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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  • 1,297
33 votes

Is this primarily a "rote computational trick" for multiplication by 9?

Anything that is just a trick leads to students having wrong ideas about what math is. But methods that help students see the patterns can help them learn the multiplication facts, along with getting ...
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  • 17.6k
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

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

How to answer a three-year-old the question "Why is $2+6$ the same as $4+4$"?

I'm nearly sure I did this with my child when she was young. First, establish that she understands that a number, like three, is equal to $1+1+1$. Hold three fingers up and ask her "how many is this"?...
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  • 7,713
20 votes

Is this primarily a "rote computational trick" for multiplication by 9?

Yes. This is also a trick that you can do on your fingers, too. For instance, let's say you wanted to calculate $9\times3$. Hold out your hands and bend your third finger down as shown. So nine ...
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  • 5,539
19 votes
Accepted

Does this property of subtraction and division have a name?

This is "left involution". ("left" because it doesn't work when you try it on the right.) \begin{align*} x \circ y &= z & \\ x \circ (x\circ y) &= x \circ z & [...
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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
16 votes

Why does the widespread erroneous definition of "irrational number" persist without being taught?

I can think of two related reasons: The characterization via the decimal expansion might be perceived more strongly like a property of the number: "This number is irrational, because this number's ...
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  • 7,592
15 votes

Adding things to bunches of things vs multiplication

Here is where it helps to get more concrete instead of more general. Have the student draw a picture of the problem and similar problems. First, you demonstrate drawing one box of pencils (a square) ...
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  • 3,684
14 votes

Adding things to bunches of things vs multiplication

Why you can multiply boxes and pencils, but cannot add? In this case, you're multiplying pencils-per-box with boxes. The units cancel and you're left with pencils. Teach students to write fractions ...
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  • 7,713
11 votes

Adding things to bunches of things vs multiplication

The thing is, it is possible to add 4 boxes to 5 pencils. 4 boxes + 5 pencils = 9 things. So start by showing the student what they can do with addition, and what its real-world application is. ...
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  • 304
11 votes

Is this primarily a "rote computational trick" for multiplication by 9?

To add on to the other answers, the reason this works is because we use the decimal system, a.k.a. the base-10 system, for our everyday maths. The multiples of the number that is one less than the ...
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  • 261
11 votes

Does this property of subtraction and division have a name?

I have never seen a name for this property specifically. When I was in grade school, I recall learning about Fact Families, which are generated by this property. The idea is that a fact family is all ...
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10 votes

Why does the widespread erroneous definition of "irrational number" persist without being taught?

Note that this definition of rational and irrational numbers is most commonly presented to high school students, who tend to have a strong and natural intuition of numbers in base $10$. At this stage, ...
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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

How to explain the motivation of parentheses in addition, subtraction and multiplication?

Why take off the parentheses? Because sometimes "taking off the parentheses" results in an easier to calculate expression. For example, $$123456789-(-9876543210+123456789)$$ is easier to evaluate ...
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9 votes

Are soroban (Japanese abacus) classes worth doing?

The article [1], according to the abstract, [claims various benefits.](https://eric.ed.gov/?id=EJ1105219 , https://scholar.google.no/scholar?cluster=12532307503119935328) However, it is not widely ...
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  • 4,681
9 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 ...
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  • 823
8 votes

Is there a numerical base that is in any way “better” for simple mathematical calculations than others?

Clearly there is no historical data that addresses this question I want to know if there are any numerical bases that are notably well-suited for humans to learn and use at an elementary or ...
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8 votes

Different ways to multiply decimals

Combining your last two methods: $3.9*7.5 = 4*8 - 0.1*8 - 4 * 0.5 + 0.1*0.5$, which can be thought of as computing the area of the big rectangle below, cutting off the two extra strips along the edge, ...
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  • 4,771
8 votes
Accepted

How do we explain to a little child that a date in 2020 and a date in 2021 are not necessarily a year apart?

I would use a number line. This is the most straight forward way to explain being "in between" integers while giving some intuition with a visual. It is possible that a school-age child ...
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8 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 ...
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8 votes

How to train facility with numbers?

If he won't practice on his own, then I suggest the following. At the beginning of each tutoring session, give him a blank 10 by 10 times table and have him fill it in. He can then refer to it while ...
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  • 6,899
7 votes

Why does the widespread erroneous definition of "irrational number" persist without being taught?

The definition of an irrational number as a "number which is not rational" is not without its own difficulties. It presumes that we have a clear definition of a real number. The audience you refer to ...
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