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: ...


33

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 a better feel for what's going on. I'd call this a way to think about 9s. (There are many.) This method shows that you add 10 for each new nine, and then take ...


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/...


20

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"? Then spread them out and ask the same question. Are we adding $1+1+1$? Try holding up eight fingers (keep your thumbs down, for example) and ask her to ...


20

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 fingers are "up" (fingers up, $9$, finger #3 down. (9x3). You have two fingers to the left of the bent finger and seven to the right, indicating the product of $27$...


19

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 & [\text{apply $x \circ -$}] \\ y &= x \circ z & [\text{simplify the involution}] \text{.} \end{align*} I would be shocked to see anyone use that term ...


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

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 decimal expansion does not terminate." The other one is rather a non-property and thus not perceived as a definition of some thing. The characterization via ...


15

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) with five pencils inside (perhaps five tally marks) and make sure they understand the picture. Then ask them to draw four boxes of pencils (four squares) each ...


14

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 with units, and cancel accordingly, just as with numbers.


11

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. But then remind them what you really want to know - how many pencils are in the boxes (which they can easily count to see it's not 9). And then you can show ...


11

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 base results in a phenomenon where the second digit increases at the same rate as the first digit decreases. $$ 9 * 1 = 09\\ 9 * 2 = 18\\ 9 * 3 = 27\\ 9 * 4 = 36\\...


11

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 of the arithmetic equations generated by the same numbers. This property in particular is really just a consequence that subtraction is the inverse of addition ...


10

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, it's much easier to understand what it means for a number to be irrational in terms of its decimal expansion rather than the statement "a number which cannot ...


10

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 the algorithm works behind the scenes, it's essentially multiplying the two numbers in base 2, so it could have more than one pedagogical use. This technique is ...


10

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 solution), generate in some way left-hand sides of equations, and then compute the corresponding right-hand sides. If you use more than one equation per variable ...


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, ...


9

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 by using $123456789+9876543210-123456789=9876543210$.


9

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 cited at all. I do not know the landscape of education journals well enough to know if the journal is of high quality. I checked the Finnish publication forum, ...


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 ...


8

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 grade-school level since we have ten fingers and humans have learned only decimal arithmetic for everyday use. I just finished four weekly sessions with fifth ...


8

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, then adding back in a copy of the corner piece since you have removed it twice, instead of once.


8

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 would be familiar with it, too. The number line is particularly useful because you can demonstrate that the distance between a point near the end of 2020 and a ...


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

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 probably does not know anything about Cauchy sequences or Dedekind cuts. So the complement of "rational" is at best only intuitively defined for them.


7

Note $9 = 10-1$ so: $$ 5 \times 9 = 5 \times (10-1) = 50 - 5 = 45, $$ and the same for all the others: $$ 8 \times 9 = 8 \times (10-1) = 80 - 8 = 72. $$ This works for $k \times 9$ where $1 \le k\le 10$. Although we always have $$ k \times 9 = (k-1)\times 10 + (10-k) , $$ this is the final decimal answer only when $1 \le k \le 10$. After the kids do this,...


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