Week two of Your Daily Wu begins thusly…
Wu on fractions
[T]he publishers mistakenly believe that intuitive arguments and analogies suffice. Thus, fractions are simultaneously (and incomprehensibly) parts of a whole, a division, and a ratio. (p. 4)
what i think he’s saying
The only reasonable explanation of the meaning of a fraction is the one provided in undergraduate modern algebra: A fraction is an equivalence class of ordered pairs (a,b) of integers, with b not equal to zero, and with the rule that (a,b) is equivalent to (c,d) if and only if ad=bc.
But wait…that really *is* what he’s saying
[A] fraction m/n (for whole numbers m and n, n≠0) is just a symbol consisting of an ordered pair of whole numbers with m preceding n. (p. 10)
Or is he?
This is from “Teaching Fractions According to the Common Core Standards”, available on Wu’s homepage.
The shift of emphasis from multiple models of a fraction in the initial stage to an almost exclusive model of a fraction as a point on the number line can be done gradually and gracefully beginning somewhere in grade 4. This shift is implicit in the Common Core Standards. Once a fraction is firmly established as a number, then more sophisticated interpretations of a fraction (which, in a mathematical context, simply mean “theorems”) begin to emerge. Foremost among them is the division interpretation: we must explain, logically, to students in grade 5 and grade 6 that m, in addition to being the totality of m parts n when the whole is partitioned into n equal parts, m/n is also the number obtained when “m is divided by n”, where the last phrase must be carefully explained with the help of the number line.
So are those multiple interpretations of a fraction (1) incomprehensible and foolish (the former adjective is explicitly Wu’s in “Phoenix Rising”; the second is implicit), or (2) where we start with students before we formalize? And then do we then return to them at some point? I’m confused.
Overthinking my teaching 
Wow, someone who gets that when it comes to K-12 mathematics education, the Emperor Wu is naked??? I’ve been trying to point this out to folks since he first came to my attention as a graduate student in mathematics education at the University of Michigan in the early 1990s. Wu was sending a flood of papers to one of the math ed faculty there, who was properly humbled by Wu’s perspective on NCTM and progressive reform mathematics. I use the word “properly” with as much sarcasm as can be injected into a single word. Since this faculty member is now deceased, I won’t sully his memory by naming him, but I will suggest that he was only too happy to find a GEN-YOU-WHINE mathematician who would support what I suspect was his own ambivalence towards the direction NCTM was heading in the late 1980s and early 1990s.
In the ensuing two decades, I’ve read a lot of what Professor Wu has had to say about teaching K-12 mathematics, generally with very little pleasure or agreement, save for the time he had the honesty to testify at a California public hearing on math textbooks that Saxon Math wasn’t very good, particularly if one wanted to get any idea of what mathematics is.
But that heresy aside, Wu has generally been well within the anti-progressive mainstream, along with folks from Mathematically Correct, HOLD, and other educationally-conservative groups (think Wayne Bishop, David Klein, Dick Askey, Jim Milgram, ad nauseum). And I keep wondering why anyone takes these people seriously, no matter how good or indifferent they may [be] as mathematicians (Milgram is certainly no slouch, until he opens his mouth about mathematics education, when he transforms himself into a perfect [redacted]). It’s good to find someone who is taking at least one of them to task. Keep up the good work.
I’m not sure if this is what Wu is arguing for, but the following seems plausible:
Stage 1 (maybe grades K-3): fractions are dealt with on an intuitive level. Three fourths of an apple is what you have if you cut an apple into four (equal) pieces and take three of them. At this stage, any “problems” that students work on can be solved by simple reasoning with intuitive notions. For example, subtracting one fourth from three fourths results in two fourths.
Stage 2 (maybe starting in grade 4): Here we begin a systematic study of fractions that is fundamentally different from stage 1. Instead of students always going back to fundamental, intuitive ideas/principles, we build up a theory that allows us to solve lots of problems relatively easily. For example, at some point we hope that when a student needs to add fractions with different denominators, they do not need to reason through, from first principles, how they can do that. Instead we hope that they think: “we need a common denominator, …..” We surely hope that they understand why a common denominator is necessary, but we don’t expect them to think through everything from first principles.
This is the great thing about abstraction: If we can deal with “general fractions” (and I don’t mean that we write on a fourth grade black board: a/b + c/d = ….), then we can save ourselves a LOT of work that would otherwise need to be done in each specific case.
But as soon as we decide that we want to reason in a general situation (where we want students to understand how to work with general fractions and not specific fractions), then it becomes REALLY helpful to have very clear and explicit meaning for what a fraction is. Then when we are discussing complicated things we can refer back to that as a “common starting point.”
This comment is quite long, and so far I’ve only arrived at: “a definition for a fraction could be a good idea starting in grade 4 (or whenever this process of understanding general fractions begins).” If there is going to be a definition, a common starting point that teacher and students can refer back to, then “point on a number line” has a strong case.
If we agree that a definition, and more specifically, the “point on a number line” definition, is a plausible way to teach about fractions, then we can get to what Wu says here (or one of the things that Wu says here): We want students to understand that the fraction m/n is an appropriate way to view m divided by n. m divided by n has a natural meaning: “You start with m units and you divide what you have into n equal pieces. m divided by n is one of those pieces.” But this is different than our definition of m/n. The point on a number line definition corresponds to this meaning: “you start with infinitely many units and you divide each of them into n pieces. m/n is m of those pieces.” (the infinitely many units is necessary to deal with fractions like 29/3) This is probably best represented pictorially:
m/n is relatively easy to find on a number line. You divide each [k, k+1] on the number line into n pieces and you take m “steps to the right” starting with zero. What about m divided by n? First find the point m on the number line (easy). Now we ask ourselves, “how can I divide the interval [0,m] into n equal pieces? How long is each piece?”
The ideal situation (if we have a definition, or common starting point for fractions) is that we clearly understand how division relates to that definition.
That was long.
The problem is that Wu seems to arbitrarily conclude that there is one correct way to present fractions to students and/or one correct way to think about what fractions are. This is simply absurd. Anyone who gives the topic a moment’s non-Wu thinking will recognize that fractions and rational numbers avail themselves of a host of semantic meanings/interpretations, and that different people are likely to find different viewpoints more or less intuitive, clear, sensible, etc. In addition, the real world most of us inhabit requires flexible thinking about various elementary and less elementary mathematical ideas, particularly as they interface with ordinary use and experience. How viewing fractions as points on the number line helps kids make sense of rations and proportions, finding 2/3rds of 4/5ths of a cup when scaling a recipe, or a host of other things is at best unclear. Only someone as divorced from actual K-12 teaching as Professor Wu could attempt to declare by fiat a single “best” way to think about or teach this extremely complex and protean mathematical topic. But then, that’s what I’ve come to expect from the whole Mathematically Correct/HOLD cabal and its allies, among whom Wu is generally numbered.
Aaron writes:
Right. Agreed.
We don’t want students to have to reason through, from first principles, how to add fractions. But we certainly want them to be able to. And this seems to be a place of tension between what Wu advocates and what I would.
Additionally, there are robust research findings about what first principles are. There are concept images and there are concept definitions. Most people-even those who can cite correct definitions-reason mathematically based on concept images. People tend to go to definitions only in a limited number of circumstances.
As a student of graduate modern algebra, I had a concept image of injective mapping and one of projective mapping, and those images are what I went to most of the time when working on problems (for the record, these images were of syringes injecting saline solution into a turkey and of a film screen too small for the projected film, respectively).
Wu seems to leave no place for these concept images in the instruction he advocates, and he seems to believe that people reason exclusively from definitions. But it’s just not so.
Now, a generous interpretation of his position would be that he simply wants more use of definitions as a basis for meaning in K-12 mathematics, not exclusive use of them. By overstating his own case, he hopes to move the field closer to his real vision. If I believed that, I’d be right there with him. I can get on board with more careful attention to definitions in mathematics teaching. But given how strongly and repeatedly he states his case in writing, and the contempt with which he seems to regard those who are not completely on board, I think this interpretation is more generous than is warranted.
Seen this here piece ‘o prose from Dr. Wu?
Click to access Wu.pdf
MPG, that’s what started all of this silliness on the blogorooni here, and the source of nearly all quotations therein. It’s a beauty, no?
Here’s my intro for the Daily Wu feature. Enjoy.
Yeah, obviously I missed your first post and assumed you were just replying to something older of Wu’s, as his basic romance with using the number line as THE magical way to teach fractions (starting in 3rd grade, no less!) isn’t new. So I started reading his phony phoenix piece and am having trouble getting too far into it. Not because it’s difficult to follow (though his prose at times is shameful), but because he’s so totally out of touch with real kids in real schools, so totally full of himself, so utterly unable to grasp that his number line model is no better (and in some ways worse) than many others that are out there, and finally because he really doesn’t get that the semantic loading onto rational numbers and fractions isn’t something made up by “mathematics educators” or “textbook publishers,” but rather a matter of all the various roles that these numbers and the concepts related to them (ratios, proportions, parts of various wholes, division, points on the number line, a particular sort of number that we need to solve certain equations and certain real-world problems, etc.) place on these numbers and hence on students who are trying to get what’s going on.
I do agree that there’s a mathematics we teach in school that differs in most ways from what mathematics is. But Wu has it wrong: it’s not “textbook math” but “school math,” and there are reasons for school math to be really bad that have almost nothing to do with his analysis. The irony is simply too thick when he tries to distance himself from the current California standards, a set of arbitrary benchmarks that HIS OWN ALLIES in the anti-progressive world of academia came up with in the late ’90s to undo the really interesting framework California had gone with in the late ’80s/early ’90s, in many ways anticipating the NCTM Standards. But of course, the Wayne Bishop/David Klein/Jim Milgram/Jerry Rosen crowd that created Mathematically Correct, and others who formed HOLD, hated what they deemed “fuzzy” math, and eventually, with the help of the governor, did a dirty backroom deal to replace that framework with something they cooked up at Stanford and Berkeley. Wu was not necessarily complicit in that cabal, but he was far more part of that camp than he was with those in California whose work was destroyed or undermined by them.
Paul Halmos wrote eloquently about the “two maths” in this country long before Professor Wu ever thought to try to address the issue (See Halmos’ “Mathematics as a Creative Art,” a beautiful essay). And John Taylor Gatto has written persuasively about why this country insists on teaching subjects in ways that make no sense to anyone, particularly kids (See, for example, DUMBING US DOWN and THE UNDERGROUND HISTORY OF US EDUCATION). Put Gatto together with Halmos and you don’t need Wu’s incoherent and ludicrous attempts to take one bad approach to mathematics education and replace it with another bad approach. It’s hardly a coincidence that Wu is now selling a very expensive book in which he proposes to cure our ills (if you’re familiar with the Bourbaki’s attempt to cure the K-12 mathematics ills of France and how well that project went, you can get a pretty good idea of what we might expect if the nation suddenly decides to turn our future over to Wu).
The more I read of his fraudulent phoenix self-promoting nonsense, the more outraged I become. This man knows nothing about kids, about how kids learn, about why they might possibly WANT to learn mathematics, or much else that is necessary to promote effective K-12 mathematics teaching. When he has the temerity to give his approach to explaining equivalent fractions and then writes, “When each of these 5 segments is divided into 3 equal segments, it creates a division of the unit segment into 3 x 5 = 15 equal segments. IT IS THEN OBVIOUS that the point 2/5 is exactly the same point as 6/15, which is (3 x 2)/(3 x 5), as shown below”[emphasis added]. Clearly, he has no clue that when he chooses to claim that something is “obvious,” he’s lost all credibility. If it were obvious, kids wouldn’t struggle with the idea. And his magical model is neither new nor sufficient for many, many students. Not to mention many elementary teachers as well. But of course, if everyone would buy his book. . .
Sorry, but I have to go shower in something akin to industrial strength disinfectant. I feel like I’m covered in bull shit.
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