Jo Boaler’s MATH-ish Boo-Boo

Periodically, someone seeks to beat up Jo Boaler. These beat ups are usually, well, beat-ups. The arguments and evidence tend to be flimsy and to be given the least generous spin. It is also difficult to feel much sympathy for the target. It is Trumpian the manner and frequency with which ill-directed attacks distract from well-directed attacks, all of which Boaler invariably responds to by playing the victim.

The Tweet

The latest round of this game began with a tweet last week, pointing out a clunker in MATH-ish, Boaler’s new book. The tweet included a not-great screenshot of an annotated page from the book, the most glaring issue being with a box indicating to “multiply” fractions:

This is of course a very bad error to have occurred in such a book. Nonetheless, errors occur and they should not be over-milked, as the tweeter shamelessly did:

The glaring mistakes in Jo Boaler’s new book “Math-ish” reveal her unsolid and erroneous conceptual understanding of fractions and simple multiplication. If anything, the mistakes suggest weak procedural fluency and vague conceptual understanding reinforce each other.

This is absurd and it gave Boaler, who quickly acknowledged the error, an easy return of serve:

The Error

Boaler had every right to respond in this manner to an aggrandized attack, even if her “there is an error” underplays the significance of the error. The problem is, there is worse in the error than the error. The second and much greater problem is, the battle over this error precludes any proper discussion of Boaler’s approach to the broader topic as outlined in MATH-ish. As I argue below, I think her approach is dreadful.

The error cannot be simply dismissed as typographical. Of course Boaler’s intention was clear, to multiply both the numerator and the denominator of a given fraction by the same number. The instruction to “multiply each fraction”, however, appears in both the graphic and the detailed caption. It suggests a more systemic carelessness (along with an inability to write clearly). There is worse.

The goal of the box and caption is to rewrite two fractions, in this case 3/4 and 2/3, to have the same denominator. So, one is seeking a (lowest) common multiple of 4 and 3. How does Boaler do this? She lists “multiples” of 3/4 and 2/3 until she hits upon two with the same denominator. Which is ludicrous. If the reader does not immediately see why, give 2/7 and 3/5 a go.

Boaler’s example is (misleadingly) simple enough that one needn’t get into prime factorisation or whatever, but you don’t then simply leave it to the Gods of Chance. Somehow, one must deal with the numbers at hand, how the multiples of 4 and 3 relate. Both the error and the intent here are very Boalerish. She is an Ideas Man. Attention to detail and proper technique are simply not her thing.

MATH-ish

After the attacking tweet appeared, we took some time to look at MATH-ish. The book is thin, physically and conceptually, and we read a decent chuck, browsed more and worked through with some care the section containing the offending fraction multiplication. It was nauseating. MATH-ish is a masterpiece of first person prosing. Every page is about Boaler. Every discussion is there to trumpet her presumed wonderfulness. She never stops.

Placing to one side the self-promotion and the graceless writing and the clunky title and the doudy cartoons, MATH-ish is concerned with “finding creativity, diversity and meaning in mathematics”. It is her familiar shtick. Boaler is 100% “reasoning” 100% of the time, which appeals to the fans of Deeper Meaning. Unfortunately, without the structure upon which to hang the reasoning, without the solid knowledge and techniques to keep it together, the sum total of all this reasoning is a jungle of semi-thought.

Chapter 5 of MATH-ish is titled, Mathematics as a Visual Experience, and includes a section on the division of fractions. It is a page from this section that was attacked in the tweet. The section begins,

One of the most contentious areas of mathematics taught in elementary schools—causing huge difficulties for students, producing low test scores, and having little practical value—is the division of fractions.

“Little practical value”.

To be fair, Boaler may be making the weaker claim, that it is the teaching of fraction division in elementary school that is of little value. She suggests that as well as fraction division being “a prime candidate for a complete rethink of the way it is taught”, the topic should be moved to “later years in the curriculum standards”. God knows precisely what Boaler intended, if anything.

We’ll come to Boaler’s “complete rethink” shortly, but first a quick diversionary question: at what age is fraction division typically taught in the US? Boaler writes,

… the operation [of fraction division] itself is too difficult to understand or reason about at ten years of age, the age at which the content is taught in the United States. [emphasis added]

It can be very tricky to understand the officialdom and the reality of another country’s schooling system but this statement appears to be false. The common core standards seem to indicate that fraction division is to be taught, pretty weakly, in Grade 6, when students are typically 11 or 12 years old.¹ Note also that Grade 6 is part of what the US refers to as middle school, not the “elementary school” to which Boaler refers. It is quite possible that I have this all wrong, but it seems to me that Boaler is either being very sloppy or is conveniently sliding the facts to be in her favour. Freud would suggest there is little difference between the latter two possibilities.²

The Bones of Fraction Division

On to the division of fractions. Fractions are very fertile ground for Boalerisation, since nothing about fractions or the teaching of them is a gimme. They are hard and non-obvious,  taking thought and practice. At least for understanding what is going on, division of fractions may be the hardest part of all. This is encapsulated by the familiar couplet,

Ours is not to reason why,
Just to flip and multiply

Boaler gets very good mileage out of this couplet and the supposed horrors for which it stands. Which is partly fair and mostly ridiculous.

It is obviously sub-optimal for kids to be performing fraction division, or any algorithm, without a sense of why it works. But it is not that sub-optimal. The very point of an algorithm is to handle efficiently and thoughtlessly a mechanical task, thus freeing up time and mind for thoughtful tasks. So, to some extent an algorithm is exactly doing its job if kids are using it without thinking why. Boaler notwithstanding, mathematics is not about 100% reasoning 100% of the time.

Still, Boaler has a point. Not thinking about why something works is not the same as not knowing why something works. If we don’t want kids distracting themselves with a redundant “why” while using an algorithm, we would still like them to have a sense of the “why” if they put their minds to it. This is important on both general principle and for specific pragmatics. And, there are probably very many students, and not a few teachers, with no good idea why “flip and multiply” works for fraction division. It is reasonable to argue about how the technique should be introduced and taught, and to question what is being done currently.

How is fraction division currently being taught, and how should it be taught? We don’t know, and we don’t know. In Australia at least, it is undoubtedly being taught appallingly, as much for being over-Boalered than being over-mechanical. And Wu, for example, is the guy to ask what to do, not us. Still, the basic bones seem pretty obvious and, if hard, not that hard.

It seems clear that the place to start is with a simple division by a unit fraction, for example,

This division question is easy to understand in the familiar “how many” manner:

How many fifths are in 6?

So, the question is,

Well, there are five fifths in 1, and then there must be six times as many fifths in 6: ³

This holds true and can be understood in the same manner for the division of any number , fraction or otherwise:

What about division by a full-blown fraction, say

Well, we’re now asking,

How many four-fifths are in 6?

The answer is, a quarter as many fifths as there are in 6:

And, more generally,

That is pretty much it. Those are the bones.⁴

Bones gathered, there are still many questions of what should be done with them. How the bones should be laid out, how quickly or slowly, and how the bones should be fleshed out, none of this is obvious, at least to us. Undoubtedly, an effective approach would depend very much on the age and maturity of the class. It is also clear that some visuals, number lines and pizzas and whatnot, would be beneficial. At this stage of fractioning and for the goal in mind, I’d be going light on the pictures, and the number line would seem more intuitive and more directed to the end goal than the pizza, but there is plenty of room for discussion and personal style.

Questions of arrangement and fleshing out, however, should not distract from the substance and the goal. Fraction division contains just a few bones and students should have some feel for these bones, and then the assembled skeleton. Then, critically, they must be able to, um, apply the skeleton. They must be able to do the flip and multiply of fraction division, efficiently and accurately, without a moment’s concern for its origins.

Boaler’s MATH-ish Fractions

There seem to be three key aspects driving Boaler’s approach to division of fractions, as described in MATH-ish. First, she wants to be as visual as possible. Secondly, she wants as much as possible to treat a fraction a/b as a number, rather than dealing with the separate components a and b. Thirdly, she wants to avoid entirely “flip and multiply”. All three aspects are reasonable concerns to a point, and only to a point.

Boaler begins her discussion with consideration of the division   This seems like a peculiar place to begin, but no matter. Boaler goes straight to a picture, three “visual approaches”:

The great value of a good graphic is that it captures attention and can provide the “aha”, the wordless geometric intuition that can then be teased out with words and expressions and equations. But there is no “aha” in the above graphics; even granted the greyscale formatting of the book, they are boring and borderline meaningless. The third, number line graphic is significantly better than the first two but is still very far from good, essentially useless without accompanying explanation.

Accepting for now that the above graphics explain why , where to from here? How does one consider more complicated examples? Boaler doesn’t offer much. She provides another picture later, for a visual approach to :

One can kind of see it. One must squint to do so.

At this point, Boaler pretty much ends this line of discussion. She does not similarly consider more complicated fraction reciprocals, such as or , let alone general fraction division. She also offers nothing on the translation of these visuals into numerical representation. This is standard Boaler. She provides a “great explanation” for overly simple examples and just ignores the fact that the returns on her great explanation diminish very very quickly. Take Boaler’s explanation off the easy road and onto a dirt track, and her vehicle will very quickly get bogged.

Boaler has a real distaste for algorithms, but of course she cannot ignore them entirely. As Boaler puts it,

I am not arguing that algorithms are not useful; I am arguing that they should not be taught to students until they understand fractions conceptually. Students need to understand what a fraction is and be given a lot of experience thinking about the value of fractions as a conceptual whole.

So, a couple of years of pictures before attempting any substantial computation. Ignoring the fact that to “understand what a fraction is”, or any mathematical concept, involves getting one’s hands dirty with many, many computations, and seeing many, many answers to these computations. Boaler’s think system is absurd.

When Boaler finally gets around to an algorithm for division of fractions, she chooses a bad one. She illustrates with the example , and her approach is to write the fractions with a common denominator. (It is this subsidiary task that contained her “multiply the fractions” error, above.) Thus, Boaler first transforms the division:

How is this latter division done? Boaler is not entirely clear. She offers a “visual approach”, and little else:

Why perform fraction division in this manner? Boaler explains:

For division of a fraction by a fraction (something most people have never used in their lives after school), I prefer to make both fractions have the same denominator first. Changing fractions with different denominators to have the same denominator is work that focuses on the relationship, which is valuable for students.

Yes, meaning is valuable. But, after correctness, by far the most important requirement of an algorithm is that it be efficient; transparent meaning is a very distant third. Obviously the algorithm proposed by Boaler is anything but efficient, or elegant, even on simple examples, and much less so on the dirt track; the reader can give a go. But Boaler simply does not care. It is not her thing. Doing mathematics, actual computing, is not her thing.

Boaler’s Vision

Boaler ends the section on fraction division with a much broader invocation:

This visual approach to mathematics can extend to any grade, any level, any mathematics, and any task.

Yes, it can. And it will almost always be wrong.

 

1.  By way of comparison, fraction division is taught in Australia (homeopathically) in Year 7, to 11-12 year olds, and in Singapore (properly) in Primary 6, also to 11-12 year olds. 

2. This is not the only instance in the chapter where Boaler makes a claim that appears to be false, or is at least unsubstantiated in the manner proposed. A good rule of thumb when reading Boaler is to never to accept any claim on face value and to never presume that a reference provided in support will provide support.

3. One could also reason that there are 6 units in 6, and so 5 x 6 fifths in 6. This would lead naturally to the framing .

4. It is off-topic, but we’ll make the point that we took a lot of time deciding upon the particular numbers used, the 6 and 1/5 and 4/5, and we changed our choice a number of times. We’re still not entirely happy with the choice, because 5 is a large denominator with which to begin, and 30 is a large output. But, we wanted some cancellation, and not complete cancellation, in the final division, and we did not want other common factors clouding things. These details really matter, and it is why when people laud AI for doing their lesson plans and whatnot, we are deeply sceptical.