Hung-Hsi Wu, emeritus professor of mathematics at Berkeley, is one of the very big shots of American maths education. His work has been brought up on this blog on a number of occasions (e.g. here and here and here and here), and Wu’s many articles can be found on his Berkeley homepage. A few days ago, Wu responded to a discussion on a mathsed Listserv group. The discussion, triggered by doings in Virginia, was on the learning of the multiplication tables (on which readers will be well aware we hammer often, most recently here and here). Wu’s comment struck a chord and he has kindly permitted us to repost his comment. Here it is, slightly tidied for clarity.
I wasn’t going to be foolish and comment on this perennially “popular” topic until I saw [a previous comment, on the difficulty of the Babylonian’s base 60]. I’d like to amplify it because I feel that in our math-teacher education, we have not sufficiently emphasized certain aspects of the history of mathematics. This is a case in point: one reason the so-called Hindu-Arabic numeral system is now universally adopted is precisely because (by being base 10) it is simple enough to be learned. We had Egyptian hieroglyphic numeral system, Greek alphabet numeral system, Babylonian base-60 (sexagesimal) numeral system, Roman numeral system, and many more, but none of them stuck around because (I believe) the people rejected them as being unlearnable. This part of history should be known to all elementary teachers because they need this assurance to firm up their resolve to teach little kids this seemingly unreasonable numeral system (place value, etc.). Teachers would be far more confident in carrying out this part of their duty if they know that this may be the best humanity has to offer and therefore those kids had better bear down and learn it (the same way toddlers must learn how to walk even if falling down and scraping their knees is no fun).
Learning the standard algorithms includes memorizing the multiplication table, of course. Groans. Why don’t they begin by rebelling against memorizing the alphabet? The latter is not easy, not easy at all! After all, there is no rhyme or rhythm to the alphabet. For example, why should n follow m or v follow u? Why not a nice number like 25 letters instead of 26? Etc. But I’d make a guess: people would say that everybody knows memorizing the alphabet is a small price to pay for being able to read and write the rest of your life. Exactly! A major sin of Textbook School Mathematics (cf. page 241 of Inservice mathematics professional development: The hard work of learning mathematics)* is that it makes all kids believe that memorizing the multiplication table is nothing more than one more drill-and-kill, and they have too many of that already. This then brings us inexorably back to why we MUST eradicate TSM from our schools. The rationale behind the standard algorithms is now all but unknown to teachers brought up in TSM:
CTS (Complex-to-Simple). Any computation with multi-digit whole numbers can be broken down into multiple steps of computations with only single-digit numbers. (See, e.g., pp. 39-40 of Chapter 1: Whole Numbers (Draft)).
It is not only teachers who ought to know why the standard algorithms are such wonderful theorems that everybody ought to learn them, but kids should also get to know it asap. The idea of reducing the complex to the simple is in fact the fundamental tenet that underlies all scientific and mathematical developments (e.g., all the complicated mechanical phenomena in the world? They come down to only 3 (three) laws enunciated by Newton). The sooner we expose kids to CTS, the better off we are. But TSM cannot teach this because maybe the authors of TSM were not aware of this themselves, but more likely it is because there are no logical explanations in TSM and therefore every algorithm is nothing more than one more piece for rote memorization. There is just no way TSM can teach CTS to kids. This failure to bring out CTS in school mathematics, as we know, has catastrophic consequences. “Down with standard algorithms” has become a popular slogan not only in the popular press but, much more depressingly, also in the education literature because it has become popular even with some professional educators (they are not entirely to blame, because WE have chosen to teach them only TSM). In this ongoing battle, the multiplication table has been singled out and some well-known educators have gone so far as to declare that multiplication tables are unnecessary. How can school math education go on?
We choose to teach every kid TSM, and we reap what we sow. Simple, isn’t it?
[By the way, the Hindu-Arabic numeral system is at least 90% Chinese.]
*) To give context to “TSM”, Wu initially linked to slides from one his presentations, Professional development and Textbook school mathematics. In response to a query in regard to the extent of TSM, Wu decided that this reference was insufficiently precise and suggested instead the book chapter linked above. Wu also clarified, that TSM “may be rephrased as the math textbooks from the major textbook publishers in the years after the New Math, roughly from 1970 to the present. My very conservative estimate is that > 80% of the school students in this era were exposed exclusively to such textbooks. I had almost a whole collection of those around 1998, and I can vouch for the fact that all of them carried the same mathematical message, to the point that I started to ask myself why they bothered to publish those books separately.”
Good to see Boaler and her disastrous garbage getting a deserved kicking in the linked article.
For those who wonder: TSM stands for textbook school mathematics.
Distributivity works in any place value system (or positional numeral system). In particular, it works in the Babylonian sexagesimal system. I teach this stuff whenever I find the time (which means hardly at all), but *IT’S GREAT*! Apparently the Babylonians taught the multiplication tables up to 20, although (by distributivity), the small version would have sufficed. But these guys simply loved computing. There’s a tablet on which the kids had to compute 9^46 by repeatedly multiplying by 9; then they copied the result on the back side and divided by 9 until they reached 1 (see https://arxiv.org/pdf/1306.5989).
That’s hilarious. Thanks, Franz.
I find Wu’s obsession with TSM not very helpful. To be sure, he has a valid complaint about how mathematics is often presented and taught in schools, but one has to be cautious in stereotyping it. All stereotypes are based on some reality, but not every instance may fit the stereotype (cf. jokes about (fill in your ethnic group) mothers-in-law, drinking habits, etc.).
What happens in the classroom depends very much on the choice of text (which really do vary; one could usefully look at some of the texts from earlier in the last century) and the capacity of the teacher. In some classrooms, children are invited to construct the multiplication table before they have to learn it which gives them a better chance to grasp the structure involved. If time is taken for kids to discover some of the patterns in the table, so much the better. Some teachers and texts will take the trouble to explain why the standard algorithms work. Even when the algorithm is learned by rote, if the teacher is careful in long division about emphasising the role of the four numbers involved: divisor, dividend, quotient and remainder, how they are related and how they can be used to check the work. In fact, having students routinely check calculations, preferably in an independent way, is something that seems to have been neglected in a lot of modern teaching. The bottom line is the level of awareness that the student has about what is going on, and this can be achieved in many different classroom regimes if the teacher chooses to deal with it.
One point Marty raises that is important and should be part of the classroom discussion is what a tremendous advance the present system of numeration is and the ingenuity behind the current algorithms. In a traditional classroom, Roman numerals offered a nice example of the limitations of a defective numeration system. However, it might be useful for pupils to learn other methods of computation, whether paper-and-pencil (e.g. the scratch method for division), stones on a counter or the abacus, to compare the convenience and efficiency of various methods. If this is not possible for kids, it surely should be for teachers-in-training.
Thanks, Ed. I have thoughts and will reply. But quickly, do you mean “Wu” for “Marty” in your final paragraph?
Thanks, Ed.
I have reservations of my own, but I’m not quite sure what your objection is to Wu’s comment. It seems to me that TSM is undoubtedly a thing, even if textbooks are not necessarily the primary culprit, and even if TSM is not the only bottomless pit in which one might fall. And, sure, a teacher may teach non-TSM even if they are saddled with a TSM text, but how many do? And a TSM text hardly makes it easy to do so.
Wu is writing of the American experience of the last fifty years or so, and I cannot judge the accuracy of his research and observations. Australia is probably different in many ways, and it certainly feels different to me. For example, there are essentially no mathematics texts for primary schools, and there haven’t been for decades. Does that mean there is no TSM? I have thoughts, but I’ll leave it.
I’ll just point out that I see plenty of TSM at the secondary level and at the university level. The index laws crap I was attacking recently is pure TSM. I was also tutoring a (smart) 1st year kid a couple months ago and the lecture slides were appalling, pure TSM. (There was no prescribed text, which, ironically, is an active encouragement to teach TSM.) It is perhaps better to not name the university, which may or may not have been the University of Melbourne.
Ed writes: “ The bottom line is the level of awareness that the student has about what is going on, and this can be achieved in many different classroom regimes if the teacher chooses to deal with it.”
But teachers can’t deal with it if they don’t have sufficient awareness about what is going on.
I don’t think Ed would disagree with this at all.
This is utterly brilliant and should be a component of all maths methods teacher training. I didn’t know this (love the reduction of the complex to simple).
Well, of course you knew it, JJ. But I agree that the Big Lights advertising of CTS is not common, and it should be.