(i) Create a diagonal and measure the angles on either side of the diagonal. A square's diagonal bisects the 90 degree angle in half, so the measure that are closest to equal would be the square/rectangle that is closest to an actual square

(j) Draw both diagonals. Measure the smaller angle created at the intersection of the diagonals. The closer it is to 90, the more square the shape is.

(k) Attempt to circumscribe a circle around the rectangle. The closer you can get all four corners to touch the same circle, the closer to a square it is.

I found today more frustrating than yesterday. I had to be a bit of an ogre about "I'm going to collect this and grade it" because otherwise, I was sensing kids giving up in response to a little frustration, instead of working through the frustration to understanding. I hate being like that because it sucks all the fun out of the room. But honestly I don't know what else to do. I get that there are more fun things to do with similar figures (go outside and measure shadows and find the height of the flagpole! etc), but honestly I find those kinds of activities a bit juvenile for tenth grade - middle school-ish, if you will. Not that they are bad, but it depends on your goals. We are going for more formality in their Geometric arguments at this level, and I don't know of any of those more-fun activities that stand up to the rigor required to work through what we did today. And honestly, once they engaged and understood the assignment, they were good to go. It just took some prodding to get them to that point.

## Tuesday, April 24, 2012

### Squareness

I was looking for a way for Geometry students to wrestle with similarity ratios. Daily, I try to escape the "this is how you do this kind of problem" mindset that plagues them, that they prefer, that is a trap we keep falling into. I suspected that Avery's "rank these rectangles according to squareness" problem would be worthwhile. I was excited by what the kids did with it.

Here's the handout I gave out, that I basically copied from Avery:

We had to really unpack 3. "devise a measure of squareness" to get them beyond "it just looks closer to a square." They were able to understand what I meant, but only after I stated it several different ways.

- Come up with a single number for rectangle E, and a single number for rectangle D, that shows that E is more square than D.

- Find a way to do a calculation for each rectangle, so that you can compare all the results, to see which is the most square.

- etc, rephrased a dozen different ways.

Ideally I would like to find a concise way to phrase this for students, that would get them closer to understanding what we are looking for by themselves.

Eventually, lots of groups came up with either or both:

- shorter/longer, and closer to 1 is more square.

- longer - shorter, and closer to 0 is more square.

No groups on their own noticed that you could break the subtraction method by choosing uncooperative rectangles (

I asked each group to summarize their method in a google doc...here is the raw output.

We had some really interesting methods, too!

- Draw a diagonal and measure the angles made by the diagonal and the sides. The closer the two angles are, the more square. (Way to draw a connection to the properties of special quadrilaterals! I pounced all over that like...something that pounces enthusiastically on something. So cool.)

- Attempt to circumscribe a circle around the rectangle. The closer you can get the vertices to all touching the circle, the more square it is. (We did not have time to delve into this today but MAN, THAT IS GOOD. I would really like to find a way to get them to all think deeply about that.)

- A somewhat crazy method involving comparing the average of the four side lengths to the length and width separately. I encouraged this group to write their method algebraically, but they ended up with an expression that always reduces to zero. However, it doesn't always reduce to zero when they do it with numbers. Would like to spend more time on this, too.

In a nutshell, I had a great time this morning. Philosophical side note: I have been engaging in these thought experiments recently, often, for some reason, about the idea "Could I be happy...?" (Could I be happy if I won the lottery and didn't have to work? Could I be happy working at Walmart? Could I be happy being a housewife? Could I be happy as a math teacher forever?) and what I have come up with so far is that I can be happy as long as I am spending a non-trivial amount of my time learning something fundamentally new - about myself, about other people, about the way things work... ("fundamentally" implying, well, fundamental...specifically

Here's the handout I gave out, that I basically copied from Avery:

We had to really unpack 3. "devise a measure of squareness" to get them beyond "it just looks closer to a square." They were able to understand what I meant, but only after I stated it several different ways.

- Come up with a single number for rectangle E, and a single number for rectangle D, that shows that E is more square than D.

- Find a way to do a calculation for each rectangle, so that you can compare all the results, to see which is the most square.

- etc, rephrased a dozen different ways.

Ideally I would like to find a concise way to phrase this for students, that would get them closer to understanding what we are looking for by themselves.

Eventually, lots of groups came up with either or both:

- shorter/longer, and closer to 1 is more square.

- longer - shorter, and closer to 0 is more square.

No groups on their own noticed that you could break the subtraction method by choosing uncooperative rectangles (

*and I don't know how to get them to do this organically*.) But after we reported out our methods, and I said "I'll tell you that the subtraction method doesn't always work. See if you can find some examples of rectangles where the subtraction method gives you misleading results." they were interested in looking for ways to break it, and lots of groups found counterexamples.I asked each group to summarize their method in a google doc...here is the raw output.

We had some really interesting methods, too!

- Draw a diagonal and measure the angles made by the diagonal and the sides. The closer the two angles are, the more square. (Way to draw a connection to the properties of special quadrilaterals! I pounced all over that like...something that pounces enthusiastically on something. So cool.)

- Attempt to circumscribe a circle around the rectangle. The closer you can get the vertices to all touching the circle, the more square it is. (We did not have time to delve into this today but MAN, THAT IS GOOD. I would really like to find a way to get them to all think deeply about that.)

- A somewhat crazy method involving comparing the average of the four side lengths to the length and width separately. I encouraged this group to write their method algebraically, but they ended up with an expression that always reduces to zero. However, it doesn't always reduce to zero when they do it with numbers. Would like to spend more time on this, too.

In a nutshell, I had a great time this morning. Philosophical side note: I have been engaging in these thought experiments recently, often, for some reason, about the idea "Could I be happy...?" (Could I be happy if I won the lottery and didn't have to work? Could I be happy working at Walmart? Could I be happy being a housewife? Could I be happy as a math teacher forever?) and what I have come up with so far is that I can be happy as long as I am spending a non-trivial amount of my time learning something fundamentally new - about myself, about other people, about the way things work... ("fundamentally" implying, well, fundamental...specifically

*not*something stupid like how to navigate the menus in a new app or what Loft has on sale this week.) In order to be happy being a math teacher forever, I need more days like today, where I am getting kids excited about learning. As ultimately cheeseball as that sounds.## Sunday, April 1, 2012

### Here Are Some Videos

So...I realize the date for this post is inauspicious, but this is really a thing. About a year ago, I was contracted to make an Algebra 1 course in the form of <10 minute videos by a company that didn't ultimately get off the ground. I completed part of a course - up through factoring trinomials. I didn't get to solving/graphing quadratics, rational expressions, radicals, or right triangle trig. I made a halfhearted attempt to finish it up on principle but MAN, there are so many more interesting things to do with my time. (And also necessary, boring things taking up my time, like getting a work visa for Argentina, a process not unlike one of those unsolvable mazes designed to induce frustration paralysis in laboratory animals.) I retained the rights to the videos so I put them up online with a Creative Commons license, because I don't have the time or inclination to do anything else with them for now.

One of this company's innovations was for the videos to ask questions and pause and wait for a response. So, within each there are questions with a short pause before the answer is revealed. Each one took me about a week of time outside of school to plan and record and cut. The quality gets better within the first few, due to me learning some things, and also acquiring a decent microphone. Also... I don't think anything like they are the answer to any real or perceived problem in math ed. I don't think they are the best way for anyone to learn (the first 5/8 or so of) Algebra 1. They are the best I could do with the format and something I did for some extra money because, hi! teacher. I'm not un-proud of them, but I know they are imperfect. I'm not inclined to go back and re-record or fix anything, so feel free to criticize but I am not likely to react.

But, they're a resource and maybe they can do some more good than they were sitting on my hard drive. I hope someone finds them useful.

One of this company's innovations was for the videos to ask questions and pause and wait for a response. So, within each there are questions with a short pause before the answer is revealed. Each one took me about a week of time outside of school to plan and record and cut. The quality gets better within the first few, due to me learning some things, and also acquiring a decent microphone. Also... I don't think anything like they are the answer to any real or perceived problem in math ed. I don't think they are the best way for anyone to learn (the first 5/8 or so of) Algebra 1. They are the best I could do with the format and something I did for some extra money because, hi! teacher. I'm not un-proud of them, but I know they are imperfect. I'm not inclined to go back and re-record or fix anything, so feel free to criticize but I am not likely to react.

But, they're a resource and maybe they can do some more good than they were sitting on my hard drive. I hope someone finds them useful.

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