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I am now tutoring a student in Grade 9, who falls behind in math study. He lacks the basic understanding of operations and inverse operations, and have trouble dealing with negative numbers and fractions.

As a consequence, he knows how to solve an equation like $2x+3=7$, but fails to solve equations like $2x+3=1$. When I asked him what number times 2, and then plus 3 equals 1, he hesitated and answered: there is no such number. Then I told him to pay attention to the last operation performed on the LHS and use the inverse operation, he then figured out the right answer. Now try $7-3x=10$. He found that the last operation is subtraction, so he added $7$ to both sides. I then told him $7-3x$ actually means $7+(-3x)$ and we need to subtract, but he seemed confused. He didn't know subtraction is addition of the opposite.

Same thing goes for $\dfrac{3}{2}x=-5$. It seems to me that he is not familiar with real numbers and used to seeking for solutions only within the set of natural numbers.

Indeed, he is not well prepared for algebra, since he's not familiar with the number properties. When he combines like terms, not only does he add or subtract the coefficients but also does the same thing to the exponents. (Sometimes he even fails to identify the like terms). I told him that was not right, and explained how the distributive property works here. Next time, he made the same mistake. He is easily confused and shows poor understanding of abstract concepts and expressions.

Now he is learning geometry (not the advanced class that requires two-column proof) and his school teacher commented he was weak in mathematical reasoning. I really want to help him build a solid foundation in algebra (since he can follow the geometry course in school, I don't need to teach him much geometry), fill up the gaps, but we don't have too much time and therefore need a clear and efficient plan. I don't know where to get started, since there is too much to learn and it takes him too much time to understand, internalize and solidify the knowledge and skills. Many times, I decide to start somewhere only to find that he lacks something even more basic.

Is there any suggestion to help him into the gate of high school algebra? To start with the most basic knowledge (like G1~G6 math) or just start where he seems to struggle the most? Is pre-algebra a good choice for him?

Additional information: I have only 2 hours with him each week, and now I'm trying to get more time. Maybe on holiday, I can make and carry out some teaching plan.

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    $\begingroup$ Start with an overview of numbers and linear equations, the meaning of parenthesis, order of operations... $\endgroup$ – Paracosmiste Dec 26 '17 at 17:00
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    $\begingroup$ It seems like there are two distinct possibilities 1) the student truly does not understand that subtraction is the inverse of addition, and 2) the student does not understand the correspondence between their intuitive understanding of quantity and equations. I have found the latter to be much more common. $\endgroup$ – Adam Dec 26 '17 at 17:51
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    $\begingroup$ Have you assessed the student on concepts and definitions [NOT procedures] involving (a) Fractions/Decimals/Percents, (b) integers, and (c) introductory algebra? $\endgroup$ – WeCanLearnAnything Dec 27 '17 at 2:49
  • $\begingroup$ @WeCanLearnAnything Yes, he seems to only know how to do it without understanding what's going on (rote learning, I guess). That's why I ask this question, where to start with? Conceptual teaching is time-consuming and may not work for some students, although the rewards may be significant in the long run. $\endgroup$ – Mathis Dec 27 '17 at 3:44
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    $\begingroup$ If you like, send me a private message and I'll send you "all concept, ~no calculation" assessment you can have the student do. It's designed so that conceptual understanding makes the questions easy and lack of understanding makes them hard to even start. It also discourages guessing by asking students to rate their confidence. I suspect your student will leave most of it blank. Then you can show that to the teacher/parent and suggest a more drastic intervention of forgetting about school work and reteaching integers, rational numbers, intro algebra, etc. $\endgroup$ – WeCanLearnAnything Dec 27 '17 at 20:32
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My guess is that he depends on "following the rules", and there are now too many rules for him, because none of the 'rules' makes any sense to him. I believe he needs to see things differently to succeed. I don't know for sure without meeting him. How old is this student? Did he hire you, or did his parents or school? If the latter, it may be an unworkable situation.

If he cares about getting it, I might start out with puzzles meant to intrigue him. It will not bear fruit quickly.

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    $\begingroup$ He is in Grade 9, now learning geometry in school. I am his private tutor and only have 2 hours a week with him. Yes, I feel it really hard to reteach him everything in limited time. $\endgroup$ – Mathis Dec 28 '17 at 3:55
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It seems he has a lot of gaps. It would be good to teach him each gap separately. For example, don't expect to teach him about adding negative numbers and identifying like terms by exponents at the same time, although you may be able to work on more than one gap in a lesson.

Identify a gap and then work on it with him. He should have a notebook just for working on his background unrelated to class. Introduce him to Khan Academy. There he can drill on what you've taught him. If he identifies you as his teacher, you can assign topics for him to work on and see his progress. You can also see what he's done. That means if you tell him to practice for a certain amount of time (e.g. 30 minutes 3 times a week or 10 minutes everyday), you can see if he's done what was assigned. The computer will give him drills until he gets a certain number right and then text him on the same concepts later in master drills.

You seem to have a clear handle on where his gaps are. It doesn't seem like you need someone to tell you what to work on, just how to get him to move forward. I hope this advice will help.

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    $\begingroup$ Great idea about using Khan Academy! He only comes for me once a week, and spends 2 hours with me each time. Long gaps between two consecutive tutoring even make things worse, and he is likely to forget what has been taught in the previous lessons and even forgets to do extra homework that I've assigned him. I really want to assign him more work to do, know his progress and give him more instructions. Thanks for mentioning Khan Academy. I will try it, Hope it'll work for him. $\endgroup$ – Mathis Dec 27 '17 at 0:29
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In addition to the suggestions about remediation and the like, consider to just teach the fellow to mechanically solve for x. Not even thinking about "what number what get me this"...I think this is more how gifted kids or at least average kids without gaps might think about the problem. But it is going to be super hard sledding (and just not work) for someone who is bad at double negatives and fractions.

Consider to just teach him to mechanically "do everything to each side of the equation" in order to solve for x. Every step written out explicitly for each side of the equation. Perhaps this gives him some progress at least with the immediate objective and over time helps him to realize the concept of negatives and fractions and the like. Some people have to learn that way.

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    $\begingroup$ I totally agree with you. I think he just fails to build the connection. Some students may discover patterns and identify subtle differences by themselves, but this one definitely isn't able to. Learning how to do it (even without fully understanding what's going on) can give him some confidence and something concrete to start with. Hope over time he will figure out the reason why things should be done this or that way. $\endgroup$ – Mathis Dec 27 '17 at 0:18
  • $\begingroup$ Good luck, man. Fight the good fight. $\endgroup$ – guest Dec 27 '17 at 0:24

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