84 votes
Accepted

Why are there two inverses to exponentiation?

The concept of an inverse operation itself is a bit tricky. Often we consider arithmetic operations to be binary operations: $\DeclareMathOperator{\add}{add}$ $\DeclareMathOperator{\subtract}{subtract}...
Justin Hancock's user avatar
59 votes

Is there a virtue to learning how to compute by hand?

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 ...
ryang's user avatar
  • 1,832
52 votes

Proof of why BODMAS (or BIDMAS) works?

It's purely a matter of how we choose to define the notation. The main reason for it is that it lets us write polynomial expressions (which are extremely common) without parentheses, e.g., $x^3 + 3x^2 ...
Daniel Hast's user avatar
  • 4,893
46 votes
Accepted

Which product of single digits do children usually get wrong?

https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children There are links to a dataset in the article. As far as I can tell, this isn't a formal study: But some new ...
Adam's user avatar
  • 5,382
33 votes

Why are there two inverses to exponentiation?

Is your variable at the base or the exponent? An exponential function is a function of the form $f(x)=a^x$ for some constant $a$. In this case, the inverse is indeed given by a logarithm $f^{-1}(x)=\...
Luiz Cordeiro's user avatar
28 votes

Is there a virtue to learning how to compute by hand?

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 ...
Ross Millikan's user avatar
25 votes

Is there a virtue to learning how to compute by hand?

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 ...
ljrk's user avatar
  • 449
23 votes

Why are there two inverses to exponentiation?

(Despite good answers already, I thought the concrete example below could be useful.) For multiplication and addition, there is exactly one inverse operation, namely division and subtraction. Yet ...
jpa's user avatar
  • 399
21 votes

Is there a virtue to learning how to compute by hand?

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 ...
Amy B's user avatar
  • 8,037
20 votes

A PEMDAS issue request for explanation

The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous. If implied multiplication works ...
Sue VanHattum's user avatar
  • 20.2k
17 votes
Accepted

A PEMDAS issue request for explanation

If the expression were, say, $48\div 4\times 12$, there would not be much disagreement (multiplication and division are performed from left to right). But an expression such as $48\div 4(12)$ results ...
JRN's user avatar
  • 10.8k
15 votes

How to teach multiplication between integers for the first time

My preferred model of multiplication with integers involves motion. Imagine you are recording a video of a car driving at a certain velocity. If the car is going forward, the velocity is positive; ...
mweiss's user avatar
  • 17.3k
15 votes

Third Grade Question -- This makes no sense to me

Here are the two previous pages from those materials (a pre-publication version found with a Google search): And here is the page containing the homework problem in question: The intent now seems ...
Nick C's user avatar
  • 9,406
12 votes

Is there a virtue to learning how to compute by hand?

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 ...
Ferenc Beleznay's user avatar
11 votes

Proof of why BODMAS (or BIDMAS) works?

Perhaps it is worth pointing out that every programming language defines an operator precedence structure to avoid ambiguities. An example table for C and C++ can be found here. Ambiguities must be ...
Joseph O'Rourke's user avatar
11 votes

How to correct visualization of mathematical expressions?

One way to see if the student understands the commutative property of addition is to have "fill-in-the-blank" questions such as $$2+3=3+\_\_$$ $$2+3=\_\_+2$$ $$2+\_\_=3+2$$ $$\_\_+3=3+2$$
JRN's user avatar
  • 10.8k
10 votes
Accepted

Has Benezet's teaching experiment ever been reproduced?

You can find some further studies that cited the original writing on google scholar: link Mahajan (PDF) remarks that: Etta Berman, one of the teachers in the program, studied it for ...
Benjamin Dickman's user avatar
10 votes

Different ways to multiply decimals

There's a fun method which I've seen referred to as Russian peasant multiplication or ancient Egyptian multiplication. (I don't know if these names have a historical basis.) If you think about how ...
Mitchell Spector's user avatar
10 votes

Generating system of equations with unique solutions

Generating systems. The same method that works for linear equations works also for polynomial equations. Starting with a solution in mind (in mathematics and computer science, we call this a planted ...
Yuval Filmus's user avatar
10 votes

Is there a virtue to learning how to compute by hand?

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 ...
Geoffrey's user avatar
  • 890
10 votes

Did Americans, before new math came in to schools, really say, 'three from two is nine carry the one", instead of borrowing ten from the tens column?

It sounds like a variation of subtracting that I learned in high school (1968). My instructor called it European subtraction. \begin{array}{ccc} & 3 & 4 & 2 \\ - & 1 & 7 & 3 \...
Steven Alexis Gregory's user avatar
10 votes
Accepted

How to convince a high school student that the $=$ symbol denotes identity?

Although some of the trouble may be language dependent, there is also the issue (here in the U.S., too, in English) that young students get the feeling, perhaps from teachers, even, that = is like ...
Sue VanHattum's user avatar
  • 20.2k
9 votes

Proof of why BODMAS (or BIDMAS) works?

To build on other answers, you might show how other conventions exist. Use an H.P. calculator for example (postfix), the LISP family of languages (prefix), and the APL language (all right-associative),...
JDługosz's user avatar
  • 191
9 votes
Accepted

How to correct visualization of mathematical expressions?

The problem is due to imprecise specification of the intended result. Here's a more precise way. $\text{Recall that the }{\bf commutative\ law}\ \color{#c00}X+ \color{#0a0}Y = Y + X\ \text{ is true ...
Bill Dubuque's user avatar
  • 1,038
9 votes

Is there a virtue to learning how to compute by hand?

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 ...
Joseph O'Rourke's user avatar
9 votes

Is there a virtue to learning how to compute by hand?

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 ...
David Mulder's user avatar
9 votes
Accepted

Analyzing an answer to the following problem: Give meaning to $\frac{4}{5} + \frac{2}{3}$

The student who designed this problem wasn't thinking about the different wholes. IN your students problem, there are 3 different wholes. Anna's flowers - The whole is 5 flowers and $\frac{4}{5}$ are ...
Amy B's user avatar
  • 8,037
8 votes

Proof of why BODMAS (or BIDMAS) works?

I understand this is not a realistic suggestion, but can you avoid "teaching" "PEMDAS" or "BOMDAS" altogether, and teach your students just the math instead? As pretty much everybody already said, ...
zipirovich's user avatar
8 votes

Why in the FOIL Method the terms are taken with their signs?

I'm going to answer with something of a polemical frame challenge: FOIL is evil, and probably shouldn't even be taught. Okay... that's a bit extreme. How's this: FOIL is a mnemonic that is, in my ...
Xander Henderson's user avatar
  • 7,697

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