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Proofs in sequences and series

  1. Dec 2, 2004 #1

    mathwonk

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    I am teaching honors calculus in college, and trying to teach something about convergence of sequences and series. my class has apparently never seen a genuine proof in high school and have no idea how to begin one (answer: with the definition). I have had students ask me what "QED" stands for, and what "lemma" means. quantifiers are a completely foreign language.

    no matter how many proofs I do on the board, hardly anyone can do one as exercise, or even begin one. they never even seem to think of starting with the definition of convergence, when trying to prove convergence.

    when i tried to help them get started by asking what is the definition of convergence, it seemed nobody had bothered to learn the definition.

    the exercise i assigned was a proof i had already given on the board in class, but still no one had any idea how to do it. i do not know if they just ignored what i wrote on the board, or tried and did not understand it.

    these are smart, curious students, and i love them, but we are both struggling. they all have relatively high AP scores but know little about calculus, or mathematics, or effective learning. still there are a lot of bright spots, i get wonderful questions, sprinkled in with mouth droppingly odd ones. they do want to understand the stuff, but seem to have no experience either at rigorous reading or writing of mathematics.

    many freshmen seem never to have seen a geometric series, which in my day was explained (without convergence details) in the 8th grade.

    I am enjoying my class, and i hope I am helping them with a difficult transition to college, but sometimes wonder if my expectations are totally out of kilter. Am I doing something strange by teaching series to freshmen honors students?

    when do other people try to teach these things? to freshmen? juniors/seniors? (i have no relevant experience to guide me, since as a freshman in the 1960's, without prior calculus experience, our class was taught rigorous sequences and series in the first semester, indeed at the beginning of the semester. the first homework exercise set included proving e is irrational using the series expansion.)

    In high school we were taught propositional logic with quantifiers and truth tables. we never had calculus, whereas nowadays most of my students, honors and non honors, have had calculus. Still most do not understand geometry or algebra or logic or proof, and are apparently much less prepared than we were for college maths.

    Is something backwards here? Why teach calculus to people who do not know any of the more basic material, and without encountering reasoning or logic? I.e.; those of us who were taught more elementary material well, seemingly were better prepared for college calculus than today's students who are taught watered down calculus badly in high school, with time for it purchased by omitting a decent algebra and geometry background.

    There is no way to turn back the clock, but there is a real challenge here for us in university to try to accomodate these extremely math deprived high school graduates in college math. What do the high school students out there suggest?
     
  2. jcsd
  3. Dec 2, 2004 #2
    i suggest talking about convergence and sequences in simple terms. What is it in basic english? Then transition into the more advanced explanation. Because what is the point of saying something with all these deltas and epsilons when the students will not understand it. In AP calculus, they stress to get everything done for the exam. Perhaps using Courant's book will help. I think tou are right in teaching basic analysis first. Like methods of proof.
     
  4. Dec 2, 2004 #3

    Tide

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    It's a very serious problem! I've resigned myself to the fact that students are utterly unprepared and take it upon myself to remediate them with lessons on basic logic and proof. "Basic" is the keyword and that includes a lot exercises in what most of us would regard as trivial proofs (such as "the square of an odd integer is also an odd integer") but it helps significantly and most do get "the hang of it." It makes the rest of a course go much more smoothly!
     
  5. Dec 2, 2004 #4

    mathwonk

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    the problem seems to be that even if one explains everything in simple terms and then transitions to the rigourous versaion, the students are trained to stop listening after the simple version.
     
  6. Dec 2, 2004 #5

    mathwonk

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    i admit i never thought of proving the square of an odd integer is odd in an honors calculus course. i thought of a basic trivial proof as something like: the limit of a bounded increasing sequence equals the least upper bound of the sequence.
     
  7. Dec 2, 2004 #6

    jcsd

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    I have to admit I'm a product of of the modern eductaion system and I don't rmeber seeing a single matematical proof before going to university (with the possible exception of the irrationality of sqrt(2) and I think also 0.999.. = 1 as well)

    Anyway I thought of a silly little theorum today that's just the right sort of level for such a class to prove:

    a sequence whose set of terms is an interval in the rationals is divergent.
     
  8. Dec 2, 2004 #7

    mathwonk

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    i seem to have thought the negation of that theorem to be true. i.e. an interval in the rationals is one point. no?
     
  9. Dec 2, 2004 #8

    jcsd

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    Oops I see what you're saying, okay then when the set of all terms is the set of all rationals in an interval and when it is not just a single point.
     
  10. Dec 2, 2004 #9

    mathwonk

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    thats a good one. it took me a little while to see why it is true, but when i saw it, it follows just from the basic idea of convergence. good exercise. it provides and reinforces, insight into the concept.
     
  11. Dec 2, 2004 #10

    mathwonk

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    along those lines, maybe little basic facts, like every convergent sequence is bounded would be useful exercises. i tend to think of stuff like that as obvious but it is only obvious after you understand convergence, which is the whole point of exercises.
     
  12. Dec 2, 2004 #11

    jcsd

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    Yep, I thoguht it was a nice little excerise that involved a little bit of thinking and illustrates an important definition, plus I've never seen it before.

    I think proving theorums that require the understanding of a certain concept is a good way to re-enforce that concept and show excatly what it means.
     
  13. Dec 2, 2004 #12
    mathwonk: I think it may be standard university practice nowadays to require (as a core class) a "transition to higher mathematics" class of mathematics majors, usually taken after or during Calculus III. For me, in calc BC in high school, it was just learning how to do problems, not proving theorems. Even when we did sequences and series, we just had a chart showing what tests to use, and if those failed go to this test, etc. Nothing was really proven for us.

    I got out of Calculus I and II when I went to college, and was able to take "Foundations of Higher Mathematics" in the Spring Semester as a Freshman. This is where I first encountered different ways of proving things and learning about quantifiers and all the symbols and stuff. In this course we also learned about set and function theory. It was a great course; I liked it.

    Now I'm a sophomore taking analysis, where all we do is proofs. It seems to me that this is how things work now: Calculus I, II, III have little-to-no rigour (emphasis on problem solving and learning formulas/theorems), linear algebra (about half and half on proof/problem solving), then the theory courses, with a "transition" course in between.
     
  14. Dec 2, 2004 #13

    mathwonk

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    yes that sounds standard, but i do not like to teach that way, as to me most of the value of a math course is in the reasoning, not in the formulas.

    so i am trying to restore some logical integrity to the courses by giving more theory in the calc 1,2,3, especially in the honors section.

    idealy i would like to see the high schools go back to teaching geometry and algebra with more rigor. we do not have time in college to teach everything over again, and at present the high schools are not teaching most kids anything of use to us at all.

    i have never met a college student in the last 10 years who knew even that a polynomial is divisible by (x-a) whenever a is a root. this used to be sophomore high school algebra, complete with proof.
     
    Last edited: Dec 2, 2004
  15. Dec 2, 2004 #14

    jcsd

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    It's a bit like te old saying:

    “Give a man a fish; you have fed him for today. Teach a man to fish and you feed him for a lifetime. Unless he doesn't like sushi—then you also have to teach him to cook".

    The abilty to derive a theorum is much more powerful than simply knowing the theorum.
     
  16. Dec 2, 2004 #15

    mathwonk

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    absolutely. a high school math teacher i know, and a superb one, said that studies show for most students, the amount of factual material retained from a math course after one year is essentially zero.

    so what should we be teaching? we might as well try to teach ways of thinking. they are not going to remember the fundamental theorem of calculus or the mean value theorem.
     
  17. Dec 2, 2004 #16
    my calc 2 professor today was just discussing putting proofs on the board today (and we are doing stuff with series, so a double coincidence). we are going through the different tests for convergence/divergence, and he finally got to the point where he said "do you really want me ot put the proof up, becaue it takes up time, and i really wonder if you even care?". a whole bunch of people in the class (most of whom seem similar to your students) responded that the proofs were indeed helpful. i was amazed, and thought they just wanted more examples. i don't know if they all said that to not make him feel bad, or not come off as idiots, or if the proof is genuinely desired.

    he also gave a mention that if you just memorize the necessary conditions for each test, like a recipe, that's fine for now, but you will never be able to apply what you know to something that is slightly modified......

    i like proofs. i do a lot of htem in matrix theory. at first, i was really bad, but, as you stated, the best approach is to start with the definition and go from there. sometimes i work from both ends, seeing how close i can get the beginning and end together. seems to help.

    i say continue with the proofs, but, i don't know if caclulus is the place for it.
    i think linear algebra is a good spot for it (and the course is designed to be the class where you learn to prove things) since you start with soem really really basic stuff, and then when you go on to vector space, you build from there. i am pretty sure, with some time, i could prove everything right up to some statements about eigenvalues and eigenvectors.
     
  18. Dec 2, 2004 #17

    mathwonk

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    thanks very much. your clas sounds unbusual to me, like an honors class. my non honors classes never ask for proofs.
     
  19. Dec 2, 2004 #18
    All of the freshmen here at Caltech begin with proofs in Math 1 and the lectures teach almost exclusively proofs and not methods. Methods are left to the recitation sections.

    I don't have much of an opinion on how to teach, since I'm not a teacher and I rarely attend lecture, but perhaps you would do better contacting someone in the math department here? Professor Wilhelm Schlag lectures for 1a, if you'd like a name.

    --J
     
  20. Dec 3, 2004 #19

    Galileo

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    Here in the Netherlands there too is a big gap between high school and university mathematics. I was lucky to be in the year just before a big high school edu. change, but even for me it was a big leap. Even more so for the new students now.

    The freshman lectures (in Leiden) start with two lectures next to analysis 1 and linear algebra 1. (The new freshman nowadays also get an extra lecture to make the high school math 'connect' better to the uni. math. Pretty pathetic this should at all happen, but a good initiative.)

    They are called 'Caleidoscope' and 'Mathematical Structures'.
    Basically, in Caldeidoscope they hope to give a view about what mathematics 'is/contains'. (there's a very small chance they got it from high school).
    It has a little of everything. Graph theory with simply proofs, 'what is a proof', what makes a good definition and why are the definitions and axioms chosen the way they are, basic logic,a bit of set theory, how to study mathematics, how to hold a presentation, etc.
    It ends with an oral exam to test your understanding.
    'Mathematical structures' is like an introduction to rigor and algebra.
    These two series of lectures are so fundamental, in conjunction with Algebra, where you have to do tons of exercises and proofs.

    The physicists also follow a lot of math courses ofcourse (but only analysis and linear algebra) and the mathematicians are allowed to follow some physics courses. Statistically, the ones who followed the math courses at the beginning score way better on average.
    In noticed a big improvement myself when I chose to also go for a mathematics degree after a few years of physics.

    Anyway, my point is that high school students lack a solid logical basis and are not pointed to the fact there is one at all. The first thing to do is to explain to them how math works, learn them some basic logic and do simple proofs so they get a hang of it.
     
  21. Dec 3, 2004 #20

    matt grime

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    Hopefully this won't get too anecdotal.

    You are fortunate that you at least get to teach honors students, a privilege denied to me in the states, and you may well be able to control their learning - are you the only teacher of this particular section? I was one of 6 lectures in my calc course (second year multivariable) and could never teach what/how I wanted.

    Even back in the UK I'm finding it odd to teach away from Cambridge, even at the my current university which has ambitions of being a leading centre of mathematical teaching. For instance (trust me I do have a point somewhere, mathwonk) I'm supervising (I think recitation or examples class is an adequate translation) number theory and group theory. 27 lectures into the course and the lecturer has *finished* number theory, which means they know what the highest common factor of two integers is, Euclid's algorithm and the *very* basics of modulo arithmetic, and started group theory: axioms, very simple example and the definition of order, so far. 27 lectures?!?!? That's just sad, really. Anyway, they struggle to prove anything with any level of rigour - we all did to start out, pretty much, let's be honest. And I've taken to ignoring the questions set in the course (well, I explain the answers) because they are far too easy, really: set this week: find the orders of the units mod 7, and show that conjugate elements (not that they use the word conjugate) have the same order, with hints. And I'm supposed to form a mark for their permanent record from that.

    The point is:

    Instead of spoon feeding them easy material as if it's hard, I quickly go through lots of examples, theorems, and definitions as if they were nothing to be scared of - and they aren't are they? This tends to get them thinking far more quickly and has the benefit of making the homework they have to do seem quite easy by comparison.

    Incidentally, what material do you think should be in a first course on group theory?

    The existence of Sylow groups (without proof); cayley's theorem that concrete and abstract groups are the same thing; orbit stabilizer theorem; definition of coset, normal subgroup? conjugacy classes?

    None of those is mentioned in this course. At the end of the course almost none of my students would realize that there is anything other than abelian groups if it weren't for me banging away that you can't start your proofs "since xy=yx".
     
  22. Dec 3, 2004 #21

    mathwonk

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    well, i am not the only teacher teaching honors, and already in the first few weeks several students dropped out of my class either back to non honors, or into a different section of honors.

    since high school AP tests are nowhere near our co=urse difficulty, we are tending to lower our difficulty level to accomodate these high school tranied sincoming students. However I am constitutionally indisposed to constantly lowering the levels of our courses, since we in the US are already reliably said to be near the bottom of the industrialized world, except at very elite places like Caltech, Stanford, Harvard, MIT, Chicago, etc... which still have high standards, for very strong, very motivated students.


    But what to do? One cannot ignore the population in the room, (although they may do this at some very difficult schools).

    So basically, what one "should" offer in a course at a standard school, where there is no queue outside waiting to get in, has to be related to what the students can accept. but i try to start strong and ambitious, then feel my way along, and usually downwards, as reality kicks in.

    I believe most students can achieve at a level far higher than either they or we think. Hence i try to convince them of this, by stretching them a bit more than they would have expected. I am deliriously happy when someone writes on my evaluation "he taught the most challenging math course i have ever had."

    on the other hand i am depressed when someone writes "he was a terrible teacher, way over our heads, and should only be teaching graduate level, or strictly for math majors!"


    I have not taught beginning undergraduate algebra lately, but did teach beginning graduate algebra sometime back, starting however from almost "scratch" with groups.


    I assumed they knew only matrices and determinants, and decided the most fun topic would be galois theory, since it relates to a topic everyone has found interesting in high school, namely formulas for solving equations. to me motivation is everything. if you can get them interested in learning it they will work hard to do so.

    I had luckily a very strong group of beginning graduate students, although not at all sophisticated in algebra, so they were the ideal audience, and the plan worked well that first year.

    we ended the quarter by presenting an example of a polynomial with no solution formula, having proved literally everything needed for it, including the existence and essential uniqueness of maximal normal towers of groups.

    this philosophy that students are more ready in the beginning for polynomials than for modules, or maybe even for groups, in a sense also guides some of the undergraduate curricula here. the younger students are taught integers and polynomial rings first, and simple field extensions, and impossibility proofs in geometry, before learning about groups.

    when i personally teach groups, i usually start with the trick of how to count the isometries of a platonic solid, by counting those isometries leaving a vertex fixed say, and then counting the vertices. then i lead into the groups of motions of objects and the more general permutation groups, and cosets as those elements taking a given vertex to another vertex, (cosets of the subgroup fixing that vertex).

    conjugate subgroups come out as the fixed subgroups fixing other vertices.

    by always starting them on groups of isometries, they meet non abelian groups immediately.

    i am surprized to hear the situation in leiden resembles ours here in US, as I am familiar with the tradition of very high level of dutch research mathematics at universities such as leiden, amsterdam, and utrecht.
     
    Last edited: Dec 3, 2004
  23. Dec 5, 2004 #22

    matt grime

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    I fully understand and sympathize with the feelings you have about how it is us that are required to lower out expectations rather than the students raise theirs. Depressing really. A quote from one of my students, upon being asked to realize that if n is an integer n(n+1) is even: we were supposed to figure that out in the first week?
     
  24. Mar 12, 2005 #23

    bfd

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    What helped me and others in my introduction to analysis class is when the professor would try to walk us through some of the thought process behind the proof in question. For example he would ask questions like "Well what happens if we do this and this" or "can we do this, why or why not?" By him asking such questions it helped put the proof in the right perspective and not seem so intimidating. Regardless I still felt much unprepared for class. I kept up with the class average but still I felt like I learned very little.

    I've read from a few textbook authors that they feel students in this country are not prepared well to do proofs. When just looking at my own academic experience, it confirms those beliefs. Before that class my only exposure to proofs was in h.s. geometry. What little I gained from that class I tried to use as much as I could for the analysis one.

    What I would suggest is maybe just having a short discussion with your students about the subject of proofs or math in general. Try to get a feel for how they look at those. I'd wager that the biggest problem is misconception rather than the material presented.
     
  25. Mar 12, 2005 #24
    Well, I'm a second year math-physics double honours student from Canada (Carleton University, which isn't a math/science-oriented school, the important one for that being Waterloo around here).

    The thing that I find most astounding relating to students entering university is lack of knowledge of fundamental facts. Yesterday a non-math student, who I know in passing, and who had, in fact, started university in math last year and then switched out, asked me for help with his (now non-honours) calculus course. He showed me one of his tests, and wanted me to explain to him how to do the problems. The first section involved evaluating various compositions of the basic transcendental functions (I think one of them was [tex]\mbox{simplify} \cot(\sin^{-1}(x)), \; x \ne 0 [/tex] or something similar). Needless to say I was a little concerned that he was asking me to help him with these questions. I asked him if he could sketch [tex]e^x[/tex] for me and he answered with a firm no, with the same result for [tex]\sin{x}[/tex]. I had no choice but to show him the properties of trigonometric functions starting from the definition in terms of ratios of triangle side lengths. He informed me that while he found it easy to take derivatives, and he could tell me what [tex]\lim_{x \rightarrow 0}{\sin(\alpha x) \over \alpha x}, \; \alpha \neq 0[/tex] is, that he really didn't know what the objects he manipulated were.

    This is obviously anecdotal and I find it very hard to believe that lack of understanding this severe can be very common, but even with classmates of mine in honours I often notice obvious logical inconsistencies in proofs. An issue which seems to make the problem worse is the lack of quality of marking of work turned in. Most of my assignments take several pages of work to get to my standards of rigour, but classmates of mine will often hand in single-page assignments, with the afformentioned obvious logical errors, and lose very little credit. Often the credit that is lost is not in the right places (the marker will misinterpret something that is correctly stated in the solution and remove marks, but miss an obviously incorrect statement later on). I am certain that this is an issue relating to the time that can be afforded to marking each assignment, and perhaps of lack of interest of the marker in the material, because of its elementary nature, and because real errors are so common. It certainly doesn't help though. Obviously I have no idea what the situation is like at other universities.
     
    Last edited: Mar 12, 2005
  26. Mar 13, 2005 #25

    bfd

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    I was just as surprised as you were when I read that. After I wondered "If that is true for a majority of math students in the u.s. how are so many able to get accepted into grad school and survive?" I just can't imagine going to grad school without somewhat of a good exposure to proofs. Wouldn't they be at a major disadvantage?
     
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