Old math

Vern Williams teaches math the old-fashioned way, writes Jay Mathews in the Washington Post. Students love it.

. . . as innovative as Williams is, he is a stubborn advocate of what has worked for math teachers for centuries: demanding assignments, hard work and some repetition. He hates fads, hates the derisive term “drill and kill” for traditional math teaching and refuses to use the new textbooks with all those colorful pictures.

“When you start telling me that you have to print books with 10 different colors on every page, with charts and stories about the rain forest and what you are going to do at Giant today because we have to make everything relevant 100 percent of the time . . . I say, no, no,” he said. “I think we are doing our students a disservice.”

Williams thinks today’s students are just as capable of learning challenging material as students in the past. His students certainly are.

1. jab says:

Speaking of teaching math the old-fashioned way:

I’ve taught calculus for a couple summers to students in the Upward Bound program… I was frustrated beyond belief that students couldn’t give a rough sketch of a function without those darn graphing calculators… I actually forbade their use in my classroom and on tests… but then I was further shocked when I heard the A.P. Calculus exam requires these darn graphing calculators… when did they do that? WHY did they do that???

More recently, I’ve been tutoring my significant other in calculus, and have been annoyed with the horrible textbook… 50% of the homework exercises
have a little calculator symbol next to them, indicating that you are supposed to use the graphing calculator for the problem…

Finally… I’ve taught freshman physics at Berkeley, and again, students can’t do basic trigonometry or calculus anymore (that is, without a graphing calculator)… its maddening.
How can you explain a difficult physics concept when the kid is struggling with basic trig?

2. Bill says:

Why does this surprise anyone, the basic concepts of math have gone by the wayside, as the left-wing whackos need to spare kids the concept of “math fear”, and thus, the calculator was born for exactly this. Perhaps students who do the work w/out a calculator can receive extra credit then the ones who use them?

3. Steve LaBonne says:

The reason the students are fooling around with calculators instead of learning calculus, of course, is that the students *can’t* learn calculus because they never mastered algebra.

4. Caffeinated Curmudgeon says:

jab wrote: “…I was further shocked when I heard the A.P. Calculus exam requires these darn graphing calculators… when did they do that? WHY did they do that???”

Amen. In my day, calculus exams only rarely had any numbers to calculate, and those were only to a couple sig figs. What the heck do precise numbers have to do with basic calculus anyhow? If you can’t express a result in terms of coefficients, variables and functions, you don’t understand the problem.

5. Eric Jablow says:

It’s been like this for a long time. In the mid-80s, we gave a calculus midterm examination at SUNY-Stony Brook. We included a problem that required the students to go back to the definition of differentiation and its physical meaning:

We drew the graph of y = x², but we did not state that it was that function; we only stated that it was the graph of an unknown function f(x). We plotted the points (0,0), (1,1), and (2,4) on the graph. We asked the students, “Can f′(2) equal 2? Explain your answer. Do not guess as to the actual form of f(x); if you do guess at what f(x) is, you will receive no credit on the problem. You can assume that the graph is not designed to trick you. There are no microscopic irregularities.”

All throughout the exam, students asked us whether we meant what we said about not guessing the form of f(x), and all throught the exam we told them that they would receive a zero on the problem if they guessed. When we graded the exam, we found that 100 of the 200 students had left the problem blank. Another 95 had guessed that f(x) was x². Only 5 students had done anything substantiative with the problem, and none had analyzed it well enough to deserve full credit.

The solution is simple: the chord from (1,1) to (2,4) has slope 3, and the tangent line at (2,4) is obviously steeper. So, f′(2) > 3, and so f′(2) ≠ 2.

In fact, this was a damning indictment of us. Our students were ill-prepared for this question because we taught them cookbook mathematics from books similar to the ones Vern Williams disdains. We showed that we did not respect our subject. Yes, the students should have been able to follow instructions. However, we should have taught them how to think mathematically.

6. Laura says:

As I’ve stated here before, my daughter was utterly baffled by some of her Algebra II homework before I shoved the (required) graphing calculator aside and showed her how to do the work the old-fashioned way. She’s since shown some other students in her class who were similarly stumped.

The teacher told us at open house that the calculators were great because the students didn’t have to “spend a lot of time” graphing functions, which meant they had time to move on to more advanced concepts. The thing is, graphing the functions is a learning exercise. That’s where you gut out the connection between the equation and the graphical interpretation. If you don’t do that in high school algebra, when are you going to do it?

7. Walter E. Wallis says:

Perhaps we should pay teachers like insurance salesfolk – A stipend up front, then a percentage of the student’s income for life of the student.

8. What struck me most about the article is that the reporter seemed surprised by the results, as if he had just seen a Model T win a car race. I wasn’t surprised since I have seen many other examples of success with similar methods, and I am sure all of you have, too.

I may be wrong, but I believe that Japn, Taiwan and similar places also use “old-fashioned” methods–and also succeed in teaching math.

9. Steve LaBonne says:

Singapore! Anybody can buy their books and see what effective- and cheap!- math textbooks look like.

10. Caffeinated Curmudgeon says:

Eric Jablow wrote: “However, we should have taught them how to think mathematically.”

Laura wrote: “The teacher told us at open house that the calculators were great because the students didn’t have to “spend a lot of time” graphing functions, which meant they had time to move on to more advanced concepts.”

Wow! Thanks for answering my somewhat inchoate question. I don’t think it’s the toys (or tools if you insist) themselves that are the problem. It’s the mistaking the tool for the subject matter.

By analogy, carpentry is not about operating a nail gun or a power saw. Carpentry is knowing how to cut the wood to fit and where to put the nails. A skilled carpenter can produce more and better work with power tools, but without the underlying skill, he’ll only make more sawdust and noise.

With calculators and computers, it’s terribly easy for students, and even some teachers, to forget that distinction.

11. Insufficiently Sensitive says:

Being an old geezer, finishing high school in 1957, I never had the enhanced learning experience of the picture-book math texts.

But we had a legendary math teacher, who took us through solid geometry and trig. No calculus. But a serious treatment of the other two subjects.

I have been an engineer and land surveyor throughout my career. The mathematics we received in high school sufficed for all of it. The many units of calculus and other exotica I took in college were demanding mental games, but the work never demanded them.

Thank you Alpheus Green.

12. Richard Cook says:

I’m not a teacher but, boiled down to my level, teaching math seems more concerned, or is settling for students knowing that 2+2=4 instead of teaching the student why 2+2=4. Parents and teachers that support this are doing such a grave disservice to students its a crime. This nicely dovetails with the entry above on “old math”.

13. This is one of those odd situations were opposing arguments are both right.

Vern Williams is right about the text books. Gosh, they’re terrible. Too many pictures, not enough practice problems.

Math books need to be written by good math teachers, not by a subcommittee of the Human Relations Task Force.

Oh, and they need to be written in two colors. Black and white.

However, contrary to widespread opinion, knowing math facts is not necessary for learning math concepts. Most math teachers can tell you that from experience (not from new age teacher training.) Most parents will not understand.

One pitfall of school choice is that parents will place their children in schools that emphasize memorization over learning. I’ll take drill and kill over the gobbledy-gook promoted by current textbooks, but they’re both off the mark.

Memorization never killed anybody, but when an 8th grader isn’t allowed to take algebra because he can’t do 75 multiplication problems under 60 seconds, well, that’s a crime. A crime of ignorance–but not his.

14. Jim C. says:

Richard Cook: I don’t think that’s the problem. To continue your analogy, the students don’t know that 2+2=4. They need a calculator to find out.

On the other hand, too much memorization isn’t good either. Richard Feynmann described the situation in Brazil in one of his books. The students could recite the law connecting refraction and polarization, but they couldn’t apply it to the real world and were astounded when he showed them a simple demonstration.

15. Wacky Hermit says:

I hate with a passion those busy algebra books. They send the message to students that (A) they are way too stupid to do any actual algebra, (B) that actual algebra consists largely of social studies problems, and (C) that the objective of a textbook (and school by extension) is to keep them entertained. I’ve always maintained that kids are smart, and that they pick up on these kinds of messages, even if only at a subconscious level.

They gave me one of these to teach algebra to 7th graders with. There are about 6 problems in any given section that I would even consider assigning, and they are all easy problems. The textbook encourages “guess-and-check” solutions, even when they take longer than standard solving algorithms. The problems have all been carefully selected for this purpose, since it is unlikely a student would guess an answer like 11/2. They all have nice integer answers. The pages are so busy that I can’t even look at them, they make me dizzy. It looks like the sort of textbook you’d get if math textbooks were written entirely by graphic arts designers.

16. Underwood Dudley, a marvellously entertaining expository math writer, wrote an article about calculus textbooks. One thing he pointed out is that they keep getting longer and longer, presumably because authors think that the reason students aren’t learning calculus is because their textbooks don’t have enough topics, or enough examples, or enough applications…he mentioned something to the effect that if the trend continues at this rate, in around a hundred years, the average calculus textbook will be around 2000 pages. And students STILL won’t get calculus.

From what I’ve seen of middle school textbooks, in a hundred years we can expect a typical one to be a pop-up book illustrated with holograms. And students STILL won’t understand fractions.

17. D. Cooper says:

Ah … Williams indicates that ‘demanding assignments’ and ‘hard work’ are key components in what has worked … now there’s a revelation !!!! All you have to do is get that out of them and you’ve got something … anyone who believes that that’s easy has never been in a classroom working with todays teens and all their distractions (too numerous to mention here but to include MTV, VH1, et. al.). Although not impossible to be sure, it is one of the factors that has become more and more difficult to achieve over the years. I’ve been doing it for 35 years and it doesn’t get any easier. I never give up but I swear there’s a ‘math gene’ .

18. ***Julie*** says:

The problem is worse, I fear. We have high school students who don’t know how to do simple double digit multiplication problems without a calculator!
I believe that technology has it’s place – but elementary school is not the place. We have to start at the root of the problem – get them to use their brains first – understand the concept of math, then show them the short cuts with a calculator. (to top it off, some kids don’t even know how to do long division even with a calculator…which number do they put in first???)

19. jv says:

I’m late to this post but one point that anti-old math advocates are missing: The best teaching would teach concepts and so-called math facts. But in the absence of IDEAL teaching, drill and kill works better. In a real world with real world teachers, the average teacher — given real world salaries and constraints anywhere in the world — will botch the New Math. Drills guarantee some degree of knowledge and force hard work.

In Taiwan, China, Japan, etc, the best teachers do go beyond sheer drill, but the main thrust of the teaching guarantees that all students will learn something. Most of all, teachers in the Far East are under no illusion that students should necessarily see math work as “easy” or “fun”.

It can be, but students need to learn that much of life is dealing with material that could be better taught but which needs to be learned, no matter how dull the experience.

20. J.Mason says:

…required to take a course in “teaching K-7 math” to renew certificate.

…NCTM is all that is being taught… “fuzzy math”, “new-new math” etc. is the garbage being taught to teachers of math. what a crok!

…Bobbs Merrill Algebra (book 1 and book 2) found at College library … Copyright 1927 -1934
These are perfect for 7th -9th Algebra and not a word about a calculator or even slide-rule.

21. Mary Ann says:

Thank you for your math wisdom which gives me the courage to bring a math workbook which I’m developing to fruition. –Mary Ann