### Show me the data.

What’s that you say? You’re *sick* of all those long-winded education rants that *never go anywhere*? Me too! However, I can’t pass up this opportunity to commend the California Commission on Teacher Credentialing for so succinctly summarizing everything that’s wrong with elementary-school-level mathematics education today. And in an exam that every prospective teacher in the state is required to write, to boot - now *that’s* efficient delivery!

Many people believe children will never learn mathematics if allowed to use pocket calculators. Having spent countless hours memorizing multiplication tables and doing long-division problems unaided by any mechanical device, many adults cannot conceive of anyone acquiring this knowledge without similar effort and practice. ______________. What many people fail to understand is that mathematics is constantly evolving; it is not a fixed body of facts. Students must still learn basic skills, but they do not need to perform the endlessly repetitive exercises that calculators largely eliminate. Youngsters can better use their time—time they would have spent performing long-division problems—to learn mathematical concepts that will enable them to become better problem solvers.Which sentence, if inserted into the blank line, would best focus attention on the main idea of the passage?

(A) It is true that mathematics is not the easiest subject in the typical elementary school curriculum.

(B) Many of you have doubtless heard about the bitter classroom experiences of students who learned mathematics this way.

(C) There is much to be said for instilling this kind of discipline in students.

(D) Although it was clearly not fun, students trained in this manner rarely forgot what they had learned.

(E) Such views, however, seem to reflect a resistance to change rather than a rational approach to mathematics instruction.

(F) Contrast this instance of common sense with the hallucination that follows.

Just kidding! (F) isn’t an option. The correct answer is (E).

Yes, yes: fish in a barrel are targets for *amateur* marksmen, I know. But what we have here is a barrelful of tranquilized guppies that been given free reign over early mathematics education, so let’s have a go at it before any more metaphors get mixed:

What many people fail to understand is that mathematics is constantly evolving; it is not a fixed body of facts.

You know, I skewered this one months ago. I even used the word *evolved*, for crying out loud. My readers found this -

Yes, if there’s one field that has evolved beyond recognition in the past quarter-century, it’s introductory calculus. Why, back when I was a tot, the area of a rectangle was length

pluswidth, the derivative of sin x was 5, and we only had whole numbers, so whenever we needed to compute the area of a circle we had to use pi=3, AND YOU NEVER HEARD US COMPLAIN.

- to be hysterically funny, *because it was so over-the-top*. Except that…it wasn’t. It was very much beneath the top. It was smack dab in the *middle* of what those enlightened educators, unlike “many people”, have *succeeded in understanding*.

Students must still learn basic skills, but they do not need to perform the endlessly repetitive exercises that calculators largely eliminate.

They don’t? Really? I’m not convinced. Let’s see what a cognitive scientist has to say about this!

It is difficult to overstate the value of practice.For a new skill to become automatic or for new knowledge to become long-lasting, sustained practice, beyond the point of mastery, is necessary.… By sustained practice I mean regular, ongoing review or use of the target material (e.g., regularly using new calculating skills to solve increasingly more complex math problems, reflecting on recently-learned historical material as one studies a subsequent history unit, taking regular quizzes or tests that draw on material learned earlier in the year).

This kind of practice past the point of mastery is necessary to meet any of these three important goals of instruction: acquiring facts and knowledge, learning skills, or becoming an expert.

But what does that guy know, anyway? The times (tables), they are a-changin’! Mathematics is *evolving*! Get with the program!

I’m curious about something, though: is this anti-repetition view unique to mathematics education? Have the musicians among my readers, for instance, noticed a similar trend in music pedagogy? *Music is EVOLVING! It is not a FIXED BODY OF FACTS! We have new-fangled technomology that enables students to bypass all that boring stuff, like learning scales!* Or…do music students still practice scales, even though scales really aren’t that much fun to practice? I know that when I teach pottery, I spend a fair bit of time focusing on the basic skill of centering the lump of clay, even though it’s more rewarding to throw teapots.

Onward:

Youngsters can better use their time—time they would have spent performing long-division problems—to learn mathematical concepts that will enable them to become better problem solvers.

I’ve spent countless minutes (that seemed like hours) waiting for students I’ve tutored to multiply 4 times 5 or 3 times 6. Oddly enough, they weren’t adept at understanding mathematical concepts or solving problems.

The idea that students who don’t waste their time learning the boring basics will have freed up valuable time and brain space to become creative problem-solvers is one I’ve seen cited often. It’s an idea that every single mathematics educator with whom I have ever communicated - and I have communicated with hundreds, across several continents, both in person and through this blog - considers to be utter hogwash. But it’s possible that we’re just outliers. It’s possible that on the whole, students who don’t learn their times tables beyond mastery are brighter, bolder, and more creative problem solvers than their predecessors. And if that’s the case, then there should be plenty of statistics to bear that out. And surely the proponents of calculator-based curricula have plenty of rigorous statistical studies at their fingertips.

Let’s see them. Put up or shut up.

*Show me the data.*

Show me a single peer-reviewed study that indicates that students who were raised with calculators show greater facility and more creativity in problem-solving than those who were raised without. I want to see test scores. I want to see comparisons of performances in university-level mathematics classes between students who were made to memorize and practice their times tables, and those who weren’t. I want to see evidence that *learning the basics* and *learning advanced mathematics* are *negatively* correlated.

Until then, I’m going to *rationally* resist change, and teach mathematics in the way that my peers and I actually *learned* it.

Mathematical pedagogy does change over time. Derivatives are explained by talking about “zooming in'’ to the graph rather than considering ever smaller secants.

I hope people start teaching trigonometry with rational numbers . I also want students to stop solving difficult anti-derivatives by hand. There is an algorithm to do it . Finally my ultimate goal would be to give require machine verified solutions from students. This would prevent students from taking ridiculous steps.

Of course this comment doesn’t have too much bearing on elementary arithmetic. But I still want to emphasize that methods of understanding mathematics have changed over time. I’d even say that the entire body of mathematics have been revised in the last 100 years or so to fit with our new standards of rigour.

I keep losing my links in my posts. Do you have any instructions on how to post comments? I mean, am I posting text/plain or text/html?

MS, there is an American textbook author who agrees (actually did agree) with you.

John Saxon wrote math textbooks with drill and repetition in mind. He even references things like piano players learning scales, and football players doing drills.

I have heard a elementary-ed major describe his textbooks as boring. But I have also seen, among the home-school community in the States, an incredibly large usage of Saxon’s math texts.

The key in that department is, the daily lessons are so well-written that a parent and child can learn the math together from the book.

Many of those students claim to hate the texts, but once they arrive in higher education, they love the results of using drill and repetition in learning basic mathematics, algebra, trig, and analytic geometry.

Of course this comment doesn’t have too much bearing on elementary arithmetic.And this is my main point, here: as far as I know, the reforms taking place in mathematics education are taking place at the pre-university-for-math-majors level are not explained, or justified, by, for instance, the concerns that introduced epsilon-delta proofs into analysis.

As for posting links - HTML is allowed here. You can either post the URL, or use regular HTML. I checked the submitted comment, and didn’t find any link there. What are you doing?

Yeah, it’s great when students don’t learn long division, because it makes it more fun to try to teach polynomial long division to them.

It’s great when students instinctivley reach for their calculators to compute 7*(1/7), because it makes it easier to see how to simplify (x-1)(x+2)/(x-1).

If I ever need to justify why I don’t care one whit about the noises that the teachers’ unions make regarding Gov. Schwarzenegger’s “defunding of the schools”, then I’ll simply point them this way and mention that I’m not for spending one more dime on K-12 schooling while everyone’s tax dollars are spent on creating a generation of calculator-dependent innumerates.

Chris Phan -

That’s a fantastic point. Maybe these lower level, more mechanical concepts that are being eschewed in favor of “concept-oriented” teaching can, you know, be used to help better understand these concepts.

Nathan — are you insane? The reason we’re in this whole mess to begin with is that it’s a fairly innumerate section of society that is ACTUALLY IN CHARGE of teaching math to our kids, because what we pay teachers means we’re not going to get the best people. Plus, without reducing teacher/student ratios, students will never get the individual attention needed to make sure they’re actually learning mathematical concepts, not just trying to get the right answers. And beyond that, you can’t blame shortcomings of the overall curriculum on the teachers union: blame that on the school boards, they’re the ones who set that kind of policy.

Just because the schools aren’t doing their jobs that well doesn’t mean we should stop paying for them. That’s ridiculous.

Now, if they’re complaining that they don’t have enough money to buy the best calculators for the students, then I’m with you.

Mango Juice beat me to my main response to Nathan’s point - talk to any math teacher with more than twenty years experience in the classroom, and odds are, they’re just as appalled by the calculator dependence as we are. But they have a curriculum to follow - curriculum that’s set by politicians, not by people who have studied either mathematics or education in any depth. (That said, it seems to me that the main problem around these parts isn’t defunding; it’s flagrant misuse of funds. I don’t know how things are in California, but ’round these parts, high schools replace textbooks every year or so (because each new edition has instructions for the

newTI calculator), and yes, schools are spending a non-negligible amount of money on buying the newest calculators. That’s thousands of dollars per classroom right there - enough, for example, to decrease class sizes somewhat right off the bat.) But anyway - I am in touch with a high school math teacher who’s a year away from retirement. She’s a talented educator - everyone loves her - and she knows her math. She has a reputation among the province’s university’s math professors for actually preparing her students for university-level math courses. And in recent years, she’s been constrained by calculator-based curricula that don’t allow her to actually teach what she knows students need to know. And when she raises this issue…she and her fellow teachers are pegged as incompetent whiners. (Because if they were competent, then there’d be better results in the schools!)Which isn’t to say that I wouldn’t sooner send my hypothetical kid to be raised by wolves than have him or her “learn” “math” in one of today’s brave new schools. But this one isn’t the unions’ fault: the Ministry of Education doesn’t belong to the teachers’ union.

Chris and Simon - you’re missing the point. There are now

computersthat do polynomial long division, and to simplify expressions such as (x-1)(x+2)/(x-1). We don’t need to teach regular long division, or manipulation of fractions, because even the stuff it’susefulfor can now be done by machines! Now we can assign students tosolve advanced math problems creatively!I wrote my links like <http://example.com/> because I thought I was submitting text. Maybe my links are lost and gone forever.

Nathan: Did you notice who printed the article quoted above which advocates “long-lasting, sustained practice, beyond the point of mastery”? The American Fedaration of Teachers.

err, Federation.

Of course, the U.S. spends more on education per pupil than most countries in the entire world, and I’m sure Canada is not far behind. As MS says, it’s not a matter of not enough total money - it’s a total misallocation of those funds. Here’s a modest proposal: let’s destroy all federal level educational bureaucrats and all state level educational bureaucrats and their attendant bureaucracies. That would be a good start.

Of course, it would be easier to defund a couple bridges to nowhere in Alaska than to dislodge bureaucrats, but a woman can dream. There’s a lot of money to be made in picking out the next year’s math texts and what the approved calculators are going to be, and making fluff pronouncements at do-nothing education conferences. Pretty sweet life, especially if noone is going to fire you because the kids aren’t actually learning.

Have you ever tried pointing out that my version of Apostol was last revised in 1967? I’ve heard that Stanford still uses that text for all their freshman calc classes and I don’t think anyone has been complaining about the results… My other favorite calculus text is Spivak, and that’s been barely revised in the last 30 years.

As for school funding, I think it should be slashed. Kozol was having tantrums about schools *only* spending $11,000 per student! If you have 20 kids in a classroom, that’s $220K/year. The fuck are they spending all that money on?

Mango Juice:

Yes, I am insane, but probably not in all the ways you think I am. Let me see if I can explain myself a bit more fully…

Agreed; I think we (in California, at least) ought to institute a good (there are ways to screw this up, certainly) system of merit pay so teachers who produce know-it-alls from mediocre or even abysmal incoming talent get megabucks…and we don’t waste large salaries on teachers who have a middling ability to teach math.

Completely agreed. I dislike having curricula set in Sacramento (or worse, Washington DC); I’d think it’d make lobbying dollars more effective if you’re from Houghton Mifflin or a similar textbook manufacturer. Especially since they seem to be intent on changing curricula as often as I change my computer desktop’s wallpaper…

No, but I think I’m right to complain that money is being taxed and thrown only in the general direction of educating children. Plus, as I understand it, our current governor isn’t exactly defunding the public school system; Davis raised school funding during the dot-com era…and that high level had to be maintained by Schwarzenegger according to Proposition 98.

If you’re interested in California politics—or simply really bored—have a look at California Connected’s Return of the Fab Four (I’d prefer the term “Gang of Four”, but that’s not fair to anyone present, Brown included…). At the 9:30 mark, the interviewer asks Davis, in a nutshell, “Are you getting any schadenfreude from seeing the guy who ousted you get beat up by the teacher’s unions?”

On to meep:

That’d be close to what I’d like; I’d abolish the Department of Education and all state bureaucracies that are neither statewide nor confined to one school district (from what I gather, there are countywide bureaucracies, at least).

Mango Juice: Does all that still sound insane?

the point i always make (aside from the obvious “we don’t scoff at practice makes perfect in athletics) is basically chris’s. if you don’t learn

and practicethe basics, you can’t learn the advanced stuff. chris’s example (simplifying 7*(1/7)) is a good one…there are plenty more:practice calculating 37*4 by calculating 120+28 and you learn the distributive property.

practice comparing 20^2 with 21^2 and you will remember that (a+b)^2/=a^2+b^2.

practice division with remainders, and you will understand why P(a)=(the remainder upon dividing polynomial P(x) by (x-a). and you will understand the basics of modular arithmetic.

calculate 2^30 by hand just once, and you will quickly appreciate exponential functions.

etcetera, etcetera, etcetera

Nathan, I propose that you know nothing about what happens in California K-12 math classrooms. The use of calculators below pre-calc is quite dramatically restricted, especially compared with other states, and on the STAR tests that students take each year, calculator use is prohibited. Likewise with the CAHSEE exam.

And having been on the front lines of K-12 education (albeit for a total of 22 weeks), let me say that the deck is strongly stacked against having good K-12 teachers of any sort:

First off, the primary mission of the teacher ends up being classroom control rather than education. This was my demise as a high school teacher. I sucked royally at it.

Second off, the work load is crushing: 5 classes per day, 5 days per week (and 40 students per class), with one period of prep time (as if one could, in one hour, prepare for teaching 2-3 different classes, not to mention grading and other administrative tasks), and a brief lunch break. And all for a salary of about $40,000 per year. It’s a lot of skills being demanded, a lot of time being demanded (I was easily working 12 hour days when I was teaching full time), and the don’t talk to me about not working summers.

And the problem, now that I’m teaching remedial students at MTU is not calculator dependence, at least not in California. Most of my students can do the arithmetic demanded of them. It’s conceptual understanding (e.g., knowing that (x+y)^2 is NOT x^2+y^2). Even in the city where I was teaching previously, I didn’t see calculator use as being the problem that MS does: Sure I had students who couldn’t calculate 20-8 without a calculator, but that didn’t really impinge on their ability to do algebra. Algebra and Arithmetic are largely orthogonal skills.

Hell, I’d say that polynomial long division is easier to learn than regular long division (and when I stopped having students do a long division problem to prep for polynomial long division, I stopped having students do an integer long division problem as prep for when I taught polynomial long division and found no loss of ability to do the procedure).

The big problem that I see is that students try to approach mathematics as a set of recipes to be memorized rather than concepts to be understood. It’s a difficult thing to overcome because there ARE some things which need to be memorized, and teaching students to learn by understanding rather than memorization is a difficult battle.

Show me a single peer-reviewed study that indicates that students who were raised with calculators show greater facility and more creativity in problem-solving than those who were raised without.Beware what you wish for. Peer review is meaningless to the ED crowd. most Ed research, and when I say most I mean over 90%, is garbage and much of it is peer reviewed. Let’s settle on a study that meets the loose minimum standards we allow for social science research: control group (demographicly matched). no selection bias, replicatable, etc.

I want to see test scores. I want to see comparisons of performances in university-level mathematics classes between students who were made to memorize and practice their times tables, and those who weren’t. I want to see evidence that learning the basics and learning advanced mathematics are negatively correlated.Not surprisingly, all the scientifically valid research shows just the opposite. See project follow through for 30 year old vindication.

Point taken, KDeRosa. And if someone did hand me such a peer-reviewed study, I wasn’t intending to say, “oh, very well; I suppose you were right and I was wrong; carry on, now, and I’ll go invest in a classroomful of TI-83+’s” without at least reading it through. It’s just that I’ve never even seen any of the pro-calculator folks even

tryto defend their case - they just state it as though it’s the gospel truth, and that obviously, anyone who’s thought deeply about education would agree with them.Thanks for the link, by the way.

You underestimate the lack of shame these people have:

Take a look at this “article” from our friends at NCTM.

Clearly, he’s banking on you not reading the research cited since many (the valid ones) contradict his actual argument.

Nathan, I think getting rid of national standards will only exacerbate the problem. Local control means more power to parents and less to people who know something about mathematics and mathematics education. It’s harder for calculator manufacturers to lobby that way, but it’s also harder for teaching strategies that work to take root, and it perpetuates a class system by defunding lower-class schools.

Local control doesn’t necessarily mean that the schools are funded solely locally, you know. There’s the matter of money, and there’s the matter of curriculum-setting (and personnel policy, etc.) I’d rather not the feds take over personnel policy, for example. And I don’t want them setting the curriculum, either.

You can’t say “getting rid of national standards”, because there are no national standards. For anything. I think that’s good for upper-level courses, but yes, perhaps there should be some kind of national standards for the basic skills one is supposed to learn in elementary school, such as basic grammar and writing, arithmetic, and reading comprehension. It’s tough to catch up in middle school and high school if your elementary school years are crap.

I memorized multiplication tables in 3rd grade, along with the rest of my class. At that age, I would think that multiplication tables are not so much an “endlessly repetetive exercise” but more a revelation about patterns. Get them in your head early, and you’ll start seeing other things. Use them enough, and you’ll barely notice the effort, so you’ll be able to focus on the issues that are new or more complex. I recently observed an algebra class (after a decade as a software engineer, I have begun a teacher certification program) in which the teacher, after walking around the room to see where students were struggling with a problem set, suggested that they use a calculator for 5.6x = 7. The teacher told me afterwards that she had not allowed calculators until recently, but now wanted students to concentrate on operations; this probably explains why upon the suggestion there was a mass retrieval of calculators from backpacks. There seem so many things wrong here: What can be done with a decimal? Doesn’t 56 look familiar? If a calculator is not necessary for any of the other problems in a set intended for practice of a new concept, why would it be necessary for this one? After the calculators emerged, the teacher noticed some students plugging in 5.6/7, and mentioned this to prevent others from making the same error. This is a charter school for engineering. As an engineer, I can say that struggle and error are a good (both large and interesting) part of the daily job. What is the aversion here?

On a somewhat related topic, I just realized that there’s a very important reason to learn long division, and to practice at it a lot: unlike the other four operations, division requires students to be able to estimate. (2452 divided by 36 - how many times does 36 go into 245? What’s a reasonable guess?) I suspect that students who have to repeat such exercises “endlessly” will develop a much better intuition for such things than students who have calculators foisted upon them at a young age. And the former will be a lot less likely to overlook careless errors that result in preposterous numbers.

Actually, division does not require that skill anymore. You just have to use “forgiving division”.

http://www.kitchentablemath.net/twiki/bin/view/Kitchen/TeacherGuideEverydayMath?skin=plain

(Yes, I think it’s silly, too.)

Okay, I lost it when I saw that picture of Spongebob Squarepants. Oh MY.

I’m a bit astonished that that question is actually on an elementary-teacher certification exam. On the other hand, everyone knows that the field of elementary education does not generally attract the best minds (either as educators of would-be elementary educators on the university level or as the would-be elementary-ed teachers themselves) so perhaps it shouldn’t be as surprising to me that the field is filled with such hogwash masquerading as prudent pedagogy. Some people, I’m sure, will be very unhappy to hear me say this, but, really, what is the percentage of trully brilliant people in the field of elementary ed as compared to the percentage in, say, mathematics or physics?

Lobster: there is a chart on this web page:

http://www.econphd.net/guide.htm

that shows the mean GRE scores by major. Please note that the only major to have a lower average score than Education is Public Administration. Figures, don’t it?

You might find this entry on long division on Siva’s Glob of Thoughts to be interesting…

actually estimating for division is really easy. take the first digit of the divisor and ask how many times does it go into the appropriate “prefix” of the dividend. that gives you a really good guess, typically it will be off by zero or one. so for example to solve 36 into 245 first ask whats 3 into 24? the guess for this problem will be two units too high but its still a good guess