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I have a college student who I've been tutoring over the past two semesters. He's a hard worker and quite bright. Unfortunately, I do not feel like I am having much success in helping him learn mathematics.

I work with him on a weekly basis, usually somewhere between 2 and 4 hours. He is currently taking college algebra, and took a remedial algebra course last semester.

There are two issues I notice while he is working:

1.) Whatever I teach him, he can't seem to retain on a long-term basis. For instance, last semester I'd taught him about factoring and how it relates to distribution, and he had gotten the hang of doing both. This semester, he seems to have almost completely forgotten all of it. He has forgotten his exponent rules, rules for working with fractions, and so on as well. I feel like he does not really understand what a fraction is either, as he cannot easily see the relationship between fractions and decimals, or percentages, or the relationship between fractions and division. So fractions are a point where we trip up. I've tried many strategies that will help him, including going over the forgotten material, and making cheat sheets to help him look things up easily when he's forgotten, but even then it's like he's forgotten the procedure for working with those aforementioned subjects. Needless to say, he's struggling with his college algebra course because it picks up assuming you know all of those things.

2.) He seems to be a bit of a perfectionist. He will spend hours on a single problem that he does not understand how to solve. I've suggested that he try a timer, and spend no more than maybe 15-20 minutes on a problem (these are problems I think are probably meant to be done in closer to 5 minutes -- his tests are 30 questions over two hours), but that suggestion does not seem to have helped.

To be clear, I think this student is bright. When I build things up for him from the ground up, he follows the argument. I ask questions while I make the argument to ensure that he understands the logic between one step and the next, and to the best of my ability he does. But when it comes time to do the work, I feel like either he did not follow the argument or simply cannot remember it. This student works hard, and I feel like it is more my fault that he is not processing this material than his, but I cannot figure out which approach I should take.

Has anyone had any experiences with a student like this? A motivated, hard-working student that just is not getting it. If there are any suggestions, they would be greatly appreciated.

Thank you.

TL;DR I have a hard-working, motivated student who just seems like he is not understanding the material from his math courses, and I need help figuring out strategies on how to move forward with him.

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    $\begingroup$ 1. What efforts does the student make on his own? One session per week is not going to be enough to build retention if he is not practicing a little each day. A long multi-week gap between semesters will exacerbate this issue. 2. Are you talking about test-taking behavior? If so, it might be good to emphasize that tests are just artificial games where each question has a finite upside (points possible) and a fixed amount of time total. $\endgroup$
    – Steve
    Feb 28 at 19:15
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    $\begingroup$ 1.) The student practices quite a bit on his own -- 15-20 hours per his approximation. I believe him based on the impeccable notes he takes and the pages upon pages of attempted problems he regularly comes to me with> 2.) I have discussed with him not to spend an inordinate amount of time on a test problem, but never phrased it as a sort of game as you suggest. I will try it. $\endgroup$ Feb 28 at 19:39
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    $\begingroup$ You mention he's having difficulty with fractions and decimals, which are typically taught in grades 3 to 6 or so, around age 8 to around age 12. You say you're checking for understanding, but what kinds of questions do you ask? Are you asking "Does this make sense?" to which he can just say "yes," or are you asking him to explain things and paraphrase his understanding back to you? $\endgroup$
    – shoover
    Mar 1 at 2:13
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    $\begingroup$ Actually this does not sound like a bright person. $\endgroup$ Mar 1 at 11:05
  • $\begingroup$ @shoover I'm probably guilty of asking "Does this make sense" from time to time and taking his word for it. But I also ask him to explain why something works (i.e.) why is $(x+a)^2=x^2+2ax+a^2$, where I would check that he understands that the exponent means to do $(x+a) \cdot (x+a)$ and that from there you can FOIL it, and I would remind him that FOIL works because of the distributive property. So I'm probably guilty of saying "Does this make sense" and leaving it more than I should, but I do try and ask pointed questions when the opportunity arises as well. $\endgroup$ Mar 1 at 19:32
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You make the (common) error of emphasizing explanation/understanding as the KEY aspect of pedagogy. However, the key is PRACTICE, not preaching. Humans are not computers, to which an instruction set, given once, can be run indefinitely and perfectly, once scanned properly.

Now, of course, nothing wrong with an explanation. And it can help. But if the trainee is not performing, you need more DRILL. For instance, you don't even mention how he comes to you with a notebook full of solved problems or the like. He needs more practice. He just does.

And if the procedures seem long/complex to him (not by words, but by demonstrated performance), then you need to drill simpler problems to start. And a more progressive (SLOWER) rate of increasing the complexity over time.

That's the actionable advice. Other than that, I would not assume he is as bright as you think...some students appear so conversationally, maybe based on being poised, but are not, in terms of brute intellect. "College algebra" is a remdial course in the US and has been so for last 50 years at least. The expectation for a stereotypical freshman is to be taking first semester calculus. If you're in "college algebra", you're behind. This is not 1940 any more, where some high schools didn't even teach trigonometry. I'm not expecting the kid to be in calculus in high school. But if he hits college and is still taking algebra he's behind a normal track.

Also, remember, "you can't win them all". Some students, you just won't help. (This isn't meant to beg the question...just that people are not roots of equations. You will not solve every one.)

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    $\begingroup$ Since you bring it up, I should mention that he does regularly attempt problems on his own, and he does have notebooks full of his attempts on homework problems. Perhaps the key is to start him on simpler problems, as you mentioned. $\endgroup$ Feb 28 at 19:36
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    $\begingroup$ You should analyze his practice. You say there are notebooks full of attempted problems. Well, does he drill with problems and then check the answer? Does he peek, or does he do the problem first and then check? When he gets it wrong, does he redo the problem? This is more like music or gymnastics. He needs to practice the skill, and then redo failed attempts. $\endgroup$
    – guest
    Feb 28 at 19:45
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    $\begingroup$ I agree that conceptual understanding is vital, and then comes the practice. The amount that he's forgetting does sound like this is somehow garbage in - garbage out for him. I wonder if he'd like Beast Academy? It's meant for gifted kids, so he's not their intended audience. But it's fun, it's very conceptual, and he'd be practicing with immediate feedback, hopefully daily. (The risk of suggesting it is that it's designed for kids. Would that offend him?) $\endgroup$
    – Sue VanHattum
    Mar 1 at 0:50
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    $\begingroup$ @guest His homework problems are online, and one of the things the site he uses allows him to do is to look at an example. The example is typically extremely similar to the homework problem, and he can usually follow along with the example well enough to get through his homeworks. I've encouraged him to stop using the examples, because solving math with a recipe for your particular problem is not helpful in my opinion. I've offered that he can call me more or less any time, and I would rather help him figure something out that he's tried on his own, rather than him use a recipe. No luck so far $\endgroup$ Mar 1 at 1:33
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    $\begingroup$ Thanks again, @SueVanHattum $\endgroup$ Mar 1 at 19:33
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The title says the student doesn't "retain," but your final summary at the bottom that he doesn't "understand." These seem like totally different things to me. From your description, it seems like the problem is that he doesn't understand, and therefore when he learns an algorithm, it's a bunch of arbitrary mumbo-jumbo, like being asked to memorize a poem in Greek if you don't speak Greek.

I think you're being unrealistic when you say that he's bright. If he had normal educational opportunities in the US in K-12, then the fact that he landed in remedial algebra in college means that he has always been a poor math student. You describe him as being able to follow your reasoning when you lay it out for him step by step, and in a comment you say that his online homework system offers him models, which he follows step by step. This is parroting, which is not a high level of intellectual activity.

More drill is not the answer. He has had plenty of drill in the past and it hasn't worked. This student does not seem like he's going to succeed unless he gains a better fundamental understanding of things like the conceptual meaning of division, and changes his mental habits. He needs to change his mental habits so that he's no longer using algorithmic crutches to get through problems that he doesn't understand conceptually. So for example, say he doesn't understand how to simplify $(a^9)^7$. You can remind him of the identity or make a cheat sheet for him, and that's probably what he wants, but that isn't going to help. So in this situation, you say something like, OK, let's do an example with simpler numbers. Say it was $(a^2)^3$. Now exponentiation is repeated multiplication, so can you write the part inside the parens that way? OK, now do you see what will happen when you do the outside exponentation the same way? How many factors of $a$ did you get? Why did it turn out to be 6?

The problems you describe with not understanding division and ratios conceptually are extremely common. Kids take a wrong turn in 4th grade, and they end up in college still not understanding these concepts. This has to be approached conceptually from a variety of angles. For example, in a word problem with some context, you can talk about trends, such as what would happen if the denominator got bigger. If he says that a 7% change is a factor of 0.07, talk him through an example with 7% tax on a \$100 TV and get him to come up with the fact that they're not selling him the TV for $7. Show him dimensional analysis so that he can see whether it makes sense that meters divided by meters/second gives units of time. Conceptual understanding of multiplication and division comes through a long process of seeing things like this from all these different angles, and developing mental habits of sense-making.

The idea of you making cheat sheets for him is really bad IMO. Encourage him to make cheat sheets for himself if he needs them. He needs to be independent and take responsibility for organizing his own knowledge. That's part of the process of gaining conceptual understanding.

So we're talking about getting this student to change his mental habits and develop better conceptual understanding of the last decade's worth of math instruction -- not the last semester. Realistically, most students at this age are not willing to put in the kind of sustained effort that would be required to completely retrain their brains. And making this kind of change takes years, so if they're already in a college STEM major, then they don't have enough time to make the change. You don't say what the student's major is or his intended career. If he's an engineering major, then it's essentially impossible to accomplish these things this late in the game. If he wants to be a pharmacist, then it's realistic to set a goal like getting him to the level where he can convert milligrams to grams without ever getting the conversion factor wrong.

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    $\begingroup$ Say it was $(a^2)^3$. Now exponentiation is repeated multiplication --- When I taught (1983-2005; includes courses taught as a graduate student), I pretty much always pointed out that for something like $(a^{2/3})^{1/4}$ you could consider $(a^2)^3,$ which can be determined without knowing exponent rules via $a^2a^2a^2 = (aa)(aa)(aa) = a^6$ (suggesting multiply exponents), but nearly always it seemed to me that those who got this and could make use of this device were already fine with exponent laws, and it didn't help those who didn't know the exponent laws. However, (continued) $\endgroup$ Mar 1 at 17:44
  • $\begingroup$ around 2003 or 2004 I had one college algebra student who was really bad at math, but otherwise seemed reasonably intelligent (especially with rote procedures, except she kept getting the "math rules" mixed up), and as she came by my office for most of my office hours that semester, I managed to get this method to work for her. Whenever she stumbled and forgot what to do, I had specific crutches she had learned (from having her write them down every time she needed one), such as expanding $a^2a^3 = (aa)(aaa)$ and $(a^2)^3 = \ldots$ (continued) $\endgroup$ Mar 1 at 17:52
  • $\begingroup$ For adding fractions, I had her double check her "cross-multiplying" procedure with $\frac{1}{2} + \frac{1}{2}$ (to make sure the process she used produces $1$ in this case), and many other things (e.g. percent to a decimal -- which way and by how much to move the decimal point? -- double-check your "guess" with $50\%$ and $100\%$ which we know to be one-half and one). She never got very fast, but this wasn't important as she only needed to pass College Algebra, and I think she actually wound up with a B or B+. $\endgroup$ Mar 1 at 18:01
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    $\begingroup$ @Ben Crowell - I couldn't have said it better myself! My only quibble is with "I think you're being unrealistic when you say that he's bright. If he had normal educational opportunities in the US in K-12, ... remedial algebra in college means he has always been a poor math student." $\endgroup$ Mar 2 at 2:52
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    $\begingroup$ There are plenty of bright people who get screwed by "normal educational opportunities in the US" because the norm isn't very good. That's why remedial math after high school is so common. Were they given math only slightly beyond what they've already mastered with a good teacher and curriculum, they would thrive. This student, motivated enough to do endless of rote drills, is likely one of them. $\endgroup$ Mar 2 at 2:57
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tl;dr Does he need to know algebra, or just to pass the course? Rant follows.

The kinds of struggles you describe suggest that he won't/can't major in a science. Perhaps college algebra is simply a hurdle he must jump in order to get on with his literature or history or psychology, where he may indeed be a good student.

If that's the case, what you need to do for him is help him pass the exams. Find some samples, then do lots of problems just like those.

The sad fact is that although algebra is an important step on the path to more real mathematics, "college algebra" is often a conspiracy. The problems on the exams are just like the ones on the book with different numbers (perhaps different letters, but usually just x and y). When a student passes the exam the teacher can pretend to have taught something the student can pretend to have learned. But it all vanishes right after the exam.

Final note. Fractions are conceptually difficult. In elementary school they occur as divisions of pies, points on the number line, and in formal manipulations. Understanding how those ideas fit together is hard for teachers, let alone for students. See https://math.stackexchange.com/questions/1127483/how-to-make-sense-of-fractions/1127776#1127776

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