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Question 1. Accessing Knowledge During Problem Solving



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1. Accessing Knowledge During Problem Solving

February 20, 1998

by Annie and John Selden

Does thinking of using things one already knows (facts, procedures, etc.) play a major role when solving mathematical problems?

Focusing this question somewhat more narrowly, does "accessing" one's "knowledge base" play a major role in helping one solve novel problems? On the one hand, success in solving novel problems might depend mainly on one's reasoning skills and the quality of one's knowledge base, and accessing it might be relatively routine, automatic, and unproblematic. On the other hand, it might be that one can know quite a lot, but often fail to solve a problem through not accessing the appropriate part of one's knowledge.

In I., we discuss the meanings of the terms we use and place them in a somewhat larger setting, i.e., we sketch a theoretical framework. In II., we discuss some related literature that suggests the answer to our initial question might sometimes be yes. In III., we narrow the question to what might be called a directly researchable question, and in IV., we suggest a way to partially answer that question.

I. However one characterizes (personal or private, as opposed to public) knowledge, it resides in memory, of which psychologists have described three main kinds -- short-term, long-term, and working (Baddeley, 1995). Short-term memory has a half-life of about fifteen seconds and holds about seven "chunks" of information. These can consist of any information that one can think of as a unit, e.g., words, patterns of chess pieces, or the Pythagorean theorem. One is aware of the contents of short-term memory and it is directly available for reasoning (Miller, 1956). The rest of memory is called long-term. It has a very large capacity, but one is not aware of its contents and these are not directly available for reasoning. Parts of the contents of long-term memory can be brought into short-term memory, i.e., activated. Finally, working memory consists of short-term memory plus reasoning and control mechanisms that swap information between short-term and long-term memory.

In the psychological literature, "knowing" and "remembering" are sometime used interchangeably. However, when considering, for example, knowledge constructed through reflection, as one does in mathematics education, one is not referring to just any long-term memory. Knowledge refers to memories of a certain kind and duration.

Most philosophers have taken a tripartite view of knowledge -- it is justified, true belief. This assumes the kind of memory that can be a belief. That is, one that can be expressed as a proposition, such as "differentiable functions are continuous." This view is very narrow relative to analyzing problem solving, which can also use knowledge in the form of, say, remembered images and procedures. Furthermore, individuals seems to acquire and use false beliefs in more-or-less the same way as true ones -- in the midst of problem solving, they are usually unable to internally detect which of their beliefs are true or adequately justified, although they may subsequently find a belief is false, e.g., by coming upon a contradiction. Even memories of direct experience can be unreliable in a way which is internally largely undetectable (Garry, Manning, Loftus, and Sherman, 1996). Thus, we do not see the tripartite view as very helpful for mathematics education. We will not now attempt to characterize which kinds of memories count as knowledge, but surely images and procedures should be included. We also do not assume one's knowledge is true, just as mathematicians do not assume sets to be nonempty.

As to the duration of knowledge, there is a difference between remembering and using something for a short while versus a number of months or years -- to accommodate this, we distinguish two subcategories of knowledge in long-term memory. We consider knowledge that lasts a number of months or years as being in one's knowledge base (and refer to less lasting knowledge as information). This includes one's knowledge of mathematics and its conventions, as well as of logic and reasoning. The knowledge in more ephemeral or partly activated memories associated with a current problem or interest, we will call local knowledge. Such knowledge might include that a proof started with "Let x be a number" or that one has recently used the triangle inequality in solving a particular problem. It is a part of long-term memory which is easily accessible to working memory. Local knowledge grows and persists during an attempt to solve a problem, but much of it may well fade when it is no longer useful for that particular problem, perhaps with part entering one's knowledge base.

There is some evidence suggesting the distinction we are making between one's local knowledge and knowledge base (a distinction concerning the mind) may be a reflection of the structure of the brain. For example, Kossyln and Koenig (1995, p. 390) describe H.M., whose medial temporal lobe (including the hippocampus) was removed in an effort to relieve otherwise intractable epilepsy. H.M. was unable to create any new long-term memories, but appeared to have normal short-term and working memory, as well as old and persistent long-term memory, i.e., knowledge base, but no local memory.

By accessing one's knowledge base, we mean activating or bringing into working memory, and hence consciousness, a small part of its contents. We see the structure of one's knowledge base metaphorically as a weighted, ordered graph in which the nodes represent units of knowledge such as concepts and relationships between concepts, e.g., theorems. The edges express which new units of knowledge a person can access soon after having activated a particular one, and the weights contribute to the probability of such access. Other major contributors to the probability of accessing a node are one's most recent thoughts and perceptions, and one's current psychological contexts. We see these contexts as subgraphs of one's knowledge base that receive additional weights (one might say they are "lit up") for awhile and slowly fade. For example, the word "red" might bring to mind apples, rather than stop lights, if one had recently been discussing gardening or fruit trees. However, the opposite would probably happen in a discussion of roads and driving. If one is aware of a logical relation between the knowledge associated with two nodes, then one's knowledge base is likely to have an edge joining them, but there might also be other origins for edges, e.g., emotional ones.

The idea of (psychological) context is consistent with the well-established psychological phenomenon of priming, of which individuals need not be even conscious. (However, priming has often been studied through word associations that appear to draw on a rather special kind of memory.) Seeing a knowledge base as a graph is an extension of the idea of concept map (Novak and Gowin, 1984, pp. 15-54) in which the nodes represents concepts, the edges represent "logical" relations between concepts, and the whole map models only a small part of one's total knowledge base. The general idea of a person's knowledge base (without the suggested metaphorical graph structure) has been mentioned by Schoenfeld (1992, pp. 348-350) and the existence of numerous links between various units of knowledge is reminiscent of what Hiebert and Carpenter refer to as understanding (1992, pp. 67-70).

What we call a novel or nonroutine problem is what Schoenfeld has called just a problem (1985). It refers to a task the subject has not previously seen and which is not closely analogous to a previously seen task. Thus, the novelty resides in the relationship between the subject and the task, not in the task alone, i.e., novelty is a property of subject-task pairs. However, many experienced mathematics teachers can point out tasks that will be novel problems for almost all students with normal backgrounds and such tasks might be regarded as novel problems, independent of particular students. All of this introduces a constraint on studies of problem solving -- there is a sense in which a novel problem cannot be administered twice to the same subject.

II. In "Even good calculus students can't solve nonroutine problems" (1994), we described the ability of students who earned A or B in traditionally taught first calculus to solve 5 nonroutine (novel) first calculus problems on a one-hour test. Immediately afterwards, the students took a second, half-hour, test of (very easy) routine problems which covered the knowledge adequate to solve the nonroutine problems. The students did very poorly on the first test and well on the second one. A number of students appeared to have adequate knowledge to solve the nonroutine problems, but were unable to solve them. This suggests that they may have had adequate knowledge bases and their inability to solve some of the nonroutine problems may have been primarily due to lack of access. However, there are other possible explanations such as the students' difficulties with reasoning and combining information.

Kieren, Calvert, Reid, and Simmt (1995) have described a situation in which access seems to play a role. They describe two college students, Stacey and Kerry, cooperating in solving an open-ended, novel problem called the arithmagon problem: A secret number is assigned to each vertex of a triangle. On each side of the triangle is written the sum of the secret numbers at its ends. If the numbers on the sides are 11, 18, and 27, find the secret numbers. Generalize the problem and its solution. The students solved the problem and extended it well beyond what was expected. Kieren et al's analysis captured their dynamically changing understanding, as that understanding and aspects of the problem interactively emerged. Kerry favored computation and easily accessed his knowledge of simultaneous equations to solve the original problem, but Stacey took the first key step in the extension of the problem solution. She serendipitously extended the construction, acting on her knowledge that experimenting without a clearly articulated goal or reason can be appropriate in mathematics -- something that did not occur to Kerry. Very probably Stacey and Kerry would not have progressed so far, without Stacey's accessing her (perhaps tacit) knowledge of the nature of mathematics. In general, throughout the extension, Stacey thought of things and Kerry checked them. That is, Stacey's accessing of her knowledge base, not just its quality, played an crucial role in their success. Our own experience as mathematicians suggests that examples of the importance of accessing one's knowledge of, say, a theorem or property could also be found, perhaps though (subject) self-initiated recordings soon after the access had occurred.

III. We now turn to the original problem of access and narrow it even more. Are there apparently normal, successful undergraduates and a set of novel problems, for which the students have adequate knowledge bases to solve them, but cannot, because they are unable to fully access the relevant knowledge?

If there are such students and problems, it might be interesting to investigate when and under what conditions major aspects of their knowledge are accessed, with an eye to improving the process. This might be done using cooperating pairs of students, as in Kieren's studies (Pirie and Kieren, 1994; Kieren, Reid, and Pirie, 1995) or by training students to "think aloud" and recording problem solving sessions and subsequent interviews.

This kind of investigation would combine well with Kieren's and others' analysis of problem solving in action and with Schoenfeld's observation that good problem solvers tend to monitor their work, occasionally asking themselves, "How am I doing?" Asking oneself such a question can be crucial when the answer is "not very well," and subsequently accessing one's knowledge based can be a useful, although not necessarily conscious or intentional, response. Mathematics graduate students "stuck" on an attempted proof sometimes deliberately search their knowledge bases. However, this phenomenon is probably not often seen in less advanced students, who may not tolerate confusion long enough before giving up.

IV. There are a number of ways to try to answer the above question. Indeed, there might be more than one answer, depending on such things as the maturity and backgrounds of the students and the kinds of problems considered. One way would be to select students, problems, and a testing situation more or less similar to those in our paper, "Even good calculus students can't solve nonroutine problems," but to repeat the first test or a similar one using novel problems, after the (routine) second test.

Suppose students could be found who did poorly on the first (nonroutine) test, well on the second (routine) test, and then well on the third (nonroutine) tests. Suppose further that the tests were constructed so that the knowledge exhibited in the second test was also what was needed to work the first and third tests. Doing poorly on the first test and well on the second suggests only that the students have difficulty with novel problems, despite having adequate knowledge. But it leaves the possibility that the students' reasoning/synthesizing abilities were weak or that their knowledge bases were of low quality, perhaps through having weak links. These explanations would be largely eliminated for student who did well on the third tests, suggesting that for them a major obstacle to working the first test was inability to access an adequate knowledge base.

Some alterations to the tests we used might be needed -- the time required for taking three tests in one sitting (approximately two-and-one-half hours) might be too long. The relationship between the knowledge elicited in the second tests and the novel problems (which was not intended to be obvious to the students) might be too complex to prime the appropriate access. Our problems, which were novel for our students, might not be novel for some others. Finally, there is a sense in which a novel problem cannot be attempted twice. At least some of the novelty will have been altered on the second attempt. The latter difficulty might be eliminated by dividing the set of problems into two parts A and B and having half the students attempt A first, followed by the routine test, then B. The other half could begin with B and end with A. This would be especially interesting if the novel problems in A and B could be solved by accessing the same knowledge, which was elicited by the routine test.

References

  • Baddeley, A. (1995). Working memory. In M. S. Gazzaniga (Ed.), The Cognitive Neurosciences (pp. 755-764), Cambridge, MA: MIT Press.

  • Garry, M, Manning, C. G., Loftus, E. F. and Sherman, S. J. (1996), Imagination inflation: Imaging a childhood event inflates confidence that it occurred, Psychonomic Bulletin and Review, 33(2), 208-214.

  • Hiebart, J. and Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 65-97), New York, NY: Macmillan.

  • Kieren, T., Calvert, L. G., Reid, D. A. and Simmt, E. (1995), Coemergence: Four Enactive Portraits of Mathematical Activity, University of Alberta, paper presented at AERA.

  • Kieren, T., Reid, D. and Pirie, S. (1995). Formulating the Fibonacci Sequence: paths or jumps in mathematical understanding. In D. T. Owens, M. K. Reed, G. M. Millsaps (Eds.), Proceedings of the Seventeenth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 1 (pp.123-128), Columbus, Ohio: ERIC.

  • Kosslyn, S. M. and Koenig, O. (1995). Wet Mind: The New Cognitive Neuroscience, New York, NY: Free Press.

  • Miller, G. A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information, Psychological Review, 63, 81-97.

  • Novak, J. D. and Gowin, D. B. (1984). Learning How To Learn, Cambridge, UK: Cambridge University Press.

  • Pirie, S. and Kieren, T. (1994). Growth in mathematical understanding: how can we characterise it and how can we represent it?, Educational Studies in Mathematics, 23, 505-528.

  • Schoenfeld, A. H. (1985). Mathematical Problem Solving, New York, NY: Academic Press.

  • Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 334-370), New York, NY: Macmillan.

  • Selden, J., Selden, A., and Mason, A. (1994). Even good calculus students can't solve nonroutine problems. In J. Kaput and E. Dubinsky (Eds.), Research Issues in Undergraduate Mathematics Learning: Preliminary Analyses and Results (pp. 19-26), MAA Notes 33, Washington, DC: Mathematical Association of America.

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