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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 converged to some number $x = 1 + 2 + 4 + 8 + \dotsb$, then clearly this number $x$ would have to satisfy $2x = 2 + 4 + 8 + \dotsb = x - 1$. Solving $2x = x-1$ ...
82
I've been using "pets" and "owners" (as in: possible pet-shelter adoptees) in recent years.
79
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
$$x^{x^{x^{\dots}}}=4.$$
Uniqueness can be proved like this: since for such an $x$ we ought to have $x^4 = 4$ we get that $4^{1/4} = \sqrt{2}$ is the unique candidate for ...
78
A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol "=" is essentially operational, not relational — they think it means "the next step" or "the answer" or is an instruction to perform some operation. Knuth et al. ("The importance of equals sign understanding in the middle grades", Mathematics ...
77
In the stable marriage problem, you can introduce the problem as it is. But then you ask your students how things change if you assume there are not only heterosexual but also gay and lesbian people (assuming that a heterosexual person will never marry a person of the same gender, and a gay or lesbian person will never marry a person of the opposite gender). ...
63
A few possibilities off the top of my head:
Students and chairs. How many ways are there for $n$ students to sit in $k$ chairs. The game of musical chairs might be fun to play around with. One can also consider natural restrictions, such as myopic students who need to sit near the front.
Replace "men" and "women" with faculty from different departments. ...
58
Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of calculus predated formal definitions of the real numbers (Dedekind cuts, Cauchy sequences) and a formal definition of continuity, individuals must first make ...
56
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 might take the square on both sides and get the equation
$$2x-1=x^2,$$
which has a unique solution $x=1$. Substituting back in the original equation we see that our ...
51
For some reason, the 'extend it to 100 doors and eliminate 98' explanation doesn't make it any clearer for me.
Rather than talk about probabilities as fractions, I explain it this way:
"If you picked the car (without knowing it) on the first choice, you'll lose it by switching, whereas if you didn't pick the car, you'll gain it by switching."
(stop here ...
50
Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations. (I should add that I'm basing this on my experience teaching calculus to non-major college students, but I think the relevant issues are similar.)
Mathematicians have a bad habit of conflating rigor with conceptual understanding. A lot of this seems to come out ...
47
When teaching linear algebra, I rely heavily on carefully constructed examples where everything can be done with rational numbers of small denominator. (It is faintly amusing that the construction of such examples often requires mathematics that is far harder than the actual content of the course.) However, I also show them the following slide:
I also ...
44
The problem you describe is well-known in mathematics education research. I cite the paper of De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50(3), 311–334. and give some ...
44
Yes, showing your students that you too can make mistakes, and that real mathematics is not a linear process, are both very good ideas.
However, this does not mean that you should come to the lecture unprepared!
If I seem a bit vehement about this, it's because I've met all too many math lecturers who seem to feel a need to "prove their manlyhood" (or at ...
43
It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu ("The Mis-Education of Mathematics Teachers," Notices of the AMS, vol. 58, no. 3, p. 376) calls the Fundamental Assumption of School Mathematics.
In terms of the nitty-gritty of classroom instruction, real numbers are handled in K–12 by what ...
40
I don't view these common mistakes as 'universal linearity' assumptions. The mistake that $(a+b)^2=a^2+b^2$ is just a visually appealing statement. It is mistaken to be correct because it looks nice. Our brains tend to like things that look nice. Similarly, $\sqrt{a+b}=\sqrt a+\sqrt b$ is visually appealing and it resembles the correct formula $\sqrt {ab}=\...
39
Such an approach seems designed to force (or at least, strongly encourage) students to learn by pattern-matching from examples. This is one of three modes of student learning in mathematics described in this article by Frank Quinn; it is the least powerful, most fragile, and most error-prone of the three. To quote the relevant passage:
There are (roughly) ...
38
I'm really wondering if this is the end of the world or the beginning of an improvement in the way we teach math.
I'll answer this first, and then talk a little bit about the rest of the question. Please excuse my bluntness. I will try to give you the opinion you ask for by weighing what I think may be most important relative to quality of math education. ...
37
There is yet another issue here, it is related to "expert blind spot", but not the same.
To be a good teacher, one needs to understand the pupils and their issues with the topic. This becomes hard if the teacher thinks in a qualitatively different way. He may know exactly what causes the problem and why, but cannot explain it well, and his messages don't ...
36
The issue is not making problems about heterosexual married couples. The issues are:
Implicitly making the assumption that all married couples are heterosexual.
Making problems about heterosexual marriages but not about other kinds of couples.
Both points are unengaging for people following other types of marriage, but they can easily be solved while ...
35
Whether one can or can not double the brilliant in real life has nothing to do with the Banach-Tarski paradox. Various mathematical objects are models of various aspects of our universe. $\mathbb R^3$ is typically taken as a model of three-dimensional space. But, it is only a model. Some things that are true in the model are false in space and some that are ...
34
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 $f(x) = 1$, we deduce $a+b=1$; taking $f(x) = x$ we deduce $b=1/2$.
Of course, there is no such formula, but this computation does hint at something useful: If ...
33
There's a lot in this brief question, and I would like to try to give a brief answer, so I'm going to pick and choose what I respond to (and others might choose different things). Here are the parts of this question I see:
Is teaching multiplication as repeated addition problematic?
Is the problem with teaching multiplication as repeated addition that it ...
33
Try objects that often occur in pairs but are distinct from each other: forks and spoons (or forks and knives), left and right shoes, salt and pepper shakers, and so on (where each fork has an obvious partner spoon, perhaps sharing the same color or design, and so on).
33
This is indeed a challenge, especially for adults. Three suggestions, none
of which is a panacea.
(1) Emphasize a growth mindset.
Make it clear to them that learning math is a skill accessible to everyone,
with effort. It is not only accessible to those with a
mythical "math gene."
(2) Compare understanding the abstractions of math and its notations as
...
32
Draw a number line and label all the integers.
Tell him that adding $x>0$ is moving $x$ units to the right and subtracting $x>0$ is moving $x$ units to the left.
Tell him that adding $0$ is not moving at all.
Tell him that adding $x<0$ is moving $-x$ units to the left and subtracting $x<0$ is moving $-x$ units to the right.
32
This is a 100% subjective opinion, but it is based on teaching in various venues for close to 20 years (although none of that teaching was pure math). Also, my college calculus courses are close to 30 years behind me, so please excuse me if my examples aren't directly related.
IMHO, one of the biggest mistakes in teaching is failure to compare and contrast ...
29
The problem with induction proofs is that too often the problem is given by "Prove that..."
After a few examples and explanations of induction, if the students know elementary calculus, the following sequence might prove interesting:
Find the first ten derivatives of $x\cdot e^x$.
What seems to be the formula for the $n$th derivative of $x\cdot e^x$?...
29
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 then you still have to check the second derivative to see whether it is, in fact, a minimum.
This is also a very physically relevant example, because it comes up ...
28
Again and again he finds $-4$ greater than $-3$.
Ask him who is richer, he who has a smaller debt $($like $3$ rupees$)$, or he who has a bigger debt $($like $4$ rupees$)$, assuming both persons have no money, just debts.
He has spent several years seeing $4$ greater than $3$.
A debt of $4$ rupees is indeed bigger than one of $3$ rupees. But the one that ...
27
Your explanation, by the way, is very elegant. As an experienced mathematician, I see immediately that it cuts right to the heart of the matter and admits no ambiguity. Unfortunately, this is precisely the quality that makes it unconvincing to others; the main confounding aspect of Monty Hall is that it ruthlessly exploits an intuitive misunderstanding of ...
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