Americans and Math

From the current issue of The New York Times Magazine:

One of the most vivid arithmetic failings displayed by Americans occurred in the early 1980s, when the A&W restaurant chain released a new hamburger to rival the McDonald’s Quarter Pounder. With a third-pound of beef, the A&W burger had more meat than the Quarter Pounder; in taste tests, customers preferred A&W’s burger. And it was less expensive. A lavish A&W television and radio marketing campaign cited these benefits. Yet instead of leaping at the great value, customers snubbed it.

Only when the company held customer focus groups did it become clear why. The Third Pounder presented the American public with a test in fractions. And we failed. Misunderstanding the value of one-third, customers believed they were being overcharged. Why, they asked the researchers, should they pay the same amount for a third of a pound of meat as they did for a quarter-pound of meat at McDonald’s. The “4” in “1/4,” larger than the “3” in “1/3,” led them astray.

Heh. It reminds of something I read in John Allen Paulos’s book Innumeracy. He described watching television with a group of people when a commercial came on. The ad claimed that its prices were a fraction of the other guy’s prices. Paulos quipped that the fraction was probably 4/3, and was mostly met with blank stares from the other folks in the room.

That’s the high point of the article, though. The rest is just the standard clichés of education reporting. American math education is terrible because it focuses too much on rote learning. We should be emphasizing reasoning and problem solving and blah blah blah. These two excerpts will set the mood:

Instead of having students memorize and then practice endless lists of equations — which Takahashi remembered from his own days in school — Matsuyama taught his college students to encourage passionate discussions among children so they would come to uncover math’s procedures, properties and proofs for themselves. One day, for example, the young students would derive the formula for finding the area of a rectangle; the next, they would use what they learned to do the same for parallelograms. Taught this new way, math itself seemed transformed. It was not dull misery but challenging, stimulating and even fun.

And later:

As soon as he arrived, he started spending his days off visiting American schools. One of the first math classes he observed gave him such a jolt that he assumed there must have been some kind of mistake. The class looked exactly like his own memories of school. “I thought, Well, that’s only this class,” Takahashi said. But the next class looked like the first, and so did the next and the one after that. The Americans might have invented the world’s best methods for teaching math to children, but it was difficult to find anyone actually using them.

It’s not so much that the main claims in the article are wrong. It’s just that this article gets written over and over again, each time presented as though the reporter has made some great discovery. Yes, if you present math to students as strictly a matter of rote memorization then, in addition to driving them mad with boredom, you are going to leave them with little ability to apply what they have learned to practical situations. After all, life does not present you with clear-cut problems in the manner of textbook exercises. This is news?

There are all sorts of reasons for why it is so difficult to change the habits of math teachers. Here is one:

Consequently, the most powerful influence on teachers is the one most beyond our control. The sociologist Dan Lortie calls the phenomenon the apprenticeship of observation. Teachers learn to teach primarily by recalling their memories of having been taught, an average of 13,000 hours of instruction over a typical childhood. The apprenticeship of observation exacerbates what the education scholar Suzanne Wilson calls education reform’s double bind. The very people who embody the problem — teachers — are also the ones charged with solving it.

I see this every semester in my own classes, especially the ones that are directed toward future teachers. Students are happy to memorize formulas, and they will apply those formulas with machine-like precision at test-time. But if you ask them to prove anything, or to explain why something is true, they feel betrayed. They will regard you as unfair for asking such a thing.

Shortly after you present a new concept to students, you can be certain they will ask, “How will you ask us about this on a test?” They are not amused by the correct answer, which is that they should focus on understanding the ideas, because if they do that they will be able to answer any question I ask. When it becomes clear that you are serious, they often become visibly nervous, even scared.

It’s very hard to break that mindset in a semester, or even several semesters. Inevitably, even at the college level, they can get pretty far just by memorizing formulas. A typical test will have a mix of computational questions and proof-oriented questions. The better students will clean up on the computational questions and, through the miracle of partial credit, generous grading, and steep curves, will pick up enough points on the proofs to get a decent grade.

For all my snark, the article is worth a look. It’s very supportive of the Common Core math standards, which I do not really know much about. I do think, though, that there are two big elephants the article ignores. One is class size, and of school resources more generally. When you have, say, fewer than twenty students in a classroom, you have the freedom to be very interactive and to get everyone involved in the discussion. Once class sizes get above thirty, and when the classroom is physically very crowded and unpleasant, your pedagogical options are far more limited.

The other problem is that, in this country, teaching is not a respected profession. Frankly, teachers are usually treated with contempt. People who wouldn’t last five minutes in a classroom protest that teachers have it easy, since they only work ten months out of the year. In most districts they are poorly paid and not given adequate supplies for their classroom. The only reason they get the crumbs that they do is that the unions fight tooth and nail to get them, and that is why teachers unions are uniformly reviled by the media and politicians. Meanwhile, there has been a wholesale abandonment of the public schools in this country, and for once it’s not just the Republicans who are responsible for it. In that environment, should we be surprised that talented people with options rarely decide to go into teaching?

In that regard, permit me a personal anecdote. My mother was a public school teacher, spending much of her career in high school special ed. Of course, when she was in college, teacher, nurse or secretary were pretty much the only options for women. Early in my junior year of college, when I was starting to think seriously about what to do after graduation, I suggested to her that I might be interested in teaching high school math. She pitched a fit. Off the top of her head she tossed off about a dozen reasons why that was a bad idea. She was so convincing that I decided that graduate school was worth another look.

I’ll close with my own favorite example of innumeracy. In 2000, in my first semester at Kansas State University, I taught a course for future elementary school teachers. Manipulating percentages was something we spent a lot of time on, and since 2000 was a Presidential election year I decided to put an election-themed question on the test. I pointed out that, at that time, Kansas accounted for six electoral votes out of 535. I then went to the census bureau and got the population figures for both Kansas and the United States generally. The problem was then to work out the percentage of the electoral college represented by Kansas as compared to the percentage of the population that lived in Kansas.

Well, it turned out that .97 percent of the population lived in Kansas. Several people, alas, made a decimal point error and arrived at an answer of 9.7 percent. Now, anyone can make a decimal point error. No shame in that. I pointed out, though, that they could have caught the error had they realized the implausibility of suggesting that one in ten Americans lived in Kansas. I noted that around three million people lived in Kansas, as compared to eight million who lived in New York City. They just stared at me blankly. It had never occurred to them that the answer should actually have some connection to reality.

Comments

  1. #1 Shecky R
    July 25, 2014

    Great piece, touching on some oft-overlooked or under-appreciated points.

  2. #2 Pete A
    UK
    July 25, 2014

    Thanks for this fascinating article, Jason. I’m really surprised that I was completely unaware of the points you’ve raised. During my secondary school and college years I was taught maths and science from first principles. Each exam contained at least one question asking for a proof or a derivation of something we’d learnt.

    That was a long time ago: there were no electronic calculators so we had to use a slide rule or books of mathematical tables; and getting the decimal point in the wrong place was totally unacceptable.

  3. #3 John
    July 25, 2014

    Use decimals not fractions to measure meat and other advertising.

  4. #4 L.Long
    July 25, 2014

    When teaching electrical theory in the NEC(National Electricians Code) and TechCollege, the biggest problem was teaching fractions and algebra and for the same reasons…they where not taught them in school but made to memorize the procedures and not the actual reasons for it to work.
    I had the same problems with calculus, I just did not get the 1st year, so I start over but this time the instructor went thru the effort of showing how and why it worked and how it related to reality. Awesome! since then when I teach I try to explain the how and why and relate the efforts to real life examples that the students know about.
    I’ve seen teachers that use elections, baseball, and money to keep the students interests high.
    Good Post!

  5. #5 Greg Esres
    July 25, 2014

    “It had never occurred to them that the answer should actually have some connection to reality.”

    And that’s the problem with the way math is taught in general..it’s disconnected from any problems in real life that people care about.

    Word problems are an attempt to make those connections, but those scenarios are so obviously contrived that they are typically resented by the students.

  6. #6 MNb
    July 25, 2014

    “It’s very hard to break that mindset in a semester, or even several semesters.”
    I begin with this when my pupils are 14 or 15; third grade secondary school. That’s to say, the kids who have chosen to study maths and physics. In fact it’s already too late. I recognize the resistance you’re talking about. “Sir, why don’t you give us the answer?” Before every test I provide a summary; my pupils very quickly learn that the shorter the summary is the harder the test, in terms of independent thinking. Two years later, when they continue their study at another school (Surinamese education system is a bit complicated) they almost all are grateful.

    “teachers are usually treated with contempt”
    Fortunately that situation is much better in Suriname.

    “It had never occurred to them that the answer should actually have some connection to reality.”
    That’s a very common problem; but also a somewhat unhealthy respect for numbers might play a role: “this is the outcome of the calculation, it must be right”.

  7. #7 Kevin
    July 25, 2014

    The student’s on/off switch “Will this be on the test?”

    Jason, you may find this blog of some interest. It is from a assoc prof in the mid-west..

    http://www.ginandtacos.com/

  8. #8 Kevin
    July 25, 2014

    intro from his previous blog post

    “Higher Ed administrators are, bluntly, the worst people on Earth. Take everything obnoxious about the MBA and law school types, give them no relevant skills other than self-promotion, pay them exorbitantly, and give them jobs consisting mostly of filling their own time with endless Meetings and Committees. It is a high six-figure “Dig hole, fill hole” job, yet by and large they seem to think they are brilliant and important because, hey, they don’t merely work at a university, they’re in charge of it. “

  9. #9 PedrBran
    July 25, 2014

    1/2 of humanity is below averge. :-)

  10. #10 rigadoon
    July 25, 2014

    When I worked for the Defense Dept. as a mathematician, several colleagues were former high school math teachers. It was the discipline problems that drove them away the most.

  11. #11 Richard Johnson
    United States
    July 26, 2014

    Well, this is my first visit to your blog! We are a group of volunteers and starting a new initiative in a community in the same niche. More biotechnology related material exists at http://www.nanowerk.com/ This site is also more resourceful with science, nanotechnology, biotechnology, space and astronomy related latest inventions and news.

  12. #12 Gerald
    United States
    July 27, 2014

    Another good example of innumeracy is the “Verizon Math” incident, in which a data subscriber was told that his $0.002-per-kilobyte plan would cost “point zero zero two cents”. (That’s actually “point zero zero two dollars.)

    After the customer had been charged exactly 100 times what he expected, multiple customer service agents continued to use the wrong wording when he talked to them. Admittedly, it can be very tricky to keep track of distinctions like that, but the frustrating part was the continued unwillingness to try and understand their own mistake.

    Some of their language in the transcribed conversations implied the deeper assumptions (as I see it): that at sufficiently tiny values, dollars and cents are somehow the same, or that $0.002 just “isn’t dollars anymore”. (As if cents and dollars were qualitatively different, not just quantitatively.) In a way, this intuition makes sense; a better math educaiton (if that’s possible) might have replaced that with better intuitions.

  13. #13 Kathleen Dowling
    Colorado
    July 27, 2014

    What America needs is for this man to write an instruction book that the common man can understand as a tool to teach their children math at home. The union controlled public school system will never do anything to fix this. Without the help of people such as this man America is doomed.

  14. #14 tomh
    July 27, 2014

    @ #13
    Seriously? You think unions are the reason for inadequate math instruction in schools?

  15. #15 proximity1
    July 28, 2014

    @ 9:

    Quiz: Please indicate the best response to the following–

    ” In a given sample, the ‘mean value’ and the ‘average value’ are

    a) always equal
    b) sometimes equal
    c) never equal
    d) none of thge above

  16. #16 proximity1
    July 28, 2014

    “Shortly after you present a new concept to students, you can be certain they will ask, “How will you ask us about this on a test?” They are not amused by the correct answer, which is that they should focus on understanding the ideas, because if they do that they will be able to answer any question I ask. When it becomes clear that you are serious, they often become visibly nervous, even scared.”

    Is this any wonder? I’m terrible at math and so I’ve examined dozens of math texts in hopes of finding clear, understandable presentations of varied topics. In each case, the author(s) present methods and examples and exercises which, to him, her or to them, are clear and well-ordered. And invariably, at some point between the more elementary material and the more advanced, I arrive at a point where my understanding does something akin to falling off a cliff. In many, many cases, the author presents simply the abbreviated formal mathematical notation using variables, constants, and all sorts of symbolic signs and operators which are devoid of any actual practical applied example and never done in a narrative which develops in a conversational way what these represent and how they are related–

    as in

    We read this expression in the following way …

    where… the symbol delta, which here represents or stands for… and which has a value derived by … etc.

    such a presentation is, while not unknown, extraordinarily rare.

    Imagine if medicine were practised toward / administered to individual patients the way math education is presented to students. The doctor (teacher) is before a patient (student) who is invariably somewhat like and unlike other patients (students) in any number of ways. The doctor is obliged to focus on the peculiaritites of his or her patient, without which he or she isn’t going to be able to advance in his appreciation of the patient’s circumstances. The math teachers’ respobsibilities are to “teach this material” to “these (extemely varied )individuals” within “this amount of time”– successfully.

    To do this, the math instructor, unless somehow possessed of almost miraculous talents and amounts of time, is going to approach the students and the presentation of the material in some routine practiced way–generally a repeat of methods and examples and techiques which have proven more or less successful for the average student over the years.

    If those happen to coincide with what any given student finds clear and understandable, that’s all very well. Otherwise, woe unto the student who finds that the presentation isn’t clear and understandable and unless he or she is lucky enough to have the time and resources to find some other alternative approach by which he or she can master the material within the alloted time, then the student’s road to further progress in math education stalls (and, for many, ends) there.

    No wonder students are scared. Never having pursued a science degree in the classroom, it came as quite a shock to me to learn that the degree of actual practical mastery of intermediate and advanced concepts in science and engineering can vary greatly in both the teaching and the applied professions. I’d naturally assumed that everyone with a comparable degree and level of training and experience had more or less the same real mastery of the concepts employed. I eventually learned–from reading–that there are competent working professionals who have in some particular arera or other only what can be fairly called a rudimentary understanding of the concepts and their applications. Usually, that is enough for them to get by with. But others of their profession would regard the same level of competence as unsatisfactory–but it passes.

    “Maurice de Broglie later recalled: ‘I remember one day at Brussels, while Einstein was explaining his ideas, Poincaré asked him, ‘what mechanics are you using in your reasoning?’ Einstein answered: ‘No mechanics.’ which appeared to surprise his interlocutor.” –Gallison (2004) cited in Brown, Harvey R., (2005) Physical Relativity (p. 147)

  17. #17 Brad
    Ohio
    July 28, 2014

    We frequently come upon situations in grocery stores where different packaging options for the same product are differently priced. For instance, a 12-pak of soda may be $3, but the 24-pak of the same beverage is $7. This is a phenomenon we like to call “Buy more and save even less!”

  18. #18 Christopher Winter
    July 30, 2014

    The other problem is that, in this country, teaching is not a respected profession. Frankly, teachers are usually treated with contempt. People who wouldn’t last five minutes in a classroom protest that teachers have it easy, since they only work ten months out of the year. In most districts they are poorly paid and not given adequate supplies for their classroom.

    Another factor is that school administrators tend to side with parents, against the teacher, when there’s a dispute over how their child is being taught. I don’t know how prevalent this is, but it’s often commented on.

    One poignant description of the general situation comes from Piers Anthony’s Macroscope. I’ve taken the liberty of reproducing it here.

    http://www.chris-winter.com/Erudition/Reviews/SciFiFic/Anthony_PDJ/Teach.html

    Although this is written as fiction, it grew out of a real teachers’ strike in Florida in 1968. And the trend persists today, as shown by the experience of Randy deVelbiss. “I left teaching because I couldn’t stand it any more,” [deVelbiss] said. “If I failed a child, the parents always complained. If I reprimanded a child, the school would threaten disciplinary action. I figured out there are better things I could do with my life.” See page 97 of Time to Start Thinking by Edward Luce (Atlantic Monthly Press, 2012)

  19. #19 proximity1
    July 31, 2014

    More broadly, reasoning skills, of which computational skill is a subset, are a talent, a natural endowment which, like all such endowments, is highly variable across populations. Fortunately, any average person is typically endowed with what we could call a basic level of computational skill. Nearly everyone can, when competently taught, learn to perform rudimentary addition, subtraction, multiplication and division. After that, things diverge greatly in people’s numerical skills.

    But numerical skills are just part of the picture. You could write a similar commentary about what amounts to the rather dismal state of Americans’ general reasoning skills and, especially, their moral reasoning skills.

    This should alert us to some important things about reasoning–but in typical fashion, it usually doesn’t. Many of the relative minority who possess extremely advanced skills in computation and other mathematical and logical reasoning are, to put it bluntly, relative morons when it comes to moral reasoning, to matters of ethics and justice. These are apparently just as much a matter of natural endowment as are mathematical skills and, like mathematical skills, most people can attain some basic level of moral, ethical competency. Unfortunately, that is not much if at all better than the simplest arithmetic level in mathematics.

    Take P.Z. Myers or Sam Harris or yourself, for example. All of you have expert talents in mathematical reasoning or in empirical sciences or both. But all of you demonstrate what strikes me as wildly deficient skills in moral reasoning and ethics. Reading Myers or you or Harris (who happens to excell in his grasp of many moral issues), all of you clearly grasp a good deal about certain moral issues and, on others, typically touching you at some personal interested level, you simply fail to exercise the same degree of reasoning competence.

    Americans aren’t only remarkable as mathematically deficient people, they’re also remarkable for their deficiencies in moral reasoning and other more general reasoning skills.

  20. #20 Robert
    PNW-United States
    August 6, 2014

    “… they’re also remarkable for their deficiencies in moral reasoning and other more general reasoning skills.”

    Broad claims and over-generalized conclusions with no specific examples; I’d posit that points to a lower level of critical thinking in itself.

  21. #21 proximity1
    August 8, 2014

    Robert @ 20 :
    “Broad claims and over-generalized conclusions with no specific examples; I’d posit that points to a lower level of critical thinking in itself.”

    There is no lack of examples.
    True, I spared the reader a citation of them– but mainly for the sake of concision and a recognition of the limitations of this medium for debate and discussion. One can only guess at the level of one’s readers’ intererests here in such discussion fora. Again, experience teaches me that most Americans simply don’t want to know. That you are perhaps an exception, I’ll grant. In that case, I wonder how you can have missed the obvious examples which everyday life presents. Maybe you’re so focused on your professional (scientific) cares that you don’t have time for such matters. I don’t know and you don’t explain (either) the basis for your implied view that the evidence isn’t there. It’s everywhere in the popular press, in discussion blogs, in films and in the generations-long living habits of Americans.

    You’d be mistaken to suppose that I regard Ameriocans’ moral failings as unique in the world. On the contrary, they are typical of what goes under the name of “Western civilisation” and they aren’t peculiar to this century or the 20th.

    If we want, we could put blinders on and keep our attentions fixed on only the morally beautiful things that people–wherever they live–have done. But the evidence is abundantly clear as to the general picture. Nor do I intend to argue that humanity hasn’t advanced morally since its earliest appearance. It has. So, if you’re content to congratulate people for the fact that a lot of Old Testament morality today strikes many today as barbaric, you may do that. The fact is that, as concerns morality , the supposedly most highly civilised societies (i.e. the technologically advanced and highly educated) of the world do not compare particularly well to the kind of morals practiced among some of the still-extant isolated tribes of the world’s most primative tropics-dwelling forest people. (See, e.g. Daniel Everett’s Don’t Sleep, There are Snakes: Life and Language in the Amazonian Jungle (Vintage Books) )

    If you’re really interested, Susan Jacoby’s best-seller, The Age of American Unreason (Vintage Books) isn’t a bad place to start. It’s amply annotated and has a bibiography for sources and further reading. Of course, it concerns more than simply deficiencies in moral reasoning. In fact, moral reasoning isn’t necessarily its focus but any attentive reader should find plenty of moral implications in the cases she presents.

    If you want more detailed debate here on the matter, that’ll be at JR’s sufferance–unless we take it to some other venue.

    In that case, I’d be interested to read a short list of those who, in your opinion, are the best moral exemplars in contemporary American life. And, similarly, the number of places at which exemplary moral behaviour and popular celebrity coincide in contemporary U.S. society.

  22. #22 proximity1
    August 8, 2014

    P.S. to Robert @ 20 :

    For more RE the moral state of U.S. society see he following on the ethos and mores of contemporary society, see the following references which illustrate what amounts to the “Made in America” ethos of our times (not to mention a contemporary reprise of long-standing American habits. Note the robber barons of the Ameican “Gilded Age” ) :

    … “Today the dominant narrative is that of market fundamentalism, widely known in Europe as neoliberalism. The story it tells is that the market can resolve almost all social, economic and political problems. The less the state regulates and taxes us, the better off we will be. Public services should be privatised, public spending should be cut and business should be freed from social control. In countries such as the UK and the US, this story has shaped our norms and values for around 35 years: since Thatcher and Reagan came to power(2). It’s rapidly colonising the rest of the world.”

    …”The rich are the new righteous, the poor are the new deviants, who have failed both economically and morally, and are now classified as social parasites.”

    Source: George Monbiot, (essay) “Deviant and Proud.” of 05 August 2014 at www(dot)monbiot(dot)com.

    and, the key-source author and book which Monbiot cites in his essay,

    What About Me? : the struggle for identity in a market-based society by Paul Verhaeghe ( 2014, Scribe Publications, Melbourne & London)

  23. #23 proximity1
    August 9, 2014

    JR :

    While you’re away (or once you return) you ought to look up the current (August ) issue of Scientific American with articles that touch on both teaching methods in higher education (including this thread’s topic, mathematics teaching / e.g. Carl Wieman, (Stanford Univ.) “Stop lecturing me”

    (excerpt)

    “With so much scientific evidence behind active learning, the obvious question is, Why are these methods so seldom used in colleges and universities? Part of it is just habit; lectures began at universities because they did not have books, and so information had to be dictated and copied. Teaching methods have not yet adapted to the invention of the printing press. A second reason is a fundamentally flawed understanding of learning. Most people, including university faculty and administrators, believe learning happens by a person simply listening to a teacher. That is true if one is learning something very simple, like “Eat the red fruit, not the green one,” but complex learning, including scientific thinking, requires the extended practice and interaction described earlier to literally rewire the brain to take on new capabilities.”

    and cosmology ( RE: Multiverse is a done deal) , with “The Black Hole That Birthed the Big Bang.”

    Also interesting is, ” Tapping Your Inner Rain Man : A[n accidental] blow to the head can sometimes unmask hidden artistic or intellectual gifts “

  24. #24 Robert
    United States
    August 11, 2014

    re: #21, 22 “But all of you demonstrate what strikes me as wildly deficient skills in moral reasoning and ethics.
    ….
    you simply fail to exercise the same degree of reasoning competence.”

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