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When people graduate with honors from prestigious universities thinking everything in math is already known and the field consists of memorizing algorithms, then the educational system has failed in one of its major endeavors.

If members of the next freshman class will take just one one-semester math course before becoming the aforementioned graduates, here's what I think I might do (and this posting is indeed a question, as you will see). I would not have a fixed syllabus of topics that the course must cover by the end of the semester. I would assign very simple but serious problems that I would not tell the students how to do. A few simple examples:

  • $3 \times 5 = 5 + 5 + 5$ and $5 \times 3 = 3 + 3 + 3 + 3 + 3$. Why must this operation thus defined be commutative?
  • A water lily has a single leaf floating on the surface of a pond. The leaf doubles in size every day. After 16 days it covers the whole pond. How long will it take two such leaves to cover the whole pond. (Here lots of students say "8 days". I might warn them against that. This is the very hardest problem assigned in an algebra course that I taught, according to most of the students.)
  • Here is a square circumscribing a circle. [Illustration here.] Here is how you use this to see that $\pi<4$. [Explanation here.] Now figure out how to prove that $\pi > 3$ by a similarly simple argument.
  • Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, ...... Multiples of 18 are 18, 36, 54, 72, 90, ..... The smallest one that they have in common is 36. Multiples of 63 are 63, 126, 189, 252, 315, 378,..... Multiples of 77 are 77, 154, 231, 308, 385,.... Could this sequence go on forever without any number appearing in both lists? (Usual answer: Yes. It will. Because 63 and 77 have nothing in common.) Is it the case that no matter which pair of numbers you start with, eventually some number will appear in both lists?

I said simple but serious, the latter meaning they will actually learn something worth learning about mathematics or about how to think about mathematics. Not all need be as elementary as these. With some of the less elementary problems I might sketch a solution or write out a solution in detail and then ask questions about the solution.

I would not fix in advance the date at which problems were to be turned in, but would set deadlines after discussion reveals that serious difficulties are overcome. I might also do some "teasing" concerning various math topics not covered.

HERE'S THE QUESTION: Which published books of problems can participants in this forum recommend for this purpose? Why those ones?

Michael Hardy
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    What do you mean exactly by a "serious problem"? Also, is this question significantly different than your previous question? http://mathoverflow.net/questions/28695/what-should-we-teach-to-liberal-arts-students-who-will-take-only-one-math-course – Jeremy West Feb 01 '11 at 01:51
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    @Jeremy: What I mean by "serious" is stated explicitly in my posting. How is my proposed statement of the meaning deficient? This posting OBVIOUSLY (but only if you read the whole thing) differs from that earlier posting in the content that follows after the words "HERE'S THE QUESTION", set in boldface type (Does it fail to appear in bold on your browser?). – Michael Hardy Feb 01 '11 at 02:13
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    Why are so MANY up-votes given to comments that prove that the commenter did not read the question? I've seen this with a number of other questions here. – Michael Hardy Feb 01 '11 at 03:58
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    I've hit this question with the Wiki-hammer. Please make an effort to communicate in a way that is less likely to appear condescending or sarcastic. – S. Carnahan Feb 01 '11 at 03:59
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    @Michael "Serious" = "Worth learning" is vague and subjective. I disagree that your examples would convince students that there are interesting and important open problems in math: none of them are open!

    I commented about your previous question because I intended to link to it and was surprised to see that you asked it. From the length of the question it seemed you saw it as something different, which made me wonder if I had misunderstood it.

    For the record, I did read the entire question.

     – Jeremy West Feb 01 '11 at 05:01
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    @Michael Also, lest I seem antagonistic, I thought the original question was excellent, which is why I remembered when I read this one. – Jeremy West Feb 01 '11 at 05:06
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    @Michael I think removing the phrase "everything in math is already known and" would fix everyone's confusion. As a student of the art of teaching, you must realize that when you communicate with people, they will sometimes interpret a minor remark as the crux of the matter. – Steven Gubkin Feb 01 '11 at 11:12
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    To answer your question Sylbia Beckmann's "Mathematics for Elementary School Teachers" is great at asking questions like the ones you do in the body of your question, and uses such questions to develop all of elementary school mathematics. You already saw my answer here: http://mathoverflow.net/questions/44983/resources-for-teaching-arithmetic-to-calculus-students/45001#45001, but I wanted to point it out again. – Steven Gubkin Feb 01 '11 at 11:20
  • +1 for the first 3 lines. – Qfwfq Feb 01 '11 at 11:46
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    The choice of 16 in the water lily question makes it a bit of a trap. Would you get as much wrong answers by choosing 14 or 31 days ? – BS. Feb 01 '11 at 14:40
  • @Jeremy: It was never part of my purpose to assign examples of open problems. Obviously we're talking about a class of students who arrive at a university not knowing that math involves anything besides applying memorized algorithms. Assigning them open problems doesn't make a lot of sense. – Michael Hardy Feb 01 '11 at 17:01
  • @BS: Why is 16 more of a trap than any other even number? With an even number, the obvious temptation would be to say it's half of that number. – Michael Hardy Feb 01 '11 at 17:02
  • I second Steven's suggestion, having taught the course at UGA from Beckmann's book. (By the way having taught that course, I noticed that the first example above omits to remark that repeated addition must be proved to be well defined, before using it to define multiplication.)

    Maybe we should use this book also for our math majors as well as our elementary teachers.

     – roy smith Feb 01 '11 at 18:41
  • OK, I'll bite: What's the well-definition problem with saying the following? $$ m\times n = \underbrace{n + \cdots + n}_{m\text{ terms}}. $$ – Michael Hardy Feb 01 '11 at 19:22
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    without proving first either associativity or commutativity of addition, how do you know that 3+(3_+3) = (3+3)+3? Probably you assumed that had been done. – roy smith Feb 01 '11 at 23:20
  • @roy - I think it should be used for everyone! A first college course in arithmetic. It seems to be needed. – Steven Gubkin Feb 01 '11 at 23:22
  • In fact, one needs these results for more than 3 summands, a result seldom proved in public. I recall reading spencer, steenrod, and nickerson's advanced calculus in the 1960's, and encountering the first proof I had ever seen in a book, that associativity for 3 elements implies it for any finite number. They used induction on the number of parentheses. I.e. any expression with ...... in it requires an inductive justification. – roy smith Feb 01 '11 at 23:25
  • I agree Steven, I just had not realized it before, and I was perhaps unconsciously biased into thinking a course for elementary teachers was somehow not appropriate for math majors. – roy smith Feb 01 '11 at 23:27
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    Obviously a college course in arithmetic is needed. Engineering students who don't know that if you round 1/3 to 0.33, then subtract from 0.34, then divide 5 by that difference, you don't get more accuracy by reporting the first 10 digits after the decimal point than if you report the first 9; undergraduates who don't understand that the reciprocal of a small number is big; students who don't know what GCDs and LCMs are, let alone what they have to do with simplifying or adding fractions; students who don't suspect that 2/6 is the same as 1/3;.... – Michael Hardy Feb 01 '11 at 23:57
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    Students who use calculators as anesthetics (instead of thinking about (how painful!!) the meaning of $$ \frac{18\times17\times16\times15\times14\times13}{6\times5\times4\times3\times2\times1}, $$ they desperately reach for their calculutors and find the product in the numerator and thereby feel excused from that horrible task; students who can't be talked out of canceling the "2"s in $(2 + 7)/2$, getting $7$..... – Michael Hardy Feb 02 '11 at 00:01
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    @roy: Do not prove associativity and commutativity of addition to undergraduates who don't like math. That's almost morally offensive. – Michael Hardy Feb 02 '11 at 00:04
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    well, I just meant that if you are going to ask them to explain why multiplication is commutative, it seems they should know why addition is associative first, to even understand your question. Of course I could be wrong, and I frequently am. – roy smith Feb 02 '11 at 00:35
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    One thing that has hindered me a lot in teaching undergrads is that I usually lacked the courage to use a high school geometry text say in a college proofs class or a college or graduate geometry class. It takes nerve, when you were yourself taught from spencer, steenrod, and nickerson, maybe as a freshman, to use Beckmann's elementary teachers book, or Jacobson's geometry book for juniors. I am proud of the current generation for having more guts to use what works, without fear of being made light of. My best graduate course was probably the one I finally taught from Euclid in 2009. – roy smith Feb 02 '11 at 00:41
  • @roy: I think it's nonsense to say they won't understand the question without that, and I'm surprised to see it suggested. – Michael Hardy Feb 02 '11 at 05:40
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    I told you I was frequently wrong. – roy smith Feb 02 '11 at 15:28
  • @Harry: I will admit I myself did not understand a similar question when I taught this course, at least not in the same way as the students, since they believed multiplication was defined by repeated addition and I thought it had to do with forming cartesian products. ( I learnt this from Hausdorff's set theory.) So to me the commutativity had to do with the existence of a bijection from AxB to BxA. Perhaps that is why they had more trouble explaining why it holds. If you have not taught a course like this I think you would find it interesting, and no doubt do a better job than I did. – roy smith Feb 02 '11 at 15:36
  • By the way if ones students understand multiplication to be repeated addition, one might ask them how they understand π^2, or if they reply that is a certain area, perhaps π^4? This has led me to discover that in spite of the quantifiers preceding the statements explicitly referring to "all real numbers", some calculus students tacitly assume the constants occurring in expressions like (cf)' = c(f'), are integers. This apparently has a long tradition regarding π, including the builders of Solomon's temple, and some Connecticut legislators. – roy smith Feb 02 '11 at 17:11
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    I agree we should ask junior math majors why 3x5 = 5x3, but should we perhaps also ask freshmen in calculus why 3xπ = πx3? This second operation is not obviously repeated addition. If students do not know what these operations mean, does it make sense to teach them a whole course which depends on even more subtle properties of reals? (Of course I could be wrong.) – roy smith Feb 03 '11 at 04:04
  • @roy: Associativity cannot become an issue until you think of addition as a strictly binary operation, which is both wrong and too sophisticated, within the context we're talking about. – Michael Hardy Feb 03 '11 at 04:06
  • you may well be right Michael. Indeed someone (Fermi?) observed the younger generation is always right. But I do not quite see your point in this case. Can you give me a little more explanation? – roy smith Feb 03 '11 at 05:42
  • @ Steven: Here is a basic question, with essentially the same answer as why 3x5 = 5x3. Why do we think when two people count up the same finite set in different ways, they will arrive at the same answer? This assumption, equivalent to assoc. and commut. of +, is tacitly used in the standard picture proof that 3x5 = 5x3, by counting the squares in a 3 by 5 rectangle by rows and by columns. If that proof is used, do you think it has value to ask the class afterwards why it is convincing? After why 3x5 = 5x3, should we ask calc students whether 3xπ = πx3? or if sqrt(2)xπ = πxsqrt(2)? – roy smith Feb 03 '11 at 16:41
  • @roy: When the theory of the integers is developed via the Peano axioms or the like, then such questions are relevant. But proving that addition is commutative when $m+n$ is defined as how many things you've got when you join a set of $m$ things with a disjoint set of $n$ things, then a proof of commutativity is a tedious exercise in proving the obvious. It will put all of the students to sleep except the stupidest and the most brilliant. After you've done that, when you address the question of commutativity of multiplication, you'll be announcing that you're about to do..... – Michael Hardy Feb 03 '11 at 18:11
  • ....yet another tedious exercise in proving the obvious. The whole point is to show them that what they thought was a trivially obvious thing is actually somewhat puzzling. – Michael Hardy Feb 03 '11 at 18:12
  • ....and you would also violate the "simplicity" requirement above. Remember that this whole enterprise is about getting the attention of people who need to be shown that there's a reason to pay attention. Their prejudice when the come in is that math is about stupid boring things, and so you propose to show them more stupid boring things because you think it's logically a prerequisite to the problem that you ultimately want to get at. – Michael Hardy Feb 03 '11 at 18:20
  • Thanks for you answer Michael. I am not advocating any particular approach, just trying to learn and discuss possibly useful ways to make math understandable. What do you think of asking students why it even makes sense to speak of a set of "n things"? This was my question to Steven, why do they believe that when two different people count up the elements of a set, they will reach the same result? This question already contains the essence of associativity and commutativity of addition. I agree with you that formal proofs are inappropriate. I hope discussing ideas maybe is not. – roy smith Feb 03 '11 at 18:45
  • That question I think they would understand. I'd guess even the smartest wouldn't be sure what could constitute an answer. But often slight rephrasings of the question can deal withthat. – Michael Hardy Feb 03 '11 at 20:57
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    You give me an idea Michael. I like also to show students they can understand what they thought they did not. Maybe one can make the tedious less so, using the easy argument to clarify the formal one. If a student believes counting by rows or columns gives the same answer, maybe he could then be induced to see how this also underlies the mysterious algebraic argument. I.e. 2+2+2 equals (1+1)+(1+1)+(1+1). Then point out that if this sum is by rows, then the three first "1's" in each summand belong to the same column. Hence adding by columns gives (1+1+1)+(1+1+1) = 3+3. could this work? – roy smith Feb 04 '11 at 23:13
  • Indeed one could stack those terms on top of each other like a rectangle, actually making the first 1's into columns. And then it looks also like an elementary school addition problem. so it becomes clear that the usual method of adding, first ones to ones, then tens to tens, ...., also uses associativity and commutativity. Thanks for inspiring these ideas. Maybe I can teach this better next time. – roy smith Feb 05 '11 at 11:51
  • so this seems to show that associativity for addition implies both commutativity for multiplication and commutativity for addition. i don't think i ever noticed that before. thanks again, and best wishes. (teaching this class also taught me to compare "regrouping" while adding, to putting bottles into 6 packs, cases of 24,.....) – roy smith Feb 05 '11 at 12:12

5 Answers5

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One good option is "The Magic of Numbers" by Gross and Harris (not to be confused with a book of the same title by ET Bell), which was written for the eponymous class Gross used to teach at Harvard. The problems include some stuff on, say, Catalan numbers, and some reasonably serious modular arithmetic (e.g. RSA encryption) with a minimum of baggage, which should recommend the book to non-mathematicians.

The Art of Problem Solving series (here) is also quite good. I learned a lot from some of those books when I was in high school--they have lots of exercises, ranging from very easy to problems I, at least, found quite difficult. And there is a lot of discussion of technique, which I think non-mathematicians often find lacking in other textbooks.

And Martin Gardner's entire oeuvre is great, and I think that recommendation probably doesn't require any explanation.

Daniel Litt
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  • Thank you. I've requested Gross & Harris via interlibrary loan. – Michael Hardy Feb 04 '11 at 00:56
  • It showed up today. It looks excellent so far. But there's something really weird on pages 114--115. For the proof of the infinitude of primes, it says on page 114 "we will have to argue by contradiction". That is of course nonsense, and then the usual proof by contradiction, which is pointlessly complicated, is given on page 115. Then it says "Another way to phrase this argument would be the following. Suppose we have any finite collection of primes" etc..... It shows that if you multiply them and add 1 and then factor the result, you get primes not in the set you started with. This – Michael Hardy Feb 11 '11 at 22:55
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    ....other way to phrase this is not a proof by contradiction. Yet it had just said "we have to argue by contradiction". This is a place where the book could be improved by omitting something. The proof by contradiction (which, contrary to the assertions of many universally respected authors, is not in the works of Euclid) should just be omitted. The "other way to phrase this argument", which proves that we do not "have to argue by contradiction" is in fact the one in Euclid's Elements. – Michael Hardy Feb 11 '11 at 22:57
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Your pedagogical approach sounds suspiciously like the one in many Math Circles. The National Association of Math Circles has problem lists on their website. Here's another problem set which looks interesting.

If your intention is to leave these students with the sense that there is fabulous ongoing research in mathematics, I'd recommend the Five Golden Rules: Great 20-Century Mathematics and Why They Matter. It's quite accessible because it focuses on the general ideas.

Anna Varvak
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Published books of problems generally mean to nudge serious, dedicated and perhaps talented mathematics students towards a research-oriented frame of mind. If you really mean a bound problem collection for general education, I expect you will have to write such a book yourself. But I would probably recommend against publishing such a book - on the grounds: don't teach until you see the whites of their eyes. A mathematics problem that will work with one cohort might variously and unpredictably either defeat or insult the intelligence of another. And, as the the response to your other question might indicate, teachers of mathematics will have very diverse views of what constitute a value partial knowledge of mathematics.

That said, if I had an audience of highly intelligent but not especially mathematically oriented students, I might focus their "last look" at mathematics on Lawvere and Schanuel's Conceptual Mathematics (which has many good problems). The authors show themselves as both wise and smart. While the book could save the soul of a stray mathematician, it does not harbor any hidden agenda that means ignoring the needs of the broader audience. And while it might accidentally remediate some high school induces confusions, even the best trained students will find most of what the authors say both very new and very fundamental.

Serge Lang's Math Talks for Undergraduate also attracts me, but where Lawvere and Schanuel help a student think about the larger world in a more mathematical way, Lang wants non-mathematicians to understand more about what mathematicians do.

Research mathematician/teachers generally hold as a sacred shibboleth the dictum that "mathematics in not a spectator sport." In the case of a class of general education students seeing mathematics in the classroom for the last time, and and the risk of blasphemy, I question this, I question whether having these students primarily trying to solve problems for themselves necessarily constitutes the best use of their time. I believe that the mathematics community has neglected developing of literature of what one might call proof-oriented spectator mathematics. But still I might recommend a book: I taught a course recently out of Ross Honsberger's Episodes in 19th and 20th Century Euclidean Geometry where I focused on close readings of complicated but elementary proofs of concrete and yet often spectacularly counter-intuitive facts, all material most mathematics majors will never see on the grounds that it isn't sufficiently modern. But as a toy model of what mathematicians do it worked very well for my students.

David Feldman
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    In the realm of what you call proof-oriented spectator mathematics, there is a very beautiful book by Stanley Ogilvy called Excursions in Geometry, that a 15-year-old who knows next to nothing can read and enjoy. (I read it when I was 14 or 15.) Non-mathematically inclined undergraduates intensely hate that book. – Michael Hardy Feb 01 '11 at 04:05
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    @David Feldman: While I agree with the sentiments of almost everything you write, and while I haven't read "Conceptual Mathematics," I notice that on Amazon its subtitle is "A First Introduction to Categories." I think that the language of categories, as much as I value it, is unlikely to do anything other than annoy the average (even quite intelligent) English major--do you disagree? – Daniel Litt Feb 01 '11 at 04:09
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    @Daniel All I can say is that I'd probably agree with you...if I'd never seen the book. But actually, I think categories do have a lot to say to an English major (think about characters and plots, etc.) for roughly the same reason they have a lot to say to computer scientists interested in the semantics of programming languages. But developing that point would probably better be done over lunch than in a MO comment. :) – David Feldman Feb 01 '11 at 04:39
  • Fair enough--I guess I'll read the book! – Daniel Litt Feb 01 '11 at 04:41
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    @ Michael The Ogilvy and the Honsberger are fundamentally different books. Ogilvy emphasizes theories (inversive geometry, projective geometry) from which theorems drop out - in that sense it seems "modern" and pre-professional. Honsberger just develops these wonderful, elegant, but seemingly ad hoc results. Ogilvy:Honsberger:: Chewable vitamins : exotic desserts. – David Feldman Feb 01 '11 at 05:06
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"Heard on the Street" by Timothy Falcon Crack is a collection of brainteasers that were supposedly put to interview candidates for Wall Street jobs. The book has many more questions like the ones you asked - most of them can be solved without any heavy mathematical machinery, but they all require a little ingenuity.

Sadly, the book has become famous enough that recent graduates hoping to get banking jobs often memorise all the problems, rendering the whole process useless.

Simon Lyons
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  • But most students aren't seeking banking jobs. – Michael Hardy Feb 01 '11 at 04:05
  • @Michael: true, most students aren’t seeking banking jobs, but nevertheless this book contains many problems along the lines of what you asked for; plus the fact that these skills by at least some non-academics may be a helpful motivator for some students. (Even if they don’t feel “This will help me, personally, get a job”, it can at least help them stop feeling “Ugh, after college no-one even cares whether you can do math; it’s so pointless!”) – Peter LeFanu Lumsdaine Feb 01 '11 at 04:53
  • When I said most students aren't seeking banking jobs, I was responding to your "sadly" comment. In other words, it's not as bad as that comment might suggest. – Michael Hardy Feb 04 '11 at 01:15
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How about The Theory of Remainders by Andrea Rothbart.

I remember back in the day I was struggling with the concept of modular arithmetic and randomly came across the book above. It's really well written in an unorthodox way as a dialogue between two people talking about modular arithmetic. The book introduces basic concepts of abstract algebra and has plenty of "simple, but serious" exercises. If I recall correctly, it did a really good job of motivating the concept of fields. Above anything, it was written with a high school audience in mind, so incoming freshmen should not be deterred by the level of difficulty. I also found the style of the book engaging. I dare say I was bitten by the number theory bug shortly after reading it.

Alex R.
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  • +1 I believe this book is where I read a discussion concerning which scores are possible outcomes in American football, which is a really fun question. – R Hahn Feb 01 '11 at 10:17