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I've recently taken on the task of helping out in my school's Math Center. The courses I assist in range from Algebra to Calculus. While I'm younger (in my 20's), most of the students at the school are 40+.

The first students I worked with were taking a Calculus Prep class. Teaching them was fairly easy - most of what I did was show them that the steps required to solve a problem were steps they already knew how to do and were familiar with - the problem was just being presented in a unique manner.

However, I recently got tasked with helping out students taking the lowest level math we offered - and I found it surprisingly difficult. Teaching a younger child math, you can tell them that a rule is a rule, and that they must accept it in math and go forward. From there, they might start to test that rule, and see that it applies in all cases.

But, when teaching adults, I've found that I can't just tell them "this is the way it's done, get used to it." I have students that are struggling with topics like adding and subtracting negatives, or determining if a function is odd or even - essentially why $-(x^2)$ is different from $(-x)^2$. I can try to explain negatives using a number line, or walk them through PEMDAS again, but it just doesn't seem to stick.

It seems like children just need to know how, but adults need to know why.

Any advice for teaching simple concepts and rules to adults, specifically working with negatives or proper application of PEMDAS?

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    $\begingroup$ Your title doesn't seem to match your question. Also, your question might have a little too much in it. Perhaps it can be separated into "how to teach adults elementary concepts" and "why do we graph functions the way we do" $\endgroup$ – DavidButlerUofA Dec 16 '15 at 19:29
  • $\begingroup$ Sorry, first time posting here. Edited. $\endgroup$ – charliefox2 Dec 16 '15 at 19:46
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    $\begingroup$ While not an educator myself, I think your problem is probably a bit broader than you realize. They may keep questioning why, but adults with math difficulties tend to have serious mental blocks that they have erected. They seem to refuse to learn anything. I doubt your problem is distinct from simply teaching the material, whatever they tell you. Just getting them to understand the material is probably your only problem. How to fix that, well, good luck :) $\endgroup$ – wedstrom Dec 16 '15 at 21:55
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    $\begingroup$ IE they are making the distinction between why and how important, when what they really mean (most likely) is i hate this, don't get it, doesn't make sense. Not so different from five year olds in reality, but they would like you to think otherwise. $\endgroup$ – wedstrom Dec 16 '15 at 21:56
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    $\begingroup$ If that's the case, then you are almost certain to succeed. The other answers besides mine are probably very helpful. Just keep them working, and keep the excitement and interest up. Never let them feel dumb or discouraged, if they enjoy it and genuinely want to be good at the subject, its just a matter of teaching through the concepts, bottom to top till they get it. More specific then that, I'll shut my trap and leave it to the experienced pros. $\endgroup$ – wedstrom Dec 16 '15 at 22:28
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But, when teaching adults, I've found that I can't just tell them "this is the way it's done, get used to it."

Good! Students (at any age) should never be satisfied with "This is the way it's done, get used to it", and teachers should never give that as an explanation. More than any other academic subject, everything in math should make sense. Scientific truth is subject to experimental verification or falsification, but in mathematics "being reasonable" is literally the only criterion for distinguishing between true and false. If something doesn't make sense, it's being taught wrong.

All mathematics teaching, regardless of its level or the specific course content, ought to be largely about the art of sense-making. The teacher’s job is to ask questions, and in so doing to model for students how to ask their own questions: not only how to solve a particular kind of problem, but also why the problem is worth solving, why a given method works, whether there are other methods that might work as well or better, and so on. Without that, mathematics is little more than a bag of rules and procedures to be memorized and (probably) forgotten.

Now, you asked:

Any advice for teaching simple concepts and rules to adults, specifically working with negatives or proper application of PEMDAS?

As I've said above, I don't think adults are any different from younger students in this regard: they all deserve coherent explanations that explain why, not just how.

For specific issues having to do with negative numbers, see (for example) How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$? and How to teach someone that $-3>-4$? and When $-x$ is positive, all of which deal with conceptions & misconceptions about signed integers.

For PEMDAS, see https://math.stackexchange.com/questions/1385549/what-is-the-reason-behind-the-current-order-of-operations-pemdas.

You didn't ask about fractions, but see How to explain the flipping of division by a fraction? for examples of that.

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    $\begingroup$ I do agree, it should always be taught why, but I do find that difficult for something like PEMDAS. I like the explanation for how to teach someone that $-3 > -4$, I'll have to use that. As for fractions, I think I'm pretty on point there. I show a couple of real world examples, and students catch on pretty quick. On the topic of -x, my problem comes when trying to explain to students that $-1 * 2^2$ is not the same as $(-1 * 2)^2$. Other than saying that we must follow the proper order of operations, do you have any advice as to how to explain that (maybe with a real world example)? $\endgroup$ – charliefox2 Dec 16 '15 at 21:07
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    $\begingroup$ I wish I could upvote this answer a few dozen times. I would have written almost the exact same words. $\endgroup$ – Daniel R. Collins Dec 16 '15 at 21:10
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    $\begingroup$ I think the key idea is that doing things in different orders produces different results. If I tell you "Drive three miles north, turn left, then drive two more miles" and instead you drive three miles north, then drive two more miles, then turn left -- well, you'll end up in a different place. Likewise, "Take 2, square it, then multiply by -1" is different from "Take 2, multiply it by -1, and square it." Maybe that's the issue: students need to practice transcribing a symbolic expression into a verbal description. $\endgroup$ – mweiss Dec 16 '15 at 21:10
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    $\begingroup$ @mweiss I like that answer... I'll see how it works out. Thanks a ton for the help! $\endgroup$ – charliefox2 Dec 16 '15 at 21:17
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    $\begingroup$ You may not always have to explain why something is true in all its detail. That can get really complicated. Quite often it suffices to show why it might be reasonable. To explain why a negative times a negative is a positive, for example, you probably shouldn't launch into a formal proof from field theory. It would probably suffice to start with, say, 2x3 = 6, and decreasing each argument in turn by 1 until both are sufficiently in the negative. It's just one example, and doesn't really prove anything, but mostly, they just want to know that it isn't complete nonsense. $\endgroup$ – Dan Christensen Dec 17 '15 at 5:09
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Any advice for teaching simple concepts and rules to adults, specifically working with negatives or proper application of PEMDAS?

This will be an adjunct to mweiss' answer, which I fully agree with (please go upvote that), but this will be a little too long for a comment, I think.

When some people go to teach these pre-algebra arithmetic courses for the first time, they frequently think "this will be easy", but it's not; it's likely to be the hardest teaching experience, because you really need all the details right; the students won't be able to meet you halfway or fill in gaps, so you really need near-perfect explanations. There are details that you might perform on sight without thinking; perhaps you simply learned them as rules to follow without reason. But this is a terrific opportunity to realize that that's generally incorrect, and should take the time to investigate, reflect, and really think through first principles and come up with good explanations for all of them. (Again, mweiss is on the right track by giving you links to explanations for particular topics; you're just going to have to dig in on a case-by-case basis and thoroughly think them through.)

With a small number of (axiomatic) exceptions: The order-of-operations is indeed an arbitrary grammatical convention that we do have to accept. Generally we also accept the meaning/properties of the equals sign (=), and also a small number of properties about real numbers (commutativity, associativity, and distribution). We really should be able to explain anything else in the course based on those original principles (given proper definitions of other new objects or symbols). Look for opportunities to explicate this, and connect later work back to these founding principles.

The other thing that I will hypothesize, based on my experience, is that some proportion of the population you're dealing with is likely to have some learning disabilities like dyslexia, dyscalculia, vision problems, ADD, etc. This is likely explanatory to them being in this class situation in the first place. It can be frustrating that it seems like they just flat-out can't remember material from day to day. In this regard you'll just have to be patient and review whatever material they ask about whenever it comes up subject to your available time for the job. There's no silver bullet, and like a doctor you have to realize that you can't save everyone, no matter how much you may want to.

While these classes are in some sense the most challenging due to needing really high-quality, fully-detailed explanations of things you may not normally think about, I also find them among the most rewarding, and the students among the most appreciative when you respond to their questions respectfully and thoughtfully, and treat them like colleagues. I feel like I've really made some discoveries and vastly improved my teaching and writing because of teaching these classes.

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    $\begingroup$ When I started tutoring basic algebra, I'll admit, I thought it would be too easy, and very dense. Boy was I wrong! I do enjoy the challenge, and of course it's very rewarding to teach someone something, it's a great feeling. I think the frustrating part for me is that I become frustrated with myself when I can't find a way to explain something that, to me, seems so trivial. I can explain why a difference quotient works to find the derivative of a function, but when I can't explain that $-3 + -4 = -7$, that irks me. I may not be able to save everyone, but I'll be damned if I don't try! $\endgroup$ – charliefox2 Dec 16 '15 at 22:06
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    $\begingroup$ Some proportion of the population you're dealing with is likely to have some learning disabilities like dyslexia, dyscalculia, vision problems, ADD, etc. This is absolutely correct, and makes it even more essential not to try to teach students to just accept and remember things -- they won't be able to! $\endgroup$ – mweiss Dec 16 '15 at 22:13
  • $\begingroup$ @charliefox2: You're totally headed in the right direction; often you'll have to reflect later on for a better explanation. One of my favorite discoveries (not stated in any book) was after failing in that exact same way (almost same topic), and sitting at the bus stop afterward thinking, "What the hell exactly am I doing in a simplification like that?" $\endgroup$ – Daniel R. Collins Dec 16 '15 at 22:16
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Break it down into it's simplest level, go over it and over it. Start with small numbers, make them do it until they can see the patterns. Gradually increase the difficulty, and when they get flustered(they will!), revert to a comfortable level and work back up again. As for the specifics:

PEMDAS - If you do things in a different order, you will get a different answer. It is important that we can tell what order to do them in, so we get the same answer. Like if you put the lid on a bottle BEFORE you put the water in. Additonal: Written math is like any language - when me(you, the teacher) or anyone else writes equations, we mean a very specific thing, so everyone can get the same (correct) answer. If I meant a different order than PEMDAS, I would have use a parenthesis, so that my meaning was clear.

-(x)2 vs (-x)2 Make this an extension of PEMDAS. The order is important, and show them an example, with it being different. They are saying "why" because they don't understand PEMDAS, exponents, or negative numbers (as you mentioned).

Negative numbers: Multiplying a positive by a negative is easy. Say a candy bar costs 1 dollar. I buy three of them. On my balance sheet, I have three negative values. 3 * -1 is negative three. Start from there and do bigger numbers. Bigger number cause irrational fear! You could immediately ask some students, what 3 * -1000 is, and many would throw up there hands in frustration! Work them through it till they see the pattern and are comfortable with it. Expect them to forget and need to be reminded, over and over. -3 * -3 is more abstract, you could try something like buying three fewer candy bars, but it's probably easier to just say at this point they "undo" each other, and work them through it until they just get used to it.

The people you are teaching have poor math skills as adults because of attitudes and fears, not a fundamental lack of intelligence. Those attitudes are going to be more challenging than simply teaching material would ever be on its own. Listen to what they ask, but take everything with a grain of salt, they just don't want to look/feel dumb.

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The way I explain order of operations is that the strongest things come first:

  • Addition and subtraction are weakest, so they come last
  • Multiplication (sometimes like repeated addition) and division are stronger
  • Exponentiation is stronger yet
  • And grouping symbols are the most powerful, so deal with those first
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    $\begingroup$ While true, I'll say that in my experience this doesn't get much traction at this level, because the students don't have enough number intuition to immediately say what counts as "more powerful" -- I also have to give concrete examples with natural numbers, like which gives the bigger result, $2+3$, $2 \times 3$, or $2^3$? $\endgroup$ – Daniel R. Collins Dec 19 '15 at 19:24

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