kitchen table math, the sequel: msmi 2010 Followup

Monday, June 21, 2010

msmi 2010 Followup

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...


Barry Garelick said...

I'm happy for all concerned that it was a success. I recall when I first read Wu's treatise on fractions--part of a manuscript he was writing for teachers. I found it nothing short of amazing, so I can only imagine what the course was like.

Catherine Johnson said...

The whole thing was amazing.

I'm still puzzling over why it was so good -- or, rather, why it was so effective given the enormous challenge of the 'boot camp' approach. ("Boot camp approach" isn't a criticism, btw, just a description.)

Maybe boot camps are good??

Most of us have absorbed the lesson that 'massed practice' is far inferior to 'distributed practice' ---- but, on the other hand, with msmi2010 we're not exactly talking about practice, are we?

We're really talking about a kind of immersion course.

Immersion math.

If Katharine is around, we'll have to get her to give us the thumbnail on foreign language immersion ---

Catherine Johnson said...

It was an amazing week, Allison.

Thank you for pulling the whole thing together!

Catherine Johnson said...

btw, I'm **especially** puzzled by the effectiveness of the course for me because I missed one morning and 1 afternoon. (Stuck in Milwaukee overnight; then had a severe reaction to an antibiotic.)

Even missing two chunks of the week's work, I was still above water.

Some of that is due to the fact that I have a **reasonably** solid level of background knowledge in K-8 math at this point (meaning all of arithmetic) & in algebra 1 & high school geometry --- but not all.

Catherine Johnson said...

It was an amazing week.

I'll definitely go back for geometry & try to recruit teachers here to attend.

Catherine Johnson said...

MSMI2010 is written in caps?

I though I saw it in small letters on the site.

I'll change it...

Allison said...

No, no, you're right.

It was supposed to be

Allison said...

Boot camps work, but the sense you've got will fade.

I think boot camps work because the sheer pressure/time/effort spent is enough for the brain to say "this is critical! Pay attention!" in a way that a normal course does not. That said, this boot camp worked because it started from first principles, so when you're stuck, you can go back and say "what do I know?" and you know the answer is: the defn of a fraction. So you don't need to get lost, if you can just walk the road again.

But while the overarching lessons will stick around, the problem solving ability will fade without distributed practice. Coherent distributed practice is the key.

Catherine Johnson said...

I wonder if there's another factor, which is an intensive rewriting of bad prior knowledge.

As far as I can tell, it's not really possible to forget bad prior knowledge.

At least, it's not possible to forget bad prior procedural knowledge...I'm not clear about more strictly conceptual knowledge -- although it's also not clear how much strictly conceptual knowledge there is in math...

The overlap between procedural learning, involving the basal ganglia, and 'conceptual' learning, involving only the hippocampus (I think) is huge.

This seems to be an area of murk in cognitive science.

In any event, the fact is: everyone in the room has Wrong knowledge of fractions. Does immersion go farther in terms of suppressing that Wrong knowledge?

Catherine Johnson said...

I've just jogged my own memory.

What you're after, when you're trying to correct wrong learning, is 'extinction.'

However, extinction really just means making the new, correct knowledge stronger than the old, incorrect knowledge. The old, incorrect knowledge is still there in your memory.

Anonymous said...

"In any event, the fact is: everyone in the room has Wrong knowledge of fractions."
Could someone who was there please elaborate on this at some point?

Catherine Johnson said...

Well, I hope Allison & Cassy will vet...but my summary of Wu's essential point is that it is confusing (and, I think, actually wrong) to tell children that fractions are 3 different things:

* parts of a whole
* division
* ration

Wu defines a fraction as a point on the number line, same as whole numbers, and from there things follow logically. Once you see fractions as a point on the number line, you don't have to tell students that addition of whole numbers and addition of fractions are two completely separate and distinct entities.

I'll go through my notes and pull things out as soon as I can.

He also has teachers teach equivalent fractions from the get-go, and said on numerous occasions: "Everything is equivalent fractions."



Anything to add or subtract?

Catherine Johnson said...

He also said, on the last day, I think, that a complex fraction has no meaning according to the definition we were using -- which was true.

We were using a definition of fractions in which both the numerator and the denominator are whole numbers.

Gotta run - will be back with more.

After telling us a complex fraction has no meaning he said there is no avoiding them; I think he said complex fractions are like death & taxes.

Catherine Johnson said...

Chapter 2: Fractions

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 not-equal-to 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

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


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.

Allison said...

Going forward from what Catherine posted, Wu argues that most of what teachers know about fractions, and teach to their students, is wrong in the sense that it is unsupported by what the students have previously been taught.

Wu makes clear that you must be ruthless with your self when teaching. you must not allow yourself to use any concept you've not explicitly given your students. And almost all textbooks violate this rule in almost every lesson on fractions.

Nearly all textbooks avoid defining a fraction. Since they aren't defined, they don't define how you operate on fractions in terms of any real definition. Beyond that, when they do introduce operations on fractions, the books don't define operations in terms of what students know either. They teach rules without ever teaching where the rules come from. Their "explanations" invoke properties no one taught the students, but are just asserted. That means logically, their explanations are wrong, as you can't prove a statement by assuming the statement true.

For example, how does one justify multiplication of fractions? (I'll answer how later, but for now, discuss amongst yourselves.)

Yes, you can learn to use the rule properly without understanding where it is from, but you'll only go so far, and more, you will mistrust your teachers and mistrust math. Math will seem to be an endless series of magical statements with no relationship, and sooner or later, you'll run out of time/space/effort for keeping track of them. Properly understood, math is coherent, because every new step you take follows from the ones before.

CassyT said...

The National Math Panel criticised U.S. teaching as ONLY teaching the part/whole interpretation of fractions and not the division, point on a number line or ratio interpretations.

Catherine Johnson said...

oh that's interesting - I had the sense Wu didn't think division should be taught as either division or ratio - right?