The Principle of Mathematical Induction: Scenarios
This task was designed by Rachel Arnold.
Norton, A., Arnold, R., Kokushkin, V., & Tiraphatna, M. (2023). Addressing the cognitive gap in mathematical induction. International Journal of Research in Undergraduate Mathematics Education, 9(2), 295-321.
What have students just seen?
Prior to working on this task, students have engaged with the L-Tile task. They are now familiar with the logic of quasi-induction:
P(1), P(1) → P(2), P(2) → P(3), ... , P(n-1) → P(n).
They have also noticed that each of these implications follows the form P(k) → P(k+1) where 1 ≤ k ≤ n.
They have not seen a formal statement of the PMI.
Productive engagement with the scenarios task should provide students with opportunities 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!
Before students begin working on this task, you should clarify that P(n) denotes an unknown statement about the integer n. You can connect this to statement in the L-tiles task so they have a mental reference as they work. When students decide a scenario's given information is not enough to prove P(n) for all n, ask them to identify which values of n, if any, that P(n) must be true for. Encourage students to work together on each scenario, rather than dividing and conquering.
Here is a video of an instructor launching the task:
For ease of viewing, here are the timestamps at which the group begins discussing each of the scenarios.
- Scenario 1 - 0:00**
- Scenario 2 - 1:09
- Scenario 3 - 1:29
- Scenario 4 - 2:03
- Scenario 5 - 2:35**
- Scenario 6 - 3:06
For ease of viewing, here are the timestamps at which the group begins discussing each of the scenarios.
- Scenario 1 - 0:00
- Scenario 2 - 0:07
- Scenario 3 - 0:52
- Scenario 4 - 0:55**
- Scenario 5 - 3:01
- Scenario 6 - 3:27
For ease of viewing, here are the timestamps at which the group begins discussing each of the scenarios.
- Scenario 1 - 0:00
- Scenario 2 - 0:13
- Scenario 3 - 0:52
- Scenario 4 - 1:07
- Scenario 5 - 1:35**
- Scenario 6 - 2:17
For ease of viewing, here are the timestamps at which the group begins discussing each of the scenarios.
- Scenario 1 - 0:00**
- Scenario 2 - 1:09
- Scenario 3 - 1:29
- Scenario 4 - 2:03**
- Scenario 5 - 2:35
- Scenario 6 - 3:06**
For ease of viewing, here are the timestamps at which the group begins discussing each of the scenarios.
- Scenario 1 - 0:00
- Scenario 2 - 1:35**
- Did not get to remaining scenarios due to extensive discussion on Scenario 2.
Navigating whole-class discussion will require some in-the-moment decisions based on students' reasoning overheard in during group work. You might begin by first polling the class for their intuitive answers (yes or no) for each scenario before choosing which scenarios to discuss (like in the video below).
Here we share select video clips of the instructor facilitating whole class discussion.
Video 1: Discussing Scenario 1
Video 2: Discussing Scenarios 5, 6, and 2 (in that order):
For ease of navigation, here are the timestamps for when discussion of each scenario begins:
- Scenario 5 Discussion - 0:00
- Scenario 6 Discussion - 5:15
- Scenario 2 Discussion - 9:43
You likely won't have time to discuss all scenarios -- this is okay. You can post solutions to the ones you don't discuss in class. Here are a few tips for navigating the discussion that you do have time for:
- When there is disagreement on whether the scenario proves P(n) for all n, ask to hear arguments from both sides. Start by asking the incorrect side to share first -- if they share after the correct side, they may be reluctant to share.
- Try using a number line depiction, like the one below, for helping students visualize the scenario you're discussing. In this diagram, an open circle at a value on the number line indicates that it is not known whether the claim is true for that value. A filled-in circle indicates the claim is known to be true. Arrows can be used between values to indicate which implications are known to be true.
