Figure 1.
Provide the systematic (IUPAC-approved) names for these compounds
Figure 1.
Most of the students successfully named the compound on the left, but not the one on the right. The students weren't much more successful at naming the compound shown on the right when this part of the question was repeated on the next exam. Or when the same question appeared on the final exam. The students' success (or lack thereof) on this question isn't as interesting, however, as their response to the question when they were interviewed. Time and time again, we heard students comment that the question wasn't "fair."
A similar phenomenon was observed when the following question appeared on an hour exam for the second-semester course.
Mg
CH3CH2Br --> CH3CH2MgBr
CH3CH2OH
When he wrote the exam, the instructor (GMB) was convinced that this was
a relatively easy question. (There is nothing wrong with the starting
material, which is a common reagent used to prepare Grignard reagents.
There is nothing wrong with the product of the reaction, which is a
typical Grignard reagent, or with using magnesium metal to prepare this
reagent. The only possible source of error was the solvent: CH3CH2OH.)
He therefore used this item as the first question on the exam - to build
the students' confidence.
When the exam was graded, he found that some of the students recognized that the solvent was a source of H+ ions that would destroy the Grignard reagent produced in this reaction. But many of them were unable to answer the question, and, when these students were interviewed after the exam, they frequently expressed the opinion that this wasn't a "fair" question.
The primary goal of this paper is to develop a theoretical explanation of the difference between the students who were successful on these questions and those who were not. A subsidiary goal, as you might expect, is to explain the source of students' beliefs that these questions weren't "fair." Before we do this, however, it might be useful to analyze student responses to one more exam question, based on the reaction shown in Figure 2.
Figure 2.
The students were asked to predict the major products of this reaction, estimate the ratio of these products that would be formed if Br· radicals are just as likely to attack one hydrogen atom as another, and use the relative stability of alkyl radicals to predict which product is likely to occur more often than expected from simple statistics.
Most of the more than 200 students in this course predicted that the reaction would give the three products shown in Figure 3, with a relative abundance of 3:2:2.
Figure 3. The most common answer.
When we discussed their answer with these students we found that they recognized that attack by the Br· radical at any of the three hydrogen atoms in the CH3 group would give the first product. They also recognized that the molecule is symmetrical, and it therefore doesn't matter whether reaction occurs on the right or left side of the molecule when the second and third products are formed. Unfortunately, they failed to recognize that there are two hydrogen atoms on each of the carbon atoms at which attack occurs to give the second and third products. They therefore failed to recognize that simple statistics predicts a 3:4:4 ratio for the three products they listed.
A small proportion of the students translated the line drawing for the starting material into a drawing that showed the positions of all the hydrogen atoms in this compound, as shown in Figure 4.
Figure 4. An alternate representation of the starting material drawn by every student who obtained the correct answer to this question.
All of the students who did this recognized that the reaction actually gives the four products shown in Figure 5.
Figure 5. The accepted answer to this question.
These students also recognized that simple statistics would give a product distribution of 3:4:4:1. More importantly, these students came to the correct conclusion that it is the fourth product - the one their colleagues missed - that is the most likely product to be formed in this reaction because of the stability of the 3o radical formed when the Br· radical attacks this carbon atom.
Before we can satisfy our goal of explaining the observations reported
in this section we need to decide how the problem solving ability of
various individuals should be compared and we need to define the term
representation.
SUCCESSFUL VERSUS UNSUCCESSFUL PROBLEM SOLVERS
Efforts to understand the cognitive processes involved in problem
solving have been underway for at least 100 years (Helmholtz, 1894).
One approach has focused on differences between expert and novice
problem solvers (Chi, Feltovich, & Glaser, 1981; Larkin, McDermott,
Simon, & Simon, 1980; Schoenfeld & Herrmann, 1982). Smith (1992) has
criticized this expert-novice dichotomy as unjustly equating expertise
with success. He argued that "successful problem solvers often share
more procedural characteristics that distinguish them from unsuccessful
subjects than do experts when compared to novices (p. 182)."
We agree with those who argue that research on problem solving should
focus on the differences between successful and unsuccessful problem
solvers (Camacho & Good, 1989; Smith & Good, 1984). Our goal is a
better understanding of the process by which individuals disembed
relevant information from the statement of a problem and transform the
problem into one they understand - in other words, how they build and
manipulate the representation they construct of the problem. This paper
pays particular attention to differences between both the number and
kind of representations built by successful versus unsuccessful
chemistry problem solvers and describes a theoretical model that offers
a possible explanation for the role that representations play in
determining the success or failure of the problem-solving process.
WHAT IS A REPRESENTATION?
The first step toward understanding the role that representations play
in problem solving in any discipline involves building an adequate
definition of what we mean by the term representation. Simon (1978)
argued that the following is noncontroversial.
Simon uses the term representation in the sense of an internal representation - information that has been encoded, modified, and stored in the brain. Martin (1982) uses the term in the same sense when he says that representations "signify our imperfect conceptions of the world."
Estes (1989) makes an important point when he reminds us that "a representation stands for but does not fully depict an item or event." He notes that representations are attempts the brain makes to encode experiences. Thus, a representation is very different from a photograph, which preserves all of the information in the scene - up to the resolving power of the film.
It is tempting to define an internal representation as the mental image that an object or event evokes in the individual who experiences it. Purists would note, however, that there is some question about whether representations can be stored as images (Pylyshyn, 1978). Within the context of research on problem solving, it is therefore useful to rely on an operational definition in which an internal representation is assumed to be the understanding an individual constructs about the problem being solved.
Greeno (1978, 1980) proposed three characteristics that can be used to evaluate a mental representation: coherence, connectedness, and correspondence. A representation is coherent when it is internally consistent. It is connected when it is related to other concepts (or schema) the individual has constructed. Correspondence reflects the extent to which the representation is accurate because it matches reality. [Proponents of the constructivist theory of knowledge might prefer a definition in which correspondence is assumed to measure the extent to which the representation fits the knowledge shared among a community of scholars working in a particular field (Cobb, 1989).]
The modifier internal is added to the term representation to distinguish
the information stored in the brain from external representations, which
are physical manifestations of this information. An external
representation can take the form of a sequence of words the individual
uses to describe the information that resides in his or her mind. In
other situations, it takes the form of a drawing or a list of
information that captures particular elements of the mental
representa tion. Within the context of problem solving in chemistry, it
can include the equation - such as PV = nRT or E = Eo - RT/nF lnQ - an
individual writes that shapes the way information is processed in
subsequent steps in the problem-solving process.
UNDERSTANDING THE PROBLEM: THE EARLY STAGES IN PROBLEM SOLVING
Ten years ago, we began a series of experiments to study the
relationship between students' performance on tests of spatial ability
and their performance on the hour exams they took while enrolled in
college-level chemistry courses. We expected to find a correlation
between students' performance on the spatial ability tests and their
performance on exam questions that involved the manipulation of
three-dimensional images. We found, however, that the magnitude of this
correlation was equally strong for all questions that probed the
students' problem-solving skills (Bodner & McMillen, 1986). Subsequent
experiments with students in both general chemistry (Carter, LaRussa, &
Bodner, 1987) and organic chemistry (Pribyl & Bodner, 1987) showed that
correlations with tests of spatial ability were strongest for exam
questions that differed significantly from those the students had seen
previously. Regardless of the type of question that was asked, the
tests of spatial ability correlated best with the student's performance
on novel problems, rather than routine exercises (Bodner, 1991).
Because the spatial tests used in these experiments were tests of disembedding and cognitive restructuring in the spatial domain we concluded that there were preliminary stages in the problem-solving process that involved disembedding the relevant information from the statement of the problem and restructuring or transforming the problem into one the individual understands. We described the goal of the early stages of the problem-solving process as trying to understand the problem or to find the problem. Larkin (1985) reached similar conclusions when she concluded:
Students with low scores on the spatial tests were less likely to do well in the course and they were more likely to write equations such as:
The results of our preliminary experiments on problem solving in organic chemistry were reinforced by the work that led to the observations summarized in the introduction. It seems that one of the differences between students who are successful in organic chemistry and those who are not appears to be their ability to switch from one representation system to another. Students who do poorly in organic chemistry often have difficulty escaping verbal/linguistic representation systems. They tend to handle chemical formulas and equations that involve these formulas in terms of letters and lines and numbers that aren't symbols because they don't represent or symbolize anything that has physical reality. Thus, they see nothing wrong with transforming PhCOOH into PhCl.
Students locked in a verbal/linguistic representation system can recognize that the verbal/linguistic representation on the left and the symbolic representation on the right in Figure 6 describe the same compound. But they aren't likely to spontaneously switch from the representation on the left to the one on the right, or vice versa. Other students - who tend to do better in the course - switch back and forth between these representation systems as needed.
Figure 6. Verbal/linguistic and symbolic representations of benzoic acid.
If this hypothesis is correct, similar external representations might be written by individuals with very different internal representations. Consider the following equation, for example.
O
||
CH3CH2CH2CCH3 + CH3MgBr -->
When they write this equation in
their notebooks, students believe it is a direct copy of what the
instructor writes on the blackboard. An observer, comparing the two,
would agree that the students' notes seem to be direct copies of what
the instructor wrote. In spite of the apparent similarity, there is a
fundamental difference between what the instructor and many of the
students write. The instructor writes symbols, which represent a
physical reality. All too often, students write letters and numbers and
lines, which have no physical meaning to them
Students for whom chemical formulas are examples of a verbal/linguistic representation system are more likely to write "absurd" formulas, such as the product shown in the following equation.
O O
|| Et2O ||
CH3CH2CH2CCH3 + CH3MgBr --> CH3CH2CH2CCH3
|
CH3
It is only when the letters, numbers, and lines used to write these
equations are symbols, which represent a physical reality, that students
recognize why this answer is absurd or recognize the the flaw in the
equation used to describe the graduate student's approach to the
synthesis of a Grignard reagent.
Although they might not be familiar with the details of 2D-NMR, most readers will recognize that FT-NMR experiments involve irradiating the sample with a burst of RF energy, which is equivalent to exciting all the possible spin-state transitions at the same time. A detector then measures the change in the magnetization of the sample as it decays from saturation back to an equilibrium distribution of spin states. The signal collected from this experiment is subjected to a Fourier analysis, which transforms the signal from the time domain - in which it is collected - to a frequency domain spectrum identical to the result of the original NMR experiment.
2D-NMR is a two-dimensional NMR experiment that plays an important role in the process by which the individual peaks in the spectrum of a molecule are assigned to specific environments within the molecule. This content domain was chosen because multiple representations not only can but must be used to understand the 2D-NMR experiment. The data obtained in this study were consistent with the notion that the ability to switch between representations or representation systems plays an important role in determining success or failure in problem solving in chemistry. Successful problem solvers constructed significantly more representations than unsuccessful problem solvers.
The two groups also differed in the nature of the representations they
constructed. Among the successful problem solvers, the most common
representations were those that are best described as symbolic. These
representations were characterized by a reliance on symbols or highly
symbolic equations that might include fragments of a phrase or sentence.
The most common representations constructed by the unsuccessful problem
solvers were those best described as verbal. These representations,
which were expressed either orally or in writing, contained intact
sentences or phrases, such as: "the number of spin orientations of a
spin-active nucleus is equal to two times the spin-quantum number plus
one."
A THEORETICAL EXPLANATION OF THE DIFFERENCE BETWEEN SUCCESSFUL AND
UNSUCCESSFUL PROBLEM SOLVERS
As you look back through this paper, you might recognize a theme that
has dominated our work on problem solving, from our first experiments in
general chemistry through our work with graduate students: Successful
problem solvers construct significantly more representations while
solving a problem than those who aren't successful.
Although our colleagues who teach organic chemistry are familiar enough with the content to successfully name the compound shown on the right in Figure 1 from the representation in which it was drawn, most of us would have to proceed the way the students who were successful on this task approached the problem. We would have to transform this Newman projection into a line structure, from which we could determine the name of the compound.
Our colleagues in organic chemistry would have no difficulty with the question based on Figure 2. For them, however, familiarity with similar tasks has transformed this question from a problem into a routine exercise (Bodner, 1987). Those of us for whom this task is still a problem would have to do what the successful students did, they would have to transform the starting material from the representation shown in Figure 2 into the one in Figure 4 - either within the minds or on the paper - before they could successfully answer the question.
Although the successful problem solvers throughout our work constructed significantly more representations than those who weren't successful, neither group constructed very many representations while solving the problems. In the 2D-NMR experiment, for example, the successful problem solvers constructed an average of about two representations per problem, while those who weren't successful constructed an average of just more than one representation per problem. A possible explanation for the difference between successful and unsuccessful problem solvers, which might provide insight into the role of mental representations in problem solving, can be found in the schema theory of cognitive structures. Schema theory views cognitive structure as a general knowledge structure used for understanding (Rumelhart & Ortony, 1977). Schema, also referred to as frames (Minsky, 1975) or scripts (Schank & Abelson, 1977), relate to one's general knowledge about the world. Schema are activated or triggered from an individual's perceptions of his or her environment and they provide the context on which general behaviors are based. Because they don't include information about any exact situation, the understanding of a situation they generate is incomplete. But, by including both facts about a type of situation and the relationship between these facts, they provide a structure that allows one to make inferences (Medin & Ross, 1992).
Within a given context, problem solving requires the activation of an appropriate schema that contains an algorithm or heuristic that guides the individual to the correct solution to the problem. The construction of the first representation is an effort by the individual to activate the appropriate schema. Thus, the first representation establishes a context for understanding the statement of the problem. In some cases, this representation contains enough information to both provide a context for the problem and to generate a solution to the problem. In other cases, additional representations may be needed since the solution may require more than one algorithm or heuristic. But the first representation provides the context in which the other representations are built.
Unsuccessful problem solvers seem to construct initial representations
that activate an inappropriate schema for the problem. This can have
three different consequences, each of which leads to an unsuccessful
outcome: (1) the initial representation doesn't possess enough
information to construct additional representations that contain
algorithms or heuristics that might lead to the solution, and the
individual gives up; (2) the initial representation leads to the
construction of additional representations, but these representations
activate inappropriate algorithms or heuristics, and eventually, an
incorrect solution to the problem; or (3) the unsuccessful problem
solver may never actually achieve an understanding of the problem, in
spite of the number of representations that were constructed in an
effort to establish a context for the problem.
IMPLICATIONS FOR THE TEACHING OF CHEMISTRY
Although most of our work on representation systems has focused on
organic chemistry, a similar phenomenon exists in general chemistry.
Perhaps the best way to illustrate this is to ask the reader to consider
the following question: "Which weighs more, a liter of dry air at 25oC
and 1 atm, or a liter of air at this temperature and pressure that is
saturated with water vapor? (Assume that the average molecular weight
of air is 29.0 g/mol.)"
Most students (and some of their instructors) are convinced that air that has been saturated with water must weigh more than dry air. (It seems reasonable that adding water vapor to air must increase its weight.) Many of these individuals change their mind, however, when they are confronted with Figure 7.
Figure 7. A symbolic representation of the difference between dry air and air saturated with water.
Figure 7 illustrates an important point: Representations differ in the
information they convey. Encouraging students to use different
representations when solving a problem might therefore simply be a way
of helping them recognize what information is important in generating
the answer to this question. The symbolic/pictorial representation in
Figure 7 prompts us to consider the implications of Avogadro's
hypothesis, which assumes that equal volumes of different gases contain
the same number of particles. Because the molecular weight of water
(18.015 g/mol) is significantly smaller than the average molecular
weight of air (29.0 g/mol), water that has been saturated with air
actually weighs less than dry air.
IMPLICATIONS FOR CHANGES IN HOW WE TEACH CHEMISTRY
One of the implications of this research on changes that might be made
in the way we teach chemistry can be understood by considering what a
typical beginning chemistry teacher would do if asked to work the
following question in class: What is the pH of 100 mL of water to which
one drop of 2 M HCl has been added?
The author's work with almost 1000 teaching assistants at the University of Illinois or Purdue University suggests that relatively few of these individuals would focus their approach to this problem around the drawing in Figure 8.
Figure 8.
This is important, because these individuals invariably focus their approach around a drawing when they encounter problems from other domains, such as the following question.
Two trains are stopped on adjacent tracks. The engine of one train is 1000 yards ahead of the engine of the other. The end of the caboose of the first train is 400 yards ahead of the end of the caboose of the other. The first train is three times as long as the second. How long are the trains?
It is important to recognize that Figure 8 isn't a drawing created before the problem is solved, but a drawing around which the solution of the problem is constructed. Each time more information is obtained - such as noting that a drop of this solution is about 0.05 mL or that HCl is a strong acid (Ka = 106) - it is incorporated into the drawing.
Most of those who read this paper won't be surprised to note that student performance on problem solving tasks improves when drawings of this nature are used when the instructor solves problem in class. They might be surprised, however, by another implication of the research described in this paper.
Imagine that you were trying to balance the following equation in class.
What would happen if the instructor approached this reaction using Lewis structures? Two electrons would no longer be added in the reduction half-reaction "to balance charge." They would be added to the system because two electrons are needed to transform the starting material into three iodide ions with filled octets of valence electrons.
Where are those electrons going to come from? They obviously have to come from the thiosulfate ion. And they are more likely to come from the terminal sulfur than from one of the oxygen atoms.
What happens to the neutral S2O3 molecule produced in this reaction? It combines with an S2O32- ion to form an S4O62- ion.
No one would argue that beginning students can use Lewis structures to
predict the product of the two-electron oxidation of thiosulfate. We
have evidence, however, that these students can understand how Lewis
structures can be used to explain the product of this reaction. We also
have evidence to suggest that students who have seen their instructor
use this approach to balancing redox equations are more successful at
similar tasks and more likely to understand what they are doing when
they balance one of these equations. In many ways, this is nothing more
than adding a symbolic representation - which carries different
information - to the verbal/linguistic representation the students build
when they read the equation they are being asked to balance.
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