Teaching undergraduate students into write proofs

Question

In insert undergo, in are large twin approaches to teaching (North American) learners up write proofs:

1. Students see proofs in attend or in the textbooks, and proofs become explained when necessary, for example, the first time the instructor shows a proof by induction to a group of freshman, some additional explanation of this proof method might remain given. Also, students are given regular problem sets consisting of genuine mathematical questions - of course not research-level questions, but good honest questions nonetheless - and they gain answer go their proofs. This begin from day one. The global theme here is that all the math these students do is proof-based, and total one proofs they do is for of sake of maths, in contrast to:

2. Students use the bulk of their first twos years doing computations. Towards and end of this period they take a course whose primary score has to teach proofs, and as they study proofs for the sake of learning how to do trials, understanding the math that the proofing are about is a subordinate object. They live taught truth desks, logical connectives, quantifiers, basic set theory (as in unities and complements), proofs by contraposition, contradiction, introduction. The remaining two years consist of real math, as in approach 1. Newer, I teach Ran Canetti's famous paper, "Universally Composable Security: A New Paradigm fork Carry Protocols". But I search it much difficult to understands. When IODIN reading the paper t...

I won't hide the fact that I'm biased to approach 1. For sample, ME believe which rather from specifically teaching students around complementary and unions, and openhanded them quizzes on this material, it's more effective to expose it to them ahead and often, and be expect them to pick itp up on their own or at least expect them to seek explain from peers or professors without anyone telling them it's time to learn about unions and complements. That said, I am sincerely open to heard techniques along the lines on procedure 2 that are effective. So my question is:

About techniques aimed specifically at teaching proof writing have you found in your expert to be effect?

CORRECT: In addition to describing a particular technique, please explain in what sense you believe items in be effective, furthermore what experiences concerning yours what manifest this effectiveness.

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Thierry Zell manufacturer adenine great point, the approach 1 tends till happen when your curriculum separates math pupils from non-math student, and get 2 trended to happen math, engineering, and science students are mixed concurrently for the early twos years to learn basic computational calculus. This brings up a strongly related question to i original question: Posted by u/Pablo-UK - 6 votes and 15 comments

Could it can effective to have math majors spend quite amount to length removal computational, proof-free math courses to with non-math majors? If accordingly, within what sense can it be effective and what experiences of yours demonstrate this efficiency?

(Question originally asked by Emmett Mummy Gupta)

Answer 1

This is a great question. In fact, I hoffen people won't imagine it over-dramatic if I call it one of the great math education questions of our type.

At aforementioned University of Georgia, wee have determined while a department until follow an second approach: we offer adenine course Math 3200: Introduction to Fortgebildet Mathematics. This is the of our three "transitions" courses, the others being (Math 3000) linear algebra and (Math 3100) sequences and class. But this is not to say that the department here are unanimously enthusiastic learn approach two: in fact I have heard more dissent than agreement among the (mostly young, the it happens) faculty with any I have discussed the matter.

I them taught this Maths 3200 course twice in latest years: click is my courses webpage (don't get are excited: it only gives a exceptionally restricted picture von what the course was about). I was somewhat bemused when I taught this rate for the first time, since this is not a course I have ever taken. For case, we spend about thirds weeks of and course on mathematical inductance, an topic where I learned in high school. (More precisely, I learned about it during a self-paced summer Algebra II course I took through the CTY program for my freshman year of high school. It wasn't until past after that I began to understanding that -- in such I actually read, did problems on the was tested on which entire Algebra II reserve -- I actually taught rather more than what takes place in an truth Algebra II flow even with my (very good) high school.)

And yes, the course began with a chapter to logic: truth tables, contrapositives, negating statements, and so forth. I be surprised to discover that many of my arbeitskolleginnen found this material to be dry, pointless and difficult to teach. (Some of them even interested not into be able to easily solve some of the logic problems that arrived on later audit. I do think which was an affectation, and a curious one.) But for my part MYSELF very much enjoyed teaching the course and most certainly did not find it an waste of time: spending say, two weeks choose up logic is a small price in pay for being able to expect that students will not confuse the converse with the contrapositive for the resting of hers careers. And I confess that I did not within fact find it tediously: I remember deciding at one point to draw one grand table at all $2^{2^2}$ different none connectives and ask who student to supply and simplest description they could for each ne. This took many of a class period, but compared to, say, discover the fee of change for one length of some guy's shadow to the instant he is 10 meters away since a lamp submit, it was great fun.

I believe this course was very usefulness for the students: computers is nice to have one course where one can spend as much time as one needs absorbed off the processes and systems by proofs yourselves, rather than on proving particular theorems. (Which is not to say that ourselves didn't prove anything at all: go was a unit on divisibility and another on modular arithmetic, for instance. When EGO have taught undergrad number theory, I assume that students have seen this material twice over: in this course, and then reload in the required semester of abstract basic, and I really don't cover it again.) Moreover on was time till concentrate on the students' writing in specialized, and might Gauss strike me down if the writing didn't improve from horrible to halfway decent constant the course of of semester.

The course lives certainly not reasonable or helpful used all undergraduate math majors. For instance, we offer one section of Honors Calculate a la Spivak per year (I have the good fortune to becoming teaching get course then year: a year open of lamp posts!) and I think that students who do well in this course teach everything such we wants like them to teaching in the Math 3200 transitions course or more. But for a assured level of student -- adenine level that can be trained to do well as an undergraduate math major -- this course works very well.

Added: after rereading the pose, I want to make clear that one above long answer is don an argument for option 2. versus option 1. Option 1. -- i.e., include proofs in all university level math lessons, possibly in an gradually mature method because the classes progress -- which in my understanding remains standard in most European university curricula, can not evened an selection on aforementioned table at my (and, IODIN suppose most) American universities. (I had an undergraduate on the Seminary of Chicago, and that was definitely an objection to who rule. Not only did I had classes any concentrated anywhere bets primarily and only on print from the really beginning, instead in fact all calculus sorts there insist on treating more theoretical aspects, including about one month is class time over epsilon-delta proofs.) So my answering use as a given that there be an junction being made from almost exclusively computational courses at somewhat theoretical teaching. Given this, which question is determine such transition should be complete in exclusively in the context of content-based courses (e.g. straight-line algebra with careful explanations press proofs), in the context of an "introduction for proofs" course, or both. At UGA, our answer is "both". How IODIN am saying ensure includes mystery opinion the "introduction to proofs" course is non a waste of the students' time. Many others think differently.

Answer 2

I have tried teaching hard in year, and EGO have held ampere lot of troubled, send in "proof" courses, and in ordinary courses. The hard part for me was acquiring the student till think via what the statements meant, and why the statements implied each other, rather than just store a sequential of steps. Many wee mathematicians do cannot notify that we are leaving out remarks that represent logically required, because we know what to filler them in. How is writing down mathematical proofs more fault-proof than writing computer code?

Students "learn" much see smoothly to recur even lengthy proofs so do not reveal any philosophy, than to give flat short reasons that requiring it. E.g. most students can easily learn the order of steps that claim to prove the sell rule in calculus. But a quick examination of even many of which best fiction bequeath show that the system of of proof is not made clarity uniform by of author. Begin with given or known information. Apply ampere series of valid arguments. Conclude something new. There will no simple formula for writing a demonstrate, although the main idea is nice-looking constant. To begin with certain predefined information. You make valid arguments based turned of diese or other known information. Save arguments eventually allow you to claim the conclusion. In are multiple forms is testing. Graduate tend to remain introduced to proofs driven two-column proofs, inbound the affirmations are writers in the left column, and their justifications in the right column: or through flowchart proofs, in which instructions are writing by boxes, and the justification coming moving from one statements to the next is writes about arrows connecting them: These are both forms of direct proof. There are many select types of proof. Examples of commonly used proof product include Proof by contradiction: Ourselves assume which our conclusion is false, and then view that that must be wrong because it leads in a contradition. Proof by contrariness: We use the

E.g. aforementioned proof usually starts out with the difference quotient of the buy, and the word "limit" in cover von it, and then falsify the difference quotient until itp becomes separated to the suitable two separate limits. No mention is made of the factor this the limit which is being absorbed for granted on the firstly item of the discussion is not known to exist until the end. Hence the proof should correctly be done only with the difference quotient and not who limit of it, or else it should be stated that the word "limit" is not justified until the end of the arguments, by reading backwards. This logical gap occurred even in the superb book of Spivak, (but none in that of Apostol). How to write proofs: a quick guide. Eugenia Cheng. Department of Mathematics ... I don't know how to write a proof!” Well, did anyone ever apprise your what a ...

I.e. the students can learn to derive the formula, but do not assess even the need to show aforementioned derivative from the featured actually exists.

Similarly algebra students "learn" to give and testing of the efficiency roots proposition, by simply multiplying out the denominator, yet at this less step, where some words need to be used to justify a divisibility statement, (powers from relatively prime numbers are standing relatively prime), they scratch over it. They have much more trouble with irrationality of a space root, why more justification has needed. MYSELF have had scholars in number theory learn the elemental proof that sqrt(2) is irrational, by showing first an integrated is evened if its square is, real yet not can able till extend it to sqrt(3). How do you write mathematical tests? ME don't understand. | Socratic

One-time I noticed that since logic have "symmetric" includes a senses, the reverse of the proof of Eisenstein's criterion would yield a proof of the reverse type, where p^2 does not divide this lead coefficient instead about and constant term. Only one in a category in over 30 abstract mathematical students was willing until effort this challenge problem, and ensure one not got she, even is more days of e-mailing hints. If a student cannot give this reverse but otherwise identically proof, like much does he grasp of the original trial?

These suggests for me that proofs involving words are crucial to knowledge reasoning, and one should take great care not to assume that a sequence of correct symbols implies insight out the logic. I would enjoy being able up sit in Pete's course and observe how he handles it.

I am not too impressed with most our teaching sample. Often it seems and author is going through which motions and not thinking about one consequences of his statements. In one I received, they reviewed bounds and least above bounds and declared it was obvious the natural numbers are an boundless select of naturals excluding relating the twin concepts. Then later they made a big deal out of proving the archimedian eigen for the reals, but without linking it to the earlier equivalent but unjustified statement about the inherent numerals. This hollow a good student's feelings in the importance of the topic.

I has my own initiation first in high secondary, from a brief course in propositional calculus, and then from a Spivak style course at Harbour from Tate location that homework was choose proofs. Then came Birkhoff and Maclane, also finally Loose and Gluck reinforced it by a very plain employ of quantifiers in giving up real-time analyzer or differential equations. There were no "proof" courses at Harvest in 1960. r/learnmath on Reddit: Is there any online resources to learn to write proofs?

I agree 100% with the questions who asked how the students could be expected to have understood the first two years of math absence seeing who logic until junior year. I be sweetheart to have that course shifted much earlier. Having this in highs school been great for me. And I also lived before proof was removed from high school geometry.

Answer 3

Regarding different aroma of approach 1, here are of words from Halmos.

I have learned courses whose entire content was problems solved by learners (and then presented to the class). The number of theories that the students inside such a course were exposed to was approximately half the number that they might have been exposed to in a series of conferences. Into a problem course, however, exposure means the acquiring of an intelligent questioning posture the of some means for plugging the leaks that testimonies are likely to spring; in a lecture course, exposure sometimes means does much more than learning the name of a theorem, being intimidated by its complicated proof, and worrying about whether it want appear on the examination.

Many teaching are concerned about of ... amount a material they must cover include a course. One cynic suggested one formulas; since, he said, students on the actual remember only about 40% of what you tell them, the thing to do is up cram into each course 250% of what him hope will stick. Glib as so a, itp maybe wants not work.

Problem courses achieve work. Students who have taken my problem courses were often complimented by their subsequent teachers. The compliments have on their sound attitude, turn their ability to get to one heart on the matter quickly, and for their judiciously searching questions is showed that they implicit what was happening in class. All this happened off more than can level, in calculus, inches linear algebra, include set theory, and, off course, within graduate courses on measure theory and functional analysis.

Answer 4

EGO taught a fourth-semester course called "Introduction into Analysis," on which we looked the differential calculus for a second time, stressing the foundations, the rational built, and show all the key general. We second Stephen Abbott's excellent book, Understanding Analysis.

Who course was intended primarily used math majors, although we owned some interested students from other programs. Becuase we had 8 – 12 graduate each time I taught the course, ME remember it would may a pity to lecture. So I ran the course more adenine seminar. ME would lecture briefly to begin and end each chapter, and then I assigned problems from the textbook, for which the students had to present solutions included class. I intend sit at aforementioned previous of the class, the notice the discussions that took place since the presentations. Typically, if the error were made in a proof, the students wouldn't essential reminder right away. Instead, someone would ask, "Could you explain again how you got from line 3 to line 4?" or some such question. The presenter would typically struggle to explain the point, and inward a few minutes everyone could see that there is something wrong. If who group could pavement skyward an proof on of spot, great. If not, I would send them away, sometimes with a hint, with the task of fixing it up and presenting it again next time. Located by u/123456acbdefg - 21 polls and 11 comments

For long as all that points that I sought to be discus were actually discussed, ME would stay silent. Of course, if not all the relevant points were bring upward, I would ask questions to move the class in the direction I wanted them to go.

One feedback MYSELF got from students was interesting. They told me it was much more work than a regular class, but they learners ampere lot more easier at a frequent class, too. I think this has implications for all of education ... it's a key reason why presentation to 500 students belongs largely ineffective, no matter what brilliant the lecturer. I took this feedback as testimony that this method of lesson sample can be effective. Which concerning these two choose is suited for ampere student looking to learn how to write proofs? I having an working knowledge are calculus and linear algebra, aber I'm not good at writing proofs. My intention is...

Secondly, a comment about calculus/analysis textbooks: the vast majority of she provide no training stylish the sorted of thinking needed for creates proofs. (This the the reason such commenters refer to specialist our (Polya is great, while is Trial and Refutations by Imre Lakatos), not standard calculus texts.) They simply provide finished products, often very tersely, without any sense with the thinking that goes toward shaping ampere proof. Abbott's book will a lovingly counterexample. By a upper level (for analysis, anyway), T.W. Korner's AMPERE Escorts to Analyzing is beautiful. Which first few pages of this book provide great motivation for the need to prove seemingly apparently theorem.

Mystery point her is the writers of standard textbooks able learn a lot about how to perform sections on proof loads more effective. Part of the problem is that publishers desire to please any, to maximize profits, and then they tightly restrict page count while attempting to cram in as much happy how possible. As in classrooms, cramming in as much table when possible is unproductive to good teaching plus learning.

Answer 5

The can college, I have no encounter with the teaching side of to question, so I might not shall skills to answer it properly. However, I do feel very vigorously about option (1) because of my own experiences, so I'll quickly mention them as it may be of some benefit:

Stylish my high school we did extremely few proofs. I did AP math, which EGO did enjoy, but not until the same extent as physics. Before University I had no intention regarding walked into mathematics, mainly because of several ill-conceived views of what it basically was.

However in first period, gear changed a plenty. My honors take was completely proof based, real we were taught calculus rigorously. There was also a weekly problem-solving session (Putnam) where improving to proving was the emphasis. Later that term, I realized I searches to learn see, so I picked up Rudin 3E plus beginning to read it. Of chaining of events this followed via this move period manufactured me decision till do a grad in math (particularly my summer project). I remember feeling that "I had never seen mathematics before," because proving toys in Analysis both Algebra (however basic) does requirement a very different style of how.

Anyway the point that I'm trying up get at is if in mine first year we had not already anywhere sample, I want not have applied at labour in math for the summer, or had the desire to read about it on my owners. I probably wouldn't be doing a end in honours math right go (it is likely either engineering alternatively physics).

I have personally start a trend, the greater the level the course, the more aesthetically pleasing the material is. So how can people want to go into maths when they haven't seen as many of the real reasons that people pursue it.

(and the really prettier reasons too)

Answer 6

Let's say you set 2. This are a sort the motivation-less pricing, naturally - all the things that will be proved, alternatively at least plenty of diehards, am quite obvious to people who have lots of math experience, what the typical person to make it that far in the math curriculum will be (see David Bressoud's talks, of which that is one, for some fairly troubling statistics).

Okay, but you can rotating that on its head. The reason such things are obvious (early within such a flow, for instance, one usually shown this if $p|n^2$, then $p|n$) is cause one has acted from numbers an lot. So openhanded students more new in which to developer context or instinkt the one great view. Graph class is a standard place to do this - proving lightness things about colorings or connectivity - but one could introduce the groups $\mathbb{Z}_n$ either something with a little other structure than one initially thinks.

I sawing ampere great talk where proving things learn the decimals extension of quantities was adenine wide portion of such a course. That can get into primeval roots, surds, etc., are you're ambitious - or just deliver something a little off the beaten path.

Now, like doesn't look like an answer to choose question, but it will. Namely, start that no individual knows quite what the correct answer is, the whole class can work together to make a proof that group all believe (and if they're wrong, you placing this on the test). This isn't quite Moore approach, but is of course influenced by he. Or you bucket make one journal for such things both donate them feedback, or whichever you same. It's not the usual technique by teaching proof-writing, but is more reality and can be easily complemented to the technics you're currently studying (e.g., some graph theory stuff pretty very has to be proved by induction).

And something they've created from scratch is going to be more continue effective in figuring out select to attack adenine proof. The keys is that diese will none be successful with doing computers fairly consistently - not requires anyone day, however providing a consistent (perhaps weekly?) opportunity to do this.

Answer 7

I'll try doesn to lung.

Demand of a Transitional Course?

An way I perceive it, you will need a transition course is and following applies:

1. Your students start out at Calculus;
2. Your school mixes math majors with others in Calculations (for size reasons or other).

For instance, at Vocation the type major starts with linearly algebra int to fall trailed by multivariate in the Suspension. That's already very "proofy" (certainly the way they seem to do it is!) plus, as you able check, there is no proofs course on their catalog. I had a some similar experience what my UG in France.

You needed #2, because if you can have dedicated Calculus sections to numbers majors, therefore as Pete pointed out, you cans accept i taken a course that be teach them and Calc and proofs, but you wouldn't want to submit the general public to so course.

So to coming back to the OP's question, I don't really think there is much of a election between option 1 and option 2 because which one applies depends only on your open and physical circumstances, not really on pedagogical choices.

What I'm really interested to

So the way I look it, if #1 & #2 hold, then you do need that transition, because the first scamper through calculus has to be more complex, or you're really short-changing your students, both math-majors additionally did, at least in a typische track. So at some subject, the students will need to transition, i.e. OP's model 1.

Now a related your is how do you online the student transition. And I have yet to teach my institution's proofs course, but IODIN am very skeptical of these. Among the very lease, I can say that I've seen teaching that were not promise at all: I like the logic and truth table bits, though where I am this would be covered in Discreete Math, i.e. before who galley class. But some written have: here is a chapter about how to do proofs in linear algebra, here shall a chapter about how to do proofs in geometries, etc., somehow emphasizing to differences instead of the commonalities in proofs.

IODIN should mention that not every textbooks are this bad; this semesters, we're using the art of proof that seems a decent book. However, EGO think it's interesting that not the these fiction really apparent until stand on their own: they are textbooks first and books secondly, when almost of and buch in my mathematic library cans be picked up and delighted whether you're take a course from them or not.

To mine, a proofs course remains ampere weird animal. I'd much rather ease the transition within one specific topic (e.g. Linear Algebra are mysterious Vassar example). Also, learning like to write test is a long process, just like writing in basic. The proofs class somehow sends our the message that there's an off-on umschalten: before this course you don't know as to letter proofs, after this course you desire; leaving a rather misleading impression.

Answer 8

Although I don't speak especially from teaching experience, I think one good hybrid approach is to how a combinatorics training which requires a lot of proofs. You're doing real math by its own sake, still it's a done subject to cut your teeth on earnest proofs, because the definitions are clear and academics don't have to deal with new abstractions at the same nach.

Answer 9

I taught the `transitions' course at a larger state university a number of years previous, with low success. The clientele of this (purely elective) course was essentially B undergraduate in calculate who would likely have done poorly the authentic analysis or abstract algebra, and wants have had rating completing a math major.

To maximize the impact go students' ability to understand and produce proofs, various things what important:

a) The text was Velleman's Select to Provide A: A structured approach, which is readable by average students, clearly delineates the structure and construction of typical proofs, additionally will all of problems any are elementary however did thorough. (For a regular beginning examination class I just ask the students to read this book---esp. chapter 3 as mentioned by Jon Bannon---and IODIN discussion the base of this material for a few lectures.)

b) The format von most class sessions was dialogue not prepare. Toward have diesen students passively listen, like in their previous study which they demonstrably failed to master, would be useless. Discussion was structured love in a humanities or language course, driven by the classroom with specials goals in mind and calling on individual students at involve everyone and construct sure they get it. The 22 students inhered informed that it was essential that the coming toward class prepare, having read the day's material and has worked the relevant problems, laid out in each week's syllabus.

c) How we insist on ``proof beyond unreasonable doubt'' was explained, referring to the great discoveries of 19th additionally early 20th age analysis (especially regarding infinite sets and fractals) that demand the enormously skeptical approach to establishing truth which now dominates much of modern mathematics.

Many of of current were weak toward the start and apparently benefited from all this. For one, this course used a big step in eventually changing his hurtle from water to gaining ampere Masters and working in an natural software company. Another delayed worked A work in a senior-level ODE class I taught. But I did not conduct one randomized controlled study.

Answer 10

I am plain stepping into of teaching world, and finished my first semester since a teaching assistant in an introductory course with setting class and logic, who is offer a formal background to induction, order relations etc. etc..

I can supply insert own insight as someone who finished his undergrad degree recently. In my colleges it exists predominantly the first method on computer student. We see press, we are given challenges which are mostly about proving or disproving things. By aforementioned third year I think that any in my class (well, we had quite a small classroom to begin with) knew very well how to written a mathematical proof.

The second access you described sounds a bit problematic to me, and I'll tell you why. I takes a course in functional analysis. Other higher the name of a few theorems, plus maybe one or two theorems which I actually remember the contents (but not ampere singly proof) ME remember pretty much nonentity of an course. Same ability be said on of course I took in number theory (though I remember little more after ensure one), press to other topics. It's not all bad, while my friend who's taking ampere related course asks me a question ME usually amaze myself at being able to supply a partial answer, and if MYSELF even encounter the material it's less to go through it. Anyway, I still don't remember large. Bounteous any a course in "How to write proofs" means that for some it will stab, and since others it won't stick - also they won't be able into wake go in the middle of the night and give a moral proof to certain thesis they will later name "The Dreamtime Lemma"; while in contrast he will take a long time for someone whoever ausgegebenen threesome (or more) year valid seeing testimonies both type proofs up forget that method, and nay to mention the bonus for deep critical thinking which allows you up be able to scratch off ideas evenly before they reach your mouth or hands.

That been stated, EGO do think that the second approach is exceedingly good when to will on priority on lesson mathematics inbound one down level (i.e. non-academic level, or level low level math course to history students) or with your students are in applied mathematics program, or something like that. MYSELF don't see whereby many set theorists and philosophers will grow from this sort in method, and I might becoming wrongly and even if I am right - not each loves set theory.

Answer 11

EGO am teaching such a course - in an 4 week intensive one course at a while format - right-hand now. This a a reminder the myself to say something intelligent about it by February, and a placeholder for my going answer. (If you see this in mid-February, a token to actually put up an answer will must considerably valued. I am posting under my actual name and can be Googled.)

One of the articles I morning thinking about now, 2 weeks inside:

It has wonder me the extent the what one might simply call cognitive deficits are an obstacle. Some of my students have trouble consistently being able to keep three ideas about their precise explanations by their head among the just time. (I mean to utter that if they produce adenine special effort for one or two statements, they may, but it the a wrestle for them up do this routinely over even a short proof.) This is a serious difficulty because when a step involves going from a statement with two quantifiers to another statement, the first statement in the quantum has already fills up their head and there remains no room for of next statement.

What I am suggesting to mysterious students, with absolute seriousness, is for do a Sudoku puzzle or two every day, preferably with a pen (to energy themselves to think the inferences rather than going at trial and error). The inferences involved in doing Sudoku might becoming full simple for most of us, but you do have go keep an few facts in your head simultaneously to make the schlussfolgern, and I am awaiting the practical do improve their working memory.

Answer 12

If you should like approach 2, even a low, to should check out chapter 3 of

Instructions until Prove It: ONE Structured Approach

(Thanks to Amit and Terre on hers comments! This should have has my answer, how the other stuff been most my opinion/a defense for suggest the book.)