How do you know when to use a specific teaching method or technique? This is a question that all teachers deal with, and I believe that a general tool for sorting some of this out can be very helpful. One idea I have been working on is a framework called "Learning Zone Analysis" or LZA for short. In this post, I'll discuss one aspect of LZA, which is useful for deciding when to use active learning and when one can get away with a mini lecture or flipping a topic outside of class.
Zone 1 contains dispositions. Dispositions include (but not limited to) problem-solving ability, learning to read and write proofs, positive attitudes about mathematics, being willing to experiment, searching for counterexamples, advanced techniques, communicating ideas, utilizing effective practices in the study of mathematics.
Zone 2 contains basic skills, factual knowledge, connecting Math to other subjects (or other disciplines within Math), motivation, organizing information or a unit of work that students have just presented proofs on, etc.
LZA can be represented in a diagram:
For Zone 1, it can be argued that it is most appropriate to use active, student-centered methods, such as IBL. Zone 1 is about dispositions, habits of mind, and cultivating higher-level skills. Such dispositions must be developed by students for themselves through sustained practice and reflection in a supportive environment. Dispositions cannot be learned by listening to others, and this is fundamentally why actively solving challenging problems is necessary.
Zone 2 can be effectively and efficiently covered through lectures or mini lectures. Learning about where your office is shouldn't be a problem-solving experience. Similarly, students could learn that Fourier Series can be applied to signal processing on their own, but it's much more motivating and useful if the instructor presents a succinct, clear exposition of the connections, providing value and motivation. Further I can envision setting the context of a unit, what students are responsible for learning outside of class, students' roles, and and should be done via direct instruction.
Motivation actually exists in both zone 1 and zone 2. In zone 2, the instructor can give explicit motivation for mathematical concepts. A different kind of motivation can be addressed by the instructor in the form of encouragement and praise. Encouragement and praise should be regular and clearly positive.
Motivation in Zone 1 is tacit. It is through individual successes over long time periods that students become ever more confident and motivated to learn mathematics. It is also arguable the the motivation from being successful at solving hard math problems is more authentic and long lasting compared to pep talks. Motivation from mentoring or coaching and from success are both necessary.
How does this all work in the practical world? For a specific topic, list the goals of the lesson(s) into zone 1 and zone 2. Then select IBL or teacher-centered to cover each zone. A rule of thumb is 75% IBL and 25% teacher-centered is a good place to start, with variation class-to-class to suit the specific mathematical landscape and how students are getting on with the material.
There exist other ways to use LZA. We could evaluate lessons or curricula to see how much higher-level thinking vs. factual or skills knowledge is present. LZA can also be used in class observations to measure how much of the visible work is in zone 1 or 2, and the relative effectiveness of lecture vs. IBL. More on these other uses in future posts.
