Story behind Story Problems 1 The Real Story behind Story Problems: Effects of Representations on Quantitative Reasoning Kenneth R. Koedinger* Carnegie Mellon University Mitchell J. Nathan University of Colorado RUNNING HEAD: Story behind Story Problems *Contact Information:
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Strategic Development in Verbal and Symbolic Problem Solving
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Story behind Story Problems 1
The Real Story behind Story Problems:
Effects of Representations on Quantitative Reasoning
Kenneth R. Koedinger*
Carnegie Mellon University
Mitchell J. Nathan
University of Colorado
RUNNING HEAD: Story behind Story Problems
*Contact Information:
Ken KoedingerHuman-Computer Interaction InstituteCarnegie Mellon UniversityPittsburgh, PA 15213Phone: 412-268-7667Fax: 412-268-1266Email: [email protected]
Figure 4c illustrates a second informal strategy for early algebra problem solving we call the unwind
strategy. To find the unknown start value, the student reverses the process described in the problem.
The student addresses the last operation first and inverts each operation to work backward to obtain the
start value. In Figure 4c, the problem describes two arithmetic operations, subtract 64 and divide by 3,
in that order. The student starts (on the right) with the result value of 26.50 and multiplies it by 3 as
this inverts the division by 3 that was used to get to this value. Next the student takes the intermediate
result, 79.50, and adds 64.00 to it as this inverts the subtraction by 64 described in the problem. This
addition yields the unknown start value of 143.50.
Although the informal guess-and-test strategy appears relatively inefficient compared to the formal
translation strategy (see the amount of writing in Figure 4b compared with 4a), the informal unwind
strategy actually results in less written work than the translation strategy (compare 4c and 4a). In
unwind, students go directly to the column arithmetic operations (see 4c) that also appear in translation
solutions (see the column subtraction and division in 4a), but they do so mentally and save the effort of
writing equations. (For other examples of such “mental algebra” amongst more expert problem solvers
see Hall et al., 1989; Tabachneck, Koedinger & Nathan, 1994). The next section addresses whether
students use informal strategies frequently and effectively.
Story behind Story Problems 20
Quantitative Strategy Analysis – Words Elicit More Effective Strategies
We coded student solutions for the strategies apparent in their written solutions for DFA1 and DFA2.
Our strategy analysis focuses on the early algebra start-unknown problems (shown in Figure 4a-d). We
observed little variability in students’ strategies on the result-unknown problems. Although some
students translated verbal result-unknown problems into equations (see Figure 4e), in the majority of
solutions, students went directly to the arithmetic. Typical solution traces included only arithmetic
work as illustrated in the lower left corner of Figure 4e (i.e., all but the three equations).
Table 4 shows the proportion of strategy use on start-unknown problems for the three different
representations. Different representations elicited different patterns of strategy usage. Story problems
elicited the unwind strategy most often, 50% of the time. Story problems seldom elicited the symbolic
translation strategy typically associated with algebra (only 5% of the time). Situation-less word
equations tended to elicit either the guess-and-test (23% of the time) or unwind (22%) strategy.
Equations resulted in no response 32% of the time, more than twice as often as the other representations.
When students did respond, they tended to stay within the mathematical formalism and apply symbol
manipulation methods (22%). Interestingly even on equations, students used the informal guess-and-
test (14%) and unwind (13%) strategies fairly often. In fact, as Figure 4d illustrates, sometimes students
translate a verbal problem to an equation but then solve the equation sub-problem informally, in this
case, using the unwind strategy.
Story problems may elicit more use of the unwind strategy than word equations perhaps because of
their more episodic or situated nature (cf., Hall et al., 1989). Retrieving real world knowledge, for
example, about a box of donuts (see Figure 1) may support students in making the whole-part inference
(cf., Greeno, 1983; Koedinger & Anderson, 1990) that subtracting the box cost from the total cost gets
one closer to the solution.
We also saw that word equations elicit more unwind strategy usage than equations. Although word
equations are situation-less, we did use words like " starting with", "and then", and "I get" that describe
Story behind Story Problems 21
someone performing an active procedure. Even the mathematical operators were described as actions,
"multiply" and "add", instead of relations, "times" and "plus". Perhaps this active description makes it
easier for students to think of reversing the performance of the procedure described. We suggest future
research that tests this “action facilitation” hypothesis, and contrasts our current "procedural" word
equations with "relational" word equations like "some number times .37 plus .22 equals 2.81" (see
Figure 1 for the analogous procedural word equation). Action facilitation predicts better student
performance on procedural than relational word equations whereas verbal facilitation predicts equal
performance (with both better than equations).
In addition to investigating differences in strategy selection, we also analyzed the effectiveness of
these strategies. Table 5 shows effectiveness statistics (percent correct) for the unwind, guess-and-test,
and symbol manipulation strategies on start-unknown problems in all three representations. The
informal strategies, unwind and guess-and-test, showed a higher likelihood of success (69% and 71%
respectively) than use of the symbol manipulation approach (51%). So, it appears one reason these
algebra students did better on story and word problems than equations is they select more effective
strategies more often. However, this effect could result from students choosing informal strategies on
easier problems. Use of a no-choice strategy selection paradigm (Siegler & Lemaire, 1997) is a better
way to test the efficacy of these strategies. Nhouyvanisvong (1999) used this approach to compare
equation solving and guess-and-test performance on story problems normatively solved using a system
of two equations and/or inequalities. Surprisingly, he found students instructed to use guess-and-test
were more successful than those instructed to use equation solving.
Error Analysis – Comprehending Equations is Harder than Comprehending Words
Our analysis of student strategies and, in particular, the differential use of informal strategies provides
one reason why story and word problems can be easier than matched equations. Our analysis of student
errors provides a second reason. Unlike the first and second grade students in Cummins et al. (1988),
who have yet to acquire critical English language comprehension skills, high school algebra students
Story behind Story Problems 22
have more developed English comprehension skills, but are still struggling to acquire critical Algebra
language comprehension skills.
A categorization of student errors into three broad categories -- 1) no response, 2) arithmetic error,
and 3) other conceptual errors -- provides insight into how students process story problems and
equations differently. Students’ solutions were coded as no response if nothing was written down for
that problem. Systematic occurrences of no response errors suggest student difficulties in
comprehending the external problem representation. Our counter-balancing for the order of problem
presentation allows us to rule out student fatigue or time constraints. Student solutions were coded as
arithmetic error if a mistake was made in performing an arithmetic operation, but the solution was
otherwise correct (e.g., Figure 5f). Arithmetic errors indicate correct comprehension and, in the case of
start-unknowns, correct formal or informal algebraic reasoning. Apart from some rare (19 errors out of
1976 solutions) non-arithmetic slips, like incorrectly copying a digit from the problem statement, all
other errors were coded as conceptual errors. Examples of a variety of different conceptual errors are
shown in Figures 5a-5e.
Figure 6a shows the proportions of the error types for the three levels of the representation factor:
story, word-equation, or equation. The key difference between equations and the two verbal
representations (story, word-equation) is accounted for primarily by no response errors (26% = 127/492
vs. 8% = 119/1465). No response errors imply difficulty comprehending the external problem
representation. These data suggest that students in these samples were particularly challenged by the
demands of comprehending the symbolic algebra representation. The language of symbolic algebra
presents some new demands that are not common in English or in the simpler symbolic arithmetic
language of students’ past experience (e.g., one operator number sentences, like “6 - 2 = ?”, used in
Cummins et al., 1988). The algebraic language adds new lexical items, like “x”, “*”, “/”, “(“, and new
syntactic and semantic rules, like identifying sides of an equation, interpreting the equals sign as a
relation rather than an operation, order of arguments, and order of operators. When faced with an
Story behind Story Problems 23
equation to solve, students lacking aspects of algebraic comprehension knowledge may give up before
writing anything down.
Further evidence of students’ difficulties with the “foreign language of algebra” comes from
students’ conceptual errors. As shown in Figure 5a, students make more conceptual errors on equations
(28% = 103/365) than on word equations (23% = 103/450) and particularly story problems (16% =
144/896)4. Figures 6a and 6b show examples of conceptual errors on equations. In Figure 6a we see an
order of operations error whereby the student performs the addition on the left-hand side (.37 + .22)
violating the operator precedence rule that multiplication should precede addition. We found a
substantial proportion of order of operation errors on equations (4.9% = 24/492), while order of
operations errors on verbal problems were extremely rare (0.2% = 4 /1468).
In Figure 6b we see two examples of algebra manipulation errors. This student appears to have
some partial knowledge of equation solving, namely, that you need to get rid of numbers by performing
the same operation to “both sides”. The student, however, operates on both sides of the plus sign rather
than both sides of the equal sign. These errors indicate a lack of comprehension of the quantitative
structure expressed in the given equations. Such errors appear less frequently on word and story
problems indicating that comprehension of the quantitative structure is easier for students when that
structure is expressed in English words rather than algebraic symbols.
Explaining Situation Facilitation
Algebraic language acquisition difficulties, such as comprehension and conceptualizing the underlying
quantitative relations, account for much of the error difference between equations and verbal problems.
These differences are consistent with the verbal facilitation hypothesis. However, we also observed a
smaller situation facilitation effect whereby story performance was better than word-equation
performance under certain conditions – namely when dealing with decimal numbers. This interaction
was statistically reliable in DFA2, but the same trend was also apparent in the smaller DFA1 data set. 4 The proportion of conceptual errors reported here is conditional on there having been a response -- solutions with no
response errors are not counted in the denominator.
Story behind Story Problems 24
Figure 6b shows the proportions of the three error types for the representation and number type factors
together. The interaction between representation and number type is caused largely by fewer arithmetic
errors on decimal story problems (12% = 46/389 on decimal vs. 5% = 17/363 on whole) than for word-
equations or equations (23% = 33/144 on decimal vs. 2% = 4/203 on whole)5. A common error on
situation-less word and equation problems was to miss-align place values in decimal arithmetic (see
Figure 5f). In contrast this error was rare on story problems. It appears the money context of the story
problems helped students to correctly add (or subtract) dollars to dollars and cents to cents. In contrast,
without the situational context (in word and equations), students would sometimes, in effect, add dollars
to cents.
Situation-induced strategy differences also appear to contribute to students’ somewhat better
performance on story problems than word equations. As we saw from the strategy analysis, students
were more likely to use the unwind strategy on story problems (50%) than on word-equations (22%) or
equations (13%). The unwind strategy may be less susceptible to conceptual errors than the guess-and-
test strategy which was used more frequently on word-equations. A particular weakness of the guess-
and-test strategy is the need to iterate through guesses until a value is found that satisfies the problem
constraints. As illustrated in Figure 6e, a common conceptual error in applying guess-and-test is to give
up before a satisfactory value is found. Guess-and-test is more difficult when the answer is a decimal
rather than a whole number because it takes more iterations in general to converge on the solution. This
weakness of guess-and-test and its greater relative use on word equations than story problems may
account for the greater number of conceptual errors on decimal word-equations than decimal story
problems shown in Figure 6b.
5 The proportion of arithmetic errors reported here is conditional on the solution being conceptually correct -- solutions
with no response or conceptual errors are not counted in the denominator.
Story behind Story Problems 25
DISCUSSION
Assessing the Symbolic, Situation, and Verbal Facilitation Hypotheses
In the introduction we contrasted three hypotheses regarding the effects of different problem
representations on algebra problem solving. The symbolic facilitation hypothesis predicts that story
problems are more difficult than matched equations because equations are more parsimonious and their
comprehension more transparent. Our results with high school students solving entry-level algebra
problems in two different samples contradict this claim and show, instead, that symbolic problems can
be more difficult for students, even after a year or more of formal algebra instruction.
Alternatively, the situation facilitation hypothesis follows from situated cognition research (Baranes
et al., 1989; Brown, Collins, & Duguid, 1989; Carraher et al., 1987; Nunes, Schliemann, & Carraher,
1993) and suggests that problem situations facilitate student problem solving because they
contextualized the quantitative relations. The prediction of the situation facilitation hypothesis, that
story problems are easier than both word-equations and equations, is not fully consistent with our
results. While students in DFA1 and DFA2 did perform better on story problems than equations, they
also performed better on word-equations than equations. Thus, it is not simply the situated nature of
story contexts that accounts for better performance.
The verbal facilitation hypothesis focuses not on the situated nature of story problems per se, but on
their representation in familiar natural language. The hypothesis follows from the idea that students,
even after an algebra course, have had greater experience with verbal descriptions of quantitative
constraints than with algebraic descriptions of quantitative constraints. Thus, it predicts that word-
equations, as well as story problems, will be easier than mathematically equivalent equations. The
prediction relies on two claims. First, students initially have more reliable comprehension knowledge
for verbal representations than symbolical ones. Second, verbal representations better cue students’
existing understanding of quantitative constraints and, in turn, informal strategies or weak-methods for
Story behind Story Problems 26
constraint satisfaction, like generate-and-test or working backwards. Early algebra students appear
better able to successfully use such strategies than the symbol-mediated equation solving strategy.
This second claim, that verbal representations help cue students’ knowledge, is consistent with the
assertion made by Nunes, Schliemann, & Carraher (1993, p. 45) that "discrepant performances can be
explained in terms of the symbolic systems being used." They found evidence that the chosen symbolic
system (e.g., formal vs. verbal/oral) determines performance more than the given one. Students
performed better on all kinds of problems, whether abstract or situational, when they used informal oral
strategies than when they used formal written strategies. Our strategy analysis revealed analogous
results for older students and a different class of problems. In addition to providing further evidence to
support the explanation provided by Nunes, Schliemann, & Carraher (1993), our results extend that
explanation. In our error analysis, we found evidence that the given representation has direct effects on
student performance beyond the indirect effects it has on influencing student strategy choice. More
students failed to comprehend given equation representations than given verbal representations as
indicated by a greater frequency of no-response and conceptual errors on the former (as illustrated in
Figure 6a).
Like Baranes et al. (1989), we did see some localized situational facilitation in students’
performance on story problems. First, students used the unwind strategy more often on story problems
than on word-equations or symbolic equations. Because this strategy is more reliable than equation
manipulation and more efficient than guess-and-test, students were less likely to make conceptual errors
when solving decimal story problems. A second situational effect involves support for decimal
alignment in the context of a story problem. Students avoided adding dollars to cents in the story
context and thus made fewer arithmetic errors on problems involving decimals in this context than in
content-free word equations and equations.
One might interpret some educational innovations emphasizing story contexts (e.g., CTGV, 1997;
Koedinger, Anderson, Hadley, & Mark, 1997), calls for mathematics reform (e.g., NCTM, 2000), and
Story behind Story Problems 27
situated cognition and ethnomathematical research (e.g., Brown, Collins, & Duguid, 1989; CTGV,
1990; Greeno & MMAP Group, 1998; Roth, 1996) as suggesting that “authentic” problem situations
generally help students make sense of mathematics. In contrast, our results are consistent with those of
Baranes et al. (1989) that situational effects are specific and knowledge related. Similarly, Nunes,
Schliemann, & Carraher (1993, p. 47) argued that the differences they observed "cannot be explained
only by social-interactional factors". Indeed, as long as problems were presented in a story context, they
found no significant difference between performances in different social interaction settings, whether a
customer-vendor street interaction or a teacher-pupil school interaction.
Situational effects are not panaceas for students’ mathematical understanding and learning. Clearly,
though, problem representations, including their embedding and referent situations, have significant
effects on how students think and learn. Better understanding of these specific effects should yield
better instruction.
Two Reasons Why Story and Word Problems Can Be Easier
Two key reasons explain the surprising difficulty of symbolic equations relative to both word and story
problems. These two reasons correspond, respectively, to the solution and comprehension phases of
problem solving illustrated in Figure 1. First, students’ access to informal strategies for solving early
algebra problems provides an alternative to the logic that word problems must be more difficult because
equations are needed to solve them. Our data as well as that of others (cf., Stern, 1997; Hall et al., 1989)
demonstrates that solvers do not always use equations to solve story problems. Second, despite the
apparent ease of solving symbolic expressions for experienced mathematicians, the successful
manipulation of symbols requires extensive symbolic comprehension skills. These skills are acquired
over time through substantial learning. Early in the learning process, symbolic sentences are like a
foreign language – students must acquire the implicit processing knowledge of equation syntax and
semantics. Early algebra students’ weak symbolic comprehension skills are in contrast to their existing
skills for comprehending and manipulating quantitative constraints written in English. When problems
Story behind Story Problems 28
are presented in a language students understand, students can draw on prior knowledge and intuitive
strategies to analyze and solve these problems despite lacking strong knowledge of formal solution
procedures.
One dramatic characterization of our results is that under certain circumstances students can do as
well on simple algebra problems as they do on arithmetic problems. This occurs when the algebra
problems are presented verbally (59% and 60% correct on start-unknown stories in DFA1 and DFA2)
and the arithmetic problems are presented symbolically (51% and 56% correct on result-unknown
equations in DFA1 and DFA2). While we have been critical of the sweeping claim made by Cummins
and colleagues (1988) that “students … continue to find word problems … more difficult to solve than
problems presented in symbolic format (e.g., algebraic equations)", we agree on the importance of
linguistic development. But, we broaden the notion of linguistic forms to include mathematical symbol
sentences. Cummins emphasized that the difficulty in story problem solving is not specific to the
solution process, but is also in the comprehension process. “The linguistic development view holds that
certain word problems are difficult to solve because they employ linguistic forms that do not readily
map onto children’s existing conceptual knowledge structures.” (Cummins, et al, 1988, p. 407). A
main point from our results is that the difficulty in equation solving is similarly not just found in the
solution process, but as much or more so in the comprehension process. Algebra equations employ
linguistic forms that beginning algebra students have difficulty mapping onto existing conceptual
knowledge structures.
Although our results are closer to those who have found advantages for story problems over
symbolic problems under some circumstances (Baranes, Perry & Stigler, 1989; Carraher, Carraher, &
Schliemann, 1987), our analysis of the underlying processes has important differences. Like the
situated cognition researchers we did find particular circumstances where the problem situation
facilitated performance (money and decimal arithmetic). However, our design and error analysis
focused not only on when and why story problems might be easier, but also on when and why equations
Story behind Story Problems 29
might be harder. The key result here is that equations can be more difficult to comprehend than
analogous word problems, even though both forms have no situational context. Kirshner (1989) and
Sleeman (1984) have also highlighted the subtle complexities of comprehending symbolic equations and
Heffernan & Koedinger (1998) have identified similar difficulties in the production of symbolic
equations.
Logical Task-Structure vs. Experience-Based Reasons for Difficulty Differences
An important question regarding these results is whether difficulty factor differences are a logical
consequence of the task structure or a consequence of biases in experience as determined by cultural
practices (e.g., presenting students with algebraic symbolism later, as in the US, versus early, as in
Russia and Singapore). The cognitive modeling work we have done (Koedinger & MacLaren, 2002)
has led to the observation that some difficulty factors are a fixed consequence of task structure, while
others are experience based.
The greater difficulty of start-unknown problems over result-unknown problems is a logical
consequence of differences in task structure. When students solve a start-unknown problem they must
do everything they need to do to solve an otherwise equivalent result-unknown problems (e.g.,
comprehend the problem statement, perform arithmetic operations) in addition to dealing with the fact
that the arithmetic operations cannot be simply applied as described in the problem. Thus, start-
unknown problems are logically constrained to be at least as difficult as result-unknown problems.
In contrast, the difficulty difference between word problems and equations is not logically
constrained. While there is some overlap in the knowledge required to solve word problems and
equations (e.g., arithmetic and operator inversion or guess-and-test skills), each problem category
requires some knowledge that the other does not. In particular, word problems require knowledge for
verbal comprehension not needed for equations. Conversely, equations require knowledge for
comprehending symbolic notation not needed for word problems. Thus, the difficulty difference
Story behind Story Problems 30
between these problem categories is not fixed. It depends on students’ relative experience with verbal
and symbolic representations.
The proportion of a student's exposure to quantitative constraints in verbal versus symbolic form
may depend, in turn, on cultural factors. Given the prevalence of natural language for other
communicative needs, early algebra students are likely to have more reliable knowledge for
comprehending verbal descriptions than symbolic ones. Thus, students may tend to find comprehending
word problems easier than equations, at least initially. However, in an educational culture where
symbolic algebra representations are experienced earlier and more frequently by students, we would
expect the difference to shrink over time. In contrast to the approach in traditional US curricula of
introducing start-unknown "number sentences" using boxes or blanks (e.g., "__ + 5 = 8") to represent
unknowns, students in other countries (Russia, Singapore) are introduced to the use of letters (e.g., "x +
5 = 8") to represent unknowns in the elementary grades (cf., Singapore Ministry of Education, 1999). If
our study was replicated in such countries, we expect that students would not experience the kind of
equation comprehension difficulties observed here and thus may show as good or better performance on
symbolic equations relative to story problems. Perhaps future cross-cultural research can test this
prediction.
Instructional Implications
We discuss the advantages and disadvantages of four alternative instructional strategies that might
appear to follow from our results. First, to the extent that people can effectively solve problems without
symbolic equations, one might well ask, why are algebra equations and equation solving skills needed at
all? One reason is that algebraic symbolism has uses outside of problem solving, including efficiently
communicating formulas and facilitating theorem proving. But even within problem solving, equations
may well facilitate performance for more complex problem conditions (Koedinger, Alibali, & Nathan,
submitted; Verzoni & Koedinger, 1997). Replacing algebra equations with alternative representations is
Story behind Story Problems 31
an intriguing idea (cf., Cheng, 1999; Koedinger & Terao, 2002), but we do not advocate eliminating
equations and equation solving from the curriculum.
Since equation solving is so hard for students to learn and because it is important, a second
instructional strategy worth consideration is a mastery-based approach (Bloom, 1984) that focuses on
equation-solving instruction in isolation. This strategy targets the syntax of algebraic transformation
rules and does not address algebraic symbolism as a representational language with semantics.
Instruction that isolates transformational rules may reduce cognitive load and thus facilitate learning
(Sweller, 1988). On the other hand, practicing procedures without an understanding of underlying
principles often leads to fragile knowledge that does not transfer well (Judd, 1908; Katona, 1940). To
be sure, students often make errors in equation solving that clearly violate the underlying semantics
(Figure 5ab; Payne & Squibb, 1990).
A third alternative takes a developmental view and suggests starting with instructional activities
involving story problems, which are easier for students to solve, and moving later to more abstract
word-equation problems and then symbolic equations. In this view, decomposing instruction to focus on
difficult equation solving skills is fine. However, such instruction should come after students have
learned the meaning of algebraic sentences, in other words, after they have learned to translate back and
forth between English and algebra. This "progressive formalization" sequencing (Romberg & de Lange,
2002) is unlike many current textbooks that teach equation solving before analogous story problem
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning, Cognitive Science, 12,
257-286.
Story behind Story Problems 44
Tabachneck, H. J. M., Koedinger, K. R., & Nathan, M. J. (1994). Toward a theoretical account of
strategy use and sense making in mathematics problem solving. In Proceedings of the Sixteenth
Annual Conference of the Cognitive Science Society, (pp. 836-841). Hillsdale, NJ: Erlbaum.
Verzoni, K., & Koedinger, K. R. (1997). Student learning of negative number: A classroom study and
difficulty factors assessment. Paper presented at the annual meeting of the American
Educational Research Association, Chicago, IL.
Wason, P.C. & Johnson-Laird, P.N. (1972). Psychology of reasoning: Structure and content.
Cambridge, MA: Harvard Press.
Zhang, J. & Norman D. A. (1994). Representations in distributed cognitive tasks. Cognitive Science,
18, 87-122.
ACKNOWLEDGEMENTS
This research was supported by a grant from the James S. McDonnell Foundation, Cognitive Studies in
Education Practice, grant no. JSMF 95-11. We wish to acknowledge the contributions of Hermi
Tabachneck who participated in early design and analysis and Ben MacLaren who did the solution
coding. Thanks also to Martha Alibali, Lisa Haverty, and Heather McQuaid for reading and
commenting on drafts and to Allan Collins, Andee Rubin, and an anonymous reviewer for their helpful
comments.
Story behind Story Problems 45
FIGURES
Figure 1. Quantitative problem solving involves two phases, comprehension and solution, both of
which are influenced by the external representation (e.g., story, word, equation) in which a problem is
presented. The influence on the comprehension phase results from the need for different kinds of
linguistic processing knowledge (e.g., situational, verbal, or symbolic) required by different external
representations. The impact on the solution phase results from the different computational
characteristics of the strategies (e.g., unwind, guess-and-test, equation solving) cued by different
external representations.
Story behind Story Problems 46
Situation &Problem Models
Problem Presentation Solution Notations
InternalProcessingExternal
Representations
ComprehensionPhase
KnowledgeSources
Symbolic
SolutionPhase
Unwind
EquationSolving
SolutionStrategies
8. After buyingdonuts at WholeyDonuts, Lauramutiplies the numberof donuts she boughtby their price of $0.37per donut. Then sheadds the $0.22charges for the boxthey came in and gets$2.81. How manydonuts did she buy?
5. Starting withsome number, if Imultiply it by .37and then add .22,I get 2.81. Whatnumber did I startwith?
2. Solve for x: x * .37 + .22 = 2.81
$.37
$.22
$2.81
+
*
X
Guess& Test
Situational
Verbal
$$
Story behind Story Problems 47
Whole Number Problems Decimal Number Problems
Figure 2. Proportion correct of high school algebra students in DFA1 (n = 76). The graphs show the
effects of the three difficulty factors: representation, unknown-position, and number-type. The error
bars display standard errors around the item means.
Result-Unknown Start-Unknown0
.1
.2
.3
.4
.5
.6
.7
.8
.91
EquationWord-EquationStory
Unknown PositionResult-Unknown Start-Unknown
0.1.2
.3
.4
.5
.6
.7
.8
.91
EquationWord-EquationStory
Unknown Position
Story behind Story Problems 48
Whole Number Problems Decimal Number Problems
Figure 3. Proportion correct of high school algebra students in DFA2 (n = 171). The graphs show the
effects of the three difficulty factors: representation, unknown-position, and number-type. The error
bars display standard errors around the item means.
Result-Unknown Start-Unknown0
.1
.2
.3
.4
.5
.6
.7
.8
.91
EquationWord-EquationStory
Unknown Position
Result-Unknown Start-Unknown0
.1
.2
.3
.4
.5
.6
.7
.8
.91
EquationWord-EquationStory
Unknown Position
Story behind Story Problems 49
a. The normative strategy: Translate to algebra and solve algebraically
b. The guess-and-test strategy.
c. The unwind strategy.
Story behind Story Problems 50
d. Translation to an algebra equation, which is then solved by the informal, unwind strategy.
e. A rare translation of a result-unknown story to an equation.
Figure 4. Examples of successful strategies used by students: (a) Guess-and-test, (b) Unwind, (c)
Translate to algebra and solve algebraically, (d) Translate to algebra and solve by unwind, (e) Translate
to algebra and solve by arithmetic
Story behind Story Problems 51
a. Order of operations error. Student inappropriately adds .37 and .22.
b. Algebra manipulation errors. Student subtracts from both sides of the plus sign rather than both sides
of the equal sign.
c. Argument order error . Student treats "subtract 66" (x – 66) as if it were "subtract from 66" (66 – x).
Story behind Story Problems 52
d. Inverse operator error. Student should have added .10 rather than subtracting .10.
e. Incomplete guess-and-test error. Student gives up before finding a guess that works.
f. Decimal alignment arithmetic error. Student adds 66 to .90 aligning flush right rather than aligning
place values or the decimal point.
Figure 5. Examples of errors made by students: (a) order of operations, (b) algebra manipulation, (c)
(a) Proportions of correct and incorrect responses for story, word equations, and equations. A verbal
facilitation is indicated particularly in the fewer no-response errors in the story and word equation
problems.
Story behind Story Problems 54
(b) Proportions of correct and incorrect responses for story vs. word equations crossed with whole vs.
decimal number problems. A situation facilitation effect is indicated in decimal problems with fewer
conceptual and arithmetic errors on story decimal problems than word-equation decimal problems
Figure 6. Frequency of broad error categories helps to explain the causes of the verbal and situation
facilitation effects observed.
Story behind Story Problems 55
TABLES
Table 1. Six Problem Categories Illustrating Two Difficulty Factors: Representation and Unknown-Position.
STORY PROBLEM WORD EQUATION SYMBOLIC EQUATION
RESULT-UNKNOWN
When Ted got home from his waiter job, he took the $81.90 he earned that day and subtracted the $66 he received in tips. Then he divided the remaining money by the 6 hours he worked and found his hourly wage. How much does Ted make per hour?
Starting with 81.9, if I subtract 66 and then divide by 6, I get a number. What is it?
Solve for x: (81.90 - 66)/6 = x
START-UNKNOWN
When Ted got home from his waiter job, he multiplied his hourly wage by the 6 hours he worked that day. Then he added the $66 he made in tips and found he had earned $81.90. How much does Ted make per hour?
Starting with some number, if I multiply it by 6 and then add 66, I get 81.9. What number did I start with?
Solve for x: x * 6 + 66 = 81.90
Story behind Story Problems 56
Table 2. Examples of the Four Cover Stories Used.
STORY PROBLEM COVER STORIES
DONUT LOTTERY WAITER BASKETBALL
After buying donuts at Wholey Donuts, Laura multiplies the 7 donuts she bought by their price of $0.37 per donut. Then she adds the $0.22 charge for the box they came in and gets the total amount she paid. How much did she pay?
After hearing that Mom won a lottery prize, Bill took the $143.50 she won and subtracted the $64 that Mom kept for herself. Then he divided the remaining money among her 3 sons giving each the same amount. How much did each son get?
When Ted got home from his waiter job, he multiplied his wage of $2.65 per hour by the 6 hours he worked that day. Then he added the $66 he made in tips and found how much he earned. How much did Ted earn that day?
After buying a basketball with his daughters, Mr. Jordan took the price of the ball, $68.36, and subtracted the $25 he contributed. Then he divided the rest by 4 to find out what each daughter paid. How much did each daughter pay?