How to teach a student algebra who misses too much previous knowledge?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

The student in the question here is very similar to many (most?) of the students that I get in remedial college elementary algebra courses. In particular, if they don't have basic arithmetic skills, including primarily signed numbers, and secondarily fractions, then it's exceedingly difficult for them to get traction -- and we definitely don't have time at that level to remediate those even-more-basic skills.

In response, I developed a quiz website (free, no ads, no login), Automatic-Algebra.org, with quick timed quizzes on what I'd identify as the prerequisite "automatic" skills needed for algebra and other courses. While mechanical drilling is not the majority of what you should be doing in a math class, I do think it's necessary for a small core of math-facts that (like the basic vocabulary in a new language) the user needs to have automated, so as to not spend any mental energy in the middle of more complex work.

In particular, for elementary algebra, I have quizzes on the arithmetic skills of (1) times tables, (2) negative numbers, and (3) order of operations (quizzes 5 questions each, time 15, 30, 60 seconds respectively; based on practical testing and some outside research). I do feel that to succeed a student needs to be able to complete these quizzes, without any errors, within the indicated times. I tell my students that they should practice every day until they never get any wrong.

Full disclosure: I think this only gets a small amount of traction in my classes. I'd guess that the students who arrive with the given level of prior gaps usually don't have the study skills to do daily drills like that, and it's overshadowed by the actual algebra content coming at them that we're working on in class, homework, and graded tests. I certainly don't have time to spend on it in class, so the student needs to be sufficiently self-motivated to do it on their own.

That said, for someone in the OP's position as a weekly one-on-one tutor, I think they'd have a much greater opportunity to check in and monitor this. Start every session with a run through of the basic quizzes (could take about 2 minutes). If the student is still deficient, encourage them again to practice more next week. Keep checking in every week like that. (And I'd also add in a rapid-fire verbal quiz of "What's the inverse of the ____ operation?" every week.)

Admittedly, some students will never be able to perform these basic arithmetic skills proficiently. (In fact, I've had students who couldn't believe that anyone could ever complete these quizzes in the indicated times.) Unfortunately, I don't think students in that category will ever really succeed at algebra or higher courses.

Answer 4

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.