By Michael Oehrtman

After a few classes in which students were working in groups, one table called me over for help. I responded to their question with my own, that I expected would point them in a productive direction. Clearly not appreciating my Socratic turn, one student burst out in frustration. “Aren't you going to teach us anything?” Several weeks later, as I was approaching that group to better observe what they were doing, the same student urgently waved me off. “No! No! Don't tell us anything. We want to figure it out!”\

For the past couple decades, I have worked with colleagues to create, test, and refine activities to support students’ conceptual learning in the introductory college calculus sequence. This design research process has produced a large volume of instructional resources that we maintain at our project website. In this post, however, I will describe some of the general features of tasks we have learned throughout this process that support productive student engagement and learning.

We have encoded the general principles derived from our project in definitions of the key words in the project acronym: Coherent Labs to Enhance Accessible and Rigorous Calculus Instruction. I provide our definitions below with some commentary on what we have learned makes these characteristics effective.

Coherence: Instructional activities are coherent to the extent that they leverage consistent meanings across different representations, contexts, and concepts.

We want our students to be able to apply mathematical reasoning beyond the limited contexts in which we present them in our introductory classes. Ideally, students will develop an understanding of a small number of powerful ideas that they can then apply flexibly in novel situations. This means students will need experience with a consistent set of mathematical meanings to identify and abstract common structure across their various instantiations and potential application. We focus on such coherence at three levels.

Coherence across representations: In a given task setting, the same mathematical objects and relationships are expressed consistently across various representations (usually algebraic, numeric, graphical, and contextual).

Coherence across contexts: The same mathematical concept is applied consistently across multiple contexts. Initially students think they are doing completely different problems when the context changes. When they recognize the key features of the problems and their work are in fact the same, they are abstracting the mathematical concept.

Coherence across mathematical concepts: Concepts are treated consistently across the development of other mathematics in which they are used. Limits are a critical example of this type of coherence in calculus since most of the other concepts in the course are defined in terms of limits.

Labs: Cooperative labs engage students in challenging problems requiring them to select, perform, and evaluate actions that support their construction of powerful mathematical ideas.

The heart of active learning is, of course, students’ activity. But what kind? While exercises that engage students in practicing what they already know how to do have their place in a mathematics course, they are not the engine of conceptual development. A high pay-off for the investment of devoting class time to active learning comes from students engaging in genuine problem-solving. This means they have a high degree of autonomy over selecting an idea to apply to the problem, performing those actions, and evaluating the outcome.

Fun fact: Most learning theories view individual or collective goal-oriented actions as the key resource for learning.

Accessibility: Powerful concepts are accessible in instructional activities to the extent that the tasks support students in identifying mathematical relationships, making and justifying claims, and generalizing across contexts to extract common mathematical structure.

Applied problems play a more important role than just motivating students or convincing them that the mathematics they are learning is useful. If students are to leverage the coherent structure and problem-solving described above, they first need to be able to meaningfully recognize and interact with it. Our primary purpose for incorporating applications is that students’ understanding of a situation can provide a foundation of meanings that enable them to make conjectures and claims, justify reasoning, and argue about ideas. Students’ need for tracking their own reasoning in challenging problem-solving settings or to communicate with others provide opportunities for them to systematize these intuitive ways of reasoning and eventually express them in mathematical language and notation.

Rigor: Instructional activities are rigorous to the extent that they require students to engage in problem-solving activity whose structure reflects that of the underlying mathematics to be learned.

Formal notation, definitions, accurate algebraic manipulation, and proofs are some of the trappings of mathematical rigor. But they alone will not support students to develop productive understandings that can serve as a strong foundation for further study in math and science. Students’ activity in informal settings can still be rigorous, however. For example, the CLEAR Calculus resources systematically leverage students’ reasoning about approximations, errors, error bounds, and methods to achieve a desired degree of accuracy. These are all ideas that students readily grasp and use effectively in ways that are isomorphic to standard epsilon-delta definitions and arguments. Whether we formalize the ideas in those terms or not, students have engaged meaningfully and deeply in the same ideas, relationships, and logic.

It is easy to assume a dichotomy between a formally sound, structurally robust treatment of calculus on one hand and a conceptually accessible and applicable approach on the other. In contrast, our definitions of coherence, (active learning) labs, accessibility, and rigor outline how we think about the work of developing and implementing instructional activities so these are mutually reinforcing principles.

Michael Oehrtman is the Noble Professor for Technology Enhanced Learning in Mathematics at Oklahoma State University. He conducts design research on inquiry-oriented learning in entry-level college mathematics, calculus, and analysis as well as on enacted mathematical knowledge for teaching at the secondary and college level. He is an author of the non-commercial instructional resources CLEAR Calculus and a leader in the Mathematical Inquiry Project which promotes student success in entry-level college mathematics courses.