### R.I.P. Mathematics Education

From the table of contents of a precalculus text - written by the same three folks who wrote the piece of shit I’m valiantly trying to teach from - that I’m using for supplementary examples:

1. Functions, Graphs, and Models….11.1 Using graphing utilities…2

Page 2! They really cut to the chase, don’t they? The good people at Texas Instruments, along with their shareholders, must be very pleased. Anyway, let’s see what the book has to say about functions, starting on page 1. In full, emphasis added:

THE FUNCTION CONCEPT IS ONE OF THE MOST IMPORTANT ideas in mathematics. The study of either the theory or the applications of mathematics beyond the most elementary level requires a firm understanding of functions and their graphs.

In the first section of this chapter we discuss the techniques involved in using an electronic graphing device such as a graphing calculator or a computer.In the remaining sections, we introduce the important concept of a function, discuss properties of functions and their graphs, and examine specific types of functions. Much of the remainder of this book is concerned with applying the ideas introduced in this chapter to a variety of different types of functions, as is evidenced by the chapter titles following this chapter. Efforts made to understand and use the function concept correctly from the beginning will be rewarded many times in this course and in most future courses that involve mathematics.

So, in other words, we’re going to put the cart *right here*, and leave the horse six time zones behind us. But don’t worry, in Section 1.2 we’ll get to meet the horse. Isn’t that exciting? In particular: note the conspicuous absence of, oh, say, the definition of “function” from the entire introduction of Chapter 1; note also the lack of references to, for example, a single instance of what we might be using functions *for*. All we know so far is, they sure are useful! And we can use computers to study them!

In the sidebar is a list of topics that students are advised to review - yes, I’m laughing too - before delving into the useful, rewarding, and tech-savvy world of functions; but then, we’re done with that intro, and we turn the page to learn that

[t]he use of technology to aid in drawing and analyzing graphs is revolutionizing mathematics education. Your ability to interpret mathematical concepts and to discover patterns of behaviour will be greatly increased as you become proficient with an electronic graphing device.

Makes you wonder what people did before there were graphing calculators. I imagine my parents and grandparents sitting around in caves, clad in fur, with fires burning in front of them, etching misshapen circles in the mud with sticks. The same circles, every goddamned time, *because they couldn’t discover patterns of behaviour* from one circle to the next. And forget parabolas! But then, along came graphing calculators, and God Himself smiled down upon the mathematics classroom.

Honestly, this is delusional. The bulk of my students come to me “proficient with an electronic graphing device”, and their mathematical skills end right there. They don’t use their calculators to help them graph functions; they use them as an excuse to whine that I make them graph functions by hand. They don’t find patterns; why would they? they have *machines* for these sorts of things.

I have yet to hear of a single mathematics educator, save an author of the new edition of a textbook - now compatible with the latest graphing utilities! - whose experience differs from mine.

I am wondering what good solutions there are to the “bad textbook” problem. I guess I have to ask a basic question before I properly understand the problem, though — what is the process for changing a course’s text? In particular, is it politically possible to assemble and use a good course reader instead of a regular textbook?

Well, in my case the process was me going to my department head, ranting about this horrible book, and him nodding and saying, “I didn’t realize the new edition was so bad, I’ll change it for next term.” He had an idea for a text for next term - it’s one that’s motivated by problems and applications. Philosophically, I much prefer the new text, but I know that my students are nowhere near ready for it. Ugh.

I wish we could use a good course reader instead of a regular text, but - in my case, anyway - the course in question is one that no one wants to teach, because it’s such thankless work. So it’s delegated to temporary faculty, like me, who have it for a semester or two until our contracts run out. And I teach four classes, in three different courses, and even if I had the energy to assemble a reader, I don’t have the experience to do so. And around and around we go…

Ugh, at least the calculus text I used a couple of years ago didn’t address graphing calculators until section 1.4, after discussing what functions are, methods of expression, what mathematical models are, and how to transform functions.

High school teachers love the calculator. Love it love it love it. Every time I get a program for a math + math-ed conference, there are many many many sessions on how to use some TI calculator to do something that doesn’t need to be done.

I was lucky, and had high school math who knew math and liked it. But I realized that

elementary schoolmath teachers love them some calculators; a few years ago, on a bus, I overheard some teachers in training speaking reverently about them: “thank god grade sixes get to use calculators in class,” said one, “otherwise I wouldn’t have known what to do with them during my practicum!” And I’m sure a lot of your preservice teachers could relate.Personally, I’m not against calculators, but I think that graphing calculators should be banished from high schools entirely; in university, they should only be used in modeling-heavy applied math classes and the like. But I’m happy with some calculator use. My grad school, after years of an “anything goes” calculator policy, decided to get rid of them entirely the second time I taught there. This was fine for the first two months of class, but then we got into exponential functions and students had to leave their answers in forms such as “therefore the bacteria population after five hours is 1000*e^1.3152.” Which all but undermined my “here’s how to check if your answer is reasonable” lessons.

I guess your “how to check if your answer is reasonable” could get a “how to estimate” lesson added to it. Learning how to estimate is useful. Saves you money when you notice errors in bills, too.

High school teachers (I’m one) do not all love calculators. Just like you college/grad teachers complain that students can’t graph because of years of using graphing calculators as a crutch, high school teachers complain their students can’t add and multiply simple problems in their head. You guys live in a relative paradise compared to my classes where some students have to use a calculator to answer 8 * 1 = ?

The people who love calculators are in the progressive educational establishment. Calculators are meant to free students from the tyranny of drill and kill. The NCTM loves calculators, so the textbooks are written that way and conferences are filled with calculator based seminars.

But I’m no fan of this anti-calculator hysteria either. I, for one, am glad that I never again have to use a trig table to find the sine of 40 degrees. I am also glad that I don’t have to use a table or slide rule to figure out my logrithms.

The same really goes for graphing. Once I have graphed a bunch of parabolas, by making tables first and then by translating the vertex, do I really have to spend the rest of my life graphing those buggers by hand? For scatter plots, I want my students to do their first dozen by hand. But for the rest of their lives, I want them to use a spreadsheet program so they can get accurate results from real data.

I think there is a general process for learning math and calculators that applies from grade school multiplication through college calculus:

- Learn by hand, with no caculator.

- Practice by hand until mastery is demonstrated.

- Somewhere during the practice stage, introduce calculators as a means to check your work for accuracy.

- Once mastery is demonsrated, use caculators freely.

No paradise here; these students you mention, what do you think happens to them when they get out of high school? They go to college.

My students use calculators to multiply by 1, too. A few weeks ago in class a hand went up when I added 1 + 1/4 and got 5/4. The student had no idea how I had done the calculation.

Yeah, I’ll have more to say about this later, when I don’t have one hour to set the review sheet that I didn’t set yesterday because I was so pissed off about this text. In the meantime - I mostly agree with you, CL, except that I too live quite far from paridise. I wrote some time ago about one student of mine (who ended up dropping the course), who was unable to give me a decimal approximation for (6/26)^10 (they were allowed to use calculators). On her test she’d written, “How do I do this? My calculator doesn’t have a fraction button.”

Which is at least as bad as using a calculator to find 8*1: this student had been weaned on fancy calculators, and had not even learned that the fraction 6/26 meant 6 divided by 26.

I’ve been writing software for math education for twenty years and discussions like this are always an eye-opener for me. Clearly, technology is of no use and only worsens the sorts of problems MS describes so vividly. But I have to ask: are there tools you would like to see to help your teaching?

I grew up with typewriters, and word processors are a clear win. They don’t magically endow people with logical thinking or clear expression, but they do simplify the process of writing. I have yet to see tools that provide a comparable improvement to the process of doing math. Is that a pipe dream?

Ron….I’m not a teacher, but it seems to me that software could help by making the concepts “come alive.” For instance, to understand what a function is, put a number in the function machine and get another number out. Play with the limits on differentiation and see what happens. But I think such software would have to be done very, very carefully to do more good than harm.

The problem with the strategy of having the students prove mastery and then letting them use calculators is that once they begin with the calculators, so many become crippled. In my honors physics class at the University of Chicago, a class which requires a satisfactory score on the Calculus and Physics APs, we had a quiz today in which it was necessary to multiply 256 and 42. Much as my father would be shamed to hear this, I didn’t do this in my head, but quickly jotted down the arithmetic on the side of my sheet. Walking back from class, one of my friends, who is certainly anything but math illiterate, said “Boy, the one quiz I could’ve used my calculator, I leave it at home. I just left it all unsimplified, you think that will be okay?” If above-average math students - which this fellow definitely is; he’s far more able than I am - are incapable of recovering from calculators, I doubt anyone else will be.

For another example, I had a four hour lab yesterday that I barely managed to finish within the allotted time, and several people went to five hours. Glancing over at their workstations, every time I finished plotting one set of data on my graph paper and began the next part of the experiment, they were still working with their fancy computer graphing programs. And this to graph a linear relationship of ten data points.

Ron: Packages like Matlab or Mathemagica are the equivalent of word processors. To use them, a student must know what she is doing, but if she knows what she would like to do, they grant enormous power. I believe calculators fall into this description as well. They are not learning aids, they are power tools.

When I was a little kid, I had a small electronic toy which quizzed me with simple arithmetic problems. It asked me to be the calculator, and that made it a learning aid. If you were TI and you really wanted to make learning aids, you would allow your calculators to run in reverse: they would ask questions to students.

So, MS, if you want to get in on the best scam going, we can start making tutorial modules for the evil beasts to allow them to actually be used as educational toys. We could flog them at teaching conferences, crank out matching textbooks and tutorial manuals, get kickbacks from TI for endorsing their latest and greatest hunk of junk…

Regarding the word processor analogy, Clippy does not say “It looks like you’re writing a Macbeth paper! Do you wish to use the Shakespeare wizard?… Choose length… choose thesis… choose writing style…” This is the sort of effect calculators have on much of math education.

There’s no reason to believe calculator programs that quiz their users would be significantly more effective than computer programs which do the same.

And even if Clippy did ask you all those questions about your essay, it wouldn’t be able to write the essay for you. But, many calculators can do an entire math problem for you, at least for common types of problems which have a direct answer (as opposed to a proof, etc.). Then, when a slightly more complicated (or different) kind of problem occurs that cannot be plugged directly into a calculator, students do not know what to do. That’s one reason why math teachers complain about calculators a lot more than English teachers complain about word processors.

DoJ: I doubt that any but the keenest of students would actually *use* their calculator as a tutorial aid. That said, I used to fiddle with the math toy when I was bored, and I really doubt I would have done that had it been a computer program. While totally invalid, writing tutorial programs for TIs is less invalid than pushing TIs themselves. Does anyone actually *use* a graphing calculator for serious number crunching? Hell no. A PC loaded with any of Matlab (or Scilab or Octave), Maple, Mathematica, or even S (or R), SAS, SPSS, or Excel makes a graphing calculator look like the flimsy plastic toy that it is. The only market for these things is the educational aid market, where they’re passed off as cheap replacements for actual mathematical power tools. If we’re going to be stuck with them, shouldn’t they at least *look* like they can help students learn math?

What Jordan said, regarding students not using their calculators as tutorial aids. In fact, since we do use calculators in my class, I make a point of teaching estimation and the general “how to tell if your answer is reasonable” techniques, all of which lie firmly outside the territory of rocket science. Nevertheless, many of my students are so unaccustomed to questioning the output of their magical $150 calculators that they just accept whatever is on the screen. Nothing I say or do seems to have any effect on this. I’ll dock more marks for students who give wrong answers and don’t check their work, than I will for students who give wrong answers, check their work, and remark “hmm, must have done something wrong.” I have probably had one precalculus student (out of 100 or so) in the last two terms (and I could tell you her name) who would use her calculator as a tutorial aid. The idea that students are plugging functions into their calculators in order to find patterns, as that text I was using suggests, is a fantasy. My class spent two weeks graphing parabolas by transformations. After giving them a dozens of practice questions where they would, say, find what transformations they would apply to the graph of y=x^2 in order to obtain the graph of y=-2(x+1)^2+5, the preferred method of choice for graphing parabolas remains the point-by-point one. (Which is why I am sneaky and have them graph parabolas containing fractions: that way they can’t get the intercepts just by plugging in the integer points.)

To expend on Carl Larson and Caddie’s points about students demonstrating mastery before going onto the calculators: I definitely agree that mastery should precede use of technology (with certain exceptions, which I’ll get to), but I’m 100% in agreement about students becoming crippled once they use the caclulators. It’s important to make sure that their pre-calculator skills don’t stagnate, as many of those skills form the

foundationfor skills they’ll be using later. For instance, it’s been years since many of my precalc students have done long division. Guess how many of them struggled mightily with long division of polynomials?Ron Avitzur, and others: intro college classes I teach, the one that lends itself most readily to use of technology is the statistics course. A few months ago, I gave a vague explanation of why we’d want a formula that looked a bit like the one for computing standard deviations, but I didn’t give a step-by-step development of it. And when we got to covariance, I just threw the formula at them and didn’t even bother to explain where it came from. (I did, of course, explain that it had properties that were useful to us.) They didn’t care about the development of the formulas, and nor do they need to: they’re social science students who will need to be able to analyze quantitative data, but few have any use for the technicalities. So I’m glad that there are calculators (and ones far cheaper than the blasted TI-83+) that allow them to do those computations quickly.

At higher levels, there’s some great geometry software out there, and I’ve found it helpful in testing conjectures. Some geometric structures get pretty unwieldy, and it’s useful to be able to move around and delete points and lines at will.

This discussion reminds me of an accident report I read several years ago in an aviation magazine. A commuter airliner was conducting an instrument approach in low cloud conditions. They were using the Flight Director, an analog computer that recommends flight paths based on data from various instruments. The Captain observed that the FD seemed to be acting erratically, and the flight recorder transcript ends as follows:

CAPT: “I don’t trust that thing. Let’s go to raw data.”

(Sound of impact)

…in this case, the decision was made just a few seconds too late.

But when people learn total reliance on calculators and their programs,like so many of the students mentioned in this discussion, I doubt if they will ever develop a sense of when to say “I don’t trust that thing. Let’s go to raw data.” In some cases, the consequences could be a lot worse than even a commuter airline crash.

Jordan, your scam is already well in place: it is the foundation of the business model for Maple TA and various other species of software occupying this niche of the software world, and all indications are that it is a highly lucrative market.

I think that saying a calculator is to math : word-processor is to essay-writing is not quite accurate. I would say a better comparaison is a calculator is to math : word-processor is to handwriting (and a calculator is to

proofs; : word-processor is to essay-writing). People who type everything let their handwriting stagnate, and collectively, our society is getting worse at having clear, legible writing. Because who needs to be able to print and write if you can just do it on a computer? That is the role a calculator plays.One of the wacky things I appreciated most from highschool was the calculus teacher who made us do the whole calculus course without using a calculator. I was waaay ahead of my peers when I got to college.

Related: in the calculator/word-processer/proof-writer analogy, where does the novel-writing software fit in?

Are you really telling me that you don’t use a calculator when you do your taxes. Or that you wish the bridge you drove over had it’s structural integrety checked by hand rather than a computer. Or the airplane you flew in, or car you drove, didn’t have computers running a lot of their systems. You can always point out the place where computers fail and cause problems but it is difficult to count the number of times computers work a whole lot better than humans, and where the human would have made a fatal mistake.

The problem is not the use of calculators. I do not think the problem that someone does not know that 8*1=8 is caused from overuse of calculators. Perhaps it is because, as John Paulos conjectures in Inumeracy, that we spend years teaching kids how to do arithmentic, until they consider all mathematics boring, and since they have been forced not to think in mathematics class, they cary this on into higher mathematics. I had to learn how to do long division, I also know how to take square roots by hand, what exact use is that information in solving an algebra or calculus problem?

Are you really telling me…Why, yes, that’s exactly what I’m saying. My rant about a college-level

mathtextbook thatbeginswith the technology rather than with the content presumably that calls for it was just my way of expressing my wish that engineers didn’t use computers.I’d like to see some indication that 1) learning how to compute and learning how to think are mutually exclusive, or 2) that abandoning the basics somehow “makes room” for students to get excited about interesting math and learn to think analytically. To the contrary: students who do not get sufficient practice working through an algorithm (such as division) by hand are unlikely to come up with their own (a common justification for abandoning the basics - “if we don’t teach long division, we give students an opportunity to come up with their own methods!”) Specifically, regarding your last question: I can tell you that learning how to add fractions by hand is

essentialto solving rational equations, which come up in many a calculus problem.