It is time to stop quivering in our boots in pointless fear of the future and just roll up our sleeves and build it.
- Ray Pierrehumbert

Saturday, October 31, 2015

Didactic Malpractice Defended

I object to this article on Medium, which alleges that 5 x 3 is correctly understood as 3 + 3  + 3 + 3 + 3, but NOT as 5 + 5 + 5. Seriously.

I don’t think the parents “jumped the gun” by posting this. If this is didactic theory in mathematics it is insane.
It’s not a new insanity — they tried to teach me set theory in sixth grade or so. It was called “the new math”. Everybody hated it, even kids like me who loved math.
The point is that we don’t learn axiomatically, we learn by experience. Trying to replicate the basis of formal analysis in teaching concepts is completely foreign to the way learning actually happens. It seems to me that anyone who actually remembers learning anything ought to know this.
A kid who fully grasped the equivalence of 5 sets of 3x1 blocks with 3 sets of 5x1 blocks would be well ahead of the class, and would be duly punished.
The lesson here is that math is not about knowledge but about pleasing the bureaucracy. I can imagine nothing more toxic to the idea of “converting YOU to a math person” as the author's bio smugly claims to do. You’re taking a budding math person and convincing him that teachers are not interested in math so much as in bureaucratic BS, and that the whole process is to be resented.
My career was set back years by an off-putting teacher in first year calculus. Make no mistake that this sort of discouragement happens.
I am not just in disagreement with this article. I am angry about it.

The article concludes:

I know it’s frustrating but 
Respect the Teachers 
They are qualified experts on child education. They have the best intentions for the students in mind. If you are confused, ask them why they did something before you slam and discredit them on the internet.

This is the wrong reason to trust any profession, be it elementary education, economics, or climate science for that matter. Trust must be deserved to be earned. It's a hard problem, but "we all think this and we're the experts" is a shortcut that can lead to very serious problems.


Nik Wycha said...

I get your point entirely. I initially saw it quite differently. The purpose for going back to this style of teaching math was the premise that our deficiencies in standardized mean test scores were do to a basic lack of fundamental knowledge of how basic math processes work. Whether there was a needs assessment to determine that is lost on me, but the main point remains. This was designed to bring mean score up, therefore, this is not for the students who are way ahead. I do think that your anger is justified, as is mine, which stems from the idea that this advanced student should be punished for being far ahead or grasping the concept without the intermediate step (assuming the first lesson is 5 3x1 boxes and the second is 3 5x1 boxes which then provides the "complete" understanding of how multiplication works). I'm not sure if I would go as far as to say it isn't about knowledge but about pleasing the bureaucracy. I'm not sure the bureaucracy even understands what it is doing and not doing at this point.

Nik Wycha said...
This comment has been removed by the author.
Michael Tobis said...

I should have said "satisfying the requirements of the bureaucracy". By definition, bureaucracies are never pleased, merely complied with or not.

It's not even clear that the student was "advanced" in any way. The student may or may not have been advanced. But to tell the student that the answer is wrong when it isn't is the opposite of helping.

One respondent says that in Spanish the verbalization of 5 x 3 is equivalent to "three instances of 5". I don't know if that's true or not, or whether it is relevant in this case. But it could well be, and that's the key point.

What this does is basically to delegitimize knowledge learned outside the classroom. And that is a problem.

bluegrue said...

If you go to Wolfram's Mathworld [1], you'll find a*b defined as "when a number a is taken b times", opposite to the authors "definition" taken from wikipedia. I expect him to run in circles, screaming and shouting, should he ever find out. I am sorry for the kids, who have to suffer through that hell hole of stupidity.


Mark said...

I agree entirely and vehemently with you, Michael.

(Before posting this, I noted your instruction: "Before you speak, ask yourself if what you have to say will improve on silence." I am not sure that a "me too" post actually meets this standard, but perhaps you can cut me some slack.)

I remember getting to university and the shock I had encountering lectures that were axiom-based rather than example-based. (Lemma: there is a class of objects such that blah blah blah...) I could see that there is a certain class of people (mathematicians, basically) that could find that stuff interesting and I could even manage to be interested in it myself up to a point. But most of the people to whom mathematics is taught, even in university, are not and never will be mathematicians.