In my last post, I wrote:
I also want to start them immediately on the weekly Problem Solving (aka PS — our parents call them PMS, haha, such funny parents we have). I might write a post on this because it’s really at the heart of why I love teaching math.
So, this is a post about my favorite geeky mathy thingy that I call a PS.
I love teaching problem-solving for a very selfish reason: I always learn something new from it. I learn from struggling with the problem, I learn deeper mathematics, I learn from my students’ different solutions and their non-solutions, I learn from other teachers, I learn that I have soooooo much more to learn.
I am not a math major. I didn’t study beyond calculus. I teach algebra and geometry. But I read a lot of math books. I even buy books that are way over my head and have no clue what I’m reading two pages in. I’m mostly overwhelmed by my ignorance.
I don’t believe one’s love of math is innate, thus I’d be remiss if I didn’t mention all the people who generously and lovingly share their love of problem solving that directly or circuitously have touched my life.
- My father: He was a math teacher for over 30 years. When he passed away 13 years ago, I was still teaching science, not math, but I knew he was proud of me because I’m the only one in our family who’s a school teacher. I’m also the poorest one.
- Michael Shaughnessy: I took a summer course from him in 1996 called Problem Solving for Middle School Teachers. It remains the best class ever. Dr. Shaughnessy was NCTM President in 2010-2012.
- Sabrina: My daughter has such a beautiful, efficient way of seeing and solving math puzzles. But she’s more passionate about drawing and dancing. Go figure.
- Martin Gardner: The wealth of his knowledge and his love of recreational mathematics are second to none.
- James Tanton: One of the coolest things about Twitter is the ability to follow, in real time, someone whose work (videos, blog, books) you’ve stolen over the years.@jamestanton‘s tweets are like little chunks of chocolate.
- Joshua Zucker: Biggest treat for me at last month’s Math Teachers’ Circle training was meeting Josh. I first learned about Josh through the Julia Robinson Mathematics Festival where he’s the director.
Whenever I present problem-solving at workshops, inevitably this question comes up, “Where do you get all these problems?” Here are three pages of PS_problems that I’d given out to workshop participants.
These are the main online sources for my PSs:
These are books on my bookshelf that I absolutely treasure because they are loaded with great PSs:
- A Passion for Mathematics, Clifford A. Pickover — Amazon, $13.13
- The Colossal Book of Short Puzzles and Problems, Martin Gardner — Amazon, $23.21
- Problem Solving Through Recreational Mathematics, Bonnie Averbach and Orin Chein —Amazon, $9.91
- The Ultimate Book of Puzzles, Mathematical Diversions, and Brainteasers, Erwin Brecher — Amazon, (only used books available from selected sellers)
- A Moscow Math Circle, Sergey Dorichenko — AMS, $35
- Mathematical Circles (Russian Experience), Dmitri Fomin, Sergey Genkin, Ilia Itenberg —Amazon, $36.86
- Thinking Mathematically, John Mason, Leone Burton, Kaye Stacey — Amazon, $61.59 (Holy moly. I don’t remember the book costing this much; it was a required book for my course with Dr. Shaughnessy as mentioned above.)
- Solve This, James Tanton — Amazon, $37.34
- How to Solve It, G. Polya — Amazon, $23.40 (You know you MUST own this book, right?)
[09/10/12: Sadie @wahedahbug reminded me of a small book that we both loveFostering Algebraic Thinking, A Guide for Teachers, Grades 6-10, Mark Driscoll —Amazon, $22.20. Driscoll also wrote Fostering Geometric Thinking — Amazon, $26.75]
If I didn’t hand out a PS on the first day of school, then it’ll definitely go out the second day. I have three types of PSs — weekly, in-class, and group.
The first time that I introduce this, the process roughly takes on this form:
- I pass out the PS. (To save paper and photocopying, student just gets a strip of paper that has the problem on it.)
- I call on one student to read the problem aloud. I then ask everyone to read the problem again quietly on their own. If it’s a particularly lengthy one, I ask them to read it again.
- Questions I tend to ask, “What are you being asked to find out?” “What information do you already know?” “What are your immediate thoughts about this problem?” “Have you seen a similar problem before?” “Do you have ideas on how to start the problem?” “Wanna give me a guess on what the answer might me?”
- I tell students that since this is the first time they do a PS in my class, I’ll walk them through the process of writing up a PS. I walk them through Polya’s four steps:
- Understanding the Problem
- Devising a Plan
- Carrying out the Plan
- Looking Back
- I blah-blah-blah about the importance of writing in mathematics, and that I don’t ever want to hear any whining, especially of this sort “but this is not an English class…,” because help-me-God if I do hear it.
- I emphasize that all write-ups must be on notebook or grid paper.
- Students have one week to turn in the PS. They get a new PS each Monday, it’s due the following Monday. This is the only assignment that I do not accept late.
That’s roughly my introduction. Then I help them begin the problem in class. This should take up a full period. Before I dismiss them, I hope for this exchange:
Me: When is this PS due?
Ss: Next Monday!
Me: When next Monday?
Ss: At the beginning of class!
Me: What if you don’t turn it in next Monday?
Ss: That’s too bad!
Next Monday comes around…
True to my promise, I ask the kids to pass forward their PS write-ups. As I quickly leaf through the pile of papers in my hand, I can expect (and you should too) to witness the following:
- Their papers: ripped holes, ripped corners, half-sheet, unlined, spilled cappuccino.
- Their write-ups: red inked, work written sideways and upside down (a complete mess), Dad’s handwriting, four papers are identical (including the non-legible and nonsensical steps), only two papers include the “looking back” step, Mom’s writing at the top of paper asking for extra time.
- Their solutions: all the numbers in the problem got added, all the numbers got multiplied, oh look, this student performed all four operations on the numbers, $850,000 for the bicycle, Victor is 48 years old (while Victor’s father is 36), the building is 756,411 feet tall.
- Their reasons for not having it done: I don’t get it, I forgot it at home, my Dad accidentally threw it out, my sister who’s in calculus couldn’t even do it, my uncle who’s an engineer couldn’t figure it out, I don’t think there’s an answer for it, I was absent when you gave it out, remember? I’m-sorry-I-forgot-to-do-it-but-I-love-your-dress-Mrs-Nguyen!
I take a deep breath. Mentally embrace these precious children. Remind myself that I’ll be with them all year. Worse, they have to be with me all year.
So at this time I pass out a PS scoring rubric and carefully go over it. I’ve always used a 6-point rubric but I really want to change it to 4-point.
I give them a new PS for the coming week and only do the first three steps — reading the problem and checking for understanding — as I outlined above. I remind them that I offer PS help at lunch time: Wednesdays for Math 6, Thursdays for Algebra, and Fridays for Geometry.
I don’t grade these. Because I encourage kids to do math with their family and anyone with a pulse, it’s nice to learn once in a while (about one per quarter) what they can do completely on their own. One class period.
I don’t grade these either. Of course this is my favorite type of PS because I get to watch the kids do the math, ask them questions, and listen to their discussions. Before getting into their groups (I almost always assign kids in groups randomly), students have about 10 minutes of quiet individual time to work on the problem. I do make a conscious effort to follow the 5 Practices whenever kids work in groups. I’d asked my principal for funds to purchase large whiteboards so I can copy what Master @fnoschese does with his kids — and I think my principal said yes!
I’m writing this post with the genuine hope that you’ll incorporate problem solving in some fashion, if you haven’t already, into your curriculum. If you don’t think you have time because you feel you have to cover x-y-and-z content standards, then please make the time. Learning math is a social endeavor, a really fun one, please provide students with lots of opportunities to think critically and struggle productively. I think it’s a beautiful thing when we can develop a classroom culture of doing mathematics so contagious that it spreads beyond school boundaries.