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: ...


46

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 data generated by pupils at Caddington Village School in Bedford sheds light on which multiplications are actually the hardest – and how kids do overall. The ...


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/...


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 ...


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 This is a somewhat long (unpublished) article (which I haven't studied carefully), but maybe the excerpt below suffices to give the gist of it.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

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 daffodils Beatrice's flowers The whole is 3 flowers and $\frac{2}{3}$ are daffodils The flowers of Anna and Beatrice combined. The whold is 8 flowers and $\...


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

The word you are looking for is mediant. The mediant of two fractions $\frac{a}{c}$ and $\frac{b}{d}$ is $\frac{a+b}{c+d}$. According to Wikipedia, It is sometimes called the freshman sum, as it is a common mistake in the early stages of learning about addition of fractions. Teachers often do this in grading papers. For example, a test has two parts: the ...


8

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 \\ \hline \end{array} You start by saying $9$ plus $3$ is $12$. You write the $9$ as shown and "carry" the $1$, of the $12$, as a subscript of the $7$ ...


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 ...


5

Calculus student had a final result of $\frac{1}{2\pi}$ Which she told me was 1.57. I immediately realized that she had calculated $\frac12\pi$ from keying in 1/2$\pi$. There’s no going back on calculator use, I realize. What I strive for is to have the student who performed a series of calculations (for, in this case a related rates problem in calculus) to ...


4

I don't know if your idea has a name, but it feels weird when you try to apply it to something like 10-4: \begin{array}{r} & 1 & 0\\ -\!\!\!\!\!\!& \! & 4 \\ \hline & & -4 \\ +\!\!\!\!\!\!& 1 & 0 \\ \hline & {\color{red}1} & {\color{red}-}{\color{red}4} \end{array} Since the sum of the ones place (-4+0 = -4) and the ...


4

The student was conflating the sum $$\frac45+\frac23$$ and the weighted average (where the weights account for group-size differences) $$\left(\frac{\color{#00F}5}{\color{#180}{5+3}}\right)\frac45+\left(\frac{\color{#00F}3}{\color{#180}{5+3}}\right)\frac23\\=\frac{4+2}{5+3}$$ of the two given fractions/rates. Clearly, the sum of two positive fractions is ...


4

You really need to be able to do sums in your head when debating or negotiating. To prove this, show your students these famous car-crash interviews: (1) Diane Abbott (Labour) floundering horrifically over numbers she could neither remember nor even estimate in her head. The coolness and speed with which the interviewer questioned her numbers must itself ...


4

By hand =/= in your head. Having some faculty for mental arithmetic is good. Having a fluency for deconstructing a 'problem' into basic mathematical operations is pretty essential. However many people learn, and continue to benefit, by writing out the 'sums'. If Fred drives at 45 miles per hour, how far will he go in 45 minutes? I expect most people ...


3

I think if we can introduce a variety of algorithm's for teaching addition and multiplication, it'd be beneficial for the student. No matter what mathematics level you are, algorithmic thinking is always important. When I say algorithmic thinking, I mean to say that we have a set of rules to tackle a problem such that applying the rules in the correct order ...


2

I remember getting my first 18 inch slide ruler. I had worked hard, poison oak filled, hours making fire breaks in the Oakland hills to make enough money to get it. It was aluminum, had a spring loaded cursor that slid like it was greased, and the C and D scales (I think) lined up perfectly. It was huge and had scales I was never going to use. I loved it. ...


2

You can't be overly reliant on technology. I once bought an item from a shop during a power outage. Cash register was non-functional. Not enough light to power a solar calculator. The cashier had to compute a 10% discount by hand. She did it.... the long way.


2

You're entirely correct and there is huge value to being able to compute by hand. For a most basic instance from the real world, here in the UK I once interviewed a candidate for office manager who had spent about 10 years working in a tax office, and been promoted three times. To me, that should have meant he ate, breathed and slept numbers. I asked him ...


2

What property of fractions or addition of fractions could they be misunderstanding, and how would you explain to the student where they have gone wrong so that they don't repeat this in the future? [Emphasis added.] Short Answer Fractions are numbers and they behave like numbers when we do operations on them. Many students never learn this. You (and the ...


2

Some additional ideas and links: Strategical thinking: https://en.wikipedia.org/wiki/D%C5%8Dbutsu_sh%C5%8Dgi Programming: https://www.turingtumble.com/ (Technically 8+ but the concept should be understandable for younger kids as well. Note: it involves small marbles so use caution whether it's appropriate to be used by a smaller child.) Any kind of ...


2

The method of addition and subtraction that you mention is not new. For now, I provide one reference, but I'm sure there are others. Note that your method of subtraction makes one big assumption that is not needed for the traditional method: you assume that students are familiar with negative integers. The website Knowledge Over Grades uses a slight ...


1

Your method is circular. Try: 1000000 − 1 −−−−−−− 100000 −1 Now what??


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