46
votes
How do I show students the Beauty of Mathematics?
Imagine you are put in jail. You are forced to paint a painting every day for 10 years. You have no choice in the subject: one month you paint dogs, another month you paint horses, another month ...
- 23.5k
42
votes
‘Lies to children’ in mathematics and statistics education
Young children 5-8 years old, are taught to subtract the smaller number from the bigger number. They are told that you can't subtract a bigger number from a smaller number. This lie has its ...
- 7,929
41
votes
Do I really need to cover solids of revolution in my Calculus I class?
An operation is born when we recognize the regularity in repeated reasoning.
Take multiplication for example. If we are living lives which involve even a modest amount of arithmetic thinking, we will ...
- 23.5k
37
votes
Accepted
How do I show students the Beauty of Mathematics?
To expand on my comment, I found that high school kids like watching YouTube videos. (I mean, they don't have to do any work right? Just sit and listen.) These are a few of my go to channels to pull ...
- 2,062
31
votes
‘Lies to children’ in mathematics and statistics education
We usually teach:
$$\int\frac1xdx=\ln{|x|}+c$$
Whereas it should be:
$$\int\frac1xdx =
\begin{cases}
\ln{x}+c_1 & x>0 \\
\ln{(-x)}+c_2 & x<0
\end{cases}$$
Why don't we teach the correct ...
- 619
30
votes
Accepted
How does one tutor an A-level student past the derivative paradox?
There is no royal road to geometry. - Euclid
Nor calculus.
The essence of calculus thinking is really the limit concept. One needs to wrap one's mind around that. Formally: it's the core technique ...
- 23.6k
29
votes
Why do we teach complex numbers?
We owe students a presentation of the Fundamental Theorem of Algebra -- that every nonconstant polynomial has a root; or, equivalently, the marvelous fact that every polynomial of nth degree has ...
- 23.6k
28
votes
How does one tutor an A-level student past the derivative paradox?
I teach calculus at a community college in the U.S. (2 year college, from which many students transfer to a university). I explain limits from about day two in an informal way ("h gets infinitely ...
- 19.4k
27
votes
How should a student's inefficient calculation be pointed out?
Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just ...
- 23.6k
27
votes
Accepted
Students understand during course but can't solve exam
Do NOT give exam questions that are intentionally more challenging than homework or in-class problems. I would recommend precisely the opposite.
The point of the exam is really a spot-check that ...
- 23.6k
24
votes
How to convince students of the implication truth values?
I make the statement "If it is raining, then I have an umbrella." Did I lie?
If it is raining and I do not have an umbrella, then I lied.
If it is raining and I do have an umbrella, then I didn't ...
- 10.7k
22
votes
Accepted
Students strictly follow the steps and notations in sample problems without understanding them
The 25% of students did not think of the problem as easier. It was harder because it did not exactly follow the form of the example problems. As such, it required some (little tiny bit) of creative ...
- 23.5k
21
votes
teach that $\frac10$ not defined properly
What is $\frac 1 a$? It is the unique (real) number such that $a\cdot \frac 1 a=1$. Does there exist a real number that multiplied by $0$ gives $1$? No. Why is this? Because if $0\cdot b=0$ which ever ...
- 1,705
21
votes
‘Lies to children’ in mathematics and statistics education
The idea that “a number” means “this decimal expansion”, rather than the expansion being a way of representing a number that has some more set-theoretic definition. It's the de facto truth for ...
- 319
20
votes
Accepted
How should a student's inefficient calculation be pointed out?
I like your second option the best:
...wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part?
...
- 8,856
20
votes
Teaching indefinite integrals that require special-casing
I'd avoid giving problems like that to students first learning indefinite integrals (either by not asking it at all, or specifying the range x>1 in the question). It's a subtle algebraic trap, and if ...
- 11.5k
19
votes
Do I really need to cover solids of revolution in my Calculus I class?
A point to consider that has not been emphasized much in other answers so far: removing a topic from a syllabus in a service course should not be done before getting input from instructors of other ...
- 3,048
19
votes
How can a teacher help a student who has internalized mistakes?
Coach them through doing it the right way. Have them repeat it the right way, several times. In front of you. And go very easy, including repeats. Gradually relax the guardrails and keep drilling. ...
- 207
18
votes
‘Lies to children’ in mathematics and statistics education
The average
where the lie-to-children is the word "the".
Ask anyone what "the average" of a set of values is, and immediately you'll be told the arithmetic mean. That's how it's ...
- 399
18
votes
‘Lies to children’ in mathematics and statistics education
Whether or not the derivative $\frac{dy}{dx}$ is a fraction. Similarly, what, exactly, are $dy$ and $dx$?
This actually goes through several iterations of lies:
We first hammer it into Calc I ...
- 1,119
17
votes
Is it a good idea to have one or two or three classes on basic logic before teaching $\varepsilon$-$\delta$ in Calculus?
Your assumption that teaching calculus needs to be backed by the $\varepsilon$-$\delta$ definitions could be challenged, but since it is not your question I won't do that here.
My recent experience ...
- 8,989
17
votes
How does one tutor an A-level student past the derivative paradox?
We routinely get questions related to pushing more rigor in early calculus. Usually from outstanding students and based on sample of one I like it that way logic.
There's a reason why things are the ...
- 187
16
votes
Why do we teach complex numbers?
I am surprised that nobody has mentioned differential equations. If you know about complex numbers and Euler's formula then there is a beautiful unified theory of linear differential equations with ...
- 1,506
16
votes
The Future of Worksheets - will they still be used or abandoned?
It's hard to tell what will be the changes in 10 years. Maybe we will all have jacks in our heads. Then again, some things change slower. (Where's my flying car?)
Most of the reasons for getting ...
- 169
15
votes
Is the current education system as bad as most critics and famous pure mathematicians try to convey?
I think this question is, probably accidentally, responding to a strawman argument. It presumes that the criticism of math education coming from pure mathematicians is some combination of "students ...
- 11.5k
15
votes
‘Lies to children’ in mathematics and statistics education
"Random variable."
...because, as we all know, a random variable is neither random nor a variable. It is a real-valued function. But if we tried to introduce the concept, the feeling, of a ...
- 1,472
15
votes
‘Lies to children’ in mathematics and statistics education
I was introduced to the real numbers as "all the points on a [two-sided infinitely long] line".
At best this is a circular definition. It's certainly very sloppy, and those words could be ...
- 413
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