Reforming Math 3

(Continuing from Reforming Math 1 and Reforming Math 2.)

Here's the response from the Head of School:

Your description of the Natural Friends mathematics program is distressing.  Your experience with Older Daughter is troubling.  Your report of the perception of other schools of our program and students is upsetting indeed.

I am working with our math specialist to examine our program with candor to insure among other things that the basic operations are effectively taught and learned.

I would contend that the goal of a constructivist math curriculum is not only to teach basic operations but to do so in a meaningful context.  I observed a part of a fourth grade math lesson this afternoon in which multiplication facts were being reviewed in the context of a math baseball game.  The product of numbers rolled on two cubes signaled an outcome in baseball terms (an out, a single etc.).  In short order students were given a second pair of cubes and a new set of ranges that indicated the value, in baseball terms, of the outcome of taking the sum of each selected-by-the-player pair of cubes and finding the product of those two sums.  One of the pair of students I was watching quickly realized that adding two cubes introduced an element of strategy into the game.  Rolling 6, 6, 3, 2 one student paired the sixes and therefore multiplied 12 X 5 to get 60.  His opponent noted that if he had paired each of the sixes with the 3 and the 2 respectively he would have been able to multiply 9 X 8 to get 72.  He said he would always add the two highest rolls with each of the lower rolls and then multiply.  I asked if this would give him a larger outcome every time.  He said he wasn't sure but he was going to find out.  The game offered a meaningful context for students to enthusiastically engage in a review of multiplication facts.  The introduction of the second pair of cubes provided a relevant novel element of mathematical strategy.

Math Trailblazers and Connected Mathematics are programs that attempt to engage students in the application of mathematics from the beginning without yielding up the also important algorithms and facts that are part of the language of mathematics.

I have recently learned of a book called A Mathematician's Lament:  How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.  A math teacher at Friends Omphalos' high school described it to me.  I have ordered the book.  He described the premise of the book to be this:  imagine students being taught to paint are told, "To learn to paint you must learn about color and composition by using the color associated with the number in the designated regions and paint that color within the lines."  As students progress the number of numbers increase and the regions grow smaller.  Eventually frustrated with this process students ask, "But when do we get to really paint?"  The teacher tells them maybe when they get to graduate school they will be ready to paint.

I don't know that I will persuade you of the value of the Math Trailblazers and Connected Mathematics programs but I appreciate your willingness to share your assessment.  You have successfully called my attention to potential problems with our mathematics curriculum and its implementation.

And my response:

Right off the bat, so to speak, you've stumbled across a pet peeve of mine -- the way that baseball is assumed to be of universal interest to kids.  There are many kids who have no interest or experience of baseball, for instance, girls, and the children of immigrants.  My two daughters know nothing about baseball, and terms like "single" would be meaningless to them in this context.  Baseball is often combined with math, which, in my opinion, just supports the stereotype that math belongs to boys.

Leaving that aside, if the fourth-grade kids are inspired to ask interesting math questions because of this game, that's groovy, but it's not enough.  I would want to know that every child in the room was able to do basic multiplication facts.  You might have happened on one particular kid who is gifted at math, or who receives tutoring or extra support that the school doesn't even know about.  (I happen to know a NF 4th grader who does Kumon math, for example.)  And will the kid you talked to have help from the teacher to follow through and answer his question? 

I'm certainly not opposed to teaching basic operations in a meaningful context, and if that's what I saw going on at NF, I would have no complaints.  But what I see is that the kids don't learn the basic operations, which is how they wind up in remedial math at their next school.   At NF, kids don't learn the standard algorithm for multiplication; they only learn the "partial sums" method, which is less efficient (it takes much more writing.)  They don't learn useful shortcuts like cross-canceling when multiplying fractions.   They don't learn how to divide one fraction by another.   They don't learn multiplication and division of decimals.

I doubt very much that you'll be able to convince me of the wonderfulness of Trailblazers and Connected Math.  From what I've seen so far I'm not impressed.   For instance, TB and CM produce some of the worst homework assignments I've seen, and I've seen a lot of bad homework.  (I don't think homework should be assigned in elementary school at all, but that's a different discussion.)  TB and CM homework routinely has directions like "explain how you would answer this" that involve a lot of writing.  Just the other day, Younger Daughter had a homework assignment where she was supposed to "write a subtraction story".  That's a huge task for a kid like Younger Daughter who has language delays, but I happen to know that Older Daughter would have hated it too.  Why?  It's tiring, tedious, and time-wasting (and alliterative!). 

There's an essay version of the "Mathematician's Lament" kicking around the web, which I've read.  I'm not opposed to this point of view exactly, I just don't think it's a substitute for learning the basic skills of mathematics.  If you can ensure that NF kids are solid on their basic skills, you can do all the  "fascinating and imaginative" stuff you want with no objections from me.