kitchen table math, the sequel

Saturday 9/25 - late afternoon:

My notes from Wu discussing the teaching of division with remainder:


















Actually, this was a big "Duh!" moment.

Prior posts from MSMI2010.

Back when Phil Mickelson was a child, he came up with a project for science class that hinted at his future: an experiment measuring which compression golf ball was best in terms of distance and accuracy.

Three decades and three Masters' jackets later, he's busy with an even more ambitious science project: the Mickelson ExxonMobil Teachers Academy. The academy is sort of a summer camp for science teachers, where third, fourth and fifth grade teachers come to learn how to improve their teaching of math and science. So the morning after finishing the British Open, Mr. Mickelson has flown into town for the start of this week's academy at the Liberty Science Center in Jersey City, N.J.

"I've always used math and science in my career," Mr. Mickelson explains over coffee, his conversation laced with references to coefficients, vectors and vortices. "It helps me know what I need to focus on. On putting, for example, at three feet the success rate is about 99%. At four feet it drops off to 88%; at five feet to 75%; at six feet to 62%; and so on.

[snip]

"I used to think that companies went overseas for cheap labor. That may be part of it. But I've learned that the larger issue is to attract people who have the engineering and other skills they need."

It's not just Google and Intel and Apple and ExxonMobil that have a hard time finding enough people with these skills. So do the public schools, with the result that, at least in the lower years, the majority of math and science courses are taught by people who never studied the subjects in college. Hence the academy program, which is designed by experts from Math Solutions and the National Science Teachers Association, and which emphasizes the link between these two fields.

JULY 20, 2010
Phil Mickelson's Science Project
by William McGurn

Too bad Mickelson didn't call MSMI.

Definition of a Fraction

Recall that a number is a point on the number line (§5 of Chapter 1). This chapter deals with a special collection of numbers called fractions, which are usually denoted by m/n, where m and n are whole numbers and n ≠ 0. We begin by defining what fractions are, i.e., specifying which of the points on the number line are fractions. The definition will be both clear and simple. If you find it strange that we are making a point of giving a definition of fractions, it is because this is something thousands (if not hundreds of thousands) of teachers have been trying to get at for a long time. Most school textbooks and professional development materials do not bother to give a definition at all. A few better ones at least try, and typically what you would find is the following:

Three distinct meanings of fractions — part-whole, quotient, and ratio — are found in most elementary mathematics programs.

[snip]

Such an explanation is unsatisfactory for several reasons. To say that something you try to get to know is three things simultaneously strains one’s credulity. For instance, if I tell you I have discovered a substance that is as hard as steel, as light as air, and as transparent as glass, would you believe it? Another reason for objection is that a fraction is being explained in terms of a “ratio”, but most people don’t know what a ratio is.2 In addition, while we are used to the idea of a division a ÷ b where a is a multiple of b (see §3.4 of Chapter 1), we are not sure yet of what 2 ÷ 3 means. So to use this to explain the meaning of 2/3 does not seem to make sense. Finally, we anticipate that fractions would be added, subtracted, multiplied and divided, and it is not clear how one goes about adding, subtracting, multiplying and dividing a part-whole, or a quotient, or a ratio.

This is why we opt for a definition that is both simple and clear.

Chapter 2: Fractions (Draft) (pdf file)
H. Wu
Department of Mathematics #3840
University of California, Berkeley
Berkeley, CA 94720-3840

msmi 2010: Institute on Fractions is now history. YAY!

I'll leave Catherine and any others to talk about their take on it. I'll have some future posts on what I think are the biggest lessons parents and teachers need to help their students understand fractions, but this is just a roundup.

In the course of 5 days, we covered: definition of a fraction, equivalent fractions, decimals, addition, subtraction, multiplication, division, decimals again, and percent. We had more to do, but we couldn't get to it.

Based on the reaction of the teachers, it was a success. I have NEVER had a class of students where so many students worked so hard. No matter what their background, everyone tried to do the problems. Their effort meant that as the week went on, the students were more engaged and more knowledgeable. The teachers also built up their camaraderie with each other.

Based on their personal comments to me and the anonymous survey I gave at the end, their overall impression was quite high. Several teachers told me that this course was the first time anyone had ever explained how to think about fractions. One told me it was "a revelation" to them, another told me this was the first time they'd see a way to visualize multiplication of fractions. Most responded to our survey saying that this material had changed how they would teach permanently. Several teachers had a different kind of revelation, too: that other people in other schools/cities/states knew and felt as they did. They were connecting the dots not only on fractions, but on the state of math education.

Not that everything was perfect. I was terribly out of practice for being a teacher--bad board technique, bad handwriting, bad short hand in my own thoughts and words, instead of being clear, specific and slow.

I made several errors in sizing up my audience too. I assumed that since I had told the principals what to expect, that they had told their teachers. I assumed that teachers, given a pointer to a web site that had, e.g. Wu's CV on it, would have read such.

The biggest complaint was that it was too much material/days too long, and not enough worked out examples. One solution to the latter is to strongly encourage the teachers to read the textbook a day ahead of time. But part of that is the nature of the beast: there is an enormous deficit of knowledge to overcome. Elementary math teachers didn't go into that field because of their stength in fractions. The breadth of math inexperience-experience even for teachers of the same grade was very large, yet being math experienced didn't quite help, because while those teachers probably followed Wu's proofs more easily, applying his ideas to actual math problems was still a new universe to them, and their skill was sometimes a hindrance, because he was asking them to think an entirely different way than they were used to.

Lastly, I'm thrilled to have met all the people involved. Wu is a delight to work with/for, and I'd do this again with him wherever we can. He was personable and charming as well as brilliant. His wife was just as delightful. CassyT, KTMer, is an exemplary woman. She's a brilliant teacher and student of human nature, and her insights into teachers saved me countless mistakes. She shared her expertise with me in countless ways, and the whole thing would have fallen apart if not for her.

So, where to go from here? First, more Wu institutes! Let's bring MSMI to your locale! Second, the really big thing is to help teachers turn what they learned here into changes in their school. That's no small undertaking. I'll talk about that more in the next post. Last, more documents for everyone: condensing of Wu for parents and teachers. I'm sure you'll see work product of that around here shortly...

paraphrasing Wu at msmi2010:

The idea that there is always more than one way to solve it is propaganda. Sometimes you're lucky to have one way.

from Have Technology and Multitasking Rewired How Students Learn? by Dan Willingham

When you encounter a new technology, try to think in abstract terms about what the technology permits that was not possible in the past. It’s also worth considering what, if anything, the technology prevents or makes inconvenient. For example, compared with a chalkboard, an overhead projector allows a teacher to (1) prepare materials in advance, (2) present a lot of information simultaneously, and (3) present photocopied diagrams or figures. These are clear advantages. However, there are also disadvantages. For instance, James Stigler and James Hiebert noted that American teachers mostly use overhead projectors when teaching mathematics, but Japanese teachers use chalkboards.³³ Why? Because Japanese teachers prefer to maintain a running history of the lesson. They don’t erase a problem or an explanation after putting it on the board. It remains, and the teacher will likely refer to it later in the lesson, to refresh students’ memories or contrast it with a new concept. That’s inconvenient at best with an overhead projector.

³³ James W. Stigler and James Hiebert, The Teaching Gap (New York: Free Press, 1999).

Having managed to follow most of Wu's lectures, I am a huge fan of this method - and a huge non-fan of the PowerPoint now-you-see-it, now-you-don't approach to teaching.

limits of working memory
working memory posts

MSMI2010 was amazing.

Amazing!

Still collecting my thoughts - will post - but in the meantime, Niki Hayes' John Saxon bio is out!
My copy arrived in the mail last week.

Professor Wu at the beginning of MSMI2010 ...










...and at the end of MSMI2010.


That's the result of 40 hours of extreme fraction action.

Notice how giddy and hard working the attendees are after 5 days!

Better sign up now for next year's institute on Geometry.

Well, not fishing, exactly.

I'll be attending the Middle School Mathematics Institute in Minneapolis!

I've finally got my reservations made!

msmi2010

I hope some of you can come -- I'd love to put faces to names after all this time.