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.

## Wednesday, June 23, 2010

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## 36 comments:

This has been a chronic problem for me, re-teaching myself arithmetic, algebra & geometry: severely fragmented knowledge. AND it's not really possible to un-fragment my knowledge using American textbooks.

I spent years (seriously) trying to figure out the relationship between multiplication of whole numbers and multiplication of fractions.

Just to take one example.

It was quite gratifying when Wu told us that multiplication of fractions is **the** most subtle and complicated of the four operations.

You don't say --- !

I emerged from the classes feeling simultaneously elated and downcast.

Remember that engineering notion: you can have things fast, cheap, or good but not all 3 at once - ?

I came away elated because for the first time in lo these many years I had a grasp - tenuous, but a grasp nonetheless - on the fundamental coherence of mathematics.

At the same time, I saw with crystal clarity native intelligence and effort simply can't substitute for a coherent math curriculum.

I have native intelligence & I've put in the time and effort.

But I'm nowhere near the level of understanding of a 10 or 11-year old Asian student learning arithmetic in an Asian country using an Asian curriculum.

After all this time.

I may have to re-teach myself arithmetic using Primary Mathematics.

Catherine Johnson wrote:

"But I'm nowhere near the level of understanding of a 10 or 11-year old Asian student learning arithmetic in an Asian country using an Asian curriculum."

Evidence from TIMSS suggests that curriculum does NOT explain the difference in math achievement between Asian and U.S. students, because Asian-American students (using U.S. math curricula) score comparably to students in Asian countries.

Quoting the Wikipedia TIMSS entry:

"Data for US students is further tracked for ethnic and racial groups, which can be tracked as the nation. As a whole, grade four students in the United States lagged the best Asian and European nations in the 2007 TIMSS international math and science test. However, broken down by race, Asian Americans scored comparably to Asian nations, European Americans scored comparably to the best European nations. Hispanic Americans averaged 505, comparable to students in Austria and Sweden, while African Americans at 482 were comparable to Norway.

Grade eight students in the United States also lagged the best Asian and European nations in the 2007 TIMSS international math and science test. Broken down by race, US Asians scored comparably to Asian nations, white Americans scored comparably to the best European nations"

Where is the evidence, adjusting for race, that American math curricula are worse than those used in Asia?

Here is an example of using a concept that has not been taught, from everyone's second favorite curriculum, Saxon Math. In grade 4 they want to teach equivalent fractions. This is where the spiral method really fails, since you just need to start at the beginning and develop the facts of multiplying fractions until you get to the end. What Saxon does is say that (for example) 2/3 = 4/6 because we can multiply by 1 in the form of 2/2, so 2/3 = 2/3 * 2/2 = (2*2)/(3*2) = 4/6. Unfortunately, this is the first mention of multiplying fractions. So it is bunk. Although correct.

EXACTLY.

Wu makes exactly this point: that if you have the theorem of equivalent fractions, you can derive multiplication.

But if you start "proving" equivalent fractions by arguing about multiplication, you've got nothing, because fraction multiplication depends on equivalent fractions.

That is, you can't go from

a/b * c/d to

(a * b)/(c * d) just because you know it's true. you have to prove it/justify it somehow. So you need equivalent fractions, but you'll have to define them without appealing to multiplication.

Wu also pointed out that kids paying attention will see this

"we just see a/b * c/d thing and the rule is ..." and they'll think "oh, you just do the *obvious pattern*: operation on fractions just means you do the operation to the top, and then the op to the bottom."

and they'll assume addition is

a/b + c/d = (a + b) / (c + d)

because it's about as justified as the above.

Do any of the teachers work in an Everyday Math environment? When EM talks of understanding, they don't mean anything like this. Did you get into any discussions about how many people have non-math ideas of understanding? To me, the problem is not that they just aren't rigorous enough about starting with equivalent fractions. It's that they are on a different planet.

broken down by race, Asian Americans scored comparably to Asian nations, European Americans scored comparably to the best European nationsI'm too lazy to look it up right now -- but do you recall the data showing that Asian students who immigrate during their school years do much better in math than Asian students who go all the way through U.S. schools in math?

The person to read on the difference between Asian curricula and ours is William Schmidt.

Schmidt describes perfectly my own experience of trying to teach myself math via a jumbled, incoherent curriculum.

IQ, motivation, and effort aren't enough.

High-IQ students taught math in a jumbled curriculum aren't going to match high-IQ students taught math in a logical, coherent sequence.

None of the teachers was teaching fuzzy math. They were all from private or Catholic schools.

They were so removed from fuzzy math they almost didn't know what it was.

Allison & Cassy should correct me if I'm wrong, but my sense of Wu is that he regards constructivist math as almost not worth discussion. He is addressing the problems in 'traditional' math.

Has there been any documentation of the percentages of Asian kids in American schools who go to Kumon, are tutored at home etc.? My kids atttended school with lots of Asian kids and that was the pattern. Lots of the white kids did the same.

I second your impression regarding fuzzy math being beyond Wu's attention. I wonder what he'd be like in a room full of EM fans?

One day during break, I showed Wu the book: Teaching Primary School Mathematics by Lee Peng Yee that is used in Singapore to train teachers. He shared what he felt were errors in the text.

The day prior, he had recommended Parker & Baldridge's Teaching Elementary Mathematics as the best of the lot teacher-prep text. Parker & Baldridge teach the lattice method as a method of multiplication. He also spent some time on fraction division talking around Liping Ma's thoughts on the topic.

I got the feeling that the reason Wu was writing his book was that he didn't feel there were any good texts out there. When his gets edited, there may finally be one. Next step, how to get a college prof to ADOPT it?

Here's how I would develop multiplication of fractions.

1. Explain the concept of a unit fraction (1/2, 1/3, etc.) using multiple representations.

2. Show that a/b = a * 1/b. (Or perhaps, define a/b as a * 1/b.)

3. Explain that 1/a * 1/b = 1/(ab). Use lots of concrete examples, such as one-half of one-fourth of a pizza.

4. Demonstrate multiplication of general fractions as follows:

a/b * c/d = a * 1/b * c * 1/d = ac * 1/(bd) = ac/(bd)

This is akin to how we multiply dimensioned quantities. The denominator is the unit, and when we multiply dimensioned quantities we must also multiply the units.

2 meters times 4 meters equals 8 square meters.

2 thirds times 4 fifths equals 8 fifteenths.

I wonder what he'd be like in a room full of EM fans?If I had to guess, I'd say probably the same as he was with us.

He'd just plow through proving theorems and dismissing 'the big one' out of hand.

And then, of course, there was the great moment when he said, "More than one way to solve a problem is propaganda. Sometimes you're lucky to have one."

Wu is not keen on pizza fractions.

I have to go through my notes and create a sequence.

What was most revelatory to me about his teaching of multiplication of fractions was that he moved from a unit on the number line to the unit square: multiplication as area.

From there is was very easy to see that multiplication of fractions is the same procedure as multiplication of whole numbers.

Interestingly, I had seen 'array models' of multiplication many, many times, and of course I had no real argument with them, but I never could see how arrays were connected to number lines -- and I knew they had to be.

The whole message of Wu is: math is a coherent, logical 'whole.'

(I don't know whether he would say 'whole' - so I've put that word in quotation marks.)

Some of you might remember, years ago, when Carolyn (co-founder of ktm) said that for her "math is a seamless whole."

I believed her, but I didn't understand her until I spent 5 days listening to Wu.

"Wu is not keen on pizza fractions."

So much for EM.

How do you crack that egg? How do you change fundamental assumptions about math? Are there any schools that use math-certified teachers in K-6? (not just certified in EM)

After spending 5 days with Wu, I wonder how many math-certified teachers could teach a coherent math curriculum.

Hi David,

3. Explain that 1/a * 1/b = 1/(ab).

This is the issue. By grade 5, there needs to be something better than your examples of pizza. For one, few people could actually show that you can divide a piece of pizza in thirds well enough to show that, say, 1/5 * 1/3 is 1/15. You need an actual definition. All math beyond grade 7 depends on using actual definitions, and deducing what you can from those definitions is how math proceeds. So we need to stop hobbling our 5th, 6th, 7th, and 8th graders by giving them a real definition and teaching them to work with the abstraction. If you do so, then you can show that fraction multiplication works for ANY fraction, not just the concrete examples that are too simplistic to help.

And there IS something better than pizza, or your examples that are non convincing that it works for *all* fractions, instead of just special ones. There's the theorem of equivalent fractions, derivable from the definition of a fraction.

A fraction a/b means a concatenations of 1/b, where 1/b is the length found by breaking the unit into b parts.

Then you show that on the number line.

Then you show the theorem of equivalent fractions. Then, given the theorem of equivalent fractions, you can show why a/b = ak/bk for all nonzero k, and you show that on the number line.

so for two fractions m/n and k/l, you can now understand what

1) m/n of k/l means.

it means you can find k/l on the number line, and then you can break it into n equal parts, and you take m of them.

You can do that by finding equivalent fractions, and show it with the number line.

And you can count up how many pieces you've got, in your new units.

And then it's easy to show multiplication, because you don't need to explain anything. It works for proper and improper fractions, it works when the numbers aren't nice, it just works.

I think Wu would have spoken more of discovery learning if his audience were using it. He spoke of Saxon because they were using Saxon, nearly all of them. Wu had quite a lot to say about to me and others at lunch about Key Curriculum Press' Discovering series, CMP, TERC, and EM.

But it wasn't really relevant to the teachers in this group. There were a couple of folks who were more into the discovery pedagogy, but the bulk of the teachers weren't aware of it, and so it wasn't helpful to talk about it.

It's really astonishing that his take is controversial. He is explicitly anti-rote learning, if by rote learning you mean "teach rules without teaching the whys behind them." He is explicitly anti-discovery learning if by that you mean you never need to have mastered basic facts. He isn't a zealot on any of the surface-issues people grab onto: he's pro calculator, if it is used just to compute AFTER all steps of a problem are written down--he hopes it will encourage math teachers to get away from single and easy digit computations. He sees no need for everyone to take a full year of algebra 1 in 8th grade. He is happy for students to work in groups or have discussion. But he thinks you need to make sure kids can know things, and think them through, too. Shocking!

Sorry, if I had realized that 'pizza' was a shibboleth then I would have used another example. If a day is 1/7 of a week, and an hour is 1/24 of a day, then an hour is what fraction of a week? (And how many hours are in a week?)

I like number lines, but I don't understand the obsession with them. It seems to be a frequent view among mathematicians that there is one right way to understand every concept, and if you learn it the right way the first time then you won't have to revisit it. This is surprising to me because it does not resemble my own mathematical experience.

Wu shifts from the number line to the unit square for multiplication.

I think he offered us a proof of why this is possible.

he's pro calculator, if it is used just to compute AFTER all steps of a problem are written down--he hopes it will encourage math teachers to get away from single and easy digit computationsyes, this was a major feature of the class

Wu fractions are entities like 98/117.

David, the point is that by 5th grade, analogies are not advancing the student properly toward algebra.

Examples like hours in a day and days in a week are fine ways to help illustrate a concept, but they are not substitutes for DEFINING a concept.

A fraction can be defined, and should be defined, so that students get used to using definitions.

The number line is the best, simplest way to define fractions because it shows the connection to what students learned about whole numbers: that fractions are numbers, just as whole numbers are. Using it, addition of fractions is the same as addition of whole numbers (not a brand new definition for no apparent reason). Using the number line, equivalent fractions is a piece of cake, too. Improper fractions aren't confusing or something separate, etc.

So, when you have the right definition, other things fall out naturally. the constant analogizing and offering too many ways of thinking is fine for *EXPERTS*. It's rubbish for a child who doesn't know mathematics.

You can use your example to illustrate fraction multiplication, but your example does not define fraction multiplication for every pair of fractions. But starting with a definition of a fraction as I stated above, you can define it for every pair of (nonzero) fractions.) It becomes easy if defined properly, and if equivalent fractions are derived.

So why shy away from it?

Teaching fractions to middle school students is about where they are headed, too. I don't know if what you said about mathematicians is true, but if you can define a fraction, and then use the definition, why would you choose instead to artificially depress the curriculum and avoid it? It's much easier if you can a definition that children can know and master, instead of constantly being stuck with analogies.

Yes, use your examples to illustrate the power of the definitions. But include the definitions.

"teach rules without teaching the whys behind them."

Ironically, educators' comments about rote learning are simply a rote parroting of what they were directly taught in ed school.

"he's pro calculator"

I'm pro calculator if it is not used as an avoidance tool.

I wonder what he'd be like in a room full of EM fans?

If I had to guess, I'd say probably the same as he was with us.

I guess I was thinking about how the teachers would be different. I have tried to talk to one who insisted EM was the "best" curriculum for gifted students. I was imagining her arguing with Wu.

Please remember the connection between number line and array!!

As my youngest is about to advance into the fifth grade, I know I'm going to be trying to teach fractions to a kid only ever exposed to pizza math.

I suppose I could dig up that paper Wu wrote on teaching fractions -- I downloaded it over a year ago and printed it out. Now I have no idea where I put it.

To my mind the simplest way to define fractions is via the rules n*(1/n)=1 and m*(1/n) = m/n. All other properties of fractions follow from these rules, together with the ordered field axioms. n*(1/n)=1 is very intuitive because it expresses the idea that a whole is the sum of its parts.

Allison, There is no one singular definition of a fraction. Fractions are used to represent multiple concepts (fraction as a number on a numberline, fraction as division, fraction as equal parts of a whole, fraction as a ratio, etc...) and students need to be completely familiar with each of these concepts, how those concepts are represented by a fraction and how they apply the use of a fraction in solving problems.

By limiting fractions to just its numberline representation, it may be difficult for students to transition to the algebraic notation for division.

David,

To be specific, n*1/n = 1 is not just "the whole is the sum of the parts". Your statement is much more specific. It says "the whole unit, when broken into n equal-length parts, and then all n are taken, is the whole unit." and m/n means "take the whole unit, broken into n equal-length parts, and then take m of them."

Which is exactly how Wu defines a fraction. He shows this with a picture on the number line.

So you and he are in agreement.

But when we use the number line to show these, we have a tool which makes the rest of the operations easy to follow.

Continuing, though, multiplication of fractions doesn't just fall out of these definitions easily. It falls out with a lot of work, unless you have a lemma in between. That lemma is equivalent fractions, which can be proved to age-appropriate kids, or shown on the number line to age-appropriate kids. And yes, with that picture, it's very intuitive.

Erin,

That's the problem--confusing two ideas, 1)that since children will need to eventually relate several ideas together, 2) they should be taught that there is no definition of that idea.

No. The best way to teach children to relate the ideas to each other, is to give them an actual definition, so they can always be sure of what they are trying to understand and know. Then, relate the various ideas back to what they know. Telling children that a fraction is several different things at once undermines their ability to do mathematics.

From Wu's paper,

"Some remarks on the teaching of fraction in elementary school",

(1) The concept of a fraction is never clearly defined in all of K–

12.

(2) The conceptual complexities associated with the common usage

of fractions are emphasized from the beginning at the expense

of the underlying mathematical simplicity of the concept.

(3) The rules of the four arithmetic operations seem to be made up on an ad hoc basis, unrelated to the usual four operations on positive integers with which students are familiar.

(4) In general, mathematical explanations of essentially all aspects of fractions are lacking.

These four problems are interrelated and are all fundamentally mathematical in nature. For example, if one never gives a clearcut definition of a fraction, one is forced to “talk around” every possible interpretation of the many guises of fractions in daily life in an effort to overcompensate.

...

Children are told that a fraction c

d , with positive integers c and d, is simultaneously at least five different objects (cf. Lamon 1999 and Reys et al. 1998):

(a) parts of a whole: when an object is equally divided into d

parts, then c/d denotes c of those d parts.

(b) the size of a portion when an object of size c is divided into

d equal portions.

(c) the quotient of the integer c divided by d.

(d) the ratio of c to d.

(e) an operator: an instruction that carries out a process, such

as “2/3rds of”.

...

Clearly, even those children endowed with an overabundance of faith would find it hard to believe that a concept could be so versatile as to fit all these descriptions. More importantly, such an introduction to a new topic in mathematics is contrary to every mode of mathematical exposition that is deemed acceptable by modern standards.

All the while, students are told that no one single idea or interpretation is sufficiently clear to explain the “meaning” of a fraction. This is akin to telling someone how to get to a small town by car by offering fifty suggestions on what to watch for each time a fork in the road comes up and how to interpret the road signs along the way, when a single clearly drawn road map would have done a much better job. Given these facts, is it any wonder that Lappan-Bouck (1998) and Lamon (1999) would lament that students“do” fractions without any idea of what they are doing?

...

There's much more. Read the whole thing.

Allison, Definitions are difficult for children to understand and apply without some grasp of the underlying concepts that the definition is trying to be used for.

And I am not clear on why you think (or Wu for that matter) that starting off defining a fraction as a point on a number line is preferential to defining a fraction as equal parts of a whole. (Or any of the various other valid definitions of a fraction).

Formalism (in defining specific terms) is essential for advancing the knowledge base of any field. But we are talking about how to enable rather young children who are not experts to understand and fluently apply quite complex concepts and procedures. How does knowing the definition help in this endeavor?

Singapore does not use formal definitions for mathematical concepts. They use visual descriptions that help build the concept. Their learning progression of "concrete --> pictorial --> abstract" does seem to well capture how students learn. Starting with a rather abstract definition of a concept doesn't seem to fit in well with their thinking. So why should we use formal definitions of the concepts in the early grades if they do not aid in the student's understanding or application thereof?

Erin,

--And I am not clear on why you think (or Wu for that matter) that starting off defining a fraction as a point on a number line is preferential to defining a fraction as equal parts of a whole. (Or any of the various other valid definitions of a fraction).

The issue is not what teachers do in k-3. It's somewhere between 4 and 8. What are you calling early grades? This was the Middle School Mathematics Institute material.

A definition is appropriate in grades 4-7, somewhere. By 8th grade, these kids are going to be introduced to the algebra of a line, and will be expected to handle abstractions. So we need to stop depressing the curricula in grades 4-7, and help students move to the abstract. Fractions really are abstract.

Knowing how to define a fraction as and then being able to go back to that fraction helps children build fluency because they can keep doing the exact same thing over and over again.

You say Their learning progression of "concrete --> pictorial --> abstract" does seem to well capture how students learn.

So, here, fractions are points on the number line, where you show the unit divided into n equally long parts, and you take m. That's the pictorial part, and then the abstract part, just as you've said you want.

So the number line facilitates the pictorial to abstract nicely, and the whole numbers are defined on the number line, and whole number operations are quite easy to understand this way, and so then, when we move to fractions on the number line we're doing the same thing: moving from the concrete (whole numbers) to the abstract (fractions), providing a representation that is pictorially to help. The operations are then defined in a coherent way based on students' prior knowledge: how they worked on the number line is for whole numbers is how they work here.

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