41 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 ...
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37 votes
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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 ...
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  • 1,834
35 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 ...
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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 ...
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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 ...
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27 votes
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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 ...
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25 votes
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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 ...
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25 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 ...
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  • 17k
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 ...
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22 votes
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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 ...
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22 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 ...
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20 votes
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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? ...
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  • 7,686
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 ...
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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 ...
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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 ...
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  • 1,426
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 ...
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  • 169
16 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 ...
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  • 177
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 ...
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14 votes
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Diagram of Methods to Solve Differential Equations

I would just like to mention that other similar flowcharts have been developed, of varying degrees of generality, which you might consult. Here is one (by Adam Monahan). And another (by Jeremy ...
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13 votes

Why do we teach complex numbers?

I think the fundamental tenet of this question is simply false. Here are some of the many encounters that undergraduate college students have in my classes: In Calculus II, as an application of power ...
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13 votes

How to teach Leibniz and Newton's notation

The reason why so many people get the wrong idea about differentials is that they aren't really taught what the notation means. They are merely taught "this is what the notation is, and please don't ...
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  • 919
13 votes

Do I really need to cover solids of revolution in my Calculus I class?

Pardon me ignoring your Calculus question, but there is some beautiful mathematics here, e.g., Cavalieri’s principle. So there is an opportunity to connect the calculus to these "fascinating ...
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12 votes
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As a TA, how to reduce imprecise notations/statements in students' exams

Let me echo Benjamin's comment that any proactive step that you take should be done with the instructor's permission. At a practical level, I think there are ways to address issues (a) and (b). For (...
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12 votes
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Is there a more intuitive way to solve combined rates of work problems?

By throwing in all of the $r$, $t$, and $d$, not to mention $x$, you're overly complicating things, especially if your brother is algebra-averse. There's really only one unknown, and you can call it $...
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  • 663
12 votes

Teaching indefinite integrals that require special-casing

This is a hard question, because students are so used to manipulation of this kind. I have found you are right that absolute values can cause the worst of these examples. Here is an example I ran ...
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  • 5,752
12 votes

Enlighten younger students about the concept of "procedural justice" in mathematics?

On the contrary, many seem surprisingly impatient when being asked to prove 1+1=4/2, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's ...
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