19

From day one. In my experience in Germany, proofs are taken seriously from day one, or even before that. We had a voluntary prep course before the first semester that was half a repetition of calculus (which is a part of the high school curriculum here) and half an introduction to proofs. And the very first homework assignments in analysis and linear algebra ...


14

It's still not known whether $$\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$$ is a rational number.


11

It takes a lot of browsing to find problems somehow related to calculus or analysis, but this is a great MathOverflow list: Not especially famous, long-open problems which anyone can understand. Here are a few from that list: Are there an infinite number of primes $p$ such that the repeating part of the decimal expansion of $1/p$ has length $p-1$? Link. ...


11

Is there any better alternative to the three-dot notation? The usual general advice is to use words instead of symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that ...


10

In my experience (U.S.), that's on the boundary between 2nd and 3rd year -- either the end of sophomore year or the start of junior year. Two years ago I did a survey of Associate in Science (2-year) Mathematics degree programs. It's not common to have a dedicated only-proofs course, but I think many use a Discrete Mathematics course as a vehicle where proof-...


10

You probably get Euler's constant $\gamma$ when you do the integral test comparing $\sum\frac1n$ to $\int\frac{dx}{x}$. Then you can remark that it is unknown whether $\gamma$ is rational.


10

On The Timer I tried a long timer (about 3-5 days) for my exams and final exam last semester. The reason for this is that my favorite assignments from college -- the ones where I learned the most -- were challenging assignments that had a long availability period and allowed incremental work over the course of several days. However, an unbelievably large ...


9

This is a bit obvious I think, but when you introduce sequences and their notation in either an algebra or calculus class, you should certainly show students the Collatz Conjecture as one of the examples.


7

A great start to doing proofs is working through Daniel Velleman's How to Prove it: A Structured Approach, 2nd Edition.. I've used it many times in teaching, usually as a supplementary text.


6

Generally about cheating in tests if you are trying to prevent cheating, trying to limit the student's ability to do it is very hard and mostly counterproductive, as it can introduce frustration , specificity in an online setting. Instead, I recommend to minimize the student's motivation to cheat, after the first time it is just get's easier, if you design ...


5

As a general introduction, the book Theorems, Corollaries, Lemmas, and Methods of Proof by Richard J. Rossi can be useful. For example, how to prove that a sequence converges? A detailed explanation is given on pages 168-170. As a general rule, to get better we have to do a lot of proofs. A lot of sentences to be proved can be found in problem books on the ...


5

In the UK, students usually learn proofs in the first year of a mathematics degree. My experience is similar to Sumyrda's answer. They also gain some exposure to proof techniques before university in A-Level Mathematics and Further Mathematics, which include proof by contradiction, trig proofs, elementary algebraic proof and proof by induction.


4

From my experience in French Classes Préparatoires, we learn proofs during year one without a specific course about it, just while we learn calculus and linear algebra, starting on day one (we actually start a bit in high school). Maths in these classes are very rigorous, and everything that is taught gets proven (with few exceptions), even requiring to re-...


4

This answer is meant to supplement Daniel R. Collins' answer, which is excellent—my goal is to draw a little more distinction between the "levels" of education and training in the US. Short Answer In the United States, "proofs based" courses (and formal proofs in general) are typically regarded as topics in "higher mathematics", ...


4

Daniel Solow, How to Read and Do Proofs. Wiley, 6th Edition, 2013.


3

Aside from any particular book, I'd say that you need a human being reviewing and giving feedback on your proofs. This is a type of writing for consumption by other people. One of the main things is that a proof should be clear, explanatory, and insightful. Partly this criteria depends on the level of expertise of the expected audience. Now, I'm not entirely ...


3

I learned more from Lakatos than any other source about what constitutes a proof, and: the roller-coaster ride adjusting definitions to clarify the proof claim, perhaps realizing that a hidden lemma has a counterexample, the need to reformulate the proof statement in light of these ups-and-downs, and on and on.           Proofs ...


3

I taught all of my classes in an asynchronous (fully online) format this term. Based on ideas posted here, I chose the following routine: Content Delivery: I made videos with my phone and posted them to YouTube for the "lecture" content. I printed copies of activities I wrote, and worked through them in the video, periodicaly asking students to &...


3

Here is my experience: Throughout the term I gave several quizzes with long timer short availability questions all at once academic integrity statement few modifications The average score was around 80 %. For the final exam, I gave a quiz with majority (around 2/3) of the questions very similar to those given throughout the term (or even the same questions,...


2

Randomize questions Other answers are good, but I think a very important aspect on online quizzes has not been covered: questions should be different for every student - as different as resources and fairness allow. Questions can be randomized in several ways: Random parameters. Slightly different versions of the same question (e.g. one version with ...


2

Ok firstly I need to debunk your misconception regarding mathematics being easier to self learn than other subjects. Firstly, mathematics does not have an experimental component for the most part, therefore peer review for content online is a natural outcome. Secondly, you seem to be of the generation that takes the internet for granted in some respects. As ...


2

I can only speak to my personal experience, but during my time in Undergrad there was a dedicated proof writing class ("Introduction to Higher Mathematics") that was coded as a 330 course. All courses coded above 330 required 330 as a prerequisite, while everything below it did not. Courses below 330 were Calculus 1, 2, 3; Linear Algebra, and (I ...


2

Naval Academy has their proofs class spring of sophomore year. It is required for either applied math or pure math. https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMA.pdf https://www.usna.edu/MathDept/_files/documents/majorMatrices/SMP.pdf They have it before linear algebra.


2

At my school, UIUC (https://illinois.edu/), we have a dedicated proofs course for CS majors (CS 173: Discrete Mathematics) and a dedicated proofs course for math majors (Math 347: Fundamental Mathematics). CS majors will typically take CS 173 in the first or second semester of their first year, and Math majors will take Math 347 before the end of their ...


1

I'm in Scotland and did rigorous proofs in the first year of my degree (Physics) as I sat the full first year maths syllabus. Although, in the final year of high school we were introduced to some simple proofs such as sqrt(2) irrational, and had discussed logic and different proof methods such as contradiction and proof by induction.


1

The context isn't entirely clear so I'll assume this is about teaching. Then, I support Pedro's answer but also want to add that doing both verbal and symbolic versions may be a good idea. For example: Theorem. A polynomial has a higher order than another if and only if its degree is higher. In other words, for any two polynomials $P$ and $Q$, we have: $$P=o(...


1

The hyperboloid of one sheet appears to be very curvaceous everywhere, and yet it is actually a ruled surface (i.e., can be traced out from a straight line moving in space).


1

Isn't the "functional equation road" the same thing as the "differential equations road"? You can take an axiomatic approach to defining sine and cosine. See Apostol's Calculus book, page 95. Many other authors have done a similar thing and the background isn't too steep. You simply declare that there exist two functions satisfying a few properties (which ...


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