ProfoundUnderstandingFundamentalMathematics

Posted on May 30, 2005 @ 17:23 by CatherineJohnson

Carolyn mentioned Liping Ma's concept of 'profound understanding of fundamental mathematics' (PUFM).

This chart is Ma's map of the 'knowledge package' Chinese teachers possess for the topic of subtraction. This is what Chinese mathematics teachers know and understand about subtraction.

I don't happen to have this knowledge package inside my own head, and neither does any other parent I know.

This is why it won't do to say:

One way to understand a math program like EM is to read through and do the exercises in the curriculum consecutively, openmindedly as a learner, not as an assessor. Play with the manipulatives, perhaps even borrow a teaching guide. These programs are much different, and much more exciting than the way we were taught. They are also very hard to describe. With some study, you might find yourself a great parent contributor to something your children's school is attempting to perfect.


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Chinese math teachers develop pedagogical content knowledge over the course of many years teaching and studying elementary mathematics.

There are no shortcuts.

How long does it take to acquire a profound understanding of fundamental mathematics?

I'm guessing 10:

Some evidence that a great deal of practice, and not just talent, is a prerequisite for expertise is the "ten year rule," which states that individuals must practice intensively for at least 10 years before they are ready to make a substantive contribution to their field. What about prodigies like Mozart, who began composing at the age of six? Prodigies are very advanced for their age, but their contributions to their respective fields as children are widely considered to be ordinary. It is not until they are older (and have practiced more) that they achieve the works for which they are known.


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No parent is going to pick up a copy of Everyday Math, read through the book, work the exercises, and be ready to teach or tutor the curriculum effectively.

That's not the way it works.

Parents have a fighting chance of teaching or tutoring effectively with a direct-instruction curriculum like Saxon Math. We have that chance because the books are written so that anyone who's been through grade school can understand what the lessons are about.

None of us is going to do a brilliant job teaching math using Saxon. Becoming brilliant at anything takes 10 years.

But we can help our children learn math.

It's not just children who need direct instruction. Parents need it, too. We parents need to be able to pick up our child's mathematics textbook, read the lesson, and know what it's talking about.

That school districts consciously select unproved mathematics curricula they know parents will not understand and will not be able to teach or tutor from is, to me, unconscionable.

It's not up to us to go begging for a peek at the teacher's guide.

It's up to our schools to bring us into the loop.

Back to main page.

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Oh man, now I've got to understand that chart: I love a challenge.

Why are those two ellipses in boldface? And what does 'the rate of decomposing a higher value unit' mean?

And why am I getting the uneasy feeling that this chart is actually over the top?

-- CarolynJohnston - 31 May 2005

Well, as you know I OBSESSIVELY read every single word of KNOWING AND UNDERSTANDING ELEMENTARY MATHEMATICS . . . and my problem with these charts was what you might call the 'associative property' problem . . . you can only do one operation at a time.

That is, how do you translate a three-dimensional 'knowledge package' into a . . . one-dimensional classroom teaching experience (the one dimension being time).

When you're speaking or teaching, you're pretty much working in a linear, sequential mode . . .

That's not completely true; you can be using manipulatives or visual representations or words that pull together more than one concept at the same time . . .

Even so, you have to start somewhere, move on to somewhere else, and end up somewhere new.

What I think is great about this chart is that she's probably done a decent job of isolating all the 'subtopics' that make up the 'big topic' of subtraction with regrouping.

I have absolutely no memory of why she has rectangles & ovals.

I think I used to know.

WHICH IS PERFECT EVIDENCE THAT CONCEPTUAL 'KNOWLEDGE' IS NEVER, EVER ENOUGH.

THE FACT THAT YOU UNDERSTAND SOMETHING TODAY DOESN'T MEAN YOU'RE GOING TO REMEMBER IT TOMORROW.

-- CatherineJohnson - 31 May 2005