First, let's look at the big picture. The major problems in Math Ed are not at the top end of the achievement distribution, where students are successful no matter what pedagogy is employed. This isn't the first priority when one is thoughtful about the landscape of issues that we face in our system. This isn't to say that we should ignore opportunities here or the students. It's just that the way to handle high-achieving students is clear and straightforward. Let's dispatch this one quickly...
General effective pedagogy includes keeping all your students, no matter the level, engaged in activities appropriate to their level. High-achieving students are a pleasure to work with. These students like math, they are motivated, and want to learn more! So there's no barrier here to deal with other than laziness. All these students need are good problems and some feedback.
The main ideas to keep high-achieving students engaged are listed below.
- Problem sets should include problems challenging to your high-achieving students.
- High-achieving students can also be given extra problem sets where they submit proofs in writing. These problems would not be presented in class normally.
- Additional /supplemental articles or chapters can also be assigned.
So there you go. Give them problems at their level, and then get out of the way. Offer feedback and support as necessary. What's the trajectory? Junior/senior students should be able to do graduate-level work after about a year of full IBL in a particular subject (e.g. a year of Real Analysis, Topology or Abstract Algebra).
Additionally, high-achieving students can become peer mentors and teachers. They can model how to write a good proof, offer constructive feedback in ways that are often better received than from the instructor, and can be deployed to help struggling students, via small group work.
When you have high-achieve, socially skilled students, you have a great, great asset that can positively affect the learning culture of your classroom. Good teaching practices tell us to keep all students engaged appropriately, and then one can redirect all this talent and ingenuity to increase the level of the whole class.