This so-called cyber cafe was the beginning and the end of free Internet access at NCSM 2010. It was like the fall of Saigon around there. Basically.
- Lessons Learned in Leadership for Classroom Change. Judy Kysh, Diane Resek.
- Coaching Teachers to Ask Questions That Provide Just Enough Help to Move Students Forward. Denisse Thompson, Charlene Beckman.
- Learning to do Mathematics as a Teacher. Deborah Loewenberg Ball.
What Is A Rich Math Problem?
According to Kysh and Resek of San Francisco State University, a rich problem …
- … is mysterious, and the mystery is mathematical.
- … has very little overt scaffolding.
- … is accessible. All students can learn and build on what they know.
- … has natural extensions.
I’m not sure if there’s some selection bias working for me here but every session I’ve attended on problem solving has approached it very courageously and with profound respect for teachers and students. Like this:
Students need to experiment and fail productively. Teachers want to be helpful to their students so they run around protecting them from failure. But you learn by failing. Failing is a productive thing. Mathematicians need a big wastebasket. Test scores can improve by teaching less.
They discussed examples and counterexamples of rich problems as well as strategies for teaching them (none of which diverged much from the template laid out in yesterday’s awesome problem solving session) but the best question came up at the end and went unanswered:
“Where do you find rich problems?”
How Do You Teach Teachers To Anticipate Student Thinking?
Such a great title here: Coaching Teachers to Ask Questions That Provide Just Enough Help to Move Students Forward.
If you can’t anticipate student thinking you end up helping students with comments like, “You might want to check on step two there.” Or, “Tell me how you went from step two to step three there.”
So ask them to justify every step, not just the wrong ones. Or at least ask them to justify enough correct ones so that your request for justification becomes a meaningless predictor of the rightness or wrongness of an answer.
In order to provide nuanced feedback, it becomes really important to acquaint yourself with all kinds of routes (mistaken and correct) through a problem.
We were at thirteen tables. We were all given the same problem. We worked through it in different ways and then passed our answers around, examining another group’s work on our table like a coroner pokes at a dead body on a slab.
“What happened here?”
“What do we do next?”
This was a useful exercise.
Edu-Hero: Deborah Loewenberg Ball
This was a monster presentation. Just incredible. I couldn’t stop tapping away. Looking back through my notes, I find it hard to believe she pushed through that much material with so much nuance.
It seems impossible to condense a presentation further that’s already so lean (a nice man with an expensive camera filmed the entire thing — I hope to find and relink that soon) but here’s a brief shot:
Math teaching is complicated work that requires a teacher to …
… become mathematically agile on her feet.
Can you quickly analyze and remediate a correct answer that came about from incorrect computation? All of these students have the right answer.
How do you nudge those students towards a generalizable method?
… choose pedagogically strategic examples.
The answer is [c]. Does anyone want to take a swing at the rationale in the comments?
… develop sensibilities about mathematical language, including a realization that not everything in K8 education needs a definition and an awareness of the overlaps and conflicts between math language and everyday language.
If a student defines a rectangle as “a shape with four square corners that’s flat and closed all the way around,” is she wrong? Will that definition lead to undesirable complications later on in her learning?
If you ask a student to define a “polygon,” will you accept anything less formal than “a simple closed plane curve composed of finitely many straight line segments.” How much mathematical fidelity will you sacrifice in order to exploit your students’ everyday vocabulary.
I just took some criticism via email of my use of the term “steep” instead of “slope” in my TEDxNYED presentation:
I’d urge against using the word “steep” to describe slope. Steep is a physical characteristic, and “rise” is very different than “increase.”
Given the remedial populations I teach, it shouldn’t surprise you to learn that I tilt towards familiar language as a starting (though not an ending) point, though I feel the tension along that rope all the time.
… recognize how subtle alterations in the wording of a problem drastically alter the mathematical work required to solve it.
- I have pennies nickels dimes in my pocket. If I pull out two coins what amounts of money might I have?
- I have pennies nickels dimes in my pocket. If I pull out two coins how many combinations are possible?
- I have pennies nickels dimes in my pocket. If I pull out two coins how many how many different amounts of money are possible? Prove that you have found all the amounts that are possible.
Along these lines, I’d like to recommend Kate Farb-Johnson’s analysis of math assessments that test students’ reading more than their math.
Here are DLB’s slides. Find the video of this talk.
Gratuitous iPad Review
- I need to teach a class on iPad touch-typing. Steve Jobs could be my TA. Seriously. Three days of non-stop notetaking is all this thing took.
- If I take my eyes off the screen, though, I fall apart.
- 95 wpm at a sprint. Seriously.
josh g.April 27, 2010 - 10:23 am -
The decimal ordering lists I think I can follow, but I’m more interested in what the analysis was on those multiplication questions. Both A and B look generalizable to me but I can’t follow that third one at all.
(Which is a good reminder that it’s useful to catch students doing this before it shows up on a test, so you can just ask them to explain their steps in person.)
Dan MeyerApril 27, 2010 - 10:28 am -
Looks to be binomial multiplication. (30 + 5) * (20 + 5)
paul thomasApril 27, 2010 - 11:00 am -
ON a different topic, DLB is one of my heroes. She has a no-nonsense approach that emphasizes results. Her organization for the types of knowledge a Math teacher needs to command is good, but I have seen more succinct versions that I really like (I will post a follow-up comment if I can find it).
Regardless, I like her implication that getting Math teachers who know as much math as possible does not guarantee getting good teachers. Having a Master’s in math does not guarantee you would be a good teacher. You need to know how to manage the question -> response -> feedback/question cycle.
Jason DyerApril 27, 2010 - 11:24 am -
On the decimal question:
A would be answered correctly even by a student that doesn’t understand decimals (.01, .5, 7, 11.4) — just ignore the decimals and you get the right answer.
B puts everything in hundredths so the student can bypass some understanding.
Again, like A, if a student compared .60 and .45 they would get the answer right even if they didn’t understand decimals, whereas if they compared .6 and .45 they would need to know about decimal place value.
Dan MeyerApril 27, 2010 - 11:40 am -
Yeah, right on. DLB noted that if a student stripped off the decimals from [A] and [B] and ordered the resulting whole numbers they’d still come to the correct answer, which makes them really, really lousy assessments. I loved that example.
Karl FischApril 27, 2010 - 1:44 pm -
I’m currently reading What’s Math Got to Do with It? by Jo Boaler and she cites some research about math and reading scores correlating at about .93 on some California achievement testing (if I’m remembering correctly). She makes the case that our assessments should often be stripped of often confusing contexts for that reason (while arguing that the work we do together in class should be full of context).
MindyApril 27, 2010 - 2:31 pm -
The third multiplication method is called the partial products algorithm.
You look at each place value in the bottom number and multiply it by each place value in the top number without carrying. Just write down the ones * ones, the ones * tens, the tens * ones, and the tens * tens. Then add down like normal.
So: (5 *5) + (5 * 30) + (20* 5) + (20 * 30)
Some kids really like this method but it’s not very efficient when you deal with larger numbers.
I even find myself using it occasionally, rather than the standard method, which is a great discussion to have with the students.
Dan MeyerApril 27, 2010 - 2:33 pm -
Karl, Boaler’s book is shooting up my reading list. She gave a pre-recorded five-minute Ignite talk you may want to check out at this page.
Karl FischApril 27, 2010 - 4:40 pm -
Yeah, I watched it (after yours, of course). Got the book after you tweeted something about her.
Here’s the stat (but no reference in the book) from p. 92:
In California in 2004, there was a staggering correlation of 0.932 between students’ scores on the mathematics and language arts sections of the tests used.
Just before that she had referenced the SAT-9, but it’s not clear if she’s talking about the same test.
Kate E Farb-JohnsonApril 28, 2010 - 11:24 am -
“Where do you find rich problems?” I don’t know the answer to this in general. However, so far, I seem to find them when I’m not searching for them, but just keeping my eyes open.
Teaching mathematical definitions by examples and counterexamples reminds me of the game Zendo. I seriously think that playing this game will improve one’s ability to rapidly think of good examples and counterexamples.
I also recommend Jo Boaler’s book.