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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
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(Ed.), The Cognitive Neurosciences (pp. 755-764),
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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.),
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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
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Schoenfeld, A. H. (1985). Mathematical Problem Solving,
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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
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Mathematical Association of America.
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