I always get a kick (as well as a wave of nausea) when I hear arguments about how math should be taught referred to as “math wars stuff”. Such criticism implies that the we are long past the math wars and that they were just trivial spats that signified nothing. In a communication I once had with Jay Mathews–who for many years has spewed his arrogant views of education in a column he writes for the Washington Post–he said that the math wars were two groups of smart people calling each other names.

I won’t comment on the word “smart” here, other than to say it’s overused to the extent that it means nothing, and has become a code word for edu-pundits who compliment each other by saying so and so “wrote a smart and thoughtful post” about whatever.

Well, I had the opportunity to write a “smart and thoughtful post” on math education, courtesy of Rick Hess who invited me to do so. It was published first at Ed Week, then at AEI, and finally at Education Next’s blog. While it has proved a popular piece, there was a recent take-down of it, also published at Education Next’s blog. The author works for TERC which publishes Investigations in Number, Data and Space, which in my opinion and others whose opinions I respect, is one of the worst of the NSF-sponsored atrocities.

I was about to defend my stance, when to my great relief, Sanjoy Mahajan, a research associate in mathematics at MIT did the heavy lifting for me on Twitter, reproduced below:

- There’s so much to say about that clever nonsense. There’s the straw man of “practicing procedures alone” bringing understanding. But Mighton’s new book _All Things Being Equal_, pp. 98-102, has a great treatment of the long-division procedure with understanding.
- There’s the sneaking in of “when division applies in solving real-world problems” onto (into?) the list of concepts underlying the long-division ALGORITHM. Sure, it belongs on the list underlying division — but not underlying the algorithm.
- After these concepts, mostly valid, comes a call to develop a “multi-faceted view of division.” But I want students to understand not all ways that one could divide but rather the long-division algorithm.
- And the method offered will not help: the “rich task” of justifying the algorithm for “any two RATIONAL NUMBERS.” That choice is either sloppy or insane. I have never used the long-division algorithm for arbitrary fractions, only decimal numbers.
- About the incessant calls for “authentic mathematics.” It’s rich coming from educators whose favorite incantation for stopping any rigorous teaching, e.g. long multiplication, is “development [un]readiness.”
- Proving this conjecture is not at all authentic mathematics. The main effort of mathematicians, not evident from the format of journal articles, is making the conjectures.
- Finally: Even if it were authentic, where’s the argument to show that acting with the outer forms, but without the inner knowledge, of mathematicians makes you understand math like mathematicians? It’s cargo-cult thinking.

I hope you enjoyed this foray into the supposedly defunct math wars.

Enjoyed indeed!

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“Cargo cult thinking”. Wow, I’ll be using that. Great piece.

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Reblogged this on Nonpartisan Education Group.

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Sigh.

Yes, let’s make sure that kids know how to count in octal and hex so that they really understand numbers. Why not have them prove everything from the start. Let’s have them explain all (whatever?) about money and different bases like 12 for time, 16 for ounces/pounds and 24 for hours in a day.

Nothing is rote if you actually use the learning. We all learned a lot of number sense by applying the traditional algorithms. Divide 27 into 5528. You quickly learn that 2*27 is 2*20 + 2*7. Then how about 27 into the remaining 128? This is all about practicing and individual homework problem sets. Those have disappeared from K-6. That’s their real disaster.

Their arguments are all vacuous. Does any of it really work? I would be their biggest supporter. I’ve tried to find any success after 20+ years of study and helping my son through MathLand and Everyday Math until he could get to proper Glencoe textbooks starting with Pre-Algebra and then continuing with proper textbooks and traditional math through AP Calculus BC. In K-8, we parents have to provide that success and enforcement of mastery of basic skills.

All of my son’s STEM-ready friends had to have help at home or with tutors. I got to Calculus in the 60’s with absolutely no help from my parents. That’s not possible today. It would be very easy for them to survey the best math students in high school about the help they got outside of class.

They want to argue vague philosophy so we don’t ask them for ANY examples of success.

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For deep understanding we need to go back even further. Kids must start with basic set theory, the foundation of all mathematics. From there, they should proceed to the set theoretic definition of the natural numbers (0=|∅| 1=|{∅}| 2=|{∅,{∅}}|, and then to the axioms of arithmetic.

Also essential, to truly grasp the nature of arithmetic: Godel’s Incompleteness theorem.

Once elementary kids grasp these essentials, everything else will flow naturally–without the mind-numbing drills or the mindless execution of algorithms.

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the weirdest part is the correlation between those who oppose teaching the long division algorithm and those proposing teaching coding in senior Kindy (okay maybe grade 2).

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And it’s play coding or hacking – guess and check. I used to have college Computer Science students who debugged programs like that. The test data set might pass, but I could look and their code and find data that would break it.

This “One Hour of Code” (etal) really annoys me.

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