As mathematicians and mathematics educators we very often see the Dunning-Kruger effect in action. Our calculus students are certain that they are masters of Calculus because they took the AP exam. To be fair, we ourselves are not immune: I often think I know something better than I do and end up choosing a more ambitious research project than I should.

A way to combat this effect in research is to check whether you can lecture clearly on a topic without preparation and fill in all of the details. In fact, an excellent research colleague of mine actually seems to write his papers this way: he lectures on his topic many times over and over again and then once he seems to have everything in his mind and can reproduce exactly how it all fits together he goes and writes the paper.

Of course it is possible to write mathematical papers without having everything about one's project on the tip of one's tongue, but the effort to do so may produce better work since the effort to compress and organize one's ideas in memory is reflected in the resulting writing. In fact, it is likely that such careful organization of one's ideas may lead to better and more surprising results, since having the entire framework of a problem or theory in mind allows the subconscious to have access to that data.

My question is the following: What does the mathematics education literature say about the effect of such memory effort on student performance in coursework? Does knowing the solution to a problem inside and out translate to better performance on other problems that are not obviously analogous?

Although this question seems a bit strange, I think it is very important to address. Implicitly, Inquiry-Based Learning Courses focus in depth on a few problems in order to encourage this sort of deep familiarity, whereas traditional instruction often requires working on many problems in order to build skill and identify analogy. Discovery methods, though, may be weak in providing the memory framework on which to hang one's discovered results. One might remediate this by requiring students in an Inquiry-Based Learning course to learn a "working knowledge outline" around which to think. Data should drive such decisions, though. Hence this question...

Also, graduate students studying for comprehensive exams often spread themselves too thin, I think, as a result of being too "expansive" with their efforts rather than instead learning some central arguments deeply. I'd like to know if there is evidence that this is a valid observation.

Edit: One may view the question as a subquestion of the following about a more general "theory" of mathematics education: If mathematics is about mathematical analogy, where a mathematical analogy requires the wholes to be analogous, the parts to be analogous and the way these fit together to be analogous (as discussed in Polya), then must the structures between which the analogy is drawn be committed to memory for their most effective comparison?

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    $\begingroup$ +1, nice question, I'd like to see some answers to this $\endgroup$ – seeker Jul 30 '14 at 13:53
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    $\begingroup$ Does knowing the solution to a problem inside and out translate to better performance on other problems that are not obviously analogous? Not necessarily. The term to comb the literature is "transfer." Math students often struggle to uncover the structural similarities of mathematically related problems (see: matheducators.stackexchange.com/a/1256/262), and frequently won't use even heuristics that they've already mastered when problem solving (I think this works as a source: Schoenfeld (1979) Explicit Heuristic Training as a Variable in Problem-Solving Performance). $\endgroup$ – Benjamin Dickman Jul 30 '14 at 14:16
  • $\begingroup$ @Benjamin: Thank you! $\endgroup$ – Jon Bannon Jul 30 '14 at 14:18
  • $\begingroup$ Do you consider the impact of memory on primary/secondary level mathematics to be already well understood, are you simply less interested? If the former, it is still controversial, if the "calculator debate" is any evidence to go by. $\endgroup$ – Opal E Jul 30 '14 at 19:21
  • $\begingroup$ @Opal: I am also interested in primary/secondary. This is very closely related to the calculator debate. My position on calculators is: "If it's not in your head then you cannot think about it"…so calculators shouldn't be to readily relied on. I'd like data on this, but maybe it would clutter this question by being to close to the calculator debate... $\endgroup$ – Jon Bannon Jul 30 '14 at 23:33

Anecdotally, based on self-observation and observation of many faculty and grad students: "if it's not in your head in some form, you can't think about it".

A funny point here is that it seems not strictly necessary to "completely understand" something, if one can keep it in one's mind. Indeed, I don't see how to make the transition from not-understanding to understanding without having the thing in one's head already at the not-understanding phase. Nevertheless, it appears to me that many people have great difficulty "remembering" something they don't already understand, or at least "feel comfortable with". This is obviously an obstacle to gaining understanding... if, in effect, one can't bear to contemplate the thing one wishes to understand?

Similarly, indeed, one cannot make analogies to things one has truly forgotten, or never known.

The more one can keep in one's head, while remaining able to manipulate the things, the better. But/and I don't think this at all requires "complete understanding", but, rather, something more like "awareness". And it would be misguided, I think, to try to force-memorize things, since the resultant clutter and fragility often makes "manipulation" difficult.

  • $\begingroup$ Your first quote is my mantra for students. +1 $\endgroup$ – Jon Bannon Jul 30 '14 at 23:28
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    $\begingroup$ There is certainly a subquestion in here about "cognitive load". Somehow a complete mental model is psychologically satisfying and fluid to think about once it's in place. Memorizing isn't the right thing to do…it's more like "construction" or working hard to store the ways things can be discovered/recovered naturally. Maybe "compression"... $\endgroup$ – Jon Bannon Jul 30 '14 at 23:30
  • $\begingroup$ I quite like the points this answer makes. The words "clutter and fragility" highlight the difference between memorizing and the sort of readily manipulated memory that results from "understanding". There is some kind of simultaneous awareness of the relative position of everything in the latter state. I don't know if we encourage the process of forming this enough... $\endgroup$ – Jon Bannon Jul 31 '14 at 13:32

In light of your edited "more general" question, I thought I would make a few remarks.

Historically, an early treatment of the subject of reasoning by analogy can be found in work on Gestalt psychology. One of the better known authors/books in this area (where mathematical examples are plentiful) is Max Wertheimer/Productive Thinking. For a somewhat more recent consideration of this book (by the author's son, Michael Wertheimer) see here, where memory is broached by contrasting "reproductive thinking" and "productive thinking." In particular, see the following excerpt:

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The younger M Wertheimer continues later on in the same piece:

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This latter excerpt bifurcates nicely into two areas of potential interest.

First, there is the discussion of the relation to "insight" or "illumination" (terminology originally due to Wallas' (1926) The Art of Thought, I believe) which suggests there is a fair amount of information that can be found in the literature on creativity. For a survey of literature related to creativity, see my earlier MESE response here. With regard to analogies, in particular, see (for example) The Encyclopedia of Creativity; a relevant bit related to Wertheimer is here, which references a section specifically about analogies, in the same book, here. (See also Fauconnier and Turner on conceptual blending.)

Second, the excerpt above mentions anagram tasks. In keeping with your mention of Polya, I thought I would record here an observation that I have not seen made so explicitly in the literature on mathematics education: Jacques Hadamard, in his investigation of mathematical insight, finds that almost no mathematicians are thinking in words. The one true exception, to whom Hadamard devotes a couple of pages, is George Polya. (See the latter half of my MESE response here.) Unlike other mathematicians with whom Hadamard conversed, Polya finds analogies with words, letters, and turns of phrase to be critical to his thinking. You may recall from How to solve it that Polya (1945) gives an example asking how to rearrange (anagram!) the letters DRY OX TAIL IN REAR. His suggestion, as best I recall, is first to separate the letters into the vowels and consonants (I won't spoil the solution here). This is but a single instance related to the more general distinguishing feature noted by Hadamard. In this respect, I find it of note that the study of problem solving within Mathematics Education has sprung forth from an opus by a mathematician whose modality of thought is quite different from that of his peers. Might this conceptual provenance have led to meaningful ramifications later on? I am not sure, and leave the reader with this question as food for thought...

  • $\begingroup$ Thanks for this, Benjamin! $\endgroup$ – Jon Bannon Aug 1 '14 at 10:56
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    $\begingroup$ This bit on Polya (in Hadamard) is fascinating!! $\endgroup$ – Jon Bannon Aug 1 '14 at 10:58
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    $\begingroup$ @JonBannon Agreed! Funnily, I wrote a short paper for a(n obscure) journal of recreational linguistics on using some Polyaesque heuristics to play the word game Boggle (digitalcommons.butler.edu/wordways/vol46/iss1/25). Only a couple of years later did I come across that particular section on Polya in Hadamard's (1945) book. It fit so well that I've been tempted to tug on the thread just to see what comes untangled... $\endgroup$ – Benjamin Dickman Aug 1 '14 at 11:26
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    $\begingroup$ I think you should tug that thread! I have personally experienced an "uncertainty principle" with regard to language and thought in mathematics. It may actually be (of course I cannot establish this) that Polyaesque heuristics can be harmful to the working of the subconscious as Hadamard discusses. Nonverbal mathematical thinking, though, seems to require the construction of tight mental models in memory that allow for fluent manipulation. I would love to see someone test this uncertainty principle, but I have no idea how one would do it! $\endgroup$ – Jon Bannon Aug 1 '14 at 13:16
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    $\begingroup$ @JonBannon (and anyone else interested!) It seems the pay-wall has been removed from the earlier mentioned Polya/Schoenfeld problem solving paper in the context of word games. It can be found here: Problem Solving Strategies in Boggle-like Games. $\endgroup$ – Benjamin Dickman Jun 7 '15 at 5:02

The strongest students often tell me that they like taking my courses because they learn so much in doing the work for my course. I lecture and give traditional, human graded, homework. One thing I do which I see being done less in other courses is to ask some difficult open-ended problems along-side the "standard working knowledge" component. If I had a recitation day, I would love to give "modular" projects which tell a story with a sequence of 4 or 5 problems which move from basics to abstraction. My brother has done this with considerable success in modern algebra courses. That said, I don't have that sort of time in the higher math courses so I use homework to encourage them to delve deeper than my tests.

As I assign homework problems for a given course I usually try to diversify the problem sets: roughly:

  1. routine problems, to encourage the building of rudimentary computational skill,
  2. global concept problems, to encourage understanding of how the theory for the course weaves together,
  3. common danger problems, to pose paradoxes to test if the students understand the need for preconditions to a given theorem,
  4. open ended problems, these lead a student in a particular direction then ask them to get the rest of the way themselves, or, more disturbingly, if there even is a way to find.

For the weak students, if they complete item 1 and some of 2 and 3 they can usually pass. Those make up about 80-90 percent of the problems. On the other hand, for the stronger students, completing all the problems is a significant achievement. After the homework the test is comparably easier. In my best efforts, the problem sets encourage both the adventurous discovery learning as well as the standard mode of learning efficient, well-known techniques. In other words, I insist everyone acquires the "working knowledge outline" and I give guided-opportunities for some open-ended questions. In short, if the students do the work, they have seen the content of the course from several different angles. All of this in homework.

For example, a couple years ago, I roughly asked a student if there was some way to transfer the cross-product to a particular set of matrices. At the time, the student just had calculus III behind him and was a week into linear algebra. This was part of my sometimes ambitious advanced calculus course. The student responded by solving the problem in the next day. He learned multiplication of matrices to solve the problem and discovered the commutator bracket for matrices. Of course, the student was excited about his solution. It gave me something quantitative and special to write for his recommendation letters.

My experience with that student is special. In large lecture sections populated by mainly degree-seeking students it would not go well to ask such questions. I tone it down quite a bit for calculus or physics. But, in courses where the students choose to take the course I try to include something where the student can take ownership of a problem or topic either by the problems assigned, or sometimes a term-paper.

All of this said, I mainly lecture during the class meeting time. I leave the discovery time for the home or dorm.

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    $\begingroup$ of course, as usual, my answer has little if anything to do with the math education literature, my apologies for its anecdotal status. $\endgroup$ – James S. Cook Jul 30 '14 at 14:55
  • $\begingroup$ Actually, James, I forgot to include in my question that I am interested in anecdotal evidence as well. Thanks! $\endgroup$ – Jon Bannon Jul 30 '14 at 15:12

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