My favorite class as an undergraduate student in math was designed as guided discovery. My professor gave us tons of examples, scaffolded proofs and put hours into worksheets that would get us thinking but also keep us on task and feeling like we had enough hints to make progress. He had us write all of our answers on whiteboards before recording it in our notes and he constantly circulated to answer questions. I learned so much in this class and I think it inspired the way that I teach. I’ve thought deeply about his methods and I while I don’t do it as well as he did, I do have some steps for getting started.
1. What should students discover? Figure out what you want your students to discover. Just like every lesson plan, you need to start with your desired outcomes. However, this is subtle. Not every learning target makes sense for a discovery based lesson. Good candidates are learning targets which
- state a rule or formula (but actually have deeper meaning that is lost when you simply state the rule)
- have many tangible examples associated to them
- can be accessed through multiple lenses
- build naturally on what students can already do
Example: In my precalculus class, one of our learning targets is to be able to find horizontal asymptotes of a rational function. There is a very clear procedure and three discrete rules for determining them: If the degree of the polynomial on top is bigger there is no horizontal asymptote. If the degree of the polynomial in the denominator is bigger, the horizontal asymptote is at y=0, if the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. This topic is very dry. Sure, I could just present these rules and then give students a bunch of examples to practice on. But then, do they really know what a horizontal asymptote is? What will they do if they forget the rule? Instead, I can have them graph tons of rational functions and look for patterns in their asymptotes!
2. How will they discover it? Choose your examples carefully.
- have enough examples so that the patterns actually start to present themselves.
- begin with simple examples that will get them to develop a rough idea about the theory
- throw in a harder example that is counter intuitive to force them to refine their theory
- consider scaffolding the way that students make their observations, especially for younger students or struggling students
Example: I give my students lots of examples of rational functions. I ask them to first guess at the what the horizontal asymptote is by using the graph. Next, I have them estimate it by plugging larger and larger values for x into the function. Finally, we justify that for large values of x, it makes sense to only look at leading terms and they determine the asymptote that way. As they work through examples they record their findings in a table:
3. What will they do when they think they’ve discovered it? Have students make a conjecture based on their observations. Then have them come up with their own examples to test their conjecture. That’s what the two empty rows are for in the image above.
4. How will they know they are right? You’ll need to tell them…I had a horrible experience as an undergraduate in which a different professor than the one I described at the beginning of the post tried to run a discovery based class. The only problem was that he never checked our work, corrected our misconceptions, or even confirmed when we had reached a correct conclusion! To this day, I’m not sure of anything that I learned in that class. It was extremely frustrating and I attribute some of my lack of confidence in graduate school to this experience. Make sure that your students are validated when they are correct and challenged when they are wrong. You could do this be showing a reveal video or simply summarizing the lesson and what they were meant to get out of it.
For another example in a science class check out this video from The Cult of Pedagogy: https://www.cultofpedagogy.com/inductive-learning/.
Good luck and have fun!
Emmification 