Sunday, January 23, 2011

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.


  1. I don't get why baseball gets linked in with teaching math either. I remember once in middle or high school running into a set of "baseball math" problems on a test and getting flustered. I had never seen a baseball game in my life and had no idea what any of the terms meant.

  2. Right. Teachers like these baseball-themed projects because they think it's fun for the kids. It doesn't seem to occur to them that plenty of kids don't know or care about baseball.

    Instead of these hokey attempts to paste fun onto learning, they should focus on teaching their subject. There's deep satisfaction to be gained from learning. Learning IS fun -- no need to paste it on.

  3. I'm always amazed that principals or teachers don't see that baseball criticism coming. It's like when a seasoned politician gets in front of the camera and makes a stupid gaffe -- you think, Even if he really thinks that, doesn't he know not to *say* that?

  4. Chris, that's a funny thing. They don't see the criticism coming, and I wouldn't be surprised if I'm the first person they've heard it from.

    In spite of what teachers like to say, it's actually very rare that they hear criticism or complaints from parents. Parents are amazingly silent on these issues. They complain behind the school's back, but to the school? Hardly ever.

  5. I've enjoyed reading your math related posts.

    I agree with you that too much writing is expected in how math is taught today, and that just increases the frustration level for the kids.

  6. Baseball math?? WTF? Now I know where fantasy baseball came from ...oy

    I pity the poor fool who would tussle with you about Math! Did you tell them you have a masters in it? You can partial sum them under the table before your morning cup of coffee! I think you have found a real calling here( besides all you other ones ) A master decree holder who is calling for kids to know actual math...unstoppable !!

    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.

    What a radical notion! Basic math skills? Their eyes must of crossed at that one .

  7. Anne, you'd be amazed at how little my qualifications matter. There's plenty of people with better qualifications than I have who have spoken out about these goofy math curricula and gotten nowhere.

  8. Thanks, Anne, for alerting us to FedUpMom's impressive secret credentials in math! (She is clearly too modest to mention them here on her blog, even when they're relevant.)

    FedUpMom, This line made me laugh out loud: "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 do think part of the problem with constructivist math is this tendency to try and make it "fun," "relevant," etc., by relating it to something extrinsic to itself whereas, as you suggest, perhaps the point should be to allow kids to learn to appreciate math for its own sake, very much as a language. One of my biggest issues with the constructivist program used in my daughters' school is that it actually discourages kids from appreciating math on that level.

    You might be interested in a guy called John Mighton. He's a mathematician (actually a playwright who went back to university and got a PhD in math) who has taken on the constructivist curriculum here in Canada with limited success. One of his arguments against the watered-down constructivist pedagogy prevalent in Canadian schools is that by eschewing incrementalism, it contributes to math anxiety. You can read more about him here.

  9. Anne, you'd be amazed at how little my qualifications matter. There's plenty of people with better qualifications than I have who have spoken out about these goofy math curricula and gotten nowhere.

    January 24, 2011 5:59 PM

    In the general scheme of things certainly and I'm sure ...but as a parent talking to the school head...they can't dismiss you on Math! I believe they are use to saying a few buzz words and the parent supposed to sling away.But you can return the lingo and more lol! In math as well as other fields...but I should allow SOME mystery about you remain.

    and yes commentators, your blog Diva, FedUpMom, is too modest

  10. I love the paint by numbers over-simplification from the Friends Omphalos math teacher (who is clearly ignoring the fact that FO has already integrated into their culture that NF students NEED remediation, but DON'T BLAME the NF approach! It's ARTISTIC! and after we bring your 'already behind' child up to our standards, you'll understand why their program is so wonderful...)

    The paint-by-numbers example simplifies the argument to that mental 'no brainer' response of 'of course college is too late!' I don't recall anyone proposing that we wait that long to teach our children the fundamentals of anything? And the example (which is happily disarming from the teacher's side of things, and nothing wins an argument faster by arguing a ridiculous position and forcing your 'opponent' to 'defend' the middle ground and 'common sense'.

    And the paint-by-numbers offers such great material to work with. in learning color composition through PBN, did they explore monochromatic themes? interpretive themes? abstract themes? did they just use paint or did they learn to 'color by numbers' with crayons (addition), colored pencils (subtraction), chalk (multiplication), and charcoal (division)? did they just put the color on or did they learn about shading, application techniques, layering, etc...

    the FO teacher would like us to just picture that cheap watercolor kit available at the local arts & crafts store and go 'Of Course! how silly of us to question your wisdom in choosing this path for our children. the fact that our children are immediately behind when they transfer to your school is of no consequence, since you recognize the problem and already target them for remedial studies. You are wise in the way of the Math and we are but mere mortals in your presence...'

    and to cross reference another post - you'd probably only receive the grade of C- as a parent, due to questioning their curriculum and not doing more to embrace and reinforce their methods, which would make them (both the school and their methods) more successful - and therefore your children more successful. Isn't that how we measure a schools success? By how well they implement their curriculum? We can't measure our schools by the children's performance and knowledge, they are too variable and unpredictable....

    first-time post - great blog! I'm a stay-at-home dad that is in his third iteration of 'going through elementary school' and wondering why how I learned it was apparently so bad. And wondering why I have to learn every technique taught in class to be able to answer questions that my children have while they do their homework, all because the the answer isn't right if it wasn't solved by the technique learned in class that day....

  11. It's great to hear from a new reader!

    I think the "Mathematician's Lament", as well as a lot of the new "Fuzzy Math", is motivated by a dislike of bad traditionalist teaching. That is, it's a backlash against making kids memorize an algorithm, with no explanation of how or why it works, and then making the kids do way too many repetitions of simple problems. I agree that this is indeed a bad way to teach math.

    Unfortunately, Fuzzy Math is also a bad way to teach math.