8

Some problems that are easy to understand (but not understand the solution to): Fermat's Last Theorem Four Color Theorem Solving the Cubic and Quartic Equations The impossibility of the Quintic Equation Archimedes' Cattle Problem Sudoku problems (17 being the minimum amount of moves needed in order to be solvable) Theorema Egregium Constructing a 17-gon ...


8

That gives a different result for the cube root of -8. It doesn't give a different result, it just gives two additional roots that are complex numbers, for a total of three roots. A physics or engineering student in the US probably first learns about complex numbers in high school, but never sees any interesting applications. Then in college classes they ...


6

I don't know the sources... but (at my R1 university in math in the U.S.) many students from abroad do tend to use this. My comment on it is that it is all too easy to misread this. That is, it's not stable/robust from an information-theory point of view. And I do advocate writing a narrative (in natural language, e.g., English) rather than a parade of ...


6

This is a complicated question, and there are a number of articles written on the topic in the math education literature. Here are some of the entries that I would recommend (taken from the bottom of this answer): Goel, Sudhir K., and Michael S. Robillard. "The Equation: $-2 = (-8)^\frac{1}{3} = (-8)^\frac{2}{6} = [(-8)^2]^\frac{1}{6} = 2$." ...


5

For problems that are beyond just calculations, if you can algorithmically create different problems for each student that are similar in difficulty, this may help with the near-guarantee that someone will be posting all of your problems to Chegg. Also, it may be done with any of the problem ideas you list above. Here is an exam question I gave this term, ...


4

WolframAlpha is not your problem. It really isn't. The students who are cheating by and large aren't good enough to figure out WolframAlpha's syntax and interpret its answers. Your problem is that any of your students can buy a membership to Chegg.com and upload an image of a question. Chegg.com his hired people in less developed countries who will do the ...


4

I taught a geometry course which counted both for math majors (sophomore level), and was required for the math education program secondary education teacher majors, and we used the text: Edwin Moise: Elementary Geometry from an Advanced Standpoint, 3rd Edition, which was amply challenging for both cohorts of students. For math majors completing this course,...


4

You should know the assigned text perfectly. Read it. Drilled it. Four dot oh. Don't ignore it. After all, it was designed to mesh with the drill problems in it. And if you have the choice, assign a structured text with good explanations, and drill problems. NOT the approach of some liberal arts topic where you just opine on things and assign a bunch ...


3

One of the things I'm currently testing is the addition of "find the error" based problems, and possible variations. For example (related but not the same topic) last semester I taught indefinite integrals, so in a partial test I decided to ask my students the following problem: The integral $$\int \dfrac{3-x}{(2+x)^2}~dx$$ can be solved in ...


3

I hold the opinion that preventing or disincentivizing cheating is practically impossible. If I want to quiz my students about partial fractions or trigonometric integrals or the sorts, I can’t prevent them from putting it into WolframAlpha or the like by just changing the phrasing a bit, because WolframAlpha can solve partial fraction integrals. Instead, I ...


2

Introduction This is some sort of “apology” for teaching homogeneous ODEs. I think there’s a certain beauty and simplicity to them. That beauty is overlooked in most textbooks, which Rota sourly criticizes. Even one of my favorite older textbooks (G. Simmons), which does a better job for conceptual and qualitative development than its contemporaries, ...


2

My copy of Speigel (which has separate chapters on applications, after each theory/calculation chapter) has one problem in "geometry" (really ray optics, finding curve that gives parallel reflection of a point source) that results in a homo first order (non separable) ODE. I didn't check overall first order applictions chapter, but it seems like ...


2

Thoughts: Student can't go back. You have to solve each question to the extent you are able, then move on. If you are unable, you can mark it "return later" But when you return you get a new version of the problem. This effectively eliminates the Chegg problem, as there system has too much latency. This requires submitting your first partial ...


2

The equation $(-8)^{1/3}=-2$ in isolation is taught in early algebra. Later, in precalculus, on learns about the Fundamental Theorem of Algebra. At this point, one starts to understand that this equation should be seen more generally in the context of roots of the polynomial $x^3+8=0$. One learns there are three roots, and falls back on the fact that $(-2)^3=...


1

When would what we learn in high school not serve an undergraduate majoring in physics or engineering? It would begin to be a problem when you enter any field where complex numbers appear. Obvious places where complex numbers appear would be in linear ODEs, Fourier transforms, etc. In quantum mechanics, wave functions are complex valued. In many of these ...


1

I know it's probably impossible for you to completely figure out what it really is that's going to make her happy and the best you can do is get some idea of it. If she could find a way to figure out more about her true feelings, she might get a better idea of it and then if this question explained it better, I might be able to get a good idea of what will ...


Only top voted, non community-wiki answers of a minimum length are eligible