Story problems are the most common kind of problem encountered by students in formal education. Although not the most innovative or the most authentic, they are clearly the most commonly solved kind of problem in schools and universities as well as the most extensively researched. Students begin solving story problems in early elementary school and often encounter them through graduate school. From simple combined problems in beginning mathematics (e.g., Tom has three apples. Mary gave Tom three more apples. How many apples does Tom have in the end? [Riley, Greeno, & Heller, 1983]) to complex problems in thermodynamics, story problems are the most common kind of problem in formal education. Many innovations in mathemat- ics and science education have attempted to replace story problems with more complex and authentic problems. Notwithstanding those innov- ations, story problems remain the most ubiquitous kind of problem solved in schools and universities.
HOW DO STUDENTS SOLVE STORY PROBLEMS?
Story problems typically present a set of variables embedded within a shallow story context. Story problems are normally solved by identify- ing key values in the short scenario, selecting the appropriate algorithm, applying the algorithm to generate a quantitative answer, and hopefully checking their responses (Sherrill, 1983). Despite our understanding of the requirements for solving and transferring story problems, learners
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usually employ a more tactical, problem-avoidance strategy to solving word problems:
1. Search for key words.
2. Select algorithm (formula) based on key words.
3. Apply the algorithm.
(Sherrill, 1983) Rich (1960) elaborated that process slightly:
1. Represent the unknowns with letters.
2. Translate relationships about unknowns into equations.
3. Solve equations to find the value of the unknowns.
4. Verify or check the values calculated to see if they fit the original problem.
The solutions to story problems emphasize the quantitative represen- tation of the problem, that is, the conversion of the values in the story into a formula, because that is what they are expected to do. The students are smart enough to realize what is rewarded, so this is what they do.
Based on that approach, it was formerly believed that children’s major difficulty in solving word problems was their inability to select and apply the appropriate arithmetic operations (Zweng, 1979). Unfortunately, it is only the unsuccessful problem solvers who base their solution plans on the numbers and key words that they select from problem (Hegarty, Mayer, & Monk, 1995). When problem solvers attempt to directly translate the key propositions in the problem statement into a set of computations, known as the direct translation strategy, they more frequently commit errors. Why? This translation process is difficult.
Converting semantic entities from a shallow story into a mathematical representation is difficult, but solving story problems requires more than the transformation of values into formulas. Rather, successful problem solving requires the comprehension of relevant textual infor- mation, the capacity to visualize the data, the capacity to recognize the deep structure of the problem, the capacity to correctly sequence their solution activities, and the capacity and willingness to evaluate the procedure used to solve the problem (Lucangeli, Tressoldi, & Cendron, 1998). Solving problems is more complex than plugging values into formulas and solving for the unknown. The complexity of the solution process suggested by Lucangeli et al. (1998) explains many of the dif- ficulties that students have when they use a direct translation strategy to solve story problems.
Contemporary approaches to story problem solving have emphasized conceptual understanding of the story problems before attempting any 28 • Problem-Specific Design Models
solution. Successful problem solving requires the construction of a conceptual model of the problem and the application of solution plans that are based on those models. It is the quality of their con- ceptual models that most influences the ease and accuracy with which the problem can be solved (Hayes & Simon, 1976). Those conceptual models, also known as problem schemas (see Chapter 15 for a descrip- tion of problem schemas), are mental representations of the pattern of information that is represented in the problem (Riley & Greeno, 1988).
In order to solve story problems consistently, learners must demon- strate conceptual understanding of the problem types by constructing a conceptual model that includes a situational model of the problem, a structural model of the problem, and an algorithmic model (formula) of the problem from the problem text (Reusser, 1993). Because students normally make no effort to construct any kind of conceptual model of the problem, they commit errors and are unable to transfer any correct problem solutions to similar problems. Sherrill (1983) found that although students can identify key words in story problems, they frequently:
•
select either the wrong algorithm or the wrong sequence of algorithms;•
select the proper algorithm, but use the wrong numbers;•
select the proper algorithm, apply the algorithm properly, and stop, not realizing that it was a multi-step problem;•
do not check their answers; and•
make little use of heuristics.These responses all reflect an inadequate conceptual understanding of the problems or the problem-solving process. Such weaknesses in story problem solving can only be resolved by employing a more conceptually oriented approach to learning to solve story problems.
That is the purpose of this chapter.
Next, I examine the role of conceptual models (problem schemas) in solving story problems.
HOW SHOULD STUDENTS SOLVE STORY PROBLEMS?
Figure 2.1 illustrates a conceptually oriented model for solving story problems. According to this approach, transferring story problem- solving skills depends on students constructing a conceptual model of the story problems they are required to solve and accessing that model when they are required to solve structurally congruent prob- lems. When parsing the problem statement, students should search for Story Problems • 29
Figure2.1Story problem-solving process.
an appropriate conceptual model of the problem. To do that, students must identify the sets of values presented in the problem and determine their situational and structural characteristics and associate them with problem schemas. As illustrated in Figure 2.1, searching for a problem schema involves identifying the sets or elements contained in the prob- lem, identifying the relationships among those elements, and identifying the situational characteristics. When an appropriate problem schema is accessed, the student can then successfully classify the type of problem.
It is important that students ignore the irrelevant situational character- istics and classify the problem type based on the relevant structural properties. Correctly classifying the problem type is facilitated by analogically comparing pairs of problems (see Chapters 11 and 16 for a description of cases as analogues and analogical encoding processes).
With a problem schema in mind, students can then access their conceptual model of that problem type, which is the key to solving story problems. From that conceptual model of the problem, students can retrieve the processing operations necessary to solve the problem, including any strategic knowledge about when to apply problem sche- mas and formulas. After assigning the sets to a model of the structural relationships between the sets in the problem, those sets are transferred to a formula directly associated with the structural model of the prob- lem. Students then estimate the size of the result, solve the formula, and reconcile it to the estimate. If the solution was successful, students should work on developing strong associations between the new prob- lem and the problem schema to better elaborate their conceptual model of that problem type.
HOW CAN STORY PROBLEM SOLVING BE SUPPORTED?
Given the description of the story problem-solving process in Figure 2.1, I next describe the essential components of a story-problem learning environment and present relevant research findings to support those components.
What Are Problem Types and Typologies?
Chapter 15 describes different problem types, each of which is described by a different schema. For example, there are three kinds of simple addition problems: combine, compare, and change. In combine story problems, the quantity is unknown and values must be combined. In a compare problem, the total is known while the student must compare values, and in change problems one of the values must be changed to calculate the total. These are simple problem types; however, larger Story Problems • 31
content domains, such as physics, can contain many different kinds of problems such as Newton’s second law, conservation of energy/work- energy, conservation of momentum, angular motion, rotational motion, kinematics, and dynamics, circular motion, center of mass, statics including conservation of angular momentum and work problems, linear kinematics, vectors, and springs.
Instruction should begin with a graphic organizer illustrating each kind of problem solved within the domain and highlighting the prob- lem type currently being solved (see Figure 15.1 [p. 243] for an example of a graphic organizer illustrating the semantic relationships in a com- bine problem and Figure 15.2 [p. 251] for a semantic map of work- energy problems in physics.) This organizer should explicitly state and contrast the structural differences among problems. Research suggests that learners must be able to classify problems based on structural relationships among the sets in the problem. Emphasizing the structural properties of problems and contrasting them with other problems in the domain enhances learners’ abilities to generalize problems within a class and to discriminate between classes. Emphasizing the structural relationships among entities within the problem also focuses the clas- sification on structural properties rather than surface-level, situational characteristics. It is important that learners understand the conceptual nature of the problem and the disciplinary operations represented in the problem. These structural relationships are manifestations of the disciplinary principles that would also be illustrated when this description is accessed.
How Does a Conceptual Model of the Problem Function?
For each problem type in the problem typology, a model of the problem type may be used to describe the structural relationships between the entities in a problem. The conceptual model must also contain a visual model illustrating the situational characteristics of the problem because integrating the structural model with the situational model is necessary.
Given that structurally similar problems often contain situationally similar characteristics, the patterns of those relationships must be made explicit. For example, Figure 2.2a displays two problems that are situationally similar (both roller coasters); however, these problems are structurally dissonant (one involves conservative forces and the other non-conservative forces). On the other hand, Figure 2.2b displays a pair of problems that are situationally dissimilar but structurally the same. Students tend to associate problems that are situationally similar (Figure 2.2a) and dissociate those that are situationally dissimilar (Figure 2.2b), even though they are the same kind of problem.
32 • Problem-Specific Design Models
Figure2.2(a) Situationally similar, structurally dissimilar problems; (b) situationally dissimilar, structurally similar problems. With permission of Fran Mateycik.
Researchers have developed environments that visually depict the structural relations among problem elements. Marshall argued that in order to learn how to assign sets to an appropriate structural model of the problem, students need methods for structurally representing the problem. Cummins (1991) found that when children drew or selected pictures that represented the problems’ structure, solution performance improved, depending on the nature of the pictures drawn or chosen. A crucial determinant was the interpretation that chil- dren assigned to certain phrases. Rather than allowing students to select their own problem representations, Marshall (1995) provides a tool for explicitly mapping problem objects and values onto a struc- tural model of the different problem types to scaffold the assignment process.
Marshall’s story problem solver (SPS) interface (Figure 2.3) provides users with a small set of conceptually distinct structural models for displaying and solving arithmetic problems (options shown in upper left). These diagrammatic depictions illustrate the different structural relationships involved in different problem types, so each problem type has its own visual depiction. Marshall believed that these visual, structural models (a) represent fundamental relational concepts within a domain; (b) suggest the existence of different problem classes;
(c) suggest procedures associated with problem types; and (d) serve as conceptual building blocks for representing complex problems.
The goal of SPS is to help students construct expert math knowledge by having them solve and analyze math story problems using schematic diagrams representing basic structural relationships. In a series of studies, Marshall showed that using her visual diagrams of model types (see Figure 2.3) improved problem classification, recall, and problem performance. Students recalled the diagram and used it to structure recall of problem information. These visual diagrams influence con- ceptual development by functioning as an anchor for the students’
models. When used to represent problem structures, these diagrams should reflect the essential components of the problem structure as simply and uniquely as possible (Marshall, 1995).
A number of computer-based tools for representing story pro- blems have also been tested. ANIMATE is a computer-based tutor that coordinates the situational rather than the structural characteristics of the problem with the equation for solving mathematics problems (Nathan, Kintsch, & Young, 1992). Based on a discourse analysis theory of story problems (Kintsch & Greeno, 1985), ANIMATE runs a simple animation of the problem situation (top of Figure 2.4) that is mapped to a solution-enabling structured equation (bottom right of Figure 2.4).
34 • Problem-Specific Design Models
Figure2.3Problem schema representation in SPS (Marshall, 1995). Reprinted with permission.
Rather than ignoring situational content, this software intentionally integrates animated situational content as exemplars and feedback.
The software was implemented for amount-per-time rate problems (Mayer, 1982) using the basic formula D = R × T (distance D equals rate R multiplied by time T). Learners begin by creating a simulation of the problem by picking characters and selecting the appropriate equation to control their behavior in the animation (see Figure 2.4). Nathan et al. (1992) conducted empirical research that showed that students experiencing ANIMATE outperformed students supported by other environments in recognizing a correct solution, generating equations from texts, and diagnosing errors. Unfortunately, the error correction did not transfer when the environment was removed. The correspond- ence between the algebraic representation and the simulation was the primary reason for success. ANIMATE did not attempt to integrate the structural and situational characteristics of the problem.
Another method for representing story problems is provided by HERON (Reusser, 1993), a computer-based tool that uses solution trees to conceptualize the structure of mathematical problem solving. Like these other tools, the tree model in HERON is designed to directly mediate the translation of the problem text into an equation without the use of a structural or situational model of the problem. HERON uses a graphical solution (conceptual planning) tree to represent the
Figure 2.4 Representation of animation and problem structure in ANIMATE (Nathan, Kintsch, & Young, 1992).
36 • Problem-Specific Design Models
operation required to solve the problem. Figure 2.5 illustrates a solution tree for representing the following problem:
Little Simon and his father are watering their vegetable garden.
The father has a 15-liter watering can. Simon’s can holds one-fifth of that. Both fill their cans twelve times. After that, there are still 24 liters in the rain barrel. How much water can the rain barrel hold?
(Reusser, 1993) The solution trees, Reusser (1993) argues, are manipulable, dynamic, and flexible means for illustrating the construction process. The objects in HERON solution trees are excellent examples of set schemas arranged to represent the process for solving the problem. They also convey some information about the structural relationships or situ- ation depicted in the problem.
Tutorials in Problem Solving (TiPS) is a more recent conceptually oriented computer environment for training arithmetic and problem- solving skills in remedial adult populations (Derry and the TiPS Research Group, 2001). Students learn to solve change, compare, group, function, and vary problems (Marshall, 1995). TiPS uses worked examples (to be described more fully later in this chapter and in Chapter 9) to present the problem statement and to demonstrate a
Figure 2.5 Solution trees from HERON (Reusser, 1993). Reprinted with permission.
Story Problems • 37
procedure for solving it. Worked examples illustrate how an expert solves the problem for the learner to study and emulate. In TiPS, stu- dents use the interface shown in Figure 2.6 to solve the story problems.
They read the problem to identify the sets and set relationships in the problem statement. Doing this helps them classify the type of problem and to select the problem diagram that best depicts that problem type (in Figure 2.5, a restate problem). Students drag the diagram onto the screen and fill it in by dragging and dropping words from the problem statement into appropriate cells in the diagram. When this structural mapping process is complete, students construct an arithmetic formula to calculate the solution.
Although TiPS uses a Bayesian student model to adapt instruction and feedback, its more important feature is the problem representation formalism that it uses. Derry and the TiPS Research Group (2001) conducted a pair of experiments on the interface. In the first experi- ment, they compared the TiPS interface with a hierarchical solution tree similar to those in HERON (described previously). Students using the TiPS interface performed better than those using solution trees,
Figure 2.6 TiPS interface for representing and solving simple mathematics problems.
Derry and the TiPS Research Group (2001). With permission of Sharon Derry, University of Wisconsin.
38 • Problem-Specific Design Models
especially among lower-ability students. In the second experiment, some remedial adult learners used the TiPS schema interface while others used an interface design based on a heuristic problem-solving model that guided student practice. Students using the TiPS schema interface made greater gains over time. It appears that among adult remedial learners, the TiPS interface provided a performance advantage.
Contemporary research on story problems confirms the importance of constructing a conceptual model of the problem prior to solving it. The research by Marshall (1995) and Derry and the TiPS Research Group (2001) focused on using structural models of problem types to help learners classify and parse story problems. Nathan (1998) focused on the situational model of the problems, and Reusser (1993) concen- trated on the solution process. The proposed model for solving story problems being described in this chapter is unique in its integration of structural and situational models of the problem with the processing operations. Because the most successful story problem solvers are those who can integrate the situational and structural characteristics of the story problems, it is essential to support the association of these elements.
How Are Worked Examples Used for Story Problems?
Research by Sweller and his colleagues have investigated a method for teaching story problem solving that emphasizes the use of worked examples. Worked examples of problem solutions that precede practice facilitate practice-based problem solving by helping learners to con- struct problem schemas (Cooper & Sweller, 1987; Sweller & Cooper, 1985). Unfortunately, the problem schemas that students construct often include only procedural models of how to solve the problem but not what the problem means, because worked examples generally emphasize only processing steps (see Chapter 9 for more detail on worked examples). When using worked examples, however, Atkinson, Derry, Renkl, and Wortham (2000) verified a number of instructional design principles such as providing multiple examples per problem type in multiple forms that feature structural components. Examples should integrate parts, use multiple modalities, and clarify the structure of subgoals. This process is afforded and supported by the story- problem learning environment described next. The worked examples of problems must emphasize each of the processes for parsing the verbal problem representation, categorizing the problem type using the conceptual model, mapping problem sets onto the situational model and the structural model, mapping the structural model onto a formula, estimating the size and units of the outcome, solving for the Story Problems • 39