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I am looking for ways I can correct fundamental math mistakes.

I am currently tutoring someone taking a course which is a cross between first year calculus and grade 12 functions. In high school he learned math by memorizing a bunch of rules and then matching them to different types of questions. This works ok for simple questions but now if he sees something he does not instantly recognize for he has no idea what to do and attempts to solve by doing things that look similar to the rules he knows. As a result the only math he can do is plug n' chugging numbers into memorized solutions with no real understanding.

Here are some examples of the mistakes he makes:

$\frac{x+5}{y+5} = \frac{x}{y}$

He has a rule that says if you see the same thing on the top and bottom of a fraction it equals 1 and can be crossed out but he fails to understand that this rule only applies for multiplication not addition.

$x^2 - x = x$

The difference between $x^2$ and $2x$ is not clear to him.

$\frac{15}{3x} * 2x = \frac{15*2x}{3x * 2x}$ or sometimes $\frac{15}{3x}*2x = \frac{15}{3x*2x}$

He simply can't wrap his head around fractions he would have no issue with this example if it was presented in the form $\frac{15}{3x} * \frac{2x}{1}$

He also struggles with basic algebra such as solving a linear systems, 2 equations 2 unknowns. He can not put together the solution and gets lost just moving numbers around until he forgets what the goal of the question is.

When I see him make a mistake like this I get him to stop the question and then give him a small example like #1 with dummy numbers and get him to solve it the right way and the wrong way to make it clear that they are not equal. Then I let him return to the question and he corrects the mistake.

I am looking for anything else I can try to fix his misunderstandings with fundamental mathematics.

Also I am not a professional tutor I am just trying to help this guy out.

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    $\begingroup$ If you have time, just go back to basics. Teach fractions, exponents, everything, starting from the beginning. I think the most important point is to emphasize that everything is a number. Almost all students understand the rules behind arithmetic, most don't understand the rules behind algebra because they don't understand that we're still working with numbers. For example, regarding $x^2$ vs $2x$, it should be fairly obvious that $3^2 \neq 2\times 3$. $\endgroup$ – Javier May 15 '17 at 20:13
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    $\begingroup$ Thats the plan, the issue is I need to help him keep up with the lectures as well $\endgroup$ – Gareth Shepherd May 15 '17 at 20:44
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    $\begingroup$ You will help him more if you work on the foundation rather than trying to help him build the third story. $\endgroup$ – Chris Cunningham May 16 '17 at 16:57
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    $\begingroup$ The student should drop this course. He needs to learn how to solve fourth-grade story problems, including the meaning of adding, subtracting, multiplying, and dividing. He also needs to thoroughly understand the algebraic "order of operations", why it matters, and how it applies to fractions. $\endgroup$ – Jasper May 16 '17 at 17:30
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    $\begingroup$ You beautifully laid out things he repeatedly gets wrong. Why not do the same for him? Make a reference of exceptions to his rules (can't cross out 5's if they are terms and not fractions) that he can refer to. Of course you wouldn't say it that way - you'd show him with examples. Then give him a few similar examples at the beginning of each session to solve make sure he is remembering what you taught him. You can also give him some of these for homework so he can practice correctly. This is more efficient than teaching from scratch, but of course he won't have the same understanding. $\endgroup$ – Amy B May 18 '17 at 6:28
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It sounds like he is not prepared for a course in Calculus. I would encourage him to go back and retake College Algebra which reinforces factoring and canceling out terms. Sadly, it is going to be an uphill battle for someone in a Calculus course that makes mistakes that most Algebra 1 students don't make.

When my students make these mistakes, the only thing you can do is reteach the concept to them. For example, for a student that is struggling to multiply $\frac{1}{3}\times3$ I would reteach multiplying fractions and representing whole numbers as fractions. Students make the mistake of canceling out 5 in $\frac{x+5}{y+5}$ frequently. You might ask them to plug in a number for x and y and see if $\frac{x+5}{y+5}=\frac{x}{y}$. Then they may come to realize that you may not cancel a value unless it is being multiplied to the numerator and denominator.

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    $\begingroup$ Regarding your last sentence: I don't have any research to back this up, but personally I like to phrase it differently: we can only do something if we have shown (or the teacher has said) that it can be done. So instead of "you may not cancel unless it's a multiplication", the mentality should be "if there is a multiplication we can cancel". It's subtle, but I think it's an important difference in frame of mind. $\endgroup$ – Javier May 15 '17 at 23:23
  • $\begingroup$ @Javier Well, the reasoning for the phrasing of "may not cancel" is that mathematics is based on certain laws, axioms, and definitions. These laws allow us to do certain things and forbid us from doing other things. But, I see what you are saying if a student confuses this phrasing with only being able to do what the teacher says you can do. $\endgroup$ – MathGuy May 16 '17 at 17:12

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