Friday, July 30, 2010

Fun Math Books: I Love Math! Series

Although these books are long out of print, there are inexpensive copies available online of most of them. (They were published in 1992 and 1993.) This list gave me the twelve titles below. (If anyone here knows of more, please let us all know.) I recently picked up a bunch of them, because my son really liked the two we already had.
  • Alice in Numberland: Fantasy Math
  • From Head to Toe: Body Math
  • How Do Octopi Eat Pizza Pie? Pizza Math
  • Look Both Ways: City Math
  • Play Ball: Sports Math
  • Pterodactyl Tunnel: Amusement Park Math
  • Right in Your Own Backyard: Nature Math
  • See You Later, Escalator!: Mall Math
  • The Case of the Missing Zebra Stripes: Zoo Math
  • The House that Math Built: House Math
  • The Mystery of the Sunken Treasure: Sea Math
  • The Search for the Mystery Planet: Space Math
My son and I especially enjoy the stories (in every volume) about Professor Guesser, a cat detective who solves mysteries using mathematical reasoning. She's featured in the title of The Case of the Missing Zebra Stripes: Zoo Math. Some of the zebras are missing their stripes, and Professor Guesser figures out what's really going on.

The books are full of stories, games, mazes, riddles, and lots of math.

Monday, July 26, 2010

Before the School Year Starts: Creating Community

Having been away from the classroom for a whole year, I'm more eager than usual to get back to it. (Countdown continues: First day of class is in 21 days.)

I've printed up my class lists. (40 students in each of my two sections of Beginning Algebra, plus 10 on each waiting list. When I taught in Michigan, our cap was 30. If California weren't in a financial crisis, I'd want smaller classes to be my union's highest priority.) I've reworked my syllabus, and thought long and hard about how to grade in a way that works well for the students' learning. For the first week, I've sought out activities for reviewing pre-algebra concepts (fractions integers, distributing, order of operations), always thinking about how to turn their notions of math around.

I'm planning to teach without reference to the required textbook, and I wanted the students to know that before the term started, so they could choose to buy a different textbook. The required text costs well over $100, and used copies will go fast, probably for over $70. They can buy older editions for under $10 online. My students are struggling financially, and knowing this ahead of time will be a blessing for them.

So I got my classlist, pulled all the emails together, put them in the bcc field (so students email addresses wouldn't be seen by other students without their consent), and sent an email welcoming them to my class, giving them the textbook scoop, and telling them about my philosophy of teaching and learning math.

I've already gotten back about 5 emails from students, thanking me, and saying things like this:
I want to say that I like ur approach toward math and it makes me feel better about taking this class. I've always struggled with it therefore it was the class I didn't look forward to, but I am looking forward to changing that perspective so thankyu.

That email I sent last week might make a huge difference in the attitude people bring to class, which might make my life a lot easier.

There was a bit of tech-trouble. I wrote the letter a few weeks ago, with lots of links to online resources, and blog posts of mine. When I sent it to my own college account, I saw that the links looked terrible. (They had changed from the usual underline form to showing the whole url.)

I thought I could solve that by sending from gmail. I didn't want to send from my mathanthologyeditor address, so I created a new gmail account using my name. Sending a message from a new account to 40 addresses, all in the bcc field, alerted gmail's spam prevention system. It didn't go to any of the addresses, and I got 40 separate messages from gmail support, explaining why. I wrote to support, but they never wrote back. I ended up sending a short message from my college account, and putting the longer letter on a wiki I'd made for the class. Here's what I wrote them.

Now I'm trying to come up with a list of reminders for me while I'm up front. Over the past twenty years, I've gotten real good at colorful, concise min-lectures on the topics they struggle with. Now I'm trying to get away from giving them answers - I want to create a 'community of disciplined inquiry' (Schoenfeld) - and I know how hard it will be to change my ways. I don't have much so far, but I'll include it here. Can you all help me add to this list? I'll keep it on my desk while I'm teaching, to remind me of what I'm trying to do differently.

Reminders to Me

Wait time!   (Mean < 1 second. Minstrell waited 9 seconds!)

Question to me? Redirect. “What do others think?” or (call on someone) “X, what do you think?”

Mistake? [Don't correct or explain what they've got wrong!] Ask what the implications are. (But ask this of ‘right answers’ too.)

“Thanks for contributing.”
"Ok, so... (repeat the idea)"
“Hmm, where should we go from here?”

Saturday, July 24, 2010

Friday, July 23, 2010

Changing the Way We Teach Math - 3 Good Books

I've been reading math teacher books for years. Most of them refer to elementary education, and I've struggled to see how I could use their insights in my college classrooms. Reading teacher blogs over the past year or so has given me lots to think about, much closer to my situation.

Of course there weren't any blogs to be found 15 years ago, when I started teaching full-time. There were two very special books though, which addressed both elementary and secondary math.  I read both volumes of What's Happening In Math Class, edited by Deborah Schifter, about 12 years ago, and have been re-reading them this week. Both books are full of wonderful teacher stories, and totally remind me of my blog reading, except that there is more of a mix of elementary and high school level teachers writing these. (I've searched for elementary teachers blogging about math, and haven't found much.)

It was wonderful to re-read these and rediscover the treasures that have been living in the depths of my memory all these years. This is where I first heard of the Green Globs program (are any of my readers using that still?) and XMania (an extended exercise in creating and using a base 5 number system, used in a number of teacher ed programs).

As I reread ‘Third Graders Explore Multiplication’, by Virginia Brown (volume 1, page 18), I suddenly remembered reading this story so long ago. Remembered the group of kids who figure out the commutative property of multiplication, that 3x9 will be the same as 9x3, and in fact this re-ordering will work for any numbers. Before reading this the first time, I had always thought that was obvious. But it’s not obvious to most kids, and it can be an exciting discovery.

These books also introduced me to good questioning as a method of teaching mathematics. Almost every one of the teacher chapters in these books includes lots of dialogue, both between students and between teacher and students.

Between the two volumes, 22 teachers tell their stories, of working to change their classrooms, and of lessons that engaged the kids and taught them more about how their students learn math. Reading these books, like reading teacher blogs, reminds me that the teacher is always learning in a good classroom, right alongside the students.

Volume 1 focuses on what a good mathematics classroom might look like, and volume 2 focuses more on the struggle the teachers went through as they attempted to change the way they taught. Ruth Heaton wrote (volume 2, page 74), "In the culture of teaching, it is unusual to find practitioners willing to discuss how confused and frustrated they feel about their work." Ahh, but that has changed dramatically with the current crop of teacher blogs! Plenty of teachers are baring their souls, discussing their bad days, their confusions, and their ongoing struggles. So many of us have been hungry for this, and technology has given us our wish. Now we can read more teacher stories than we have time for, and can then narrow it down to the ones that really speak to our personal struggles.

I sometimes find that I think better when I'm reading a book than when I'm reading online. So these two books have been a great resource. I found one more gem, written by Deborah Schifter and Catherine Fosnot, Reconstructing Mathematics Education: Stories of Teachers Meeting the Challenge of Reform. Like volume 2, this book shows teachers struggling to change the way they teach, although this time through the eyes of Schifter and Fosnot. All three books came out of the SummerMath for Teachers Institute, held at Mount Holyoke College. This institute is described (page 106 of Reconstructing...) as an entirely new experience for the elementary teachers attending:
At SummerMath Institute … many [teachers] experience mathematics for the first time as an activity of construction, evaluation, and exploration, rather than as a finished body of results to be stored away. And for the first time they sense that mathematics instruction can be an invitation to the exploration of ideas, rather than a laying on of facts, rules, and procedures.

Let's all explore ideas in our math classes this fall, and if you're struggling, these books are good company to have (along with all your blogger friends, of course).

Thursday, July 22, 2010

Day One of Class: Beliefs About Math

I'm starting my countdown - the first day of class is in 25 days. This is actually my first time starting out a semester as a blogger! I'm excited about all the changes I plan to make, and a little nervous. I'll be posting lots of my plans and materials here in the next few weeks, to share and to get feedback. Once the term starts, I'll rejoin the ranks of classroom teacher bloggers.

Carol Dweck's research, on how people's mindset affects their ability to learn, seems powerful to me. And I want to share it with my students in a powerful way. I thought I had seen quizzes on this, but hers only has 4 questions, that to me all sound pretty much the same. So I mixed that question in with some math belief questions. I plan to give this 'quiz' to my students on the first day of class.

I'd like a tenth question. Got any good ideas? I'd also like feedback on the other questions. This parallels my list of math myths - it's just framed a bit differently. My goal is to get students to realize that the way they've seen math up until now has been skewed, and gets in the way of learning it. I also want them to know that if they can really commit to this, they can change their relationship with math dramatically.

Beliefs About Math and Learning

This is set up as a quiz, but it won't be graded. Think of it as one of those magazine quizzes that you do for fun. After you complete this alone, we'll vote to see how many believe each statement is true versus false, then you’ll talk with your group about it.

T F 1. I'm not good at math, so I can't expect to do well.

T F 2. It’s genetic, men are better at math than women.

T F 3. Intelligence is fixed – it can’t be changed.

T F 4. Math is mostly about memorizing.

T F 5. Intuition and creativity are not useful for math.

T F 6. It’s bad to count on your fingers.

T F 7. There’s one right way to do a math problem.

T F 8. Math ability is fixed – it can’t be changed.

T F 9. To learn math, I need to focus on getting the right answer.

T F 10.

Monday, July 19, 2010

My First Wordle: Math Map

[This is the introduction to my book, Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet, which I hope will come out in early 2011 - turned into artwork.]

Wordle: Words for Math

I like Jackie's wordle from her question: How do you learn?

I'd like to do one too now. My question is ... What do you like about math?

Say as much or as little as you'd like. (Please give me at least one word.)  I'll feed it all to the wordle.

Sunday, July 18, 2010

Black and Red TripleMatch: A Card Game for Adding Integers

I saved Ben's post on this idea, and now I've used Kate's idea for turning his problem into a card game. I've added scoring and some rules of play.

My Beginning Algebra course (community college) has an 'hour by arrangement' associated with it, and I intend to have lots of content-related games available during that time. During our first week, we'll be reviewing. [I decided I like the acronym FIDO for all of the important pre-algebra concepts that students usually have trouble with: fractions, integers, distributive property, and order of operations. FIDO is a math student's best friend; if the FIDO skills are learned well, they will faithfully help with algebra work. I hope I'm not being too silly...]

So for the first week I want fraction games (I have a game book with a deck of cards, called Fraction Jugglers, that we can use) and integer games. I know they like card games. So here's Kate's game, elaborated.

Red and Black TripleMatch

Use a regular deck of cards. 2 to 5 people can play. (More people could play with a double deck.)

Card values
Ace = 1
Jack = 11
Queen = 12
King = 13
Black cards are positive numbers, red cards are negative.
Joker changes the sign of the card it’s played with.

Dealer shuffles and deals out 8 cards to each player, and then lays down three target cards in the middle. Your goal is to match each target separately. On each turn, you discard a card and draw a card from the discard pile or the deck, and then discard.*

When you have all three targets, say ‘Triplematch!’ and lay down your cards in 3 piles for the other players to check.

If you'd like to lay down cards to match one or two piles before you've got all three, you can do that. (It could make the final match harder - those cards may not be changed after being laid down.) The advantage is that you have less cards in your hand to count against you if someone else wins first.

  • Winner gets 30 points.
  • Everyone else counts all cards in their hand for negative points (reds and blacks both count negatively at scoring time).
  • Game ends after each player has had a turn to deal. High score wins.

I’m excited about this game, but I think it might need some tweaking. I’ve just dealt myself a few hands. On the first, I only had to discard and draw once to make TripleMatch, so I thought it might be too easy. On the second, I had to draw 26 cards. Now I’m thinking it might be too hard. Does anyone have any ideas for making the triplematch easier?

If you try it, please let me know how this game works for you.

* Drawing first increases your chances of a proper total dramatically.

Saturday, July 17, 2010

SBG: Wading Into the Water

Dipping My Toe In
I teach at a community college in California. We're being pressured to develop student learning outcomes (SLO's) for each course. It's a top-down development, and there's lots of faculty resistance. We don't see a clear benefit from this process.

But I could see we'd be stuck with it, and I wanted to put in my own twist. About two years ago, I figured it was a good time to introduce Mastery Tests to my Beginning Algebra course. I continued to have my regular tests that covered two chapters of material. But I also decided on what I thought were the most important things I wanted students to learn, and made short mastery tests on those that could be re-taken. Students had to earn at least 85% on each one of those 5 mastery tests to pass the course.

I'd never heard of Standards Based Grading (SBG) at that point. I had worked at a college that taught Beginning and Intermediate Algebra (pretty much equivalent to high school Algebra I and II) as independent study, though, and they used something like what I was developing.

Re-testing until you achieve mastery is a core part of SBG, but my mastery tests only counted for a quarter of the grade. I liked how they worked. The first semester I tried using them, no one failed just because of not passing those, and many people improved their understanding of algebra substantially because of them. So now it's time to wade further in. I'm still not ready to change over completely, but I'm going in much deeper now.

Wading Further In
I've been working on a Mastery Tests tracking sheet for the students (current draft below). I still have numbers on the tests, but have given each one a short name, too, so we can refer to them by name instead of number. I'm still grading on a percentage scale (rather than 4 or 5 points), because I'm used to it, and I don't think using points is vital to the spirit of SBG. I've also grouped more together than I think the SBG folks do. It's what feels right to me. I'll re-consider this point over time.

I think I'm willing to stop grading homework. It was working well to give credit for completed homework, but the post on teacher happiness compared to homework grading policy swayed me. (Who posted that? I can't find it now.) My biggest concern is that students have limited time, and giving them credit for regularly doing their homework helps them develop that good habit. I'm not sure yet on this...

I'm still giving a comprehensive final exam, and credit in the final grade for various ungraded assignments (internet research, writing, attending hours by arrangement, ...). Right now I'm thinking 45% for the mastery tests, 25% for the final, and 30% for the rest. I've thrown out chapter-based tests.

Ok, all you SBG advocates and gurus, what haven't I thought about yet, what do you think I should change, what sage advice can you give me?

Mastery Tests (tentative)

______ Mini-Test 1:                    Multiplication Facts
______ Mini-Test 2:                    FIDO (the Fractions, Integers (negative numbers), Distributive Property, and Order of Operations will faithfully help you with future math work)

______ Mini-Test 3:                    Solving Equations
·       Solve equations
·       Solve for a particular variable
·       Use algebra to solve real problems (#1)
______ Test 4:                            Graphing Basics
·       Graph a line given equation (slope-intercept form)
·       Graph a line given equation (standard form) 
·       Find slope given two points
·       Find equation of a line given two points
·       Find slope given equation (any form)
·       Find slope given graph
·       Find y-intercept given two points
·       Find y-intercept given equation
·       Find equation of line perpendicular to given line
·       Explain meaning of slope and y-intercept in real problems
______ Mini-Test 5:                    Graphing Applications
·       Explain meaning of slope in a real problem
·       Explain meaning of y-intercept in a real problem
·       Create an equation based on a real problem
·       Make a graph for a real problem
______ Mini-Test 6:                    Systems of Equations
·       Solve a system of two equations in two variables
·       Use algebra to solve real problems (#2)
______ Mini-Test 7:                    Use scientific notation

______ Mini-Test 8:                    Factoring
·       Multiply polynomials
·       Pull out a common factor
·       Factor a polynomial
·       Solve equations using factoring
______ Mini-Test 9:                    Solving Quadratics
·       Simplify square roots
·       Complete the square
·       Use quadratic formula
·       Graph parabolas

Epilogue                                   Graph y=1/x and y= absolute value of x;  Simplify rational expressions;  Solve and graph inequalities

                                                [Final Exam covers all course material.]

Wednesday, July 14, 2010

The Math Circle Summer Institute...

... was fabulous! We had morning circles led by Bob and Ellen Kaplan, Amanda Serenevy, and Leo Goldmakher, and afternoon sessions where we led math circles attended by local kids. We also worked on math problems in the evenings, sprawled around our dorm lounge.

I finally led a pretty good math circle there, with high school aged kids, using the Magic Pancake problem. (All the pieces magically change to your favorite kind of pancake, and grow to give you as many yummy bites as you want, perfectly filling your tummy. One cut makes two pieces, two cuts can make 4 pieces, how many pieces can you get with N cuts?) The first year I went, I told a story that used fractions, with kids who were too young for it. The second year, I did base 3 or 8, with junior high aged kids, who I couldn't get to talk. This year my circle finally went well, though I still have lots of room for improvement. My big goal is to learn how to ask more open questions, and let the students guide us along, keeping my mouth shut more, and giving less hints about where I think we should go.

Our morning math circles continued to be my favorite part of the day. I came in late on Monday, and got to hear Bob leading the group as they worked on finding the circumcenter of a triangle. I've never taught geometry, so I don't know this stuff - it was a blast to think about. Next came the incenter, and then the orthocenter. We worked on the topic a bit more on Tuesday, and again on Friday. I'm looking forward to playing with it on my own.

Also on Tuesday morning, Amanda told us a story with Sona Drawings, and then asked: For what size rectangles can we draw one continuous lines around the dots? I knew where that would go, so I got to quietly watch other people thinking it through. I like telling stories, and would love to learn how to do those story drawings. That will be a long-term project.

On Wednesday, Leo led us in thinking about divisibility by 3 and 7. I was very familiar with the first part, and got to watch people thinking again. That was lovely. (I'm not a very good observer, and it's a skill I really want to improve.) The second part was tantalizing. He showed us how you can find out whether 7 is a factor of a large number by removing the last digit, doubling it, and subtracting it from the number formed by the remaining digits.
Example: Is 36,666 divisible by 7? Double the last 6, getting 12. Subtract 12 from 3,666, getting 3654, still too big. Do it again. Double the 4, getting 8, subtract 8 from 365, getting 357, still too big. One more time. 2*7 is 14, 35-14=21. Yes, 7 goes into 21, so that means 7 goes into 36,666.
Buy  why? That's the question! I solved it on Wednesday evening. When I'd done that, Amanda showed me her proof which I liked much better, because it involved modulo 7. I've been playing with modular arithmetic lately, and I like the power it has. She challenged me to make up another way to break a big number up to see if it's divisible by 7. I did it, using mod 7. And then I got excited, thinking about how you could mix and match different techniques. I know that 7*11*13 = 1001, so any number over 1000 can have multiples of 1001 subtracted before using the other techniques. (30030 is a multiple of 7, subtract that from 36,666, getting 6636. Now subtract 6006, getting 630. Yep, multiple of 7. Oops! I didn't even need the other techniques for this one.)

On Thursday, Ellen led us in thinking about Euler's formula for polyhedra (V-E+F=2), as an example of something whose proof is very subtle. Once again, I got to watch people thinking about the beginning steps. If you're interested in this, Dave Richeson's book, Euler's Gem, is a great treatment of it. (And Michael Paul Goldenberg wrote about this session in his post about the Institute. Thanks, Michael, for the reminder that I wanted to write a post.)

What else did we do? My swimming buddy didn't make it back this year, so I only managed to get up early once to swim in the lovely pool. (Missed you, Ellen. Thanks for the cap, Linda.) I ate less than in previous years, but the food was still amazingly good for a big food services operation. And I brought games to share once or twice during our evening gatherings at the dorm lounge. My friend Linda had just given Dizio to my son, and I brought it along. We had a great time with it. We also enjoyed Kataminos, which I mentioned in a previous post.

Come join us next summer!

Saturday, July 3, 2010

Vacation Post: Squarus

Abbie, my niece (once removed, or something like that), says her family made their own game up using Blokus pieces. They each have a color and a corner, and the first to make a 9x9 square (uses all but 8 of the little squares possible) wins.

Do you play your own family variation of a good board game?

Vacation Post: Pentominoes

There are two other posts I'd like to write, but they'd take more effort, and I'm on vacation. My son is playing down at the beach with his second and third cousins, with another mom watching. I'm on the porch, with a cool breeze  at my back.

I've just been playing with Katamino, a lovely game I bought a few days ago at Mackinac Kite & Toy, in Grand Haven. It's a way to play with pentominoes in a two-person game. The X and I pieces are left out, each player gets three single squares and a two-square rectangle, and then players take turns picking the ten pentominoes. Then the players race to put their pieces into their side of the board. The whole game takes about two minutes, which is part of its charm.

When I had no one to play with, I started working the solitaire puzzles, which involve putting 3 of the pieces in a 3x5 area of the board, and then 4 pieces in a 4x5 area, and so on. (The board can change size from 3 to 12 rows.)

I've owned a set of pentominoes in a nice 8x8 case for quite a while, and I know there are many ways to put them in so the 4 extra spaces will be symmetrical. But I've never managed to come close to doing it. I just now realized that I should have long ago listened to George Polya's problem-solving advice: Try a simpler problem that has a similar structure. Katamino had me do that. The game includes a booklet with pictures of 7 different workable combinations of 3, and 19 different workable combinations of 4. (12C3 = 220 total combinations of 3, so it's good to know the ones that work.) I've done all the 3-piece puzzles, and I'm almost through the 4-piece puzzles. What fun!

I'm also enjoying some light summer reading. Blue Balliett has written three young adult novels in which Calder Pillay and his two friends solve mysteries surrounding works of art, and marvel at the coincidences in their lives. Calder plays with pentominoes, using them like an I Ching, for inspiration. The third book, The Calder Game, came out in paperback recently. And here's what made me write:
He'd decided on the Start Small, Move to Large approach that his Grandma Ranjana had taught him. It seemed to work with almost everything in life. After all, you couldn't leap to making twelve-piece pentomino rectangles when you'd never made a five-piece one.
Coincidence rules!
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