103 votes

Unique candidate that fails

The obvious example I immediately thought of is that, if the divergent geometric series $$1 + 2 + 4 + 8 + \dotsb = \sum_{k=0}^\infty 2^k$$ converged, it would converge to $-1$. Proof: If the series ...
85 votes
Accepted

Unique candidate that fails

For a positive real number $x$ consider $$x^{x^{x^{\dots}}}$$ or formally (the limit of) the sequence $a_n= x^{a_{n-1}}$ (and $a_0= 1$). Determine the value $x$ (if it exists) such that $$x^{x^{x^{\...
  • 7,572
80 votes
Accepted

Why are induction proofs so challenging for students?

The following list comes from a combination of reading various research articles and my own experience helping students in my Maths Learning Centre for the last seven years. Some reasons why ...
59 votes

Unique candidate that fails

Here is a more elementary example. Sometimes, extraneous solutions in algebra are of the type you describe. For instance, to solve the equation $$\sqrt {2x-1}=-x, \quad (x \in \mathbb R)$$ we ...
59 votes
Accepted

What is a good method for drawing a Möbius band on the blackboard?

Draw the bottom three-quarters of an oval: Flesh that out to make the bottom half of the strip: Connect one of the open ends at the top to the bottom on the other side: Now draw a straight line ...
52 votes

What is the point of teaching variance?

Actually, your definitions are backwards: the standard deviation is the square root of the variance. In other words, one defines variance first --- it has a simpler formula, and it has simpler ...
46 votes
Accepted

How to deal with a "protest" assignment?

You need to slam him on the grade. That is what he earned. Don't be so easily manipulated by his comments on your teaching. Also I would not have sent an email apology. Just offered to meet with ...
  • 484
42 votes

Why do we teach even and odd functions?

One of the major themes of precalculus is what I call “connecting geometry to algebra”. Being able to translate between an algebraic statement like $f(x)= f(-x)$, and the geometric statement that the ...
40 votes

How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?

In a comment, the OP has suggested that he actually wants a practical example convincing students that the product of two negative numbers is positive. This is related to, but psychologically ...
40 votes

Should college mathematics always be taught in such a way that real world applications are always included?

I have worked with a lot of students coming out of courses such as yours who: passed the course by blindly memorising proofs, theorems, and algorithms; learnt nothing (lasting) except solving some ...
  • 2,411
38 votes

Unique candidate that fails

If there were a linear formula for $\int_0^1 f(x) dx$ in terms of $f(0)$ and $f(1)$, it would be $\int_0^1 f(x) dx = \frac{1}{2}(f(0) + f(1))$. Proof: Suppose $\int_0^1 f(x) dx = af(0)+bf(1)$. Taking $...
36 votes

Why are induction proofs so challenging for students?

In my experience, the biggest issue is that students don't have a clear grasp of quantifiers, so they don't see the distinction between "for all n P(n)" and "consider an n such that P(n)". This leads ...
36 votes

What is the best way to intuitively explain the relationship between the derivative and the integral?

You start by noticing that the Riemann sums (multiplication followed by addition) and the difference quotients (subtraction followed by division) undo each other. Their limits -- the integral and the ...
35 votes
Accepted

Beautiful planar geometry theorems not encountered in high school

Given that I had a good time coercing Mathematica to give me the pictures below I might as well promote my comment to an answer. My favorite such a result is the so called Steiner's porism. It is a ...
34 votes

Unique candidate that fails

A very simple example (if you know calculus) is finding the local minimum of a downward-opening parabola, like $y = -x^2$. The unique candidate comes from setting the derivative equal to zero, but ...
  • 509
34 votes
Accepted

How should I grade true-or-false questions if the student's writing is unclear?

I read them as TTTFT. But the fourth is very hard to tell, the first somewhat hard to tell, and all show poor writing, perhaps indicating a lack of engagement. You could tell him that such terrible ...
  • 356
33 votes
Accepted

What is the proper way to ask a "find the domain" question?

It's not really a question about functions and domains, but about valid expressions. The question is, "For which real numbers $x$ is the following expression valid/well-defined/well-formed?" So, for ...
  • 4,793
31 votes
Accepted

How to Teach Adults Elementary Concepts

But, when teaching adults, I've found that I can't just tell them "this is the way it's done, get used to it." Good! Students (at any age) should never be satisfied with "This is the way it's done,...
  • 16.5k
31 votes
Accepted

Is there a simple example that empirical evidence is misleading?

There are some collections of such examples at sister sites: Conjectures that have been disproved with extremely large counterexamples? at Mathematics Stack Exchange. Examples of eventual ...
  • 2,191
31 votes
Accepted

Why is the concept of injective functions difficult for my students?

I think you will find that almost everyone has this problem when they first starting to learn rigorous mathematics, and many students will never overcome this difficulty. The following three ...
30 votes

Why are induction proofs so challenging for students?

There is a fair amount of research on students' understanding of (and difficulties with) proof by induction. Some good places to start: Palla, M., Potari, D., and Spyrou, Panagiotis. (2012) ...
  • 16.5k
29 votes
Accepted

What do math majors (actually) do after graduation?

From the UK: Prospects offer some very basic stats. This CMS report has some graphs broken down a bit further, from p36 onwards. For instance, the graph below shows that of majors in math, ...
  • 5,576
28 votes

Unique candidate that fails

The "Only Critical Point in Town" test Suppose I have a nice function $f : \mathbb R^n \to \mathbb R$. Suppose it has only one critical point, and that is a local maximum. Then (of course) it is ...
  • 6,418
28 votes
Accepted

What is a good reason to change calculus texts?

I would like to encourage consideration of a free textbook. The conventional textbooks are outrageously expensive. (Actually, if your department insists on one of the choices you mentioned, I'd want ...
  • 18k
28 votes

Tutoring a recalcitrant/awkward/exasperating student---special needs?

How do I reach [this] kid? Let me be blunt: You probably don't. This is a person who is so intransigent that you effectively need to black-tag them. A hard lesson is that you can't save everyone. At ...
28 votes

(How) Do American undergraduate math programs teach complex numbers?

The following is anecdotal, but based on experience as a student and instructor in American high schools in three states, as well as my undergraduate and graduate experiences. High School: In the ...
28 votes

Getting students to actually read definitions

In my experience, students are often predisposed to "learn" by memorizing facts; that's how much of their early education worked, so that's what they're used to. When you give them a ...
28 votes

Combative students in proofs classes

You don't say in the question what kind of school this is. It must be a four-year school rather than a community college, but there is no indication of what its admissions standards are like. If this ...
  • 297
28 votes

What are some research-level opportunities in mathematics that do not focus on proofs?

I think a more correct view is that proof is the LAST of several stages involved in researching something in math. What follows is a quickly sketched out scenario of what is often the case. Before ...

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