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Background

I sometimes tutor high school students and I have come across various problem types that are best represented by the following two problems.

  1. They are unable to keep track of the correct direction of inequality symbols. That is, if I ask then to write "5 is bigger than 4" they will guess between $5 >4$ or $5<4$.

  2. They are unable to set up a division problem. That is, if I ask them to "write 10 divided by 2", they are tempted to often write $10\overline{)2}$ or $\frac{2}{10}$. However, if I ask them "how many times does $2$ go into 10," they are able to provide the correct answer, but they cannot perform long division.

Question

Keeping in mind that I have no professional training as an educator, how can I help students keep these two ideas straight?

Here is what I have tried

  1. I vaguely recall a math teacher from elementary school telling me that the inequality sign is like the mouth of an alligator and it likes to eat the bigger number. The students have no problem recognizing which number is bigger but for some reason this explanation doesn't help. Since the student doesn't seem to have a problem with "solving" an inequality problem with algebraic manipulation, i.e, a problem like $x+7<2x$, their teacher has told them (and the rest of the class) that there will be no "word problems on their tests or quizzes. That is, an inequality will always be set up for them to solve. Should I not worry too much about whether or not a student can properly write an inequality and be satisfied that they are able to perform just the algebraic steps?

  2. For the division problem, their teacher told them to just type it in the calculator. That is, $10$ divided by $2$ translates directly to $10÷2$ on a calculator. Should I also not worry about this?

Further considerations

I work at a tutoring center so sometimes the kids will come in with their own homework and sometimes they will work on assignments we give them. The reason why I feel that the students should learn the two concepts above is because their teacher next year (or the year after) may have a different policy for setting up inequalities, etc. Furthermore, the students are often faced with other problems like the two listed so maybe helping the student work through them will help them sort out other ones like it. Note that I only have an hour a week (sometimes 2 hours) with the students.

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    $\begingroup$ Just to add a mnemonic (even though I generally avoid them): You could practice having them read aloud, e.g., $3 < 4$ by saying "Three is less than four." The mnemonic I have in mind is that the $<$ symbol looks a bit like an $L$, for "$L$ess than." $\endgroup$ – Benjamin Dickman Oct 20 '14 at 22:28
  • $\begingroup$ Mnemonics are all too easy to forget since they still rely on remembering something arbitrary (e.g., perhaps the user will remember the 'L' for less but think that it means that the numbers in the sequence become less, than in order to go from 3 to 4 you have to go down); it's like writing something on a sticky note and forgetting to ever look at it. Maybe that's why he didn't recommend it either. More natural is that the wider/larger side of $<$ corresponds to the larger value, and vice versa. $\endgroup$ – Vandermonde Nov 23 '15 at 5:17
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This is somewhere between a comment and an answer, but it's too long for a comment.

When my students have similar difficulties, I usually want to know what they're thinking about as they do the problem, because there are (at least) two things that might be going on that call for very different approaches.

It's possible that the students have a heuristic in mind (perhaps the alligator story or something similar) but consistently get it mixed up. ("Which side eats which one again?") In that case the right approach is probably to find a more useful heuristic; finding the right one may depend on the student, and other answers have some good suggestions.

But the second possibility (and some of your post suggests this to me) is that they aren't getting to the point of really trying to answer the question at all. They may have been told some rule for remembering the order, but faced with the question "is 5<4 or 5>4?", they guess based on a gut feeling, and never try to apply the rule.

(At least with my students, I've noticed that once they've decided a problem should be straightforward, they're very reluctant to approach it slowly, even when they're consciously aware that they're struggling with it. So having decided that they're supposed to automatically know whether 5<4 or 5>4, and realizing that they don't, they feel lost and make a guess. Backing off---"I don't know this, but I know how to find it out"---isn't something they try.)

Your question suggests that they might not think it's important for them to get the problems you're talking about, which might contribute: having realized that they don't immediately know the answer, they may feel like it's not important enough to really try to get it right.

In the second case, if you decide it's worth pressing them to get it right (and on behalf of their future teachers, someone will one day appreciate your efforts), the issue isn't teaching them a rule, it's teaching them to recognize when they need to stop and consciously apply the rule they do know.

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For the larger and smaller: I do not find the aligator explanation intuitive. If this is a mouth it could be just as well the larger eats the smaller, which would give the wrong thing.

What I was told long ago was it is like a beak of a bird, and the larger pecks on the smaller.

However, in all these stories there is a risk of confusion. In my opinion the symbol is rather intuitive in itself. Where it is wide(r) open there is the larger thing, where it is closer together there is the smaller thing. To me it makes sense, you give more space for the larger thing. If you want more plasticity you could say, the equality signed got widened up where the larger thing is.

(I should perhaps add as a disclaimer that I have little to no experience teaching children that age.)

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[I agree with all of the previous answers on the inequality symbol: Big side, big number/value. It seems intuitive to me. My guess is that those stories are what messed students up in the first place.]

I have trouble with keeping the symbols straight for division myself. (I have never been diagnosed with dyslexia, but I believe I am mildly dyslexic.) To remind myself of the proper orders of symbols, I always use what I see as the simplest example:

6 ÷ 2 = 6/2 (picture this vertically) = 2 goes into 6 (I can't write this symbol) = 3. This grounds me, and I can move on.

When I'm working with a student, I might start by asking "How many twos in six?" And then ask them how they'd write that as a problem. Most of my students feel sure of the answers to all three problems (6 ÷ 2, 6/2, and 2 goes into 6), and that helps them see the structure. I sounds like yours aren't. Go as basic as you can, and discuss the meanings with them. If a student wrote 2/6 instead of 6/2, I'd ask them to tell me what 2/6 means, and try to get at the fact that its value is less than one.

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the larger side of the greater sign always goes to the larger quantity, so since 5>4, then when you see x>y, x is greater than y.

for division, read the division symbol as "over", so "ten over 2"

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The direction of the inequality symbols should be intuitive enough. The big end of the symbol corresponds to the side with the larger quantity. The small pointy end indicates the smaller side. Perhaps the mistakes are attributable to dyslexia rather than confusion.

For division, it may help to remind them of more familiar fractions like $\frac{1}{2}$, which is 1 divided by 2.

The long division setup is less intuitive, though. I'm not sure that there is a good explanation, other than that that is how the convention is, and it takes practice to learn the algorithm.

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I vaguely remember having this problem with inequalities (so this will be anecdotal). The problem I had with teacher is the way they would read it from left to right then double check it by reading from right to left.

Kids already know which number is bigger.

So just agree on reading it from left to right. Let them use their own words if they like smaller or bigger.

If you feel the need to reverse check (reading from right to left) then physically write the numbers in reverse for example 5>4 or 4<5. Do not perform reverse reading verbally.

Never ever verbally read 5>4 as 4 is less than 5 (which is what my teachers used to do when they reverse check). I know its true (we can read is as 4 is less than 5) but I think the children do not rotate the numbers and inequality symbols in their minds when we read it backwards verbally.

Just agree on the left to right reading order, since the greater than sign (>) and less than sign (<) are both tied to a left to right reading order

or read it like this (which still follows the consistent left to right reading order):

the number on the left is (__) than the number on the right

(again anecdotal)

easier with gif

enter image description here

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