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It's pretty rare these days for anyone to actually do division by hand. Most people reach for a calculator. Given those realities, I would question whether it even makes sense that we spend such a vast amount of time teaching young children even one algorithm for division. Maybe we should postpone it, deemphasize it, or replace long division with a slower or ...

11

A US specific answer: The Common Core State Standards define $\frac{1}{b}$ by saying it is one of $b$ equal parts making up a whole $1$. $\frac{a}{b}$ is then defined as $a$ of these. Connecting $\frac{a}{b}$ to $a \div b$ requires some reasoning. For instance $\frac{5}{3}$ of a candy bar means you take your one candy bar, divide it into 3 equal sized ...

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If you are tutoring, it's important to value whatever algorithms work. Your frustration with the (new to me) lovely algorithm you show concerns me. It shows why each step makes sense, which is much better than the form of long division I learned. "In fact a lot of kids find it [short division] amazing to learn." If you show an individual a new process at ...

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Not a teacher here, but I noticed when my kids went to school there was far less emphasis than I remember on techniques that require above average insight or intuition. I think there's more pressure these days toward making sure most students achieve a predetermined minimum performance, and less on helping high-performing students stretch their capabilities. ...

5

A key idea in maths education, that at least becomes more visible at university level, is that people tend to start thinking of new ideas as processes, but to do more advanced maths they need to move on to thinking of the same things as objects. That is, we are taught to think of $2+3$ as an instruction to take $2$ and add $3$ to it. Most people will write $... 4 By definition, a rational number is a number that can be expressed as the quotient of two integers. This quotient is called fraction and is written as$\frac{a}{b}$. Hence, division and fraction are the same, at least in the context of dividing integers and turning them into rationals. This concept is expanded in middle school for dividing irrationals and ... 3 Round. Check. Correct. What we need is$436\div48$. Rounding to remove the last digit reduces the problem to$44\div5$. This gives us an educated guess of either$8$or$9$. From here we then multiply and check, either one. Take$8$for an example.$48\times8=384$and$436-384=52$. By looking at what we have left, we know that we need only one more$48$... 3 Both of my sons are learning division right now (or rather, just finished the section), one in a public school using Eureka Math (in 4th grade level math), one in a Montessori (in Primary, at 1st grade age). The public school focused on long division certainly, but (either because of the curriculum, or more likely because of the teacher, as I don't ... 1 Doing a quick google search for "why do students struggle with division?" made me realize myself why long division is difficult for a lot of students. Common reasons claimed include: The long division algorithm is long. Remembering the steps can be difficult. The long division algorithm may not be intuitive. We can tell them that it's just "how many groups ... 1 You might explain, at any level, that for any real number$x$, we have$0x=0$. So the equation$0x=0$has no unique solution. And$0x=1$, for example, has no solution whatsoever. For these reasons, neither$0/0$(with no unique solution) nor$1/0$(with no solution) can be real numbers. 1 "divided by$x$" means "times$x$inverse". "$x$inverse" is the number such that$x$times$x$inverse = 1. 0 times anything = 0. 0 times nothing = 1. Nothing is 0 inverse. "0 inverse" does not refer to anything. "$x\$ times 0 inverse" does not refer to anything. "x divided by 0" does not refer to anything.

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