Reinventing the Principle of Mathematical Induction
Task 1. "The L-Tiles Task." Productive engagement with this task should allow students to:
- be generative in building the next case from the previous case
- assess whether their argument between specific cases generalizes to work between any two consecutive cases.
- test and adapt their argument as needed when the square that is removed is arbitrarily chosen.
- reflect on the logical structure of quasi-induction.
- provide students with the space to make sense of and summarize the general argument from one case to the next.
- promote student noticing of the role of the base case and how the next case depends on the previous case.
- begin to establish students’ intellectual need for the logical components (hypotheses) of PMI.
Task 2. Investigating the Components of Mathematical Induction. Productive engagement with this task should allow students to:
- play with various logical components to determine whether they combine to prove that a statement P (n) is true for all natural numbers n.
- explain why the base case is essential in an inductive argument.
- explain how the quantification of the inductive implication impacts the values of n for which P (n) is true.
- analyze the role of the initial value of k in linking the base case with the inductive implication.
- notice that additional base cases and/or additional hypotheses for the inductive implication might also be used to construct a valid inductive argument. This primes them for strong induction!
Task 3. Abstracting the Role of the Inductive Implication. Productive engagement with this task should allow students to:
- analyze mathematical language and practice connecting it to mathematical logic.
- abstract the underlying mathematical logic of a written proof.
- determine what mathematical statement is being proved by a given argument.
- Relate the vacuously true implication that was proved to the missing link between the base case and the inductive hypothesis.
Task 4. Understanding the Role of k. Productive engagement with this task should allow students to:
- notice that n remains fixed throughout the proof, while k varies over values ranging from the base case up to and including n − 1.
- raise their awareness of the limitations of induction: it cannot prove the infinite case where n → ∞.
Task 5. Practicing Proof by Mathematical Induction. Productive engagement with this task should allow students to:
- engage with the cognitive demand that domain-specific knowledge imposes on proving the inductive implication.
- frame their approach with demonstrating the hypotheses of PMI.
Below you can find clips from Rachel's class where students engage with this task from the start of group work to writing up a complete, formal proof by mathematical induction.
What happens when we finally give students the opportunity to write a proof by mathematical induction? After working on 4 tasks over the course of 3 class days, the PMI is finally stated and we give students a (heavily scaffolded) opportunity to write a proof. In the menus below, you can watch how Rachel sets up this task and recaps some fine details for initializing a proof by MI.
Here's how students reason as they discuss how the inductive implication could be shown to be true (to be added later).
We always want to establish the heuristic that students solve for the proof before they begin writing it. In this clip, Rachel leads a discussion to generate the important details of the proof before writing a complete example for students. Notice how she frames their approach in terms of satisfying the hypotheses of the formal PMI.
Once the class has produced a high-level, logical outline for the proof, Rachel leads the class in writing down a formal proof. Notice how students' ideas are leveraged in this discussion.
Tasks 2 and 3 give students several opportunities to abstract the role of the base case in linking to the inductive implication. In this clip, a student returns to this idea by asking, "If we didn't show the base case, does this argument still prove something?" We see this as an example of a student repeatedly engaging with an EO over the course of several class meetings and tasks in order to address it.