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There is always a risk that the teaching of translation is neglected. Meaning: there is a process to go through to get from a word problem to a collection of variables, equations and constants. Very often, it’s neglected to cover this key step – resulting in students who are extremely adept with manipulating numbers or formulas, but who are utterly unable to apply their skills outside the math classroom.

Having a class with first-year (university) math students of varying ability levels is not really a good position to start out with, as there tend to be gaps in prerequisite knowledge (or each student's feelings towards their own ability to learn math)… and teachers must decide how to meet the needs of the individual students in their classrooms, without ignoring the rest of the students.

I know there is no “ideal” solution to this as it depends on individual situations, but what approach(es) would be recommendable to handle/balance such mixed-knowledge classes? Are there any references or papers that academically analyzed this issue and present potential solutions to the problem?

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There are a great series of papers by or involving Dylan William including Inside the black box, Mathematics inside the black box and Assessment for learning: why, what and how?, in which he examines strategies that actually have the largest measurable impact in education. The title "Assessment for Learning" has been rather abused since then; to quote him, "if your approach to assessment for learning involves a spreadsheet in any way, you're doing it wrong!"

(Much of the evidence comes from a wide age range, but I don't believe people become fundamentally and radically different at age 18, for example.)

In one of his papers he references a study that examined the effects of peer teaching on different ability groups. (Sorry I can't find which paper, nor the study he references, but the example was memorably compelling.) They felt that peer teaching was beneficial for students who were doing badly but held back students who would otherwise get further ahead and excel.

They categorised the students into strong, medium and weak, then had them work in mixed ability pairings with explicit peer teaching tasks over the course of weeks, and reassessed at the end. The weaker students did indeed improve ahead of the control group as expected, but the surprising result was that the stronger students improved compared to the control group by a greater margin - peer teaching actually helps the helper even more than it helps the one being helped!

Perhaps this isn't surprising considering that teaching forces you to verbalise things clearly, answer difficult questions, reiterate fundamental material etc, and generally puts you in a position where you strongly want to have the answers ready.

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Conclusion
We conclude then that the best way of dealing with mixed ability groups, for both ends of the spectrum, is to pair them up in deliberately mixed-ability pairs and give them joint-responsibility rich collaborative tasks, and explicitly teach things like coaching over telling.

We've found this to be very effective indeed in practice.

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[Aside: If you're stuck for ways to ensure collaboration instead of copying during class, you can do it in class with Dr Spencer Kagan's "Kagan Strategies". On the face of it, you could worry that such an approach would be off-putting to serious older intellectual students, but our experience at our 16-18 college in the UK is that the older, brighter and more mathematically confident they are, the less they mind joining in with what appear at first to be silly activities. (The least prepared to take part are the groups of younger remedial students.) The key is to explain why they're doing the wacky thing and to show them an example before they start to break the ice. We've tweaked a few, but largely find that the original is best for promoting learning, even if it seems wacky at first. Certainly it's important to do this from the start of the year - students will accept almost anything when they're new but moan like crazy half way through term if you change the way they learn.]

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  • $\begingroup$ Meh... "The best way to learn is to teach" was something I heard a lot from professors I respect when I was studying... $\endgroup$ – vonbrand May 6 '14 at 22:09
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    $\begingroup$ @vonbrand For a reason, yes - and something that became vividly clear to me. My hardest undergraduate lecture course was Algebraic Topology, but I got 99% because I was obliged to run the problems class once a month. This is the first year I've taught the top Further Maths group without making peer teaching the core activity (at the insistence of managers who felt peer teaching was holding them back). The lessons have been much better explained than usual, but the level of learning has been a disaster by comparison. Putting students on front of their peers to explain beats tests completely! $\endgroup$ – AndrewC May 7 '14 at 5:58

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