54

Some of your students will become engineers, and engineers use complex numbers all the time, e.g., to represent impedance. This kind of thing is by far the most common application. Complex numbers are also used in quantum mechanics. after algebra II, they never use complex numbers until pretty much complex analysis. I assume you mean "they never use ...


27

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 occurred. Otherwise I would let them complete their process (unless obviously totally infeasible) and then compare to a faster way afterward.


26

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 precisely n roots (including multiplicities). When made, it serves as a capstone and culmination of all the work that the student has done in elementary algebra. Of ...


25

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 students know the basics and aren't just faking their way all through the class. If there is a time-limit, then that is already a lot more pressure/high-stakes ...


24

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 lie. If it is not raining, then it doesn't matter whether or not I have an umbrella; I still did not lie.


22

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 thought and understanding. This was an accident. I recommend learning from this, and asking more such questions. The students who are actually thinking ...


21

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? Instead of just mentioning the easier way, however, you could first applaud them for using good algebra to find the right answers, then remind them of the new ...


19

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 the goal is to teach students the mechanics of integration, it's going to be distracting rather than helpful. It might be an interesting question in a more ...


18

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 about a few logic classes first has been disappointing. It took much hard work, and the outcome seemed good at first, but vanished as soon as we got to the main ...


18

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 is $b$. This is about not being defined. Still... why is $\frac 1 0=\infty$ not so completely wrong? Because they can see that the smaller is $a$ then the ...


16

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 constant coefficients in terms of the characteristic equation. Without complex numbers the theory becomes somewhat ad-hoc, with different solutions depending on ...


15

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 aren't learning sufficiently advanced math", "students aren't sufficiently prepared to become mathematicians", and "students don't appreciate the beauty of math"....


14

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 Higgins): And another (by Enrique Areyan):      


13

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 series, we discuss Euler's identity: $$e^{i\theta} = \cos(\theta)+i\sin(\theta).$$ Still in Calculus II as an application of the preceding example, we derive ...


12

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 (a), make a rubric (either in advance or a running one as you go) which lays out the criteria for awarding points. This allows you to be consistent with how ...


12

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 $k$. It's the number of hours it would take Karen to paint $1$ house by herself. Everything else can be expressed in terms of $k$. Don't complicate things with $...


12

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 into recently, which I hope will help your thinking. Observe that there are two different limits here: $$\lim_{x\to\pm\infty} \frac{x}{\sqrt{x^2+1}} = \pm 1$$ ...


12

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 working knowledge. The practice of starting students out with trivial arithmetic proofs like proving 1+1=4/2 seems to be pretty common, but I'm very skeptical of it. ...


11

There are two formulations for definite integrals: $$\int_{\phi(a)}^{\phi(b)} f(x)\, dx=\int_a^b f(\phi(t))\phi'(t)\, dt$$ and the one you state: $$\int_{\phi([a,b]}f(x)\,dx=\int_{[a,b]} f(\phi(t))|\phi'(t)|\, dt$$ In the second, you do need $\phi$ to be monotone. In the first formulation, you do not need this assumption. Of course when you apply the ...


11

Often in courses like this, the problem is that students are almost exclusively asked to do problems that involve rote computation using procedures they've been taught, not problems that require reasoning about the concepts involved or even constructing their own examples. So, how about assigning problems that require students to come up with examples ...


11

You are right to be concerned that the students are "missing something", but IMO the real problem here is that the question is completely artificial. In any application of this type of integral, most likely $x$ will be known to be either positive or negative, but not both, and only one part of the "either-or" answer would apply. And there had better be a ...


11

One of the charms of graph theory is that people of all ages often enjoy learning graph theory ideas and tools. One place one can read about these ideas is in the book called For All Practical Purposes (I am a co-author) which has gone through 10 editions. The book was designed for college liberal arts students who might have almost no proficiency with ...


11

I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring $k[x]$ than most answerers are. Here are two reasonable definitions: $k[x]$ is the ring of formal expressions of the form $\sum_{j=0}^{\infty} p_j x^j$ with $p_j \in k$ and we require that $p_j$ is $0$ for $j$ sufficiently ...


9

I have a colleague who is fond of asking people this question: As you might guess, a lot of people make the same notational mistake with the Pythagorean theorem, in that the $c$ in the customary formula does not map to the $c$ in this problem. The issue of free and bound variables is legitimately tricky. I might prefer to discuss this complication ...


9

Another classic is the following: A rectangular floor measures $300 \text{ cm} \times 195 \text{ cm}$. What is the largest square tiles that can be used to cover the floor exactly?


9

When introducing functions to a student, I usually give thought to two main methods, each with its pros and cons. Method 1: Use the set definition of the function. This is what you're attempting to do at the moment. The set definition of the function states that a function is a relation between a set of inputs and a set of permissible outputs with the ...


9

It sounds like he is not prepared for a course in Calculus. I would encourage him to go back and retake College Algebra which reinforces factoring and canceling out terms. Sadly, it is going to be an uphill battle for someone in a Calculus course that makes mistakes that most Algebra 1 students don't make. When my students make these mistakes, the only ...


9

This is an answer to the title. Defining APOS & RME framework would make answering the question easier. As Massimo Ortolano mentioned in a comment, l'Hôpital's rule is one tool in a box. Maybe you use it often, maybe rarely, but it is very nice when you can use it. Just the other day, when trying to understand how ultrasound mediated electrical ...


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