52 votes

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

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 ...
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  • 1,301
49 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 ...
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  • 4,793
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 ...
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  • 4,784
38 votes
Accepted

Dividing by zero

You asked: "How do/would you explain why division by zero does not produce a result." Any such explanation that is not rooted in student understanding would be talking to ourselves, not to ...
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  • 7,405
28 votes

Dividing by zero

Have the students tell you that division by zero is a non-sequitur. This is possible at any age where division is understood at all. Teacher: If there are eight cookies and four children, how many ...
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  • 4,890
24 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 ...
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22 votes

Dividing by zero

In third grade we taught division using repeated subtraction. To divide 6 by 2, subtract 2 until you get to 0. 6-2=4, 4-2=2, 2-2=0. It took 3 steps so 6÷2=3. This can also be shown on a number ...
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  • 6,899
22 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 ...
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  • 419
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 ...
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  • 17.5k
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 ...
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  • 10.2k
17 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 ...
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  • 6,899
15 votes

Repeated addition: standard notation?

While there may be legitimate reasons behind the convention In $a \times b $ the $a$ denotes the number of terms and the $b$ denotes the individual terms the larger issue is the mismatch between ...
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  • 16.4k
14 votes

Is there any other procedure to find the square root?

You can use (basically) Newton's method. Take $x_0>0$, and define $$x_{k+1} = \frac{1}{2} (x_k + \frac{n}{x_k}).$$ The seqeunce converges to $\sqrt{n}$. And if one starts with $x_0= n$ one has,...
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  • 7,592
14 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; ...
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  • 16.4k
13 votes

Repeated addition: standard notation?

A reason why this form might be preferred is the way one says it: $5 \times 3$ is read out "five times three" so it says take $3$ five times, hence it "is" $3+ 3+ 3 + 3 + 3$. However I doubt there ...
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  • 7,592
12 votes

Dividing by zero

Dividing $1$ disk into $\frac{1}{n}$-ths ($\frac{1}{3}, \ldots$), leads to $\frac{1}{1/n} = n$ pieces ($3,\ldots$): As $\frac{1}{n}$ approaches $0$, the number of pieces $n$ grows without bound. The ...
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12 votes

Teaching arithmetic operations ($+ - \times \div$) to a 3 year old

My immediate response is 'wait a few years'. I've spent a fair amount of time with 3 year olds, and most of them are busy learning how to be a person in their own right, how to have a conversation, ...
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  • 5,568
11 votes

Repeated addition: standard notation?

One place in math where this issue actually does come up is in defining ordinal multiplication. From an ordinal perspective, the ordinal $5$ is the order type $a<b<c<d<e$, the ordinal $3$ ...
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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 ...
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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 $$2+3=3+\_\_$$ $$2+3=\_\_+2$$ $$2+\_\_=3+2$$ $$\_\_+3=3+2$$
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  • 10.2k
10 votes

Dividing by zero

This answer is intended for the second category of students: How do/would you explain why division by zero is undefined for Algebra students? Begin by introducing the reciprocal of a real number as ...
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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 ...
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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 ...
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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 ...
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10 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 ...
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10 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 ...
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9 votes
Accepted

Parse out 2/3 of 30 minus 11

it sounds like the textbook has some poorly written and vague questions. there is no reason why either of the answers would be "better" for the first example. It is simply dependent on how you ...
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  • 4,840
9 votes

Is there any other procedure to find the square root?

There is an algorithm to calculate square roots with paper and pencil (like the long division algorithm). See How to calculate a square root without a calculator.
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