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I've often wondered about the "devise a plan" part of Polya's "How to solve it" outline. What we call "problem solving" can be thought of as what to do when you have no idea what to do. From this standpoint, the plan making activity, particularly the "Do you know a related problem?" heuristic, seems to beg the question a little. (If we really didn't have any idea what to do, we would not know a related problem...since this would be an idea of what to do.)

This observation is not new. If we have never seen anything like a given problem before, then we perhaps have no real chance of a solution to that problem. The belief that we do have a chance despite knowing no related problem is, probably, some kind of belief in genius. Most mathematicians would probably not believe in such discontinuities and would say that the solution of a new problem comes from a connection with a dense neighborhood of nearby solved problems.

I wonder about the above heuristic in connection with Inquiry Based Learning. In an IBL context, the dense neighborhood of previously solved problems are arranged to be discovered by the learner. Ideally, the "first intuition" building block problems are very close to common sense.

Contrasting IBL to standard methods of instruction, the main difference may be efficiency of recall: an IBL student may have greater perceived ownership of solutions to nearby problems, allowing easier recall when they are needed to solve new problems, whereas a student of more standard methods may not have the emotional investment in solutions to previously encountered problems to be able to recall them when needed.

Said another way, IBL methods give students a smaller store of more "deeply experienced" problems to draw from when using the "nearby problem" heuristic, whereas a more content-driven approach exposes students to a larger store of "nearby problem" candidates.

It seems that the ability to plan a solution to any problem is related to the ability to recall relevant related problems.

Question: Has there been any study of the correlation of the ability of students to apply Polya's "related problem" heuristic with the intensity of the students' emotional engagement with the "related problems" recalled? (Here, emotional intensity is admittedly ill-defined.)

This question aims at helping sort out whether it is best to have students work on many, many problems or to focus on solving a small number of problems with the hope of developing better recall of the techniques learned. This is, of course, a matter of personal taste in problems...but I wonder about what general capacities are encouraged by either approach.

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    $\begingroup$ If you are interested more broadly in research on the related problem heuristic, you might check references 1 and 2 here and those who have cited them since. $\endgroup$ – Benjamin Dickman Mar 11 '16 at 16:45
  • $\begingroup$ Thanks, Benjamin! I am interested in this heuristic more broadly. $\endgroup$ – Jon Bannon Mar 11 '16 at 17:38
  • $\begingroup$ Since no one has answered in the intervening month and a half, I gave it a shot... $\endgroup$ – Benjamin Dickman Apr 29 '16 at 6:33
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(This answer has two parts: The first one is about existing research, and probably relevant, but succinct; the second one is about a problem solved in practice, and possibly relevant, but definitely rambling. I will leave the determination of what constitutes a "related" answer to the reader!)


Part I

As I perceive Polya's (1945) How to Solve It, the chief weakness is precisely that which you point to: how does one "devise a plan" in the first place, and how helpful is the advice to use a heuristic such as "think of a related problem" when one is stuck from the outset?

As far as I know, the (an) answer is that even with the explicit suggestion to think of a related problem, students will struggle if they have an ill-conceived notion of relatedness, which, in fact, is the case for many students: cf. the comment I provided above to my earlier answer (MESE 380) and the sources cited there. Specifically, work by Silver and Schoenfeld on looking at problem relatedness. The latter went on to write a book, Mathematical Problem Solving, a few years later (1985) and has a more recent book, How We Think (2010), on similar information from the perspective of decision-making.

Rather than phrasing this answer using your term ("emotional engagement") I point out instead that Schoenfeld's contribution to the problem solving literature included the argument that our knowledge (factual, procedural, etc: called "resources") and strategies (called "heuristics," although grouped under "resources" in the 2010 book) are insufficient with respect to determining how well students will problem solve. More precisely, his assertion is that there are two more categories that are both necessary and sufficient for identifying when students will be successful problem solvers: those categories are "beliefs and belief systems" and "control decisions and metacognition" (to the latter end, see also: MESE 2397 and my pointer -- sorry for all the links! -- to Schoenfeld's What's All the Fuss about Metacognition?). I suspect that work on beliefs (especially) and metacognition (perhaps) can essentially subsume what you refer to as intensity of emotional engagement.

The above contains references to four substantial papers and two books; below, I recount a brief experience from my course on problem solving in mathematics (taught to graduate students in mathematics education) in the hope that it will be illustrative of ... something relevant! Still, I separate these two parts in case only the top half is of interest.


Part II

I asked students for their observations around the product of four consecutive natural numbers; this is a list that begins with $1\cdot 2\cdot 3\cdot 4 = 24$ and $2\cdot 3\cdot 4\cdot 5 = 120$. One student observed that each is one short of a square, i.e., $24+1 = 5^2$ and $120+1 = 11^2$. We checked a few more examples to see if this continued to be the case (yes) and then decided we had a conjecture worth pursuing. Since the OP contains a reference to IBL, I believe it is fair to say that "IBL" is a roughly accurate description of the context and educational environment at hand.

So: How do you solve this problem? (i.e., prove or refute the conjecture that the product of any four consecutive natural numbers, plus one, is a perfect square)

The classroom setup is something like 4 tables of 5 students each, so they went about their attack on this problem in different ways. I sometimes write down the hashtag #WWGPD (What Would George Polya Do?) on the whiteboard, and did so for this problem. What seems like a related problem here?

All students began to use notation (recall Hadamard: "[Polya] finds that a proper notation – that is, a properly chosen letter to denote a mathematical quantity – can give him similar help") although their approaches even in this respect differed, slightly. All but one table denoted the product as $n(n+1)(n+2)(n+3)$. I happen to believe that even this step is quite non-trivial; it indicates a number theoretical or algebraic approach from a mathematical perspective, but it also indicates an I can get something out of this problem approach from an "emotional" (to use your term) perspective. (I still think of this as "beliefs" and "metacognition" to use Schoenfeld's terms, though a search for "mathematical mindset" could yield some useful papers, as well.)

One student multiplied $n(n+3)$ and $(n+1)(n+2)$ separately, in an effort to find some sort of balance among terms; indeed, this yields the equivalent expression $(n^2 + 3n)(n^2 + 3n + 2)$ whence a $u$ substitution for the first parenthetical term gives $u(u+2) = u^2 + 2u$, which is factored as a square when $1$ is added, "QED".

But most students did not have such an approach. One table tried to prove it by mathematical induction. I think this is a reasonable idea; the conjecture is about natural numbers, and we had proved other propositions about squares using induction, e.g., the sum of the first $n$ odds is $n^2$. This turned out to be a bit of a mess, though, so that table abandoned the strategy. My own wonder was whether there was a belief that the related problems solved with induction were neat enough to indicate that it was not a viable strategy here; of course, induction could be carried out for this problem, but it might not be so quick. (For the aforementioned example around summing odds, one summarizes that $1 = 1^2$ and an inductive hypothesis gives that the first $k+1$ odds sum to $k^2 + (2k+1)$, which is $(k+1)^2$ — "QED" rather succinctly!)

The subsequent approach of that table was to multiply everything out, and then use Mathematica to factor it. This worked. A student at another table also multiplied everything out, but factored it by hand. This can be done by observation/inspection, by matching up coefficients in a procedural way, or by noticing that the factored quartic expression is of the form $(n^2 + an + 1)^2$ and using the initial observation that $n=1$ yields $24$ to find $a$ rather quickly. That is, $(1^2 + a + 1)^2 = 25$, so that $2+a=5$ gives $a=3$ (should the conjecture hold). To speak again of my own wonder: Did the table that resorted to factoring by Mathematica (e.g.) feel emotionally worn out by the involved nature of the arithmetic in their (reasonable) attempt at mathematical induction, thereby priming them for a calculator-style solution? Or are they tech-savvy and able to recognize the problem (or "similar" i.e. "related" ones) as quickly reduced to a computation from that point on? I do not know.

But I spoke of one table that wrote the expression differently. There was a problem that might be construed as relevant that asked the class to prove that for a prime $p \geq 5$, the expression $p^2 - 1$ is divisible by $24$. In fact, some number of problems around divisibility were solved using a difference of squares, and so the last table re-wrote the given expression as $(n-1)n(n+1)(n+2)$. From here one multiplies the first and third expression, and the second and fourth expression, to find: $(n^2 - 1)(n^2 + 2n)$. The table managed to finagle a solution (proof) using this approach, although they articulated their initial thinking as trying to balance the terms by shifting the natural number denoted as $n$; note that this draws from a related problem, uses a reasonable heuristic (the first student mentioned also tried to balance terms), but ultimately leads to a matching of two products that is both non-routine and (in terms of brevity) non-ideal.

If one believes in drawing from related problems to balance terms, then perhaps a different shift would be better: Substituting $n=m-1.5$ yields $(m-1.5)(m-0.5)(m+0.5)(m+1.5)$. This latter factorization, to my eye, makes the seemingly opaque nature of why it is beneficial to multiply the first and last term transparent (or, at least, translucent); here, we end up with $(m^2-2.25)(m^2-0.25)$ from which another substitution of $w$ for the first term gives $w(w+2)=w^2 + 2w$, so that adding $1$ gives a square as desired.

No student came up with that last substitution; indeed, the other approaches can be found in Zeitz's (2006) The Art and Craft of Problem Solving as Example 1.2.1. The book has a few approaches, my class had a few approaches, but there was no shift by $1.5$. Why not?

I hypothesize that the phrasing around natural numbers and the use of $n$ (rather than $x$) are to blame. If you think of the problem using $x$ instead, then it looks a bit like a function; in that context, one may consider zeroes -- perhaps even a graph! -- and then the idea of shifting the (induced) function so that it is even (i.e., the substitution suggested) fits in more reasonably. In the context of natural numbers, I do not think that shifting by a non-whole number is a "nearby" heuristic; indeed, one may believe some justification is needed for why the $(w+1)^2$ factorization is really the square of an integer. (One way to do this, of course, is to unravel the substitutions and write everything in terms of $n$.)

I do not have a grand point with all of this; I record the above in part as an anecdotal record. But in thinking about the role of "emotional intensity" and relatedness, I wonder whether there was a way to scaffold instruction in non-routine problem solving so that the last strategy described would arise organically. How can the $1.5$ shift be viewed as a "nearby" heuristic? The problem was within (at least almost) everyone's zone of proximal development, and the notions around balancing terms and factorizations using the differences of squares were all floating in the ether. Is it possible, or probable, that the "emotional intensity" attached to words like natural numbers or notations like '$n$' is enough to sway solvers so that their ultimate choice of heuristic is doomed only to whiff the heuristic described? (Relegating "nearby" to something akin to "so close yet so far"...) I do not know!

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    $\begingroup$ This is certainly helpful. Thanks, Benjamin! $\endgroup$ – Jon Bannon Apr 29 '16 at 11:12
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    $\begingroup$ Nice answer, but I must say that I don't like looking at the 1.5 shift even after it's fully worked out. $\endgroup$ – Daniel R. Collins Apr 30 '16 at 3:42
  • $\begingroup$ @DanielR.Collins In what sense of "looking at"? E.g., aesthetically? I wrote it out using $m$ rather than $x$, which makes it look rather unappealing to me, too. This can be remedied by switching to $x$ or writing $3/2$ instead of $1.5$. But perhaps you dislike something other than its appearance? (Note that I am not advocating for this as the ideal heuristic; in that direction, my preference is only that students' toolbox of strategies continues to grow and be put to good use.) $\endgroup$ – Benjamin Dickman May 10 '16 at 21:45
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    $\begingroup$ @JonBannon For some related material see: Karp, A., & Roberts, D. L. (2014). Interview with Alan Schoenfeld. In Leaders in Mathematics Education (pp. 143-165). SensePublishers. $\endgroup$ – Benjamin Dickman May 12 '16 at 0:41
  • $\begingroup$ For completeness, I observed another method of solution from a high school student (grade 9). Given that $1 \cdot 2 \cdot 3 \cdot 4 + 1 = 5^2$ and $2 \cdot 3 \cdot 4 \cdot 5 + 1 = 11^2$, she conjectured the expression could be evaluated by multiplying the first and last terms, adding one, and squaring. She predicted $3 \cdot 4 \cdot 5 \cdot 6 + 1$ (correctly) to be $(3 \cdot 6 + 1)^2$. After observing this pattern, it is routine to verify the algebra by hand for the general case: $n(n+1)(n+2)(n+3) + 1 = (n(n+3) + 1)^2$. $\endgroup$ – Benjamin Dickman Mar 20 '17 at 2:47

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