20

I always organize my courses with the totality of the course set from the outset whenever I can. I see this as being closely tied to your question concerning homeworks assigned. The benefits I see: Keeps me and my students on track. The semester invariably gets busy, it's nice to have a go-to place where everything is set from the outset. I can always ...


19

If $\gcd(a,b)=1$, there exists a multiplicative inverse for $a$ modulo $b$. (Otherwise, look at the $b-1$ multiples of $a$, namely $a,2a,3a,\dots,(b-1)a$. They must fall into congruence classes that aren't 0 or 1, but there are only $b-2$ of those.) $R(3,3)\leq 6$, and other Ramsey-style arguments Give any domino tiling of a $6\times 6$ checkerboard, there ...


16

I would not recommend putting false proofs onto the board unless you immediately (within the same class period) point out their falsity, and make an assignment to find it. For smaller mistakes that I make unintentionally, I give the students 'donut points'. 30 donut points (over the semester) and I bring in donuts for the class. I tell them that my main ...


16

It seems that the key term here may be the somewhat non-specific-sounding special functions. By googling for a few examples (Erf, Si, Li) I came across a Table of Special Functions and, on the Lists of integrals wikipage, there is a sub-section on Special Functions. As a related remark, one reason that functions may be presented and/or defined in terms of ...


16

The gamma function is very useful in counting problems (among others) and is seen as an extension of the factorial function into the reals. It is defined as: $$ \Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,. $$ (Incidentally, this is the example of how to use MathJax in the help section.)


15

Since your question is very broad, here is a somewhat broad answer: Read about problem posing. Three key pieces are: Silver, E. A. (1994). On Mathematical Problem Posing. For the learning of mathematics, 14(1), 19-28. and the book Brown, S. I., & Walter, M. I. (2005). The art of problem posing. Psychology Press. The latter is a re-print of a ...


14

Edit (Dec 2016): Encouraged by a few comments on SE, and a few direct emails about this post, I wrote up the ideas below for a journal of math education. The citation, and linked pre-print, are: Dickman, B. (2017). Enriching Divisibility: Multiple Proofs and Generalizations. Mathematics Teacher, 110(6), 416-423. Pre-Print (no pay-wall). (Adapted from a ...


14

A place that Lagrange Multipliers comes up is in the proof of the real spectral theorem. Namely, let $A$ be a symmetric matrix. Define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(v) = v^\top A v$. If you maximize $f$ on the unit sphere in $\mathbb{R}^n$, the Lagrange Multiplier condition will show that the maximum is achieved at an eigenvector of $A$. This is ...


13

Yes! I have used these a lot in an "intro to proofs" course. Typically, each weekly homework assignment has at least one problem of this variety, and I've written many like this for assignments and a text. Some thoughts about your posed questions and other ideas: Does this help? I think yes in several ways, but it's hard to tell. I don't put questions like ...


12

Dirichlet's theorem that an irrational number can be approximated to within $1/q^2$ for a sequence of rationals $p/q$ exemplifies this principle.


12

Application 1: Every rational number has a repeating decimal expansion. Application 2: Each infinite decimal expansion has the property that there exists a $10^{100}$-length sequence of digits that is repeated infinitely often in the expansion. Application 3: If $x$ is irrational, then at least two digits appear infinitely often in the decimal expansion of ...


12

My first take is $$ \ln(x) = \int_1^x\frac1t dt. $$ Granted, some texts introduce the natural log of the inverse of $\exp$ but other texts define $\ln$ as above and the $\exp$ as the inverse. If I remember correctly, the definition of the logarithm by the integral was historically first.


11

Problem Each point in the plane is colored one of $n$ colors. Prove that there exists a rectangle whose four vertices are the same color. Both the problem and the solution are very simple, yet those unfamilar with the Pigeonhole Principle would likely be at a complete loss to solve it. Solution Consider a grid of points with $n+1$ rows and $n^{n(n+1)/...


11

The heuristic described here is one manifestation of what Polya (1945) and others thereafter refer to as trying a special case. I do not know of a more specific term for the context that you have put forth, but this is often how one approaches a problem if you are initially unsure as to how to solve it in generality. It is a good way to get an initial foot-...


11

Here are some suggestions for problem sources in English. Some of them are appropriate for very bright students studying geometry or Algebra II, but might nonetheless prove too difficult for students accelerated to this extent. -Mathematical Circles, Fomin et al. -Mathematical Problems: An Anthology, Dynkin et al. -Problems in Elementary Mathematics, ...


11

The function $\displaystyle \text{Li}(x) = \int_2^x \frac{1}{\log t} \,\, dt$ comes up in the study of the distribution of primes. Specifically, the number of prime numbers less than $x$ is asymptotic to $\text{Li}(x)$ and a major consequence of the Riemann Hypothesis would be the sharpest possible bound for the difference between these two functions as $x \...


9

Here is one application in introductory abstract algebra: A finite integral domain is a field.


9

In the example you mention of computing truth tables rather than use algebraic manipulations to answer questions about boolean expressions/sets, it's actually a wonderful situation where the students chooses the long and tedious way, not noticing a much more convenient method works much more nicely. In such cases, I let the student waste time getting the ...


9

There are a number of resources I like out there. I tend to use a variety of different sites for different needs -- I don't think there is one that "does it all" but here are some of my favorites. Most of them are pretty random, though the first few below that have lots of customization options tend to make very usable worksheets, and for some applications ...


8

My method for creating a concept-testing question is as follows. Start with the most open-ended question possible on the topic, then slowly refine it until it has a "correct answer." I've outlined an example for a conceptual derivative question below. Start with the generic question: "Tell me about the derivative," and recognize that the generic question is ...


8

I think there are several good problems that can be explored (e.g., using the Moore method) by beginning with a word or term and trying to axiomatize it. I happen to think that assembling several of these words/terms and axiomatizing them would make for a nice textbook, but I digress... Back to the question: Some examples. What should "bigger than" mean? (...


8

I did this for an introductory calculus course at a US state university. My reason was that I wanted to assign more homework than my colleagues typically do. On the first day of class I warned students that there was a lot of homework, explained why I think it's important, and told the students that they could see exactly what was expected of them. The same ...


8

Nice question! Let me add one reference to your list: Silver, Edward A. "On mathematical problem posing." For the learning of mathematics (1994): 14(1) 19-28. (PDF download link.) Silver cites Hadamard's famous book, The Psychology of Invention in the Mathematical Field, as recognizing that isolating key research questions is a sign of exceptional ...


8

Wondering Why are induction proofs so challenging for students?, I thought of trying this as a possible assignment after introducing induction: Find the flaw in this induction proof. Claim: $3n = 0$ for all $n \ge 0$. Base: for $n=0$, $3n = 3(0) = 0$. Assume Induction Hypothesis (IH): $3k = 0$ for all $0 \le k \le n$. Write $n+1 = a + b$ where $a > 0$ ...


8

No one has still mentioned Fresnel functions: $S(x)=\int_0^x \sin(t^2)dt$ and $C(x)= \int_0^x \cos(t^2)dt$ They are (of course) very relevant in signal analysis and in studying diffraction. What is less often mentioned is that the parametric function $x\mapsto (S(x),C(x))$ gives you a beautiful curve, the Cornu spiral, which is used by engineers in roads ...


8

A possibility, requiring one definition: What is a tiling of the plane with an infinite supply of congruent copies of a single tile (technically, a monohedral tiling). This can go as deep as you'd like, perhaps in stringing together several mini-sessions. Can every triangle tile the plane? (Yes.) Form parallelograms, then argue that a parallelogram tiles ...


7

For me there are perhaps three main types of problems which I assign: Routine skill building: either are modeled on a computation which I have shown similar problems solved, or, are a proof problem which is just a natural consequence of definition with little extra technique required. For a proof course, many problems are little more than an invitation to ...


7

This only works in some cases: Ask not only for the final result but also some intermediate results which are only produced by the new method. This way the new method becomes more feasible, since the students are required to use it anyway to produce the intemediate results. In your example on finding extrema, you could, e.g., ask your students to also ...


6

Here are a couple problems to shatter misconceptions about homomorphisms, while introducing the student to constructive thinking in group theory. Are the following statements true? Prove them or provide a counterexample. If $K_1$ and $K_2$ are isomorphic subgroups of $G$, then $G/K_1$ is isomorphic to $G/K_2$. If $\varphi:G\rightarrow H$ is a ...


6

I disagree with the "diagonalize a matrix" type exam questions. Such questions call for accurate, longish computation, not understanding. That I leave for homework. In an exam I'd ask why you'd diagonalize a matrix (e.g. explain how to do something using this, step by step, not do it). Set questions up so that computation mistakes don't invalidate the work. ...


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