An Eloquent Argument Against Rating Teachers

Over a Bridging Differences there's a good article on the dangers inherent in rating teachers by student test scores. This paragraph was striking:

I recently met with the principal of an elementary school in Brooklyn who hates the city's teacher data reports. She told me about an excellent teacher of a gifted class whose students started the year at the very top, near perfect. At the end of the year, their scores had dropped 5/100 of a percentage point. Given the margin of error on all tests, this is a meaningless difference, but the "drop" in the scores caused the teacher to be ranked in the bottom sixth percentile of all teachers.  

What is being done to teachers and schools right now is so wrong!

Summertime, and the Livin' Is Easy

Ahh, what a relief to be done. I felt a bit overwhelmed this year. Once that feeling sets in, I don't have much energy for being creative. But the semester is over, the grades are pretty much turned in, and just like every year, I've already started thinking about what I'd like to do in my classes this fall. Two are repeats from this semester, and will get some small tweaks. The third class, Pre-calculus, is one I haven't taught for a few years, so I'm looking for lots of good material for it. Those RISPs I mentioned yesterday are a great starting point. But I'll be hunting for more.

I also got this crazy idea a few weeks ago that I'd like to put out a weekly math gazette, with copies available in our department each Monday. I've been working on the first issue today. There's so much I want to tell new students each semester, and I thought this would be a fun format for all that. In between planting a few flowers and veggies, I put this first draft together. What do you think?

Math Mama’s Gazette

Issue Number 1, August 15, 2011

(Yes, most of it repeats things I've written here, but I feel pretty confident there aren't many of my new students reading my blog.)

Bits and Pieces (Mostly Links)

When I'm going through my google reader and find stuff I like, I often leave tabs open for days, waiting for me to find the time to think about a post more. Then I clean up all the tabs, and wish I could post about everything I'm reading. Here's a collection of things I've noticed in the past month or so...

• Jason says this proof, that only perfect squares have rational square roots, is insanely simple, but it's not simple enough for me to figure out with this cold I have. 
• "The answer is π²/6: What’s the question?" (at Republic of Math) ends with this cool idea:
  

  Here’s a simple yet revealing question to ask people at all levels of mathematical attainment: “The answer is 10. What is the question?”

  Try it on a few people, preferably in groups: the answers may amaze you.

  I can guarantee people will try, and you and they will be amused by the different answers given.

• Mathematician as Explorer. I saved it 2 or 3 weeks ago, and just now read it. I like stories about how interconnected mathematical topics are.
• Measuring the Measuring Device
• Seems like I'm always marking articles in Plus Magazine to save. This one on bones has more science than math, so here's an older one I liked, about ants finding their way home.
•  Here's another critique of Khan, perhaps the most useful I've seen so far. I think Salman Khan explained things well enough to help his cousin, and then went wild putting content online. That content has been very helpful to students wanting explanations right when they're ready for it. But that doesn't mean Khan is an especially good teacher. Alexandre Borovik thinks carefully about how to move a student forward, wherever they are at the moment. His alternate hints are so much better than Khan's.
• Too bad PiFactory doesn't seem to be posting these days. He has some good stuff on his site. How to Think Like a Mathematician describes some good classroom interaction, and I liked this game described in Wizard Math:
  

  There are 35 players standing in a circle. As the games wizard walks round the circle she kills every second player until only player survives. The players are numbered one through 35. Which player lives?

•  Some thoughts about the different kinds of memory, at Republic of Math.
• A new blog that I'll be following, by Rebecca Hanson, has this:
  

  I want my students to learn to fight for what they instinctively feel is correct.  I want them to experience how arbitrary mathematical vocabulary is.  Once we decide that answer that fight is curtailed so we never do.  I'm making a point.  "Who cares what anyone else says? - If it's not true to you don't accept it."

This list by Seth Godin reminds me of another list I once saw, that included things like cook a meal, keep a checking account balanced, and fix a toilet that's running:

What's high school for?

Perhaps we could endeavor to teach our future [students] the following:

• How to focus intently on a problem until it's solved.
• The benefit of postponing short-term satisfaction in exchange for long-term success.
• How to read critically.
• The power of being able to lead groups of peers without receiving clear delegated authority.
• An understanding of the extraordinary power of the scientific method, in just about any situation or endeavor.
• How to persuasively present ideas in multiple forms, especially in writing and before a group.
• Project management. Self-management and the management of ideas, projects and people.
• Personal finance. Understanding the truth about money and debt and leverage.
• An insatiable desire (and the ability) to learn more. Forever.
• Most of all, the self-reliance that comes from understanding that relentless hard work can be applied to solve problems worth solving.

RISPs: Rich Starting Points

Ahhh, I like these. I think they'll be a good addition to my practice, mainly in pre-calc, which I'll be teaching in the fall.

Jonny Griffiths, whose pages I've been exploring, gives these  definitions:

Rich: "pregnant with matter for laughter."
Starting: "making a sudden involuntary movement, as of surprise or becoming aware..."
Point: "that without which a joke is meaningless or ineffective."

- Chambers Twentieth Century Dictionary

Jonny seems to be a maths teacher in the UK. He uses these problems in his classrooms. RISP seems to be a common term in the UK, from what I can tell. Each RISP is in a pdf, so you have to dig in to find the ones you like. 

Odd One Out gives a list of 3 numbers or functions or ..., and the student's job is to give a way in which each one of the 3 could be considered the odd one out. If the list were 2, 3, 9, the students could say '2 is  the only even number', '3 is the only triangle number', '9 is the only composite number'.

Building Log Equations uses cards, which students use to build equations, and then the students decide whether their equations are always, sometimes, or never true.

I'd love to get the first book on his list: Starting Points for Teaching Mathematics in Middle and Secondary Schools, by Banwell (and others). But it starts at about $50 used. Maybe the UC library has it.

Have fun digging!

Vi is Fast, Etienne is Slow

Flottille (detail) from Etienne Cliquet on Vimeo.

Live in Illinois? Call Now to Support Teachers

House vote on Senate Bill 512 at 9AM Thursday morning. Call now.

I've been reading Fred Klonsky's blog for a few weeks now. He's been following the proposed bill that would gut teacher's pensions in Illinois. I've paid for my pension throughout my working career, as have the teachers in Illinois. If this happens there, it will happen on other states.

He tweeted the number to call: 1-888-412-6570.

Escape from the Textbook Group Met on Saturday

I was winding down my semester this past weekend, creating final exams (to give this week), hoping to get it all done by Thursday. The past two weeks have been pretty hectic, so it was a lovely change of pace on Saturday to attend another meeting of the Escape from the Textbook group, closer to home this time (at UC Berkeley).

Avery started us out with a discussion of the Habits of Mind summary he's working on. I felt like we did some great thinking together. Here are my notes (which are only on the things I was intrigued by):

Avery mentioned a few distinctions he wanted to clarify:

• Problems versus exercises was familiar to me. I think most people reading this blog agree that we need to spend more time with students working on real problems, and help students to see the difference between problems and exercises. (Yes, exercises have their place, and can help us solidify our understanding.)
• Problem-solving versus habits of mind. I had thought of these as the same thing, framed just a bit differently, and didn't really understand the distinction Avery was making. He sees the habits of mind as broader than just problem-solving strategies.

Bits and pieces:

• Avery: "There's an overemphasis on math as a noun. We need to get past defining content, so we can treat math as a verb."
• Someone else said: "Students think that guessing is the opposite of math." We talked about how to encourage guessing and then checking whether your guess makes sense.
•  In our discussion of pattern finding, I mentioned Ben Blum-Smith's collection of pattern-breaking problems, and promised to link to it here. It's in two parts, here and here.
• Christine (I think) mentioned reviewing a book on math anxiety titled Managing the Mean Math Blues, by Cheryl Ooten. I've seen the book, but already had two favorites like it (Overcoming Math Anxiety, by Sheila Tobias, and Mind Over Math, by Kogelman and Warren), and never really looked it over carefully enough to compare. Christine thinks it's better than either of my favorites. I'll look all 3 over carefully this summer. I've been buying lots of used copies of Overcoming Math Anxiety, and selling them to my students. Maybe I'll switch.
•  Avery: When we're estimating what reasonable answers a problem might have, establishing upper and lower bounds is a great technique. Students can feel some success even when they can't finish a problem.

We took a break at 10:30, and worked on problems in small groups after that. I loved the thinking our group did together.





Devising a Measure of Squareness

Our first step was to label the picture we'd been given, so we could discuss it more easily. I've included our labeling below. Before you read any further, would you please rank the pictures from squarest to least square?

We all came up with pretty much the same ordering. I thought a few pictures looked about the same, but still wrote the equal ones in the same order everyone else did. I wrote E before B, and A before D. As we discussed whether they were the same, Gretchen pulled out a ruler and checked. Yep. But everyone else had thought E was 'squarer' than B, and I had noticed it first in my quest for 'squarest' rectangles. We wondered whether excess horizontally 'bothers' us less than excess vertically. It was fun wondering how visual perception works.

When we got down to creating a measure, there were basically two proposals:

• Long side over short side (L/S)
• Long side minus short side (L-S)

I thought at first that the ratio measure was the only one that made sense. But Katie was defending the subtractive measure, and together we fixed up the problems we found with it, and created a subtractive measure that, so far, seems as good as the ratio measure. I'm curious how they compare, and want to think about this more later.

Either measure created the same ordering for a particular group of boxes. The subtractive measure depends on the units used to measure the sides. So a 3 inch by 1 inch rectangle has a measure of 2, meaning 2 inches off square, I guess. But if that same rectangle were measured in centimeters, it would become 5.1 cms off square. The ratio measure is 'unitless', and gives 3, meaning one dimension is 3 times as big as the other, regardless of the units used to measure the sides.

I gave 102cmx100cm = 2cm off, though it seems square, versus 1cmx2cm = 1 cm off, as an example of this measure not 'working' to my satisfaction. Katie suggested adjusting for total area, by dividing by L*S. I was still thinking about the resulting units and suggested using the square root of L*S. It's a bit ugly now, but I can't see a real problem with (L-S)/sqrt(L*S), and I'm intrigued that we have two different measures of 'squareness', with no clear benefit to the simple ratio version.

You can check out the other 6 problems at Avery's blog.

Want to join the Escape from the Textbook group? Check them out here.

Not Quite Math: Playing Chess

(berkeleychessschool.org)

I talk a lot about playing with math, and learning math through games. I'm not sure whether playing chess has anything to do with learning math, but a connection is often claimed. I think that playing chess can help people develop a taste for deep thinking, and a willingness to approach hard problems, both important to learning math.

My dad taught me chess when I was in elementary school, and I eventually beat him once in a while. So when there was a tournament at my  junior high, I entered. It was double elimination, meaning you were out after two losses.

I lost my first game after just a few moves, and I remember wondering how that had happened. I had never read any chess, knew no 'openings', and had no easy way of working out what each move had been. So I didn't really 'get' what had just happened. I played my second game, and lost in the same exact way. This time, I wasn't going to let it go. I asked my opponent to show me what he'd done. He did. I also learned (apparently incorrectly) that it was called Fool's Mate. I just learned last night that the name Fool's Mate is (now?) reserved for a mate in two moves that only happens if one person makes some pretty bad opening moves.

What had happened to me is called Scholar's Mate. I started hearing that name for it when I began taking my nephew to chess school, but thought it was just a sweet alternate name. I sat in on his first few lessons, and loved the teaching. The teacher talked about Foolish Freddy and Sneaky Sam, asked lots of questions, and kept the kids excitedly trying to figure things out. He showed them the Scholar's Mate opening, and explained how to defend against it. I also remember him showing the kids (and me) why moving King's bishop's pawn in the beginning is a bad move. (Of course Foolish Freddy kept thinking he'd start that way and Sneaky Sam kept beating him a different way each time.)

Back to that long ago junior high tournament. I had learned how the Scholar's Mate worked, and could find a way to avoid it, but I was out of the tournament. Or so I thought. There were maybe 30 or 40 boys in the tournament, and only 7 girls. Every girl had lost her first two games. So the organizers decided to create a girls' tournament, and we got to do a round robin (everyone plays everyone else). I wanted to keep playing chess, so I didn't question it. I played all 6 of the other girls, and I think I beat all of them. I probably used that new trick on a few of them. I got a first place trophy, which I found terribly embarrassing. I threw it in the back of my closet, and may have lost interest in chess at that time.

[I did try to learn a bit more at some point, and loved learning from the book Bobby Fischer Teaches Chess. It had pictures drawn of the chess board, which made it easy to see what he was talking about. When I wanted to read more, I couldn't find another book that used pictures. They all used the standard 'algebraic' notation, which may be a good way to remember your game, but which sucks for trying to learn. Hmm, is there a lesson for math educators there?]

I was angry about that trophy. I'd lost fair and square in the original tournament. Why would I want a trophy for winning in a consolation tournament? The word 'sexism' wasn't in general use yet, and even now, it's a little complicated - feeling angry about getting a positive treatment. (Kind of like men opening doors for women. I was never comfortable with it, nor with complaining when it happened. I developed the habit of trying to open the next door for the guy who'd opened a door for me, hoping to raise his consciousness a wee bit, or at least to return the favor.) I never connected my loss of interest in chess with what happened at that tournament until now, but it would make sense...

I've been taking my nephew J. to chess school on Friday nights since January. He's had trouble in school (bounces around too much, talks back to teachers, gets in fights, ...), but he's a good kid, and I've wanted to help him connect with the world in positive ways. He was already interested in chess when this opportunity came up, so I jumped on it. Class meets from 5:30-7, and then there's tournament play from 7 to maybe 8. I knew nothing about chess tournaments, and had never seen a chess class before that first night. When I wanted to follow J. into the tournament room, I was told parents weren't allowed. I flashed on the scene in Searching for Booby Fischer where the parents are kicked out of their kids' tournament room because the parents are behaving badly. It made me giggle.

After 4 months of carting J. to class each Friday night and hanging around North Berkeley for a few hours with my son, I was eager to join in the fun when a Friday night adult tournament started up at the same time (and place). Last week I hadn't signed up properly yet, and was paired with someone for a 'casual game'. I ask people their ratings to try to understand the system. He hadn't played in many years but used to be rated around 2000. I knew I'd lose each game at first, and lost as I expected. It was the most delightful loss I could have imagined.

I often beat J., and he's winning trophies in the kids' tournaments, so I know as much as a kid who's getting good. But that's nothing, apparently. One of the things I use to measure how I'm doing is how many of my pieces have crossed the midline of the board. I think I never managed to get a piece past that line that night. I loved thinking about the game as it progressed, and thought of his pieces as exerting a kind of pressure. It was fascinating.

Last night I had gotten my U.S. Chess Federation membership, and signed up for a game. I was paired with Gerl (rated 1484), and lost after we'd each made 12 moves (faster than in my previous game). We went over to the 'analysis room' afterward, and talked through the game. I was once again fascinated, totally intrigued by it all.

Would you like to see how much of a rank beginner I am? Our moves follow. I absolutely can't make sense of this without putting the pieces on the board and moving them. If you're interested enough, try playing this out. Rows 2 and 7 are where the pawns start, and the lettering left to right from White's point of view; pawn moves just show the square moved to, N is for knight; x means a piece was taken, + means check. White's second move was bishop to C4.

		White	Black
1		E4	E5
2		BC4	BC5
3		NC3	C6
4		NF3	D6
5		D3	NF6
6		H3	o-o (castle)
7		G4	G5
8		BxG5	A6
9		QD2	B5
10		BH6	RE8
11		QG5+	KH8
12		QG7++

  

I used to play to capture pieces. In the last two games I've played, it has become clear to me how silly that is. Both times the checkmate was almost bloodless.

I'm hoping I can get the guy I played last night to come give a few chess lessons at my son's school.

What do you think - does playing chess help us learn math?

Math Relax: Getting More Comfortable with Taking Tests

I just had the most wonderful experience. I heard a student in the math lab say, "... It's called Math Relax. It really helped me get more comfortable taking math tests." I popped my head out, and saw a student I don't know talking to two other students, who were listening intently. She was telling them about the meditation cd she listened to.

I said, "Did you know I made that?" She looked shocked and said no. I said a bit more about it, and then emailed the link (always available to the right) to the two students she had been talking to. I think she had just convinced them it was the answer to their problems.

So I just revised the blog post that goes with the recording. I realized that I have never posted my edited version of the questions George Polya suggests that we should ask ourselves at every stage of the problem solving process. Here it is:

Problem Solving

Learning how to solve problems is the most important part of math. When a problem stumps you at first, pull out this sheet. It will help you break down the problem-solving process by suggesting things you can ask yourself each step of the way. [Revised from How To Solve It, by George Polya, 1945.]

Step 1: Understanding the Problem

First, you have to clearly understand the problem.

• What are you being asked to find? (It usually helps to write:   “Let x =”  this quantity.)

• What information are you given?  It might help to organize the information, maybe in a table.

• Draw a picture. (Try to show the relationships, don't worry about good artwork.)

Step 2: Devising a Plan

Second, find the connections between the information given and what you're being asked for. You may need to consider other problems you've done in the past that are similar, or that would solve part of the problem. You want to come up with a plan for the solution.

• Can you restate the problem in your own words? (If not, discuss with someone else.)

• Have you seen a problem like this one before?

• If you can't solve this, can you solve an easier problem with the same structure?  (If so, make one up now to solve. That will give you insight that may help you with this one.)

• Can you add information to your picture that will make it easier to solve your problem?

• Are there definitions that might help you?

• Can you solve part of the problem?  (If so, do it and state a new problem from what's left to solve.)

• Can you figure out anything interesting from the given information?

• Can you say: "If I knew _____, then I could solve this" ?  (If so, state a new problem in terms of trying to find _____.)

• Are you sure you've used all the given information?

• Is there information that may be implied but not stated outright?

Step 3: Carrying Out the Plan

Third, carry out your plan.

• While carrying out your plan, check each step.

• Can you see clearly that the step is correct?

• Can you prove that it is correct?

Step 4: Looking Back

Fourth, examine your solution.

• Can you check your answer?

• Can you check the steps and the reasoning?

• Is there another way to figure out the answer?

• Can you see it at a glance now?

• Can you use your answer, or these methods, for some other problem?

Giving Ourselves Permission to Fail

I don't much mind making mistakes. Perhaps that's why I'm comfortable taking risks.

I've been enjoying Steve Miranda's blog about Puget Sound Community School (titled Re-Educate Seattle), and especially liked his summary of an article on making mistakes. 

Just to give you an example, in one study I conducted a few years ago with my graduate student, Laura Gelety, we found that people who were trying to be good (i.e., those who were trying to show how smart they were) performed very poorly on a test of problem-solving when we made the test more difficult (either by interrupting them frequently while they were working, or by throwing in a few additional unsolvable problems).
“The amazing thing was, the people who were trying to get better (i.e., those who saw the test as an opportunity to learn a new problem-solving skill) were completely unaffected by any of our dirty tricks. No matter how hard we made it for them, students focused on getting better stayed motivated and did well.

It sounds like the people who were internally motivated  to solve the problems were able to keep their cool, while the people who just wanted to 'do well' got more and more frustrated. This is an important aspect of test-taking. I like solving problems, and I'm never too worried about how I do on a test. I try to help my students develop this sort of relationship to tests with my Math Relax audio track, but I don't know how effective it is.

What is an Ellipse?

There are 3 different ways to describe an ellipse:

from Wikipedia ellipse article

• The conic section created when a plane passes through a cone as shown to the right,
• The set of all points (in a plane) whose distances to two fixed points (the foci, plural of focus) add up to a constant, or
• The points (x,y) that satisfy  (x-h)²/a²+(y-k)²/b² =1.

How do we know these are the same thing?  Well, in pre-calculus courses (and sometimes intermediate algebra), there are generally proofs that the second definition leads to the third (like this one). But the connection to the conic section is always stated and never proved. For years, I've found that annoying, but hadn't looked into it until recently. I may be working with conics just a bit in my intermediate algebra course, and will be discussing them in my calc II course, so I'm brushing up this weekend.

Last year, I found a lovely proof that the first definition of the ellipse is the same as the second. It involves something called Dandelin's Spheres. It's so simple and elegant, I was surprised I'd never seen it before. Everywhere I've found it online shows the proof for ellipses, and most mention that the proofs for parabolas and hyperbolas are quite similar. I couldn't work those out on my own, though, so I went searching online again. Here's a cute rendering that does show the parabola and hyperbola, but it's still not helping me see the relationships. I'll have to sleep on this, I think.

Math Fairs

In a telephone chat with Gord Hamilton, I found out his initial inspiration for his Math Pickle work was a SNAP Math Fair. I had never heard of these, and went to their website to check it out. What a great idea!

Paul Giganti runs some great math festivals in California, but I've wondered how we could help spread the good work he does. The SNAP Math Fairs have the kids running the math festival, while their parents wander around trying to solve the puzzles. Each kid is the expert on one puzzle, and can help visitors to their booth solve it if needed.

The website includes a puzzles page, a resource page with more puzzle ideas, guidelines for organizing your own SNAP (Student-centered, Non-competitive, All-inclusive, Problem-based) Math Fair, and lots more that looks useful. They make it look easy.


From the puzzles page:

Number wheel

In the figure on the left, numbers have been placed in the circles. For every pair of neighbouring numbers, the sum of the pair equals the sum of the opposite numbers.
The problem is to place the digits 1 through 6 into the circles using each number as few times as possible. In the picture on the left, we used the number 3 twice.
In each of the figures below the 1 and 5 are already in place. In each case, finish the puzzle by putting the numbers 2, 3, 4 and 6 in the proper places.

(This is a simpler version of Order the numbers from The Moscow Puzzles by Boris Kordemsky)

If your child is in school, this would be a good way to help move their math program in a kid-friendly direction.