54

"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each other. Does that seem like it would be a good idea?" is both more inclusive and more correct. I don't even think it's that hairy to talk through the FTA with ...


54

Colloquially, there's a lot of conceptual overlap between all of these terms, but "sameness" is not a well-defined mathematical property. Congruent shapes need not be "the same" or "equal" in all respects - they can be rotated differently, or be in different positions, or be different colors, or have different names, or differ ...


51

I couldn't agree more with @Steve's comment. The following response is written with elementary-to-high-school mathematics in mind. A lack of a decent number sense really does encumber making sense of and parsing word problems, as well as the process of exploring solution strategies. It is akin to interpreting a passage written in a not-so-familiar dialect: ...


34

There was a multiplication table posted on the wall. Like this \begin{alignat}4 1 &\quad 2 &\quad 3 &\quad 4 &\quad\cdots\\ 2 &\quad 4 &\quad 6 &\quad 8 &\quad\cdots\\ 3&\quad 6 &\quad 9 &\quad 12 &\quad\cdots\\ 4&\quad 8 &\quad 12 &\quad 16 &\quad\cdots\\ \vdots&\quad \vdots &\quad \...


30

The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually "the same" or "equal". Usually when we say two things are "the same", we mean that they are not just indistinguishable, but that they are literally the same exact thing. "Equality" means two numbers are the ...


24

I find the ability to estimate calculations quite useful and I think you need to be able do do calculations to estimate them. If you are keeping a grocery budget, I would suggest you should know what your groceries will cost within $10\%$ before they are rung up. To know that, you need to be able to add and multiply in your head. You don't need many ...


22

Yes! But the virtue doesn't lie in being able to do the calculation but in gaining a feel for numbers as well as algorithmic thinking. I teach Computer Science freshmen and one of the first things we need to do is introducing base 2 as well as number systems working with modulo (two's complement). We also introduce basic circuitry to do addition/...


19

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

I taught at the elementary and high school levels. At times we used calculators and at times we didn't. Students benefit from experience both ways. Students need to learn that calculators are only a tool and they still have to think. Students also need to learn that having a calculator doesn't guarantee that their computation will be correct. Finally ...


16

Here's a silly example: Give students collections of the same type of thing, where each collection contains "good" objects and "bad" objects -- for example, a stack of Pokemon cards with both rare and common cards. We might assume common cards are worth $1$ and rare cards are worth $5$. Have ready another stack of cards that are all common -- call this the "...


14

I found out about the Prisoners' Dilemma as a kid from a book about the Harry Potter phenomenon, which had a chapter about the problem, but presented as a story about Harry and Draco being accused of breaking school rules. Each was offered the same deal as in the original problem, formulated with House Points being taken away instead of a prison sentence. ...


13

I wouldn't do that. The parenthesis in use are also used for legal expressions within equations. So you can end with one line containing the same parenthesis meaning different things, what looks like a seed of confusion to me. If you prefer, you can replace the GeoGebra parenthesis with some others i.e. <>, or [] etc. to keep the notation similar to ...


10

The problem with $(4x+7=6x+2)-6x$ is that there is no subtraction operation that involves subtracting a term from an equation. Subtraction involves subtracting a term from a term. So the correct notation is $(4x+7)-6x=(6x+2)-6x$


10

Speaking as someone who has taught college precalculus several times, I have an intense dislike for the way that Geogebra writes this step. In my opinion, it is very important to emphasize to students that we are subtracting 6x from both sides of the equation, which means of course that we are subtracting 6x from two different expressions. If we write "-6x"...


10

FYI: here's some pro and con: http://primefan.tripod.com/Prime1ProCon.html One was originally considered prime. It is prime with the most convenient ("natural") definition. It got excluded from prime-ness because many other higher theorems would be complicated by leaving it as prime. Essentially "prime" -> "prime*". The definition of primeness was ...


10

Brian D. Rude, "The Case For Long Division." 2004. HTML link. This is a somewhat long (unpublished) article (which I haven't studied carefully), but maybe the excerpt below suffices to give the gist of it. Before this excerpt, among his closing sentences are: "But a calculator should be more than a paperweight. Let’s teach for understanding.&...


10

I attended the Computer based math educational summit back in 2016 and found their ideas interesting. I agree with some of their points and disagree with others, but it is certainly interesting to look at the following diagram from their website. They argue, that if we let computers (proper ones, not handheld calculators) do the repetitive calculations, ...


10

In Olympiad geometry, $a$, $b$, $c$ is the so-called standard notation for the sides of a triangle, so it makes sense to use it consistently when referring to a triangle (in isolation). However, in any case the general principle is introducing all your notation. Writing Pythagoras' theorem states that $a^2+b^2=c^2$. or Pythagoras' theorem states that $...


9

It might work better, with this age level, not to be concerned about how large n is when the apparent pattern falls apart. My favorite example is the problem of making all possible straight-line segments between n points on a circle, and then counting the regions. 2 points makes 2 regions, 3 makes 4, 4 makes 8, and 5 makes 16. It sure looks like doubling...


9

I second Sue Van Hattum's suggestion that you should not be so concerned with how large the $n$ is where the pattern eventually fails. I'll go one step further and recommend an example where that $n$ is not only fairly "small" but also such a situation where the students can see why that $n$ makes the pattern fail. Consider the function $f(n) = n^2+n+41$. ...


9

I have thought a lot about this question since posting it, and having read the other answers and the many comments, I want to add a perspective that no one else seems to have given. Most of the real work in doing math is understanding and conceptualizing the problem rather than in computing an answer. This will be apparent to anyone who has read a ...


9

Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet. Don't denote it algebraically at all! Draw a picture instead For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the ...


8

Your question is kind of two parts: one about a convention Is the constant term a "coefficient" and one about a philosophy, which I perhaps find to be a more important question to answer. Isn't mathematics supposed to be non-arbitrary and consistent? Different fields of mathematics have different conventions; this can lead to some mathematicians ...


8

I like Nick C's idea more than modifying the typical formulation. The notion of snitching on a friend, regardless of the severity of the "crime", has real-world ramifications beyond the punishment put out by the authorities. Depending on your student population, that is possibly going to spur a conversation that will overshadow the objectives of your ...


8

How should one talk about the question of 1 or 0 being prime ... with primary or middle school children? Depending on what you did before you will have an easy or a hard task: If the children were told: A prime number is a natural number which cannot be divided by other numbers than by 1 and by itself. ... you will have problems explaining why 1 is not ...


8

Beyond having worked as a programming teacher I have no experience with math education, but this is a topic I have been fascinated with for years. Arguments in favor of mental/manual arithmetic can be typically categorized as: It's important in daily life The argument typically goes that you need to be able to do arithmetic a lot in daily life with the ...


7

You might explain that BEDMAS is not the whole story when it comes to the order of operations. There is an operation called negation. It reverses the sign on numerical quantifies. It gives the additive inverse of any number, i.e. for all $x \in R$, we have $x + (-x) = 0$. Unfortunately, most textbooks use the same symbol for both subtraction and negation. ...


7

I'm going to use GeoGebra to teach equations. Is it OK to let the students write the steps just like GeoGebra (I mean, with parenthesis)? I would not allow this in my class, but it would depend what you think is good/clear/consistent notation for your students. The developers at GeoGebra made a choice for showing algebra steps, but that doesn't mean we ...


7

I would say that the word of is pretty important here, and as long as you get that correct, the rest is less important. Using the word of consistently when a function is being applied to an input, and talking about the fact that you are doing this, will help ward off a lot of the following kinds of mistakes that I see in my (much later) classes: Question:...


Only top voted, non community-wiki answers of a minimum length are eligible