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What are the most successful analogies that are used to teach the concept of mathematical induction?

To clarify, I am not looking for a formal explanation of the principle of mathematical induction, but analogies that can be used to help students visualise and understand it.

I am particularly interested in answers that explain how successful the analogy is and/or why it is so helpful.

[Note, a recent question asked for strategies to teach mathematical induction, but it explicitly asked not to give analogies but skills for dealing with the principle when doing a proper mathematical problem.]

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    $\begingroup$ The best answers in the StackExchange framework are determined by votes, so saying "the best answers will explain how" doesn't sound right to me. How about this instead? "I am looking for explanations of how..." $\endgroup$ – user173 May 17 '15 at 2:14
  • $\begingroup$ Good point @MattF. Edited here and in my other recent question. $\endgroup$ – DavidButlerUofA May 17 '15 at 2:25
  • $\begingroup$ (1) Are there any commonly used ones besides toppling dominoes and climbing ladder rungs? (2) What about analogies for the well-ordering principle instead? (3) Here is Polya's Induction and Analogy in Mathematics as a PDF. $\endgroup$ – Benjamin Dickman May 17 '15 at 6:06
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    $\begingroup$ (1) Even if there aren't any other commonly used ones, having an answer here on MESE that describes them and how useful they are in practice would be a good thing. (2) As to the well-ordering principle, it is not so commonly taught as mathematical induction, and I daresay a lot of schoolteachers have never heard of it, so I think it would deserve a whole other question. (3) An answer here that references Polya's book and describes some of his thoughts on the matter wouldn't be a bad thing either. $\endgroup$ – DavidButlerUofA May 17 '15 at 6:35
  • $\begingroup$ math.stackexchange.com/questions/423513/… $\endgroup$ – JP McCarthy May 18 '15 at 21:38
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When you set up a 'domino train' you need two things to ensure all the dominos will fall:

  1. You can knock the first domino.
  2. The dominoes are set up such that if one falls the next will fall.

I present both the monkey on the ladder and the domino analogies. I start with "how can I prove that the monkey climbs the ladder?" and "how can I ensure that all the dominoes fall"?

I take suggestions from the students and usually we get close enough to the answer with some prompting. The students are far better at coming up with $P(k)\Rightarrow P(k+1)$ but I press home the importance of $P(1)$ --- be it with a scared monkey, an empty room of dominoes but more likely actual examples of induction 'proofs' where $P(k)\Rightarrow P(k+1)$ alone allows you to derive an absurd conclusion.

If I could be blunt, I am very concerned if a student fails to make the connection between these analogies and an inductive proof.

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  • $\begingroup$ He also says "I am not looking for a formal explanation of the principle of mathematical induction, but analogies that can be used to help students visualise and understand it.". I do believe this answer does the question some service. $\endgroup$ – JP McCarthy May 18 '15 at 8:50
  • $\begingroup$ @JpMcCarthy yes it does service the answer, but it would still be great if you added some discussion about how helpful this is with students, even if it's just your own experience as a student. $\endgroup$ – DavidButlerUofA May 18 '15 at 19:46
  • $\begingroup$ @DavidButlerUofA Edit coming. $\endgroup$ – JP McCarthy May 18 '15 at 20:06
  • $\begingroup$ Thank you, that's much better. +1 (though I just edited to make one point clearer, if that's ok) $\endgroup$ – DavidButlerUofA May 18 '15 at 20:20
  • $\begingroup$ I didn't say it a month ago, but I feel the need to say it now: I don't like the last sentence, mainly the tone. Sure it would be a surprise if a student didn't make the connection, but I wouldn't go so far as to say it's a concern, as if there's something seriously wrong with the student if they don't. $\endgroup$ – DavidButlerUofA Jun 18 '15 at 22:08
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"Is it possible to climb this whole ladder? I'm stood on the first rung."

"Well, could you climb any one of the rungs?"

"I guess so."

"And could you climb a rung straight above that one?"

"Sure..."

"There you go. And hey look you climbed onto the first rung. You can climb the whole ladder."

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    $\begingroup$ As support for this, permit me to link to my answer to the question, "How to teach Mathematical Induction mathematically?", which also uses the ladder analogy. $\endgroup$ – Joseph O'Rourke May 17 '15 at 17:39
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    $\begingroup$ Note that the OP writes: "I am particularly interested in answers that explain how successful the analogy is and/or why it is so helpful." $\endgroup$ – Benjamin Dickman May 17 '15 at 17:46
  • $\begingroup$ When I do this it is a monkey climbing the ladder. $\endgroup$ – JP McCarthy May 17 '15 at 19:47
  • $\begingroup$ As @BenjaminDickman says, it would be good if you could add some discussion as to how successful this analogy is with students. $\endgroup$ – DavidButlerUofA May 18 '15 at 19:45
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The staircase metaphor (I prefer a rope ladder) is useful when associated with the concept of enumerable sets, the prototype of which is $\mathbb{N}$.

I usually couple my explanation with a (very simplified) reference to Peano's axiomatization of natural numbers, to shake off the conventional understanding, and to create room for the idea that we are dealing with enumerations, regardless of how exactly they are represented. I would be nice if it is possible to suggest (provocatively) that not all sets can be enumerated, even if we allow the process to be infinite.

Only then I´ll introduce the rope ladder, without ever losing completely the connection to the symbolic level of description:

enter image description here

Remind the students that we have the "left rope" (the basic enumeration). What we need is to "attach the right rope to the left, one rung at a time".

It is important to emphasize that we are doing is to prove a property $P$ over a set of objects that are enumerable. We are not performing a calculation, but a demonstration (a "logic thing", not a "regular math thing").

All along, it is crucial to avoid isolating symbolic reasoning and explicative metaphor. If these are kept together, you can simplify and reiterate at will, without having to count on the students to "just get it".

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  • $\begingroup$ If I am understanding it right, your analogy is not about climbing a rope ladder, but actually constructing a rope ladder. You have a rope on the left knotted at 1, 2, 3, 4, ... and a rope on the right knotted at P(1), P(2),... If the knot k on the left is joined to the knot P(k) on the right, then you are allowed to join the rung from k+1 to P(k+1) too. Is that right? $\endgroup$ – DavidButlerUofA Jun 2 '15 at 20:13
  • $\begingroup$ Yes, constructing. The induction step is the red arrow: if you can always get the next knot on the right side (if you get from $P(k)$ to $P(s(k))$), then you will always be able to fix the next rung (the "step" rung). $\endgroup$ – André Souza Lemos Jun 2 '15 at 20:22
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I tell roughly eight students to line up. Let's say the second student in line is named Natalie. I say to the class "Natalie knows a secret. Whenever someone in this line knows a secret, they whisper it to the student behind them. Which students will eventually know the secret?" It's up to the class then to realize that Natalie and everyone behind her will know the secret. Sometimes a student will guess that everyone in the line knows the secret, because the first person in line tells Natalie, and Natalie tells the student behind her, and so forth. However, other students in the class will usually argue persuasively that the first student in line does not necessarily need to know the secret. After this icebreaker, I begin the discussion of induction a little more formally.

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One analogy I have is for the induction step itself. I say that the induction step is like a machine that transfers the truth of the proposition from one number to the next. The machine takes as input the fact that the proposition is true for $k$ and spits out as output the fact that the proposition is true for $k+1$.

induction step machine

You proving the induction step is like constructing this machine. You technically haven't actually shown that anything is true, you've just made the machine that will show that something is true if you put a true statement into it.

proving the induction step

The idea of induction is to start with a statement that you know is true, and then imagine continually feeding the output back into the input funnel.

I only thought of this idea recently, but so far it has had some success with students. It seems to help them understand the part that the induction step plays in the whole proof, and the fact that it works on statements rather than on numbers.

Another picture that helps is to draw the line of numbers as open circles, and colouring them in when you know they're true. The machine can be thought of as transferring the colour from one circle to the next. This GeoGebra activity illustrates the idea.

enter image description here

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