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 ...
- 1,802
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 ...
- 20.1k
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 ...
- 487
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 ...
- 449
23
votes
Accepted
Is it meaningful to add a number to itself a fractional number of times?
For the product $a\times b$, I intentionally don't use the phrase "add $a$ to itself $b$ times", but rather I prefer something like "start with zero and add $b$ (copies) of the number $...
- 9,194
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 ...
- 7,999
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"?...
- 9,194
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 ...
- 5,609
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 & [...
- 641
18
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 ...
quid♦
- 7,652
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) ...
- 3,996
11
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, ...
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.
...
- 314
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 ...
- 271
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 ...
- 495
11
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 ...
- 1,228
11
votes
Is it meaningful to add a number to itself a fractional number of times?
Frame challenge: I think your verbiage "adding (whole number) to itself (whole number) times" is misleading and incorrect and exhibits an off-by-one error. Think about the example $(4×1)$. ...
- 815
10
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 ...
- 10.8k
10
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
...
- 863
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 ...
- 231
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 ...
- 226
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 ...
- 880
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 ...
- 6,751
9
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 ...
- 10.3k
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 ...
- 29.5k
9
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 ...
- 7,999
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, ...
- 5,192
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 ...
- 798
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