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In a college math course one is bound to find a fairly broad range of students in terms of their quickness in understanding the material. This is due to many reasons, including differing mathematical backgrounds. Inevitably, there are always a handful of very bright students who don't show up to lecture/discussion because they don't feel like they need to in order to pass the class (honestly, they're often right in this regard. In fact, they most often score the highest on exams).

However, just because they're quick to understand the necessary material to score well in the class doesn't mean that we don't have anything to offer them. I don't think we should be content with just letting them skate by because they have the natural ability to do so. This leads to my question:

How can we maintain and grow the interest of the brighter students in a math classroom, even if the bulk of our attention and effort is (rightfully so) spent with the rest of the class.

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  • $\begingroup$ Relatedly (since this is about keeping students interested in a more general sense as opposed to keeping them coming to class) I have been considering the notion of having 'substitute' problems in problem sets. That is, for certain problems, give the students the choice of solving a harder, more interesting problem (this is like extra credit in some sense, something like 'substitute credit'?) So for example, if the original homework set questions asks to perform some computations, the substitute harder question might ask for a proof of a general fact from which the computations follow. $\endgroup$ – Aru Ray Mar 14 '14 at 13:42
  • $\begingroup$ This worries me a little, since in some sense, the students are not doing the same problem sets any more, but since the students have a choice in the matter, I'm more at peace with this. I also think that students might be more likely to do 'substitute credit' problems than 'extra credit' problems (particularly for bright students it might give them the opportunity to avoid having to answer questions they find tedious) $\endgroup$ – Aru Ray Mar 14 '14 at 13:46
  • $\begingroup$ As an ardent hater of group work in my student days, any digression from well-thought lecture, seemed like a gimmick to me. I would not have appreciated many of these "modern" techniques. But, whatever, I guess I'm a dinosaur. $\endgroup$ – James S. Cook Jul 18 '14 at 15:22
  • $\begingroup$ I would be careful about pushing this especially if you give a marginal benefit to a few students and not much to others. Also, it is not clear to me that adding enrichment is as useful for smarties as if they just accelerated. Also, they do have other courses. Can use free time on those if yours is a gut fir them. $\endgroup$ – guest Dec 26 '18 at 6:25
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Once upon a time I was one of those brighter students. Two things helped.

One was the opportunity to help other people. I got started tutoring mathematics when people would come to me for help with their homework, and I learned that the effort to explain a concept helped me understand it better. A couple of times, I found that I didn't understand a concept well enough to explain it successfully.

Another was for the teacher to suggest additional or "extra" topics that I could work on independently. for instance, my HS Freshman algebra teacher loaned me a book on Computer programming.

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    $\begingroup$ A bonus here, is that having the brighter students help the weaker students frees up time for the teacher. Some of which they could spend on aforementioned brighter students. $\endgroup$ – Nico Burns Mar 14 '14 at 23:47
  • $\begingroup$ Can I add, being asked to solve the same problems in other ways, and being asked to look at the assumptions hidden in the problem and try changing them. As someone who is still often asked to do mathematics, I have found both of these things have helped keep my engagement high. $\endgroup$ – David Wees Jul 22 '14 at 10:59
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Differentiation is one of the hardest parts of teaching, for sure. One thing that has helped me a lot is reframing my questions so that they are more "open."

For instance, instead of asking my students to factor a quadratic, I'll ask them "Say that I've got $x^2 + bx + 9$. What can $b$ be if this expression is factorable?"

What's great about more open questions is that kids will approach them differently, depending on where they're coming from. A weaker student will wonder what the easiest binomials to multiply and get 9 are. A stronger student might start getting clever and think of some sort of generalization. We can have a good discussion with the class with a lot of different perspectives after giving the kids a bit of time to work their thoughts out.

You might check out this book from your library: http://www.nctm.org/catalog/product.aspx?ID=13782

Another way to give your top students what they need is to experiment with non-lecture or whole-group discussion formats. You might consider giving your students a great deal of class time to work on problems. Your faster students will naturally find their way to problems that are challenging and worth their while.

Check out the Park City Math Institute's problem sets from their educator's course: http://mathforum.org/pcmi/hstp/resources/course2009.html

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Eric Mazur's influential book Peer Instruction advocates a number of teaching techniques, one of which is to use short, easy, multiple-choice tests at the beginning of each meeting to enforce a requirement that students read the book before coming to class. The point is that if you don't do this, students will never read the book, and then you'll be forced to spoon-feed them every topic from scratch in lecture. Even weaker students can get some kind of feel for what's going on by reading. This allows you to spend class time on the topics that almost everyone has trouble with, because they're just plain hard.

Even so, this problem is probably inevitable to a certain extent in a freshman calc course. It's traditionally structured as a one-size-fits-all course for everyone from biology majors to math majors. Furthermore, you will inevitably end up with students who, for example, took AP Calculus in high school but for one reason or another are taking calc again in college.

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Some ideas:

Talk about why you find things interesting and about the cool stuff, even if it is at a high level. I run a maths learning centre, so the majority of students I talk to are struggling. However I still talk about the higher-level stuff in there and the "weaker" students say they like to hear people talk about those things even if they don't necessarily understand. They key is to do it because you think it's cool/interesting/fascinating/exciting. The student who doesn't get that bit needs to still catch your enthusiasm.

Be extremely clear with your explanations. This tempers the above point so that the students with less experience don't get lost. However it is worth noting that some concepts are hard for almost everyone, and there is nothing like a quick learner to get distressed when they don't understand! So be very very clear about all the steps involved in things and give lots of thinking strategies. I find the clever ones appreciate you being clear, even if they've heard it before. You just might say something they've never thought of.

Arrange for lots of time to work on problems in class time. If you have tutorials/workshops as part of your program, then set them up where students solve problems in groups, and give them a range of problems to choose from. This will benefit everyone, but the quick ones will either self-select into groups where they do the harder problems, or they will take it upon themselves to help teach the other students. This will be a win-win for everyone. You do need to make sure there is a rule that they need to make sure everyone in the group is ok to move on before moving on!

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