Algebra

Undetailed question, but here is some general advice:

Write out *every* step, *all* your work. Writing it out is part of the thinking process, and allows you to look back and see where you are and plan what to try next.

In fact, before even writing out the equations, if reading a problem, start by summarizing the information, question being asked, etc. Then visualize — draw a diagram, graph, or chart; or make a table of values. LABEL everything. USE the format “Let c = ___”; this helps you get a thnking process going, and allows you to come back and see what you have.

Very little is ever gained by staring at a blank sheet of paper until drops of blood stand out on your forehead (one author’s description of his personal writing process.) Start writing. **Make a point of writing something intelligent about every problem.** If you stall, at least you get part marks for starting; but most of the time writing down, or better diagramming, one thought leads naturally to the next.

When you finish one step, stand back and look at where you’ve gotten to. Ask yourself “What do I know now? Does it relate to anything else I know? What operations can I do next? Can I simplify, and is it time to do so yet? Where am I trying to get to? What do I need to get where I’m going? If I stall here, is there any other information elsewhere that I haven’t used yet?” One of these questions will almost certainly lead you to a useful operation.

Also ask yourself “Does this result make sense? Have I contradicted myself or written something untrue? If so, did I lose a sign? Miscalculate? Forget to distribute/do the whole side/ do the whole fraction?” These questions avoid most of the so-called “careless” errors and make multi-step work possible.

I sincerely hope you have taught mathematical operations. These are add, subtract, multiply, divide, square, root, exponent, etc. If you do the same *mathematical* operation to both sides of an equation, the entire side, then the equation will still be true.
Things that are NOT mathematical operations include “pick up this number here and move it over there” (This is what you do with a kindergarten felt board, not adult logic), “This number/sign becomes something else” (That’s what Harry Potter does with his wand, not mathematics), “Just shove the numbers in this formula and don’t ask why, hurry up and get the answer”; “Use this quick trick here, it’s not supposed to make sense, just use it” (This negates the whole foundation and purpose of mathematics, which is logic and real-world measure) and “Guess the answer because it just looks right” (more magic, and by definition unlearnable; if you knew what looked right, you wouldn’t be asking the question.)
If your student is using mathematics, the next step can be found by logic and experience. If he’s using kindergarten thinking and magical mystical recitation of abracadabras, he will need some re-teaching and unlearning.

When tutoring, I sit down and guide the student through this step-by-painful-step, asking the above questions and stressing the mathematics of the operations each and every line. It may take an hour to do one question the first time through. But then the student has actually *done* some algebra, not copied it from someone else. The next question goes a little faster, and so on, until the “What do I know, what do I want, and how do I get there?” habit is natural to him.

Thank you for your detailed message. Your insite on this subject was very helpful! We have tried some of your advice with this student already however on his tests he performs 3 or 4 steps and then makes errors. He is LD and ADHD but has overcome most of his attention difficulties. He is organized, and very concerned about his test grades. He also has extended time but even that may not help. Any other thoughts?
RW

The three or four steps and then error problem is common. There are just so many places to make errors that sooner or later errors creep in, for one thing; and fatigue and frustration set in which makes errors more likely.

ALL of us make what are (wrongly) known as “careless” errors. ALL of us lose signs, ALL of us add when we meant to subtract, ALL of us drop a term now and then. I keep telling my students that *everybody* makes mistakes; the difference is that smart people know they make mistakes and they go back and correct them.

Your student needs to learn to proofread as he goes and USE the algebra check system to verify his own answers.

A lot of kids are taught by a self-defeating policy of hurry-hurry-hurry, fill in as many formulas as possible as fast as possible,so they don’t take the time to check their work and they wait for the teacher to mark it and tell them their errors. This takes the responsibility for learning out of the student’s hands and loses the whole point of algebra, which is that math is a coherent system of logic and problem-solving.

This takes some work to turn around, but it can be done. It’s exactly the same process as turning around a guessing reader. You sit down beside the student as he works and after each and every line you ask him if he has it complete, if he has all the terms, all the signs, all the operations, all the parentheses. You ask him to check the sign rules for his operations and the accuracy of his arithmetic. Thenat the end you require him to substitute the answers back in to see if they work out. He is going to complain this is so slow; tell him it’s not a race, he doesn’t have a hundred problems to finish or else, you want him to understand rather than practice doing it wrong; and remind him (and yourself and other teachers) that there are no prizes for fast mistakes. So you take half an hour to do one problem; OK, it’s the first one he’s worked all the way through and gotten right — this is cause for celebration! After a couple of really really slow ones he’ll catch on and gradually speed up and end up going very much faster than before (same as the guessing reader — the constant errors are slowing him down the more he tries to hurry, and the solution is *not* to speed up, but to *slow down* and get it right first; fast comes later.)

It is also a good idea to get him interested in real-world problems or puzzle-solving or both; this stuff is really dry if it has no punchline at the end.

A side note: I ban pencils. (a) this is not kindergarten, and we are grown-up enough to use pens. (b) You can actually see what you write. (c) If you think you do something wrong, you just draw a line through it. If it was wrong, you can look back and see where the problem was and learn from your mistakes; if it turns out to have been right after all, you go back and re-start from where you left off. A huge time-saver. (d) Erasing takes three times as long as writing and after you have written and erased three or four times you have a sheet of something that looks like toilet paper and absolutely no math for all your work — a complete time-waster (e) Multiply erased pencil and paper is filthy and gets all over hands and books and tables and makes them unpleasant to touch. No wonder kids feel math is distasteful and ugly! They are taught to make it that way.
Also, please use clean fresh paper **with lines to keep work straight**, not stuff fished out of the trash. I am not trash, my subject is not trash,and my class is not trash. We deserve a few cents’ worth of clean paper. This issue came up at a college where I was attempting to teach: I arranged for the exam center to use blue exam notebooks for my class and told the students I would not accept trash paper. They came back to me very upset and said the exam center had refused the books. I went over and the director told me well, if it had been *English* or *history* or something like that, they would spend the money on books, but it was JUST math, and they only used scrap paper. Gee, wonderful message to send, no? Ask why this district had a huge math failure rate and a huge college dropout rate. Ask why I’m not wanted to teach there any more.

The person sending these response appears to be a great math instructor *but* not one for a students with LD/ADHD. If I was that student I’d shut down. As a matter-of-fact I am that student. My dad is a math professor and attempted to teach me in the fashion depicted here. Oh, I do use a mechanical pencil and I know many Harvard MBA’s that also use a pencil.

The problem may be that this child can only keep so many airplanes circling the airport at anyone time. The more planes that circle the airport the more likely he is to forget about the planes that began circling the airport first. It is about how much stuff this child can keep in his head at the same time and keep track of the path. I learned to do this through meds and vision therapy. In this childs case, since you are the teacher, these may not be an option. So, when all else fails have the student write down the steps needed to solve the problem and tick each one off as it’s complete.

Good luck R Willis wrote:
>
> I have a student who has a difficult time with multiple step
> equations. Can anyone give me advise?

Well, I’ve worked a *lot* on reading and basic math with some very seriously LD students, and I’ve taught algebra to some moderately LD students, and I’ve also attempted to teach algebra in colleges that let students work whatever way suited them, with disastrous results. I’ve also worked for the past eighteen years as a private tutor getting students of all abilities through high school and college algebra and calculus and linear algebra. When my students’ grades often go from 70’s to 90’s with one to three hours of tutoring, I *know* these methods work.

What I am suggesting is to land all those airplanes, organize them, and then only take up one at a time under control. This is the whole point of working in a strictly organized system. It certainly isn’t because nit-picking is fun! It’s because it works to get your thoughts on the issue at hand.

If you write down a line and make *one* *mathematical* change (+, -, x, or / ONLY) on the next line, and write that down copying everything as you go, then you have only one thing to think about. It is then possible to concentrate on which of the four operations and why, and to make sense out of the whole affair.

On the other hand, if you are fishing through your mental files through a hundred “quick tricks” (which cease to be quick and become exceedingly slow when your memory is overloaded), trying to remember if this is the trick where the sign magically changes or the one where things turn upside down, trying to decipher your scrap paper with numbers every which way on it, and madly erasing and overwriting because you think you made a mistake and then you decide it was right after all and then it isn’t neat so you erase correct work and rewrite it, well, you *do* have ten or more planes in the air and of course you come to a crash.

By the way, I personally have very bad eyesight (mistreated amblyopia, farsightedness, and astigmatism) and coordination and balance issues and a terrible ordering problem — at age 51 still have to think a second to get left and right straight, frequently have to re-dial the phone because of number reversals, proofread every post here three times to get the typos out — the concept wasn’t known in my youth but I sure look a lot like what is now called NLD. The systematic approach I recommend here is what got me through high school, two bachelor’s degrees, and two years of grad school math, so I *know* it works to simplify all those issues and get those planes lined up for takeoff.