Some thoughts on Devlin and Boaler

For those who have read and heard Keith Devlin, he is pretty close with Jo Boaler who you may also have heard about. Keith Devlin, you will recall, writes a column called Devlin’s Angle in MAA and also is known as “that math guy” at NPR.

He made a big name for himself some years ago when he claimed that multiplication¬† “Ain’t no repeated addition”.

Well, yes, in formal, higher level mathematics, there is a general definition of multiplication that must meet several conditions. Technically, it is a function which maps two objects (numbers, functions, even shapes) from a set, into one object, (e.g., f(2,4) = 8 ) and the function is commutative, associative, distributive, and has an identify function called “1” in which a*1 = a.

What he seems to miss is that this general function does in fact include repeated addition as a means of informing the particular relations between numbers. For example, the times table chart showing the various facts is formally taken as an axiomatic definition; i.e., 4 x 2 = 8. That is, it equals 8 because we define the function that way. The inconvenient fact that Devlin likes to dodge is that the definition is informed by repeated addition.

Devlin and others of like mind think that teaching multiplication as repeated addition results in confusion when we teach fractional multiplication. Actually it isn’t that confusing, and using an area model incorporates the ‘repeated addition’ form of multiplication to get the end result.

Devlin and others who repeat his refrain got to their higher understanding by starting with repeated addition, which they now disdain. Remove the ladder much, Keith?

It is a throwback to the 60’s new math in which multiplication was defined formally. The joke was that kids knew that 5 x 4 = 4 x 5, but didn’t know that it also equals 20.

Funny that the top performing nations like Singapore manage to teach multiplication as repeated addition.

Devlin came under a lot of criticism for his series of articles, yet he stood by them and defended his stance vigorously in subsequent articles.

Boaler doesn’t go quite so far, but she is more for “fluently deriving” the math facts than straight memorization. So 9 x 8, should be looked at as 9×9 – 9. That way, kids who don’t know all their facts can derive them. And also satisfy her idea of what “deeper understanding” is. Really, folks, multiplication isn’t that hard. Nothing against the strategy she talks about, but the disdain for memorization (because it supposedly eclipses understanding) is just more money-making nonsense for Boaler and her ilk.

In my opinion, of course.


5 thoughts on “Some thoughts on Devlin and Boaler

  1. It is interesting that both Devlin and Boaler occupy highly visible seats at Stanford and neither has anything to do with Stanford’s prestigious Department of Mathematics.


  2. Do you know how Devlin would try to teach that a + a + a = 3 x a = 3a ?

    I’d be interested to see how one would approach that without using multiplication can be thought of as repeated addition. He’s bound to be big on “understanding” so you wouldn’t be able to merely say “because I say so”.

    Does he oppose teaching powers as repeated multiplication? Because, again, that would make understanding it very difficult.

    (I’m not going to read him myself. It would just make me angry.)


    • Yes; and interestingly I recall when Devlin’s articles were causing a stir, one defender at a math forum on the internet said that it is obvious repeated addition isn’t the only explanation when we are faced with an equation like 3a=12, because we then divide 12 by 3 and use the axiomatic definition of 4 x 3 = 12.

      Uh, yeah, OK.


  3. Full inclusion academic classes have increased the range of willingness and ability in K-6 and their assumption is that natural differentiated learning will handle it all. Expectations have been lowered to a statistical low CC slope to no remediation in a college algebra course. This starts in kindergarten. K-6 is now officially a NO-STEM zone, and our state CC provider has officially stated that. The College Board recognizes this so they instituted their Pre-AP algebra math class in 9th grade to FINALLY emphasize individual mastery of knowledge and skills and P-sets. It’s too little and too late, and they know it. They’re just providing cover for the low CC expectations of K-8 – that they helped define. Students expect that they can get to AP Calculus, but what does the College Board tell them when nobody can squeeze in 4 years of traditional math into 3 years? Oops? It’s your fault? You didn’t have enough passion or engagement? You didn’t learn how to struggle productively? It’s you, not them?

    Many of us parents have taken up the slack in K-8 and they don’t dare ask us what we do at home. But they expect us to come to their math open houses (I went to one for MathLand, a curriculum so bad it was erased from the face of the web – except for the old really bad reviews) so we can help them make their poor math curricula work. Really? Schools now expect help from parents in math at home? Do they know what that means for the academic gap they hope to eliminate? It’s happening and they don’t collect data to see it.

    We do it, but ignore their curricula. Their talk of vague conceptual understanding is just a cover for low expectation and non-working full inclusion. High schools, long an area of mixed ability, successfully use a full inclusion environment and all real “college prep” classes are traditionally taught. Our high school pushes everyone to take at least one AP class, even if it’s in music or art. Students have to learn the excitement and engagement of working hard and doing well in proper content and expectation courses -finally!

    Nobody has ever explained why K-6 is such a different learning environment except for strictly social and philosophical natural learning reasons. This is the education world academic turf, but high school AP/IB content teachers keep quiet, focus on individual mastery of P-sets, and get the job done. Once I helped my son past K-8 to get to the AP Calculus track in high school, I didn’t have to help one bit. He graduated last year with degrees in music and in math while competing against some of the best math students from around the world. There is reality and there is what these educationalists push – which shows no proof of working except for maybe a low non-STEM statistical level – helped by what we parents now have to do at home or with tutors.

    They talk vaguely about unquantifiable “understanding” to attempt to argue on an equal content-level basis, but it’s all a cover for low expectations that magically go away in all proper high school math classes. They might attempt to connect their ideas with what goes on at Harkness Table schools, but they know that’s wrong. MIT pushes “hacking”, but that’s built on top of years of traditional P-sets and traditional classes, not the other way around with PBL. They didn’t learn differential equations or electronic hardware design by turning the Green Building into a large Tetris game.

    Education is not about statistics. It’s about individual educational opportunity and equality, and that gap is being increased by the work we parents now have to do at home. We help support their statistics and they really don’t want to add parental help in as part of their “big data” answer to all problems. Anecdotes like my son contain all of the information and understanding and different ones might point to a number of problems and different solutions. When you break anecdotes into “big data”, a lot of understanding is lost and cannot be reconstructed. The only understandings people find from big data are the ones they are looking for, and they usually look for just one explanation.


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