### This is negligence.

We have the technology, I’m sure, to design an inexpensive, hand-held device that sounds out phonemes for us. Type in - or scan, or whatever - a word, and it’ll tell you that the letter *b* makes the *buh* sound, and that a *t* and an *h* together make that sound that you get by biting gently on your tongue and breathing out. “This thing does your reading for you!” users would rave. “As long as you know the meaning of *sounds* of words, you don’t need to read or anything!” And elementary schools would begin hiring illiterate teachers who were well-versed in the operation of the reader. “I could never read,” they’d say with nonchalance, “but you never need to *read* in real life - you can get your reader to do that for you. And my six-year-olds, they understand what words *sound* like, and with the reader, that’s enough.”

You know where I’m going with this, don’t you?

One of my finite math students came to my office the other day, sat down, and announced that she didn’t get matrices. So I walked her through one of the examples, assuring her that row reduction was the exact same thing as solving a system of equations by elimination, but with different notation. “So here, for example,” I said, “You multiply the second row by three. So the coefficient of the *y* term is now three times eight.”

“Right,” she said slowly.

“Which is…?” I prompted.

She stared blankly at me. “I left my calculator in my car.”

There are aspects of this job that could - should - earn me Academy Awards. For instance, at the news that computing three times eight was not only something that couldn’t be done instantaneously, but was in fact something that could not even be done with a minute to think about it - I did not weep, or choke. I did not tear my hair, rent my clothes, or curse the heavens. I merely stared ahead for a second.

Or perhaps more than a second, because my student then giggled - ostensibly to lighten the mood - “Seriously, it’s really bad, I use my calculator to do five times one.”

The staring ahead on my part must have continued, because my student went on: “Like, my niece is six years old, and comes home from school showing us what she can do on her calculator. It’s *terrible*!” she declared righteously.

It really is.

My student was resigned to never learning her times tables; she’s too old for that, she informed me unapologetically, and besides, she’s in university now.

The rest of the day’s events included - but were not limited to - an adult student taking issue with the line “5/2+5/2=5″ on the blackboard - “isn’t that 5+5 over 2+2?”

Again with the poker face on my part, followed by, “What’s a half plus a half?”

She narrowed her eyes at me, the simple question a distraction from the real issue, the serious and difficult matter of adding 5/2 to itself . “A whole,” she replied.

“So five halves plus five halves?”

A pause. “Oh. Five wholes.”

(This, by the way, is the job at which I’m merely *temping*, as I lack a Ph.D. Exactly how that additional qualification would enable me to educate students who can’t add fractions or multiply single digit integers is beyond me.)

These are two of my students. There are others, many others, students who cling desperately to their calculators, those little black boxes.

These students were promoted through grade three, and then grade four, and then grade five, and then grade six, and all the way up to grade twelve lacking the ability to assimilate the simplest of numerical data on their own. All manner of quantitative data must be mediated through a machine before they can deal with it in even the most basic of terms. If apples are two for a dollar, and they want to buy seven of them, they have no concept of how much they’ll end up paying. If they need to double a recipe that calls for a third of a cup of sugar and a half a cup of flour, they’ll end up wasting their time with quarter-cups and sixth-cups (do those exist?) only to wonder why their cake is so small, given that they used two sixths of a cup of sugar and two quarters of a cup of flour - twice what the recipe called for.

Promoting those students through eight, ten years of math classes is absolute negligence, and I point my finger at every teacher, parent, and school administrator who didn’t notice or didn’t care that the children they were charged with educating could not deal with numbers in any sense whatsoever. Being unable to multiply eight by three without a calculator is like being unable to understand a Japanese movie without subtitles. But I never passed a Japanese class.

And, alas, there are those other four fingers pointing back at myself, as protocol here is to pass virtually everyone who puts in an effort. But again, I’m temping, and the insufferable cynic in me is tempted to assign everyone a grade of 40%, and explain - yes, that’s 10% for quizzes, 60% for tests and 30% for the final, and your grades were 70%, 65%, and 55% respectively, so that’s 40%. I submit that students who don’t argue don’t deserve to pass.

I never learned my times tables. I can multiple 3 by 8 without a calculator, though. I think we need a whole other approach to elementary math education — both teaching kids how to use calculators so you won’t have to teach them to think, and focusing on rote learning to the exclusion of all other things (ex. — kids learning to read read stories, if very simple ones, from an early age, but when do kids learning math start doing proofs?) are both signs of an unimaginative teacher.

Wow, I have a lot of things to say about this entry.

First, this is probably the proper place for me to admit, in keeping with my candidness on this dark and mysterious blog — the same candidness that prompted me to confess, in response to an earlier entry of yours, that I never memorized the imperial system of weights and measures — that, like Kirsten, I never memorized my multiplication tables. Fortunately, I can do all the multiplications quickly in my head, saving me a lot of embarrassment. (The problem with memorization is more difficult to surmount. How to explain to people that I don’t have my

my phone numbermemorized?)And yeah, it

isstrange that someone could get through high school without being able to do basic multiplication. It is sort of the mathematical analogue of being promoted through the grades without being able to read. If the problem weren’t widespread, I would almost be tempted to think that the student whom you talked to had some form of dyslexia that prevented her from understanding or readily learning numerical concepts.Can the students in your classes use calculators on tests? I guess they must be able to, because if they have multiplication difficulties they would not be able to pass without calculators (??) (I minored in math and where I went to college the department policy was no calculators were allowed on exams, though that particular policy may have been just as much a result of the programmable capacity of today’s calculators as of their capability to do basic arithmetic.)

It occurs to me that my earlier admission that I bring my Palm PDA with me everywhere to function as an extension of my mind because it has a little “converter” program for when I need to convert quarts to pints (What is a quart, again? What is a pint?) or kilometers to miles, or whatever, is dangerously close to one’s bringing a device with oneself everywhere to, for example, function as a hand-held phoneme “reader”, or even to function as a calculator.

If I have any claim to absolution it’s that my problem is one of memorization whereas the students’ problem is one of

thinking. Yet on scrutiny maybe even that distinction disappears. In addition to telling me, for example, what a quart and pint are, my hand-held “converter” program will convert between the two, doing the algebra for me as I puzzle in the grocery store in front of a rack of bottled juices, trying to decide which container is the better value. It’s easy to feel that our education system has failed when one has to resort to a calculator to do basic multiplication, but what about when I resort to using what in effect amounts to a calculator when I have to do volume conversions that involve weird fractions and more difficult multiplications — not because I have to, but because it’s quicker, easier, more painless? The critical distinction seems to lie in the complexity of the computations that one relies on electronic devices to do — basic, in the students’ case, and more complicated in mine.All of this is not to say that I don’t agree with you about how incredible it is that people can’t do basic multiplication without calculators — I *do* agree. And I also agree that in those cases there has been a wholesale failure of their past educators. I just think the whole train of thought raises some interesting general questions about how reliant we may be (or how reliant I specifically may be) on electronic appendages to the mind. This all may go a long way towards explaining my vague guilt and uneasiness when I draw out my trusty Palm in the supermarket.

One of my students told me that she didn’t need to know much math because she is majoring in Special Education, so her students weren’t going to understand it anyway.

Your students use actual calculators? Mine were upset because I asked them to divide 140 million by 2000 and the calculators on their cell phones could not take that many digits…. This was in a science class by the way.

What I have trouble with is the idea of using the results of this N thousand times and not learning it from that. I never memorized my times tables, but I’ve multiplied three by eight enough times that I know the answer without stopping to work it out.

(wes: Not memorizing your own phone number isn’t a problem; you never use it, after all. Not memorizing phone numbers you *do* use, that’s the one you need to be creative with your excuses for.)

My philosophy on the subject is that a calculator (or, more generally, tool-that-does-math-for-you) should never be used to do something you can’t do without the calculator, but if you understand what’s happening and it’s the result you’re interested in, well, there’s a reason humans learned to be tool-users. (I think I’ve calculated exactly one nontrivial square root longhand in my life.)

You “did not tear my hair, rent my clothes, or curse the heavens” … which was a good thing (even for a Moebius Stripper). Renting your clothes at the time, especially the ones you were wearing, might have left your naked astonishment more literally naked.

Then again, rending your clothes might have done that too, especially if you got really torn up about their astonishing lack of basic arithmetical skills.

Welcome to teaching today. I can promise you that in short order nothing, and I do mean nothing, will surprise you anymore.

It doesn’t suprise me, but I should point out that blaming the students is wrong. When you teach, your students is what you start from. If they can’t eat with a fork, well, they can’t eat with a fork. Complaining about it won’t make the problems go away.

So, maybe teaching should evolve. We pretty much teach in universities exactly the same way we did in 1910, but the world is a very different place.

I can honestly say I learned very little from classes at university… very little… and I was a model student… pretty much everything I learned, I learned on my own, painfully. Attending university was more of a social experience, really, and the social pressure was what made me learn. Also, now, it is very nice because I have proof that I learned something.

Let’s face it: we (collectively) can’t teach. That’s why we find these problems all over the map. If we start from this realisation that we can’t teach, instead of complaining that the student haven’t learned, then we might focus on the right thing.

For example, maybe it is about time we put the student in charge of his learning. eLearning is one approach that puts back the student in charge.

With due respect Mr Lemire I think you are fundamentally wrong.

We CAN and SHOULD teach arithmetic and spelling, and many other topics young people need to join society as responsible reliable people. The level where that teaching is crucial is NOT a level where the student can be in charge of their curriculum. What 8 - 10 year old would ever choose to learn multiplication tables and vocabulary lists?

Students are far too in charge of their learning now, especially in high school, and - surprise surprise - often choose paths that leave them bereft of basic skills they have to painfully recoup in later life, or that they never learn.

Students at university are totally in charge of what they learn.

If you learned very little from classes at university, that suggests the possibility that you didn’t read beyond the basic requirements of the courses much less ask a lot of questions of the instuctors. I have been at 6 different universities, and have seen my fair share of bad instructors, but none so bad I couldn’t learn SOMETHING from them.

Web resources are a faint and feeble echo of the vigour and stimulation of a well run seminar/ class / lab at any level.

Talking about this issue in blogs (especially articulate ones like our gracious host’s)and complaining in letters to the editor, and during political campaigns is an essential step in developing the public will to restore basic skills and standards to education.

They are sorely needed. The countries that are pushing those skills now (China and India) in an ambitious and hungry work force are going to leave us in the dust.

I second littoral zone’s disagreement. Students who are able and motivated to learn will learn simply as a result of being around intelligent people, however much or little control they have over their formal education, and students who *aren’t* motivated to learn should have as little control over their education as we can manage.

If you only look at the book knowledge I picked up as a result of sitting in class, my (partial) university education was a waste of time and money… but chasing down interesting side topics (and more depth in main topics from class), combined with being in a place with a high density of intelligent people (I learn quite well by osmosis), made it the most productive two years of my life by pretty much any standard you care to apply. (My tendency to focus on the interesting bits a few at a time combined with the course load requirements to get the degree made it unsustainable, but that’s a different rant.)

I think that Daniel does raise an interesting point : the practical art of teaching is very much in its infancy.

It seems to me that the material in a basic mathematics course (some pre-calc kind of course, with maybe some practical statistics of the kind you need to read the paper) is significantly less difficult by any measuring scheme than mastering a human language. And yet I feel as incompetent to teach that class as a native speaker of Spanish would be to teach me Spanish.

Foreign language learning has had a huge amount of study put into it, for both military/diplomatic and business reasons. There is serious work that has been put into developing methods to learn Mandarin, say, in an intensive 12 week course. These programmes have an incredibly high success rate… much higher than the success rate for any pre-calc-type class I’ve been involved with.

Dave wrote,

“students who *aren’t* motivated to learn should have as little control over their education as we can manage.”

There are so many things wrong with this statement that I don’t know where to begin, but for one, where does “not being motivated to learn” come from? Is a person born with or without motivation to learn? Or does it, perhaps, wax and wane depending — among other things — on how much control one has over what one learns?

In general, people have the motivation to learn when they think they’ll get something out of it… whether it’s acquiring a skill they want (e.g., playing a musical instrument) or some monetary award (say, a raise). But some people are aghast at people who want to learn stuff just because they want to =know=.

I just came back from a professional seminar, where the payoff for many of us there was that we would be able to do our jobs better and possibly make more money in doing it. That surely concentrates the mind. At a dinner, I mentioned that I had read a book on the history of Bananas in America. These are all highly educated people. They thought I was nuts (indeed, I believe for all the Mathcamp banana-fetish, I was the only person who read that book. It wasn’t that long, and it was very interesting.) They wondered how I had the time. But then I noticed that all these guys spent lots of time watching sports games. Well, I read books about weird history instead of watching football games. That’s my entertainment.

Back to the matter at hand. One can’t blame the students to a certain extent - it’s not their fault educational malpractice was inflicted on them as children. However, once you’re an adult, you’re responsible for making sure you know what you need to know. There were things I didn’t learn in school very well - I didn’t quite catch history the first time through, and I certainly didn’t pay attention to Dickens. As an adult, I’ve read through all of Dickens’ novels and sought out lit crit books on them, and I’ve amassed quite the odd history collection. My cousin Pat was basically illiterate when I met him (I was a little kid, he was in his 30s) - when he saw how much stuff I read just because I wanted to know stuff, he struggled through, but learned how to read in his 30s. No one is too old to learn a basic skill. It’s not a matter of memorization so much as constant work - people don’t realize how much work goes into learning stuff when they’re children. Some teachers don’t feel like dealing with all the work, and so pass the kids up to the next level. When Pat was learning to read, he had to work on it alot - it’s just like doing regular workouts if you want to run a marathon (or even half-marathon) or practicing scales and finger exercises to get good at piano. It can be excruciating at the beginning, but with practice and determination, almost anybody can learn the basic skills.

As for the special ed teacher, I must say that she was right, if she was going to be teaching even the mildly retarded. The best you’re going to do, usually, is addition and subtraction. Reading comprehension is pretty impossible, too. My grandmother taught special ed for over 20 years, and people don’t realize that it’s not simply that these people have the intellects of 6-yr-olds or some such, but that their brains simply don’t function normally and have some pretty bad handicaps when it comes to reasoning. This does not mean that a person teaching them must only know the level that is being taught. One has a life outside the classroom, and being ignorant simply makes it easier for people to cheat you in simple ways. I told that to my students, especially with dealing with percents, about how they can be used to fool you into thinking you’re getting a deal when you’re not.

However, I don’t mind students that know when they don’t know something. It tires me, but it doesn’t infuriate me like calc students who think they’re smart, but can’t do basic algebra, or think they know it all, but can’t do a simple first derivative like x^x.

Wow, my readers kick ass. Coupla replies:

was, she did not know what 3x8meant. (She told me that she needed a calculator to multiply five by one; she didn’t know that that was just one five.) And this is why I think that learning multiplication tables is important - not to give kids something to memorize, but because seeing the times tables allows students to see patterns in the numbers they’re dealing with, and think about them that way. If you’re never seeing that 3x8=24 and 4x8=32 - if both of those questions are just calculator fodder - then that’s one missed opportunity to notice - hey, add 8 to get from one to the next. (Daniel Lemire had some related thoughts on this topic on his blog.)Kirsten - It’s not even just a matter of teaching proofs versus not teaching proofs. I’ve never taught proofs outside of Mathcamp, but I give the instructions “explain” or “justify your answer” regularly, both in class and on tests. Why are we allowed to cross-multiply? (”Because it’s a RULE!” more than one student has told me) How do we check that t=5 is an answer? (I also find that the word “prove” intimidates students, whereas they will happily (well - sort of) “show” that something is true.) And expanding on Dave’s belief that philosophy that students should not be allowed to use calculators for anything they couldn’t (in theory, given enough time and scratch paper) do by hand, I often make students earn their math privileges: for instance, they can use the quadratic formula (instead of completing the square) if they can show me where it came from. What’s frustrating is that one of the texts I use tosses out formulas left and right, with no justification - not even a heuristic explanation. So it shouldn’t surprise math teachers when students get the impression that math is all about memorizing formulas, when in fact I find there’s a hell of a lot less memorization in math than in other subjects.

[…] : Miscellany — Moebius Stripper @ 11:35 am

The other day, I wrote about my university math students not knowing their times tables or how to add fractions. (By the way, y […]

I guess this analogy was so obvious that I couldn’t see it: Nail-Tinted Glasses compares (sorta) my actual students to Isaac Asimov’s freaky dystopic futuristic ones. (Ping, folks, or I miss this stuff!)

[…] ou’re interested, it’s available online - all 3.7 MB of it. For a mere $150, students who think that 5/2+5/2=10/4 can (in theory, anyway) multiply […]