On Teaching Math

Periodically some social scientist notices that math is abstract and difficult. Thinking that math educators have overlooked this fact, he breathlessly reports his findings as a great discovery he has made.

The latest example is Andrew Hacker, a political scientist at Queen's College. In a new book, The Math Myth and Other STEM Delusions, he presumes to lecture us all on the proper way of teaching math. It turns out that what people really need is “quantitative literacy,” as opposed to a lot of abstract argle bargle about logarithms and trigonometry. Gosh, who knew? (“STEM,” incidentally, is an acronym for Science, Technology, Engineering, and Mathematics.)

The New York Times has published an op-ed by Hacker based on the book. Here's a sample:

HERE'S an apparent paradox: Most Americans have taken high school mathematics, including geometry and algebra, yet a national survey found that 82 percent of adults could not compute the cost of a carpet when told its dimensions and square-yard price. The Organization for Economic Cooperation and Development recently tested adults in 24 countries on basic “numeracy” skills. Typical questions involved odometer readings and produce sell-by tags. The United States ended an embarrassing 22nd, behind Estonia and Cyprus. We should be doing better. Is more mathematics the answer?

In fact, what’s needed is a different kind of proficiency, one that is hardly taught at all. The Mathematical Association of America calls it “quantitative literacy.” I prefer the O.E.C.D.'s “numeracy,” suggesting an affinity with reading and writing.

I'm not sure why Estonia and Cyprus are being singled out, and I flatly don't believe that 82 percent figure. But I'm certainly all in favor of quantitative literacy. What does he have in mind?

Calculus and higher math have a place, of course, but it's not in most people’s everyday lives. What citizens do need is to be comfortable reading graphs and charts and adept at calculating simple figures in their heads. Ours has become a quantitative century, and we must master its language. Decimals and ratios are now as crucial as nouns and verbs.

It sounds simple but it's not easy. I teach these skills in an undergraduate class I call Numeracy 101, for which the only prerequisite is middle school arithmetic. Even so, students tell me they find the assignments as demanding as rational exponents and linear inequalities.

It may not be easy, but it's certainly very basic. So basic, in fact, that most of us learned this stuff in elementary school. I remember spending a lot of time as a schoolkid studying graphs and charts, discussing, say, the relative merits of bar graphs, line graphs and pie charts, and discussing ways graphs can be misleading. These topics all figured prominently on the standardized tests we were required to take at the end of the year. And my math teachers all through elementary and middle school emphasized the importance of good estimation skills and order of magnitude calculations. If these things are no longer being taught then Hacker has identified an important lacuna in elementary education.

I suspect, though, that these topics are still routinely taught. And this leads to me one of the great errors in Hacker's argument. He notices that students lack a particular skill, and he thinks the solution is to design a class in which that skill is taught. This is misguided.

Take Hacker's lead example. If it is true that large percentages of adults cannot carry out a simple computation about carpet prices, it is not because they never learned how to do it in school. Elementary school math involves little else but calculations of that sort. The problem is that if you learn something in a class and then do not use it for a while, you quickly forget it. If school is just a matter of passively receiving information and then regurgitating it on tests, then the most cleverly designed syllabus in the world is not going to make any difference.

School is supposed to be the beginning of your education, not the end of it. The way it's supposed to work is that after school you go home to parents who take an active interest in your education, and who provide a stimulating environment that reinforces what you learned there. And in many school districts, that is precisely what happens. Where it doesn't happen, though, poor curricular choices are pretty far down on the list of problems. Far more important is the general abandonment of public schools over the last two decades, and numerous other social pathologies that are not going to be fixed by tinkering with the curriculum. The real problems in education these days have little to do with anything the schools are doing, and much to do with the social environment in which those schools are forced to exist.

There is plenty else wrong with Hacker's op-ed. His book has recently been the subject of a fawning review in Slate, so we might have to look at that too. But in the interests of keeping things to a reasonable length, we will save that for later in the week. Looks like I'll have plenty of blog fodder for a while!

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I've heard that on standardized IQ tests, half the people score below average ;-)

If you don't use it, you lose it.

sean s.

By sean samis (not verified) on 07 Mar 2016 #permalink

He notices that students lack a particular skill, and he thinks the solution is to design a class in which that skill is taught. This is misguided.

That is, however, the idea behind the "argument" many are making about how education should be reformed: education should be geared toward teaching only the things business (and some pundits) believe useful: everything else should be cut away in the name of cost saving.

You should see what they're doing to the music curricula in high school and college. There's a huge contingent saying that music literacy is old fashioned and unnecessary, and excludes some people from a subject that is supposed to be universally enjoyed. For some that includes even the ability to read music (that old parlor trick).

By Another Matt (not verified) on 07 Mar 2016 #permalink

I think the attention the Hacker book is getting is a nightmare, and I agree that a large part of the real problem is the "social environment in which [the] schools are forced to exist." But I don't think I can fully agree that "the real problems in education these days have little to do with anything the schools are doing." You might argue that this is really the social environment, but when geometry classes replace basic introduction to proof skills with playing around with a computer geometry system (which seems to have been the case for most of my current college students), there is something wrong with the curriculum. The thing that Hacker gets wrong is that abstract thinking ability is precisely what the students need MORE than rote numerical manipulation. I've noticed that calculus students (and often even upper division students) just have no conception of what the "If...then..." construction means, and little to no ability to reason logically. I'm seriously considering spending the first week or two of my next calculus class on logic problems and reading comprehension. We need more abstract reasoning, not less, and I think the current lack of such skill development likely is a curricular problem.

By Greg Friedman (not verified) on 07 Mar 2016 #permalink


I agree. I've found throughout my life that the greatest benefit I've derived from my mathematics education ( pure math at Penn State in the late '70s ) is that it taught me how to think logically and critically and to be able to follow and construct log chains of reasoning. The final for my algebraic topology class was a take home with just 3 proofs. It took all of the whole week we were given to complete.

My first introduction to this was my high school geometry class which did consist of nothing but proving theorems. This is still an important skill. Having a computer to manipulate objects would have been fun and would have provided insight and motivation but it certainly shouldn't replace the boring old theorem-proof method. Learning is sometimes hard - but valuable. No wonder students don't want to struggle through a difficult philosophy book or dense piece of literature. They've never learned the skill or developed the patience.

"Calculus is so Last Century" is an opinion piece by Li & Bishop in last weekend's Wall Street Journal. They argue that, except for physics and engineering, no one uses calculus on the job. Instead people should focus more on multivariate analysis, probability and linear algebra to analyze "large data sets."

I largely agree with them, but think at least first year calculus is also needed. An organic chemist told me he uses no calculus personally, but much of his lab equipment is programmed with advanced mathematical algorithms. He personally has to do a lot of data analysis on the results from the machines. He mentioned Fourier Transforms as a procedure used in the equipment.

I am especially interested in other opinions on this study.

Personally I think Hacker is wrong.

I find many of my students doing a complete mind-wipe at the end of each quarter - so that nothing is retained. I find college seniors in biology struggling with graphs mainly because that are trying to memorize rather than understand general principles. It is as if induction doesn't exist.

By Michael Fugate (not verified) on 08 Mar 2016 #permalink

I think the emphasis on proofs, versus problem posing/solving skills may be one of the problems. Only a small fraction of people find proving theorems to be stimulating, and they end up simply trying to survive then forget whatever math they are forced to take. So we end up with a majority of innumerate math-a-phobics. What is a good background for a professional mathematician isn't going to be a good background for the other 90plus percent.

By Omega Centauri (not verified) on 08 Mar 2016 #permalink

You guys are all being too kind to the Hacker.

Pretty ironic name given the following comment he made in an interview, which IMHO qualifies as one of the dumbest I have ever seen

"Coding is not based on mathematics ... Most people who do coding, programming, software design, don't do any mathematics at all."

Huh?? What universe is this guy living in? As someone who "codes" for a living, I use math all the time. Don't know where he is getting his ideas from. A "hacker" he clearly isn't!

Keith Devlin blog (http://devlinsangle.blogspot.com/) has another prize quote from his book

"One of them is pi, whose 3.14159 goes on indefinitely, at least as far as we know.”

Apparently, he isn't aware that pi was proved to be irrational some 250 years ago.

Sounds like he is pretty clueless about math. Anyone who is this clueless has no business writing a book about math - let alone formulating curriculum.

- RM

By Raghu Mani (not verified) on 08 Mar 2016 #permalink

Raghu Mani wrote:

"As someone who “codes” for a living, I use math all the time. "

Well, so do I, but I do know many professional coders who have almost no math skills. Very many business applications are mainly database retrieval and storage tools, along with some UI stuff.

By Greg Esres (not verified) on 08 Mar 2016 #permalink

@Greg Esres. Even those coders who don't officially use "math" still need to be able to determine what their code is going to do, and that takes logical reasoning skills, i.e. math.

And the problem with saying that "In an ideal world, people would confront problems they have a stake in solving and then be taught the appropriate skills" is that real life problems are hard and in order to tackle them you need to be proficient at more basic skills which aren't always fascinating in their own right. It's also pretty hard to come up with skills that third graders really need. They don't really need to read either in order to hold down a job or get to school, but no one would argue that teaching first the alphabet (not intrinsically very interesting) and then basic reading and basic grammar and so on isn't an important thing to do. Yes, perhaps one can by without abstract reasoning just as some people get by being illiterate, but what sort of world are we creating that way?

By Greg Friedman (not verified) on 08 Mar 2016 #permalink

In reply to by Greg Esres (not verified)

Michael Fugate wrote:

" find many of my students doing a complete mind-wipe at the end of each quarter"

This. Those who think that adult skills depend on high school or college curricula are deluded to think that their own particular discipline can save civilization. Most people cannot transfer what they learn in school to real life. Most jobs don't require complex chains of reasoning and critical thinking skills make you unpopular in business meetings.

Part of the problem, I think, is that people are taught math when they don't have a particular need for those skills. The word problems presented aren't actually problems they wish to solve. In an ideal world, people would confront problems they have a stake in solving and then be taught the appropriate skills.

By Greg Esres (not verified) on 08 Mar 2016 #permalink

Greg Friedman wrote:

"and that takes logical reasoning skills, i.e. math. "

No. Logical reasoning and math skills aren't the same. Math requires logical reasoning skills, but logical reasoning can be done without math.

By Greg Esres (not verified) on 08 Mar 2016 #permalink

Greg Friedman wrote:

"but no one would argue that teaching first the alphabet (not intrinsically very interesting) and then basic reading and basic grammar "

Children are confronted every day with stuff they can't read, so they are pretty motivated to learn. Not being able to read is crippling. Math, not so much.

By Greg Esres (not verified) on 08 Mar 2016 #permalink

I think he’s got it exactly backward. If a person has a need to do the same mathematical operation over and over again without error, that is what computers and apps are for. That’s why cashiers have cash registers. Human thinkers are valuable precisely because they have a flexible and deep skill set that allows them to solve problems which they don’t regularly encounter. Education is about developing skills you can use in unusual circumstances just as much (if not more) than it is about skills you use in normal circumstances. Limiting math education to ‘everyday’ applications is sort of like teaching a barber how to give everyone a buzz cut. Or teaching prospective chefs how to follow a recipe. True human innovation and analysis – and the sort of white collar jobs we really want to train people to be able to achieve, be that lawyer, professor, scientist, engineer, stock broker, etc. – requires much more than the mere consistent application of well known rote processes.

Even those coders who don’t officially use “math” still need to be able to determine what their code is going to do, and that takes logical reasoning skills, i.e. math.

Indeed. I once volunteered to write problem-set solutions for an algorithms class that my boss was TA'ing for a terminal master's program in computer science. Her reports were that the students were already struggling mightily by big-O.