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 ...
Steven Gubkin's user avatar
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 ...
Amy B's user avatar
  • 8,037
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 ...
Steven Gubkin's user avatar
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 ...
ruferd's user avatar
  • 2,091
32 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 ...
Dan's user avatar
  • 841
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 ...
Daniel R. Collins's user avatar
30 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 ...
Sue VanHattum's user avatar
  • 20.2k
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 ...
Daniel R. Collins's user avatar
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 ...
Daniel R. Collins's user avatar
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 ...
Daniel R. Collins's user avatar
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 ...
JRN's user avatar
  • 10.8k
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 ...
Steven Gubkin's user avatar
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 ...
Nicola Ciccoli's user avatar
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 ...
LizWeir's user avatar
  • 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? ...
Nick C's user avatar
  • 9,406
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 ...
Henry Towsner's user avatar
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 ...
KCd's user avatar
  • 3,456
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. ...
guest's user avatar
  • 207
18 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 ...
guest's user avatar
  • 197
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 ...
Graham's user avatar
  • 409
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 ...
erfink's user avatar
  • 1,129
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 ...
Benoît Kloeckner's user avatar
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 ...
John Coleman's user avatar
  • 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 ...
guest's user avatar
  • 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 ...
Henry Towsner's user avatar
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 ...
Alecos Papadopoulos's user avatar
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 ...
Patrick Stevens's user avatar

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