Timeline for Simple but serious problems for the edification of non-mathematicians
Current License: CC BY-SA 2.5
47 events
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| Mar 10, 2019 at 13:00 | review | Close votes | |||
| Mar 10, 2019 at 15:12 | |||||
| Feb 5, 2011 at 12:12 | comment | added | roy smith | 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,.....) | |
| Feb 5, 2011 at 11:51 | comment | added | roy smith | 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. | |
| Feb 4, 2011 at 23:13 | comment | added | roy smith | 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? | |
| Feb 3, 2011 at 20:57 | comment | added | Michael Hardy | 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. | |
| Feb 3, 2011 at 18:45 | comment | added | roy smith | 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. | |
| Feb 3, 2011 at 18:20 | comment | added | Michael Hardy | ....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. | |
| Feb 3, 2011 at 18:12 | comment | added | Michael Hardy | ....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. | |
| Feb 3, 2011 at 18:11 | comment | added | Michael Hardy | @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..... | |
| Feb 3, 2011 at 16:41 | comment | added | roy smith | @ 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)? | |
| Feb 3, 2011 at 5:42 | comment | added | roy smith | 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? | |
| Feb 3, 2011 at 4:06 | comment | added | Michael Hardy | @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. | |
| Feb 3, 2011 at 4:04 | comment | added | roy smith | 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.) | |
| Feb 2, 2011 at 17:11 | comment | added | roy smith | 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. | |
| Feb 2, 2011 at 15:36 | comment | added | roy smith | @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. | |
| Feb 2, 2011 at 15:28 | comment | added | roy smith | I told you I was frequently wrong. | |
| Feb 2, 2011 at 5:40 | comment | added | Michael Hardy | @roy: I think it's nonsense to say they won't understand the question without that, and I'm surprised to see it suggested. | |
| Feb 2, 2011 at 0:41 | comment | added | roy smith | 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. | |
| Feb 2, 2011 at 0:35 | comment | added | roy smith | 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. | |
| Feb 2, 2011 at 0:04 | comment | added | Michael Hardy | @roy: <b>Do not</b> prove associativity and commutativity of addition to undergraduates who don't like math. That's almost morally offensive. | |
| Feb 2, 2011 at 0:01 | comment | added | Michael Hardy | Students who use calculators as anesthetics (instead of <b>thinking about</b> (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 $7$..... | |
| Feb 1, 2011 at 23:57 | comment | added | Michael Hardy | 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;.... | |
| Feb 1, 2011 at 23:27 | comment | added | roy smith | 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. | |
| Feb 1, 2011 at 23:25 | comment | added | roy smith | 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. | |
| Feb 1, 2011 at 23:22 | comment | added | Steven Gubkin | @roy - I think it should be used for everyone! A first college course in arithmetic. It seems to be needed. | |
| Feb 1, 2011 at 23:20 | comment | added | roy smith | 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. | |
| Feb 1, 2011 at 19:22 | comment | added | Michael Hardy | OK, I'll bite: What's the well-definition problem with saying the following? $$ m\times n = \underbrace{n + \cdots + n}_{m\text{ terms}}. $$ | |
| Feb 1, 2011 at 18:41 | comment | added | roy smith | 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. | |
| Feb 1, 2011 at 17:02 | comment | added | Michael Hardy | @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. | |
| Feb 1, 2011 at 17:01 | comment | added | Michael Hardy | @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. | |
| Feb 1, 2011 at 14:40 | comment | added | BS. | 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 ? | |
| Feb 1, 2011 at 11:46 | comment | added | Qfwfq | +1 for the first 3 lines. | |
| Feb 1, 2011 at 11:20 | comment | added | Steven Gubkin | 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: mathoverflow.net/questions/44983/…, but I wanted to point it out again. | |
| Feb 1, 2011 at 11:12 | comment | added | Steven Gubkin | @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. | |
| Feb 1, 2011 at 5:29 | answer | added | Alex R. | timeline score: 2 | |
| Feb 1, 2011 at 5:06 | comment | added | Jeremy West | @Michael Also, lest I seem antagonistic, I thought the original question was excellent, which is why I remembered when I read this one. | |
| Feb 1, 2011 at 5:01 | comment | added | Jeremy West | @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. | |
| Feb 1, 2011 at 4:53 | answer | added | Anna Varvak | timeline score: 6 | |
| Feb 1, 2011 at 3:59 | comment | added | S. Carnahan♦ | 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. | |
| Feb 1, 2011 at 3:58 | comment | added | Michael Hardy | 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. | |
| Feb 1, 2011 at 3:56 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Feb 1, 2011 at 3:47 | answer | added | Daniel Litt | timeline score: 6 | |
| Feb 1, 2011 at 3:40 | answer | added | Simon Lyons | timeline score: 3 | |
| Feb 1, 2011 at 3:39 | answer | added | David Feldman | timeline score: 4 | |
| Feb 1, 2011 at 2:13 | comment | added | Michael Hardy | @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?). | |
| Feb 1, 2011 at 1:51 | comment | added | Jeremy West | What do you mean exactly by a "serious problem"? Also, is this question significantly different than your previous question? mathoverflow.net/questions/28695/… | |
| Feb 1, 2011 at 1:35 | history | asked | Michael Hardy | CC BY-SA 2.5 |