# Tag Info

56

I'm a LaTeX user, but I'll stake out a devil's advocate position against this proposal. Reasons: Quality of mathematical thinking neither causes nor results from using a certain piece of software. Tech ed belongs in tech ed. K-12 education should mainly be about enriching people's intellectual lives and creating the level of education that makes it possible ...

50

Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations. (I should add that I'm basing this on my experience teaching calculus to non-major college students, but I think the relevant issues are similar.) Mathematicians have a bad habit of conflating rigor with conceptual understanding. A lot of this seems to come out ...

42

You're right. The random, anonymous person you met online is not competent. This is basic mathematical literacy, as taught in every freshman chemistry and physics class.

39

The product of two numbers should be given with as many significant digits as the least precise of the numbers multiplied (see https://www.nku.edu/~intsci/sci110/worksheets/rules_for_significant_figures.html). 1.60934 km/mile has six significant digits (or, if a mile is defined to be an exact number of km, then the conversion factor has an infinite number of ...

30

I would say the standard implementations of the Rational Root Theorem (make a huge list for the sake of making the huge list) indeed feel like a complete waste of time. However, the theorem can sometimes combine nicely with technology tools. I'll provide an example of this. Consider the following algebra puzzle: Solve for $x$: $7x^3 -39x^2+52x+30 = 0$. If ...

28

This is not an answer to the posed question, but only an anecdote. This semester, teaching US college students (Discrete & Computational Geometry), I prepared all my assignments in LaTeX, and made available a .zip file of the .tex, .bbl, .bib, Figure/ directory constituting the assignment. Students could submit assignment answers in any form—from ...

27

Here's a joke I like to tell when people could use a reminder about precision vs accuracy: A tour guide at Giza was explaining how the Pyramids were 4507 years old. Someone in the crowd asked: "That's oddly specific. How do we know this?" "Well. I was told they were 4500 years old when I started working here 7 years ago." I'm not sure ...

24

The assumption is, I believe, wrong, it depends on the distribution within the range. There are no reasons for the distribution to be symmetric. It's not wrong, it's an approximation. It's equivalent to the rectangle rule for approximating an integral: https://en.wikipedia.org/wiki/Numerical_integration It's good in my opinion that your kid's text does an ...

22

As an answer so that I can paste in a picture: I think that the problem with the one on the left is that it is possibly "good enough" that someone might think that it is to scale and that apparent regularities are real. (Well, maybe not one of those "right angles".) In particular, someone might become under the impression that the length ...

21

Hilbert's Hotel is a nice thought experiment for explaining results about cardinality of infinite sets and the aleph numbers. I have also used plastic bags to explain the difference between $\varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\},\{\{\varnothing\}\}\}$ etc. to kids. Let an empty plastic bag represent the empty set. Then a plastic bag ...

20

I suggest you discuss the Seven bridges of Königsberg problem (the problem that essentially started the field of graph theory), then discuss the Three utilities problem. For each, discuss the problem first, then introduce the definitions, then perhaps give a sketch of a proof. The first problem allows you to introduce the concepts of vertex, edge, walk, ...

19

There is no value in drawing the figure exactly to scale, but the left-hand figure is inaccurate to the point where it is positively misleading. Since the angle marked 30 degrees is actually drawn greater than 45 degrees, it gives the impression that a is less than 3 (or maybe equal to it, if the student takes the "30 degree" angle as being shown ...

18

Common knowledge The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea. Connections to other mathematics The notation with AB, CA and BC might be something the students have used or will use in less analytical ...

17

Just to play the devil's teacher's advocate here: one can make a point that rounding should be generally avoided but measurement uncertainty instead be expressed explicitly. Specifically, rounding errors should always be much smaller than measurement errors. Now, if you have a figure of 22 miles, I'd interpret this as $(22\pm0.5)\mathrm{mi} = (35.4\pm0.8)\... 16 Here's my advice. I have no teaching experience. Remedy that first before you lay out plans for a 6-month course of study. Find some way where you can teach just for a single day in some way at the high school that you're targeting. After that, find some way where you can teach for a week (i.e., say, 4 interlinked days). I'd suggest that you restart your ... 15 I don't think it should be a base skill for students in general. Have held jobs in engineering, chemistry, military, and finance and never needed it. Nor did my colleagues. Didn't need it for a thesis or science papers either. Just MS Word was fine. (I think I did enable the MSFT equation editor since it helps with typesetting sub and super scripts on ... 15 When a tutoring student asks me about rounding, I tell them that absent specific instructions from a teacher, common sense should apply. For a conversion, 22 miles isn’t 22.0000 miles, there’s the assumption it’s been rounded. You can’t convert and find yourself with 6 digits of accuracy beyond the decimal. As you note, there’s a number of digits that result ... 14 Is there any better alternative to the three-dot notation? The usual general advice is to use words instead of symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it. A good attitude to the preparation of written mathematical exposition is to pretend that it is spoken. Pretend that ... 12 I study pursuit-evasion games on graphs, so I will recommend using the cops & robbers game as a way to introduce graph theoretic terminology, concepts, and examples. It should also keep the tone informal and recreational, which will do far more (I believe) to actually inspire the students to study more mathematics. Below are the rules of the game, so ... 12 I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion. I will try to come back and write a more elaborated version of this answer later, with diagrams and proper notation, but briefly: (a) Don't let on that you are going to prove the Pythagorean Theorem -- don't ... 12 I think an important learning outcome of this part of high school Algebra 2 (theory of polynomials) is developing mathematical capacity to juggle several disparate abstract tools all at once, similar to how in calculus, one uses first derivative test, second derivative test, concavity, asymptotes, intercepts, end behavior, etc., all at once, in some ... 11 My experience is that weak students latch onto absolute value of a number is always positive. They are fine working with constants. When you introduce a variable, it all falls apart. To these students it is clear that: $$\lvert-x\rvert= x$$ After all the absolute value sign takes away the negative sign. If you want students to understand the absolute ... 11 I found the formula connecting the union and intersection of two sets useful at school. $$n(A\cup B) + n(A\cap B) = n(A) + n(B)$$ Say you wish to find how many numbers from 1- 1000 inclusive are multiples of 10 or 25. This may be phrased as which money amounts up to$10 can be made just from dimes or just from quarters. Let A be the set of amounts that ...

11

This is clearly an opinion based question. I will answer based on my experience and opinion. No, LaTeX should not be taught at high school. It is a skill that is costly to learn with essentially no benefit at high school level. Even at college level, it is not particularly useful. Homeworks are typically submitted handwritten which works perfectly fine. And ...

11

One of the skills that math education (or any education, for this purpose) includes is the ability of the student to learn on his own. Judging by your description, your student lacks this skill. I have also found from my experience with teaching teens that these students often also lack retention skills and do not ask meaningful questions during the ...

11

My technique is pretty low-tech. I distribute solutions to the homework after it's due, so students can mostly tell what they did right or wrong by looking at the solutions. Then I reply to each student's email with any additional comments that they need in order to get feedback that they can't get just by reading the solutions. E.g., #37 -- What went ...

11

I would split the difference by creating an accurate diagram in Desmos and then giving the students a hand drawing based on that diagram. That way, students could estimate their answer before solving and check their answer against that original estimate. But they couldn't conclude that $b=6$ by measuring the length of the known side. Pedagogically, the two ...

11

Agreed...it's one of the less useful parts of high school algebra. But not because "you could use a computer"--you could say that about almost everything. And then we get some of the same people who push the "use a computer" who are surprised when their kids flounder because of lack of manipulational ability in calculus. ;) The reason ...

11

Let $x = \sqrt 2 + \sqrt 7$ prove that $x$ is irrational. \begin{align} x - \sqrt 2 &= \sqrt 7 \\ x^2 - 2\sqrt 2x + 2 &= 7 \\ x^2 -5 &= 2\sqrt 2x \\ x^4 -10x^2 +25 &= 8x^2 \\ x^4 -18x^2 + 25 &= 0 \end{align} So $\sqrt 2 + \sqrt 7$ is a root of $x^4 -18x^2 + 25$. According to RRT, if $x$ is rational, then $x = \pm 1$ or \$x=...

10

Regarding "How do I recruit students:" You should start here -- you have started with this cool thing you want to do, and are wondering how to do it. But you should instead try to find some of these students first. What do they want? Where are they? If you have a core of 1-2 excited parents and 2-3 excited students, then you can start planning. I ...

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