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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.

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 ...

13

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):

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

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: $$\lim_{x\to\pm\infty} \frac{x}{\sqrt{x^2+1}} = \pm 1$$ ...

11

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. ...

9

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 ...

7

To me, the key point here is that the integral runs over a singularity. If you naively calculates a definite form that runs over the singularity you get the wrong answer. This is something I have done enough so that I have taught myself to be careful in this case. I am more a physicist than a mathematician, so what I care about is the connection to a ...

7

Part 1: Do they really understand? My first thought is that you are running into the limits of working memory. As students try hard to understand step 5, they are pushing previous thoughts about step 1 and 2 out of their working memory, before having really processed that information. That ~guarantees they will forget steps 1 and 2 almost immediately, ...

6

One way to introduce group theory to a general audience is to talk about how to flip a mattress so that it is in "a different position each time, so as to pound down the lumps and fill in the sags on all the various surfaces." Brian Hayes has a good discussion of this in his September-October 2005 Computing Science column in American Scientist. You'll ...

6

...what are efficient structures for a revision lecture? Things I've actually tried, sorted by increasing student engagement: Merely provide list of topics for students to study on their own time. This was simple and fast, and it made it clear that they should expect questions from the entire course's material. It also put the review entirely on them to ...

6

I like the idea here, but I agree that it misleads students, and might have the opposite of intended effect. Why not hand out a paragraph to the students, and ask them to critique it. Say that the paragraph is a fake student response to an exam question. One sentence in the paragraph could be something like Since $A \cap B$, there must be an $x$ so that $... 6 Parallelograms are useful for understanding: Paths taken by light, especially through a layer of a medium with a different refractive coefficient Shear, and related deformations Area = height * width (but not necessarily the product of the sides' lengths) Dot products Surface integrals Paths taken by light are useful for understanding which routes people ... 5 Maybe you can motivate the value of proofs by showing seemingly true claims which are in fact false, justifying the need for a proof of even an "obvious" claim. For example, this appears to be a dissection of an$8 \times 8$square to a$13 \times 5$rectangle:$64$vs.$65$unit squares. It takes some effort to uncover the flaw. ... 5 Surprise them. Especially in a (mathematical) culture where "getting the right answers" is prized and excellence on such examinations is valued in general, it is reasonable that students will value this above other expressions of mathematics. (I have a little teaching experience in that context, though most of my experience is in the "just get through it" ... 5 A point (say, in$\mathbb{R}^n$) is a vector. Vectors and points are really no different. They are both$n$-tuples in$\mathbb{R}^n$. The difference between two points (in$\mathbb{R}^n$) is a vector, but a vector has no fixed position. Points are positions in space. Vectors are displacements. It makes no sense to add two points, but it does make sense to ... 4 I think there's a lot of variability in publishing rates between different STEM fields and this can lead to administrators imposing unrealistic demands on Mathematics faculty. Anecdotally I've heard publishing in Biology once a year is pretty straight-forward: just collect a bunch of new data from a new location and publish the results. Maybe even get a ... 4 I am just now close to completing my first time teaching geometry (at a community college, a course that parallels the typical high school course) Using the fact that parallelograms (and hence rhombuses) have bisecting diagonals makes a short proof for why the line segment midpoint construction works. Other than that, I don't see rhombuses as a big part of ... 4 For a much lower-level topic, consider explaining to beginning algebra students why "like terms" can be combined. On a few occasions, I have resorted to reasoning with students that adding algebraic expressions is like adding quantities with units. [Our curriculum begins with units and geometry before algebra, so this is usually safe ground in my class.] If ... 3 This sort of example is not artificial and needs more careful treatment than it often receives. The inherent difficulty of the example is compounded by the tendency to discuss primitives without discussing their domains. The badly named "indefinite integral" of a function is really an incompletely defined primitive of a function. By "incompletely defined" I ... 3 You could say it means "Whenever P is true, Q is true". So "If it rains, I will bring an umbrella" means "Every time it rains, I bring an umbrella". It's not possible to disprove this statement by looking at what happens when it doesn't rain. 3 This depends on how you teach. If you stress creativity rather than memorization and regurgitation then you should be fine with take home. Of course it can be difficult to come up with problems that require creativity and whose solutions can't just found on the internet. Just don't put forth rules that can't be enforced and are likely to be ignored. Assume (... 3 I have not done this, but now that I consider it I think it might be helpful. I have certainly had students who, when applying the quadratic formula, will look at an equation like$3x^2+5x-10$and write$A=3x^2, B=5x, C=-10$. Then they plug those monomials into the quadratic formula and stare helplessly as they try to figure out what to do with all of ... 3 You might be interested in Brian C. Hall's book Quantum Theory for Mathematicians. The author writes on his webpage about the book: This book aspires to be a self-contained and reasonably comprehensive treatment of quantum mechanics (excluding quantum field theory) from a mathematical perspective. No prior knowledge of physics is required, but only the ... 3$A\cap B$is not true or false; but it is true that the intersection of two disjoint sets is the empty set, that is, that$A\cap B = \varnothing$is true. This comes down to a question about whether A\cap B is a boolean statement (a proposition): that which can be answered with "true or false". Asking if$A\cap B\$ is true or false is no different than ...

3

Schaum's outlines are very practical in general and cheap. Well suited to an older learner. Often the answers are right after the problems versus at the end. And you get all the answers, not the odd/even gyp. Thus suited to self learning. I like this one, overall and own it: https://www.amazon.com/gp/product/0070026505/ref=...

3

Maybe a visual approach could supplement your study? There are many such resources available on the web, not in textbooks. E.g., Trig Intuitively:                     Note: the labels show where each item "goes up to." Another: Interactive Unit Circle. Another: Inverse Trig Functions.

3

I personally would prefer a textbook recommendation I can download or pick up that is [preferably] not old and does not make trigonometry intimidating to approach (especially one that emphasizes understanding proofs behind properties/theorems). I don't have textbooks to recommend, but I can recommend an approach to doing trigonometry that facilitates ...

3

Here are GeoGebra materials: Graph Theory for Kids, inspired by Joel Hamkins' notes, to which @A.Goodier pointed.                     Four-color challenge.

3

I shall post my humble and incomplete list of bad explanations I've given or heard over the years: A function is continuous if you can draw its graph without lifting your pencil As you mentioned this is bad, but, depending on the level of the student, it can be a reason for big or small misunderstandings. At a high school level, this simply ignores the ...

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