Analysis Of Precurrent Skills In Solving Mathematics Story Problems

21
JOURNAL OF APPLIED BEHAVIOR ANALYSIS 2003, 36, 21–33 NUMBER 1 (SPRING 2003)
ANALYSIS OF PRECURRENT SKILLS IN
SOLVING MATHEMATICS STORY PROBLEMS
NANCY A. NEEF
THE OHIO STATE UNIVERSITY
DIANE E. NELLES
OAKLAND SCHOOL DISTRICT, OAKLAND, MI
BRIAN A. IWATA
THE UNIVERSITY OF FLORIDA
AND
TERRY J. PAGE
BANCROFT, INC., HADDONFIELD, NJ
We conducted an analysis of precurrent skills (responses that increase the effectiveness of
a subsequent or ‘‘current’’ behavior in obtaining a reinforcer) to facilitate the solution of
arithmetic word (story) problems. Two students with developmental disabilities were
taught four precurrent responses (identifying the initial value, change value, operation,
and resulting value) in a sequential manner. Results of a multiple baseline design across
behaviors showed that the teaching procedures were effective in increasing correct per-
formance of each of the precurrent behaviors with untaught problems during probes and
that once the precurrent behaviors were established, the number of correct problem
solutions increased.
DESCRIPTORS: precurrent behavior, problem solving, mathematics, story prob-
lems, developmental disabilities
The principal goal of mathematics in-
struction is to teach students to solve prac-
tical problems (National Council of Teachers
of Mathematics, 1989, 2000). Toward that
end, teachers often present tasks to students
in the form of word (story) problems to il-
lustrate a wide range of everyday activities
(e.g., cooking, shopping, budgeting, time
management) that require competence in
basic mathematics skills. Nevertheless, large
numbers of students of all ages fail to dem-
onstrate grade-level proficiency in solving
Address correspondence and requests for reprints to
Nancy A. Neef, PAES, College of Education, The
Ohio State University, 1945 N. High St., Columbus,
Ohio 43210 (e-mail: neef.2@osu.edu) or to Brian A.
Iwata, Department of Psychology, University of Florida,
Gainesville, Florida 32611 (e-mail: iwata@ufl.edu).
story problems (Cawley, Parmar, Foley,
Salmon, & Roy, 2001; National Assessment
of Educational Progress, 1992). Students
with disabilities have performed particularly
poorly on these tasks, which are important
to achieving goals of successful employment,
independent living, and successful integra-
tion into school and community settings
(Butler, Miller, Lee, & Pierce, 2001).
Solving story problems is often difficult
because it requires both reading comprehen-
sion and mathematics skills as well as the
ability to transform words and numbers into
the appropriate operations. Research sug-
gests that difficulties in discriminating the
correct operation, the order of operation
(when placement of the unknown within the
problem differs), and extraneous informa-

22 NANCY A. NEEF et al.
tion, as well as problems with computational
speed, are factors commonly associated with
poor performance in solving story problems
(Zentall & Ferkis, 1993). Most behavioral
research on mathematics instruction for stu-
dents with disabilities has emphasized basic
foundation skills (e.g., counting, number
recognition, matching number words and
numerals, concepts of quantity) and math-
ematics facts (accuracy and fluency in com-
putation) that are necessary but insufficient
for higher level problem solving (e.g., Harp-
er, Mallette, Maheady, Bentley, & Moore,
1995; Horton, Lovitt, & White, 1992; Lin,
Podell, & Tournaki-Reid, 1994; Mattingly
& Bott, 1990; Miller, Hall, & Heward,
1995; Whalen, Schuster, & Hemmeter,
1996; Wood, Frank, & Wacker, 1998;
Young, Baker, & Martin, 1990). By contrast,
relatively few experimental investigations
have examined procedures for teaching sto-
ry-problem solving, especially to students
with moderate to severe disabilities.
A common instructional strategy involves
the use of manipulatives, in which a student
uses objects (e.g., blocks) to represent parts
of the problem-solving equation and then
adopts a ‘‘counting all’’ strategy for addition
problems and a ‘‘separating from’’ or
‘‘matching’’ strategy for subtraction prob-
lems, depending on whether the operation
involves a change, comparison, or combi-
nation. Although manipulatives have been
shown to enhance performance on story-
problem tasks (e.g., Marsh & Cooke, 1996),
they are limited aids because they do not
ensure discrimination of the appropriate
strategy for solving a particular type of prob-
lem, and are not readily applied to problems
involving either large quantities or abstract
concepts such as distance or time.
Several investigators have used cue cards
or checklists outlining problem-solving steps
to teach students with mild disabilities to
solve addition and subtraction story prob-
lems. For example, Cassel and Reid (1996)
taught students to use a cue card with the
mnemonics FAST DRAW (corresponding to
the steps of finding and highlighting the
question, asking what the parts of the prob-
lem are and what information is given, set-
ting up the equation, tying down the sign of
using addition or subtraction, discovering
the sign, reading the number problem, an-
swering the number problem, and writing
the answer), which were then translated into
a checklist, to solve story problems. Results
showed that students’ ability to generate cor-
rect equations for story problems improved
following training. The effects of the pro-
cedure on correct solutions, however, were
not clearly demonstrated, and prompting
use of a strategy does not ensure discrimi-
nation of the relevant concepts and opera-
tions necessary to solve a problem.
Another approach involves teaching stu-
dents to translate problems into equations
whose components are inserted into parts of
a diagram. Diagrams have been used to sup-
plement explicit teaching of steps of a prob-
lem-solving chain in the context of comput-
er programs (Jaspers & van Lieshout, 1994)
or direct instruction (e.g., Jitendra & Hoff,
1996). Jaspers and van Lieshout, for exam-
ple, compared text analysis, external model-
ing, a combination of the two procedures,
and a control (practice) procedure on the
ability of students classified as educable
mentally retarded to solve story problems.
Text analysis involved a four-step procedure
consisting of reading the problem, determin-
ing the question being asked, locating the
relevant sets, and producing the numerical
answer. External modeling involved a five-
step procedure consisting of reading the
problem, representing the first set with
squares, locating the second set on the com-
puter screen, representing the second set
with squares, and locating the answer set on
the screen. Results showed that students in
the external-modeling group performed bet-
ter than those in the other three groups

23
PRECURRENT PROBLEM SOLVING
when they were allowed to use diagrams to
represent the word problems, but that the
text-analysis group performed better on pa-
per-and-pencil tests. Using a similar ap-
proach, Jitendra and Hoff taught students
via direct instruction to use diagrams to
transform story problems into equation
form. This procedure was effective in teach-
ing 3 students with learning disabilities to
solve simple addition and subtraction story
problems when they were provided with a
cue card stating the steps to solve the prob-
lems.
From a behavior-analytic perspective,
problem solving involves a precurrent oper-
ant contingency, which functions ‘‘mainly to
make subsequent behavior more effective,’’
whereas a current operant may be character-
ized as the solution (‘‘effective behavior . . .
a response which is likely to be reinforced’’)
that is made more probable by the precur-
rent contingencies (Skinner, 1968, pp. 120,
124). Precurrent behaviors may need to be
taught, but once they are established, they
can be maintained by their ultimate effect
on the current behavior that directly pro-
duces the reinforcer (Skinner, 1968). Parsons
(1976), for example, examined the effects of
precurrent behaviors on the solution of
quantity-matching problems (circling a sub-
set of comparison stimuli equal to the num-
ber of symbols comprising the sample) by 5
preschool children. The precurrent behaviors
consisted of overtly counting and marking
the symbols as they were vocally enumerat-
ed. Results showed that before precurrent
behaviors were taught and when they were
prohibited, continuous differential reinforce-
ment of the current behavior (terminal so-
lutions) failed to produce accurate problem
solving. Once established and when allowed
to occur, precurrent behaviors resulted in ac-
curate current behavior (problem solution),
and both precurrent and current behaviors
were maintained when reinforcement was
contingent only on the latter.
Analysis of precurrent behaviors has im-
plications for students who have difficulty
with story problems. First, as illustrated by
Parsons (1976), effective precurrent behav-
iors may not develop when instruction fo-
cuses only on the completion of a complex
sequence of responses. As Skinner (1968)
pointed out, differential reinforcement of so-
lutions alone without access to the variables
that control them ‘‘does not teach; it simply
selects those who learn without being
taught’’ (pp. 118–119). Second, instruction
on aspects of a problem might occasion be-
haviors that are irrelevant to, or that fail to
control, problem solution. Thus, cognitive
strategies such as those that prompt ‘‘self-
reflection’’ in the form of ‘‘What did you
notice about how you did your work?’’ (Na-
glieri & Gottling, 1995) or instructions to
‘‘read, paraphrase, visualize, hypothesize, es-
timate, compute, and check’’ (Montague,
1992) may not be effective with students
who lack precurrent skills that relate directly
to problem solution. Third, mastery of com-
ponent precurrent behaviors, albeit neces-
sary, is not always sufficient for problem
solving (Mayfield & Chase, 2002). In ad-
dition to learning how to apply a mathe-
matical operation such as addition and sub-
traction, for example, students must also dis-
criminate when to apply it to different prob-
lems. Finally, precurrent responses that are
covert may be too weak to occasion the so-
lution behavior (Skinner, 1957) and, because
they are unobservable, are not available for
correction if faulty. It is not surprising,
therefore, that teacher knowledge of individ-
ual problem-solving skills was found to be a
strong predictor of math achievement, and
that teachers who lacked such information
were more likely to rely simply on checking
and monitoring student work (Peterson,
Carpenter, & Fennema, 1989).
In the present study, we examined the ef-
fects of teaching overt precurrent behaviors
(identifying the initial value, change value,

operation, and ending value) on the current
operant of solving mathematics story prob-
lems with 2 students with developmental
disabilities. The precurrent behaviors were
designed to facilitate problem solving when
the components were arranged in varying se-
quences within the story problems, when the
operation consisted of either addition or
subtraction, and when the unknown value
was in different positions in the resulting
equations.
METHOD
Participants and Setting
Two individuals enrolled in an education-
al program for students with developmental
disabilities participated. Saul and Lucien
were 19 and 23 years old and had IQs of
46 and 72, respectively, as measured by the
Wechsler Adult Intelligence Scale. Prerequi-
site skills for inclusion in the study consisted
of the ability (a) to compute addition and
subtraction numeric equations in which the
sums were 10 or less and with the unknown
in any position (e.g., 2 1 6 5 ?, ? 2 4 5
3), and (b) to read the vocabulary words
used in the word problems (second-grade
level). These skills were assessed informally
(math achievement scores were unavailable).
All training and testing sessions were con-
ducted individually in a classroom.
Stimulus Materials
The general equations A 1 B 5 C or A 2
B 5 C were used to generate word problems
indicating whether an individual gave away
or received objects. One of the variables (A,
B, or C) was unknown, and the task was to
solve for the unknown. All problems repre-
sented variations of six specific equations
generated by changing the operation to be
performed and the location of the unknown
variable. Each of the equations also was pre-
sented in two different sequences, yielding a
total of 12 problem sequences. The rationale
for using these different sequences was based
on the assumption that generative skill ac-
quisition would be enhanced through ex-
posure to a variety of problem formats. A
list of the equations and word sequences is
presented in Table 1.
Testing and training materials consisted of
a series of worksheets, each containing five
different word problems drawn from the 12
basic sequences. The problems involved ad-
dition and subtraction in which the sums
were 10 or less. Words used in the problems
were drawn from 10 proper names, 20 verbs,
and 20 object nouns. These problem se-
quences, numeric values, and words were
combined in different ways to formulate a
sufficiently large pool of problems such that
neither student was exposed to the same
problem more than once during the study.
An example of a typical worksheet is shown
in Figure 1 (formulas are shown for illustra-
tive purposes and were not included on the
worksheet).
Each problem was divided into five com-
ponent parts. The initial set (1) was deter-
mined by words indicating the number of
objects in the person’s possession at the out-
set. Verbs stating which objects were either
added or subtracted specified the change set
(2), and these same verbs indicated the op-
eration (3) to be performed. A phrase de-
noting the final number of objects in the
person’s possession was the resulting set (4).
One set (initial, change, or resulting) was
unknown and provided the question of the
problem. The answer to that question
formed the solution (5). Given the word
problem, ‘‘If Sam began with 7 pens and
ended up with 5 pens, how many did he
give away?’’ as an example, the various com-
ponents were as follows:
Initial set: Sam began with 7 pens.
Change set: How many did he give away?
Operation: Give away (subtraction).
Resulting set: Ended up with 5 pens.

25
Table 1
Equations and Sequences Used to Generate Word Problems
Equation Sample word sequence
A 1 B 5 ? 1. If (name) started out with A objects and was given B objects, how many did
he or she end up with? (A 1 B 5 ?)
2. How many objects did (name) end up with if he or she started out with A
objects and was given B objects? (? 5 A 1 B)
A 2 B 5 ? 3. If (name) started out with A objects and gave away B objects, how many did
he or she end up with? (A 2 B 5 ?)
4. How many objects did (name) end up with if he or she started out with A
objects and gave away B objects? (? 5 A 2 B)
A 1 ? 5 C 5. If (name) started out with A objects and ended up with C objects, how many
was he or she given? (A 1 ? 5 C)
6. How many objects was (name) given if he or she started out with A objects and
ended up with C objects? (C 5 A 1 ?)
A 2 ? 5 C 7. If (name) started out with A objects and ended up with C objects, how many
did he or she give away? (A 2 ? 5 C)
8. How many objects did (name) give away if he or she started out with A objects
and ended up with C objects? (C 5 A 2 ?)
? 1 B 5 C 9. If (name) was given B objects and ended up with C objects, how many objects
did he or she start out with? (? 1 B 5 C)
10. How many objects did (name) start out with if he or she was given B objects
and ended up with C objects? (C 5 ? 1 B)
? 2 B 5 C 11. If (name) gave away B objects and ended up with C objects, how many objects
did he or she start out with? (? 2 B 5 C)
12. How many objects did (name) start out with if he or she gave away B objects
and ended up with C objects? (C 5 ? 2 B)
Solution (unknown): How many did he
give away?
Experimental Design
A multiple baseline across behaviors de-
sign (Baer, Wolf, & Risley, 1968) was used
to evaluate the effects of training on stu-
dents’ problem-solving skills. Following the
collection of baseline probe data on each of
the five component skills (initial set, change
set, operation, resulting set, solution), train-
ing was begun on the first component and
proceeded as necessary across the other four.
Procedure
Probe sessions. Probes were administered
during baseline and following the comple-
tion of training on each of the separate pre-
current problem-solving components. A
probe consisted of 10 word problems (two
worksheets). The student was requested to
‘‘Read each problem aloud. Find the answers
and put your work here [as the trainer point-
ed to the boxes and circles].’’ Correct re-
sponses were defined as having the correct
numbers in the appropriate boxes for known
quantities, an X above the appropriate box
for the unknown, the correct operation sym-
bol in the circle, and the correct solution in
the box for the unknown. Other responses
(or no responses) were considered incorrect.
Students were allowed 20 min to complete
each probe, during which intermittent praise
was delivered for on-task behavior indepen-
dent of the occurrence of precurrent or cur-
rent behaviors.
Training. Following the completion of
baseline probes, students were taught to
identify the five components of a word prob-
lem in the sequence described previously.
The prompts and definitions of correct and
incorrect responses for each of the five com-
ponents are described in Table 2. Training
on each component was preceded by a dem-

Figure 1. Example of a typical worksheet. Formulas (in bold) are included only for illustrative purposes
and were not part of the worksheet.
onstration with five practice problems. The
trainer read each problem aloud, presented
the verbal prompt (e.g., ‘‘How many prob-
lems did [name] start out with?’’), and de-
scribed and modeled the correct response.
Training trials then began. Each trial con-
sisted of one complete word problem, and
10 trials (excluding remedial trials) were pre-
sented each session. One to two sessions
were conducted per day. Correct responses
were followed by praise. When an error oc-
curred on either the component being
trained or any previously trained compo-
nent, the trainer modeled the correct re-
sponse directly on the worksheet using a dif-
ferent-colored pen. A remedial trial using a
different problem of the same type was then
presented. This sequence was repeated for

27
Table 2
Prompts, Correct Responses, and Incorrect Responses for the Five Problem Components
Initial set
Prompt
Correct
Incorrect
‘‘How many objects did (name) start out with?’’
Appropriate words underlined; number in first box if known or X over box if unknown
Incorrect underline, number, or X; no response in 10 s
Change set
Prompt
Correct
Incorrect
‘‘What happened next?’’
Appropriate words underlined; number in second box if known or X over box if unknown
Operation
Prompt
Correct
Incorrect
‘‘Was that number added or subtracted from the first number?’’
Finger placed under words indicating the operation; correct symbol in circle
Incorrect pointing or symbol; no response in 10 s
Resulting set
Prompt
Correct
Incorrect
‘‘How many objects did (name) end up with?’’
Appropriate words underlined; number in third box if known or X over box if unknown
Solution
Prompt
Correct
Incorrect
A question pertaining to the unknown, as in ‘‘How many objects did (name)
(start out with, end up with, get, lose, etc.)?’’
Correct answer placed in box with the unknown (indicated by X)
Incorrect answer; no response in 10 s
subsequent errors on remedial trials. A cor-
rect response on a remedial trial produced
the next training trial.
During instruction on the initial, change,
and resulting sets, three training phases typ-
ically were conducted unless the participant
displayed consistent correct responding be-
fore the planned training phase was initiat-
ed. First, problems were presented in which
the component being trained was always a
known quantity, and responses were preced-
ed by a specific trainer prompt. Next, prob-
lems were presented in which the compo-
nent being trained was randomly unknown
(i.e., in approximately half of the problems,
that component constituted the solution).
Finally, problems were presented in which
the component being trained was randomly
unknown, and no prompt was provided.
During training of the operation, only two
phases were used—prompted and un-
prompted—because one of the components
always was unknown. The criteria for ad-
vancing from one training phase to the next
were 100% correct responses on all trained
components (including previously trained
components) for one session under prompt-
ed conditions and 100% correct responses
for two consecutive sessions under un-
prompted conditions. When a student met
the unprompted training criterion for a giv-
en component, a probe session was con-
ducted and training was begun on the next
component.
Interobserver Agreement
A second observer independently scored
students’ written responses for both training
and probe sessions at least once during base-
line and each training phase. Interobserver
agreement was calculated by dividing the
number of agreements on each component
part of a problem by the agreements plus
disagreements and multiplying by 100%.
Mean agreement scores were 99.9% and
99.5% for probe and training sessions, re-
spectively.

Figure 2. The number of correct responses across training phases for each precurrent component.
RESULTS
Figure 2 shows the training data for each
student, expressed as the number of correct
responses during training phases for each
precurrent component. Both students re-
quired fewer sessions to reach criterion as
training progressed across components. Saul
reached mastery criterion in 26 sessions for
the initial set, in 35 sessions for the change
set, in 17 sessions for the operation, and in
two sessions for the resulting set. Lucien met
criterion in 28 sessions for the initial set, in
16 sessions for the change set, and in seven
sessions for the operation.
Figure 3 shows probe results for both stu-
dents across baseline and training conditions

29
Figure 3. The number of correct responses during mathematics probes across baseline and posttraining
conditions.
for each of the five problem-solving com-
ponents. Data are expressed as number of
correct responses. The results show (a) an
increase in the number of correct responses
on probes following instruction on the re-
spective precurrent (problem-solving) com-
ponent and (b) an increase in the number
of correct solutions (current operants) after
the precurrent responses for all components
had been established. Saul’s mean numbers
of correct precurrent responses (of 10 pos-
sible) during baseline and posttraining

probes, respectively, were 4.5 and 9.6 for the
initial set, 2.2 and 9.5 for the change set,
2.0 and 9.0 for the operation, and 0.2 and
9.0 for the resulting set. His correct solu-
tions increased from a mean of 1.2 to 8.0
after the precurrent responses in all compo-
nents had been established. Lucien’s mean
numbers of correct precurrent responses dur-
ing baseline and posttraining probes, respec-
tively, were 5.8 and 9.3 for the initial set,
3.9 and 9.5 for the change set, and 5.8 and
10.0 for the operation. After the precurrent
responses in those three components had
been established, generalization occurred to
the resulting set without further training;
correct responses increased from a mean of
1.4 to 10. His correct responses on the un-
trained solution set increased from a mean
of 3.1 to 10 after the component precurrent
behaviors had been established.
DISCUSSION
Results of this study showed that teaching
students with developmental disabilities a set
of precurrent behaviors consisting of the
identification of each of the component
parts of a story problem resulted in the gen-
erative solution of problems. As such, the
study represents one of the few experimental
investigations of a systematic procedure to
teach generative story problem-solving skills
to students with developmental disabilities,
addressing a principal goal of mathematics
instruction.
As discussed by Skinner (1953, 1966,
1968, 1969), problem solving involves an
interresponse relation in which the emission
of precurrent operants makes the solution
(current operant) more probable. In the
present case, the number of correct solutions
increased for both students after the succes-
sion of precurrent responses had been estab-
lished for each component. The precurrent
contingencies can be interpreted as a re-
sponse chain, but in situations in which it is
possible for the current behavior to occur
and be reinforced without the emission of
precurrents, they can nevertheless function
to alter the probability of reinforcement
(Polson & Parsons, 1994).
Although contingent praise was used dur-
ing training sessions to promote the acqui-
sition of the precurrent behaviors, it is pos-
sible that reinforcement derived from prob-
lem solution may have served to maintain
both the precurrent and current responses
during probes (when contingent praise was
not used). In other words, the precurrent be-
haviors may have been indirectly reinforced
by the solution they prompted. This is con-
sistent with the results of investigations by
Parsons (1976) and Parsons, Taylor, and
Joyce (1981), in which precurrent collateral
behavior was maintained when reinforce-
ment was made contingent on a subsequent
current operant. Alternatively, the precurrent
and current responses may have been main-
tained during the probe session because
praise was intermittently delivered for on-
task behavior (rather than for the precurrent
or current responses) and the participants
may not have discriminated the different
contingencies in effect between the training
and probe sessions.
The acquisition of effective precurrent be-
haviors may have been facilitated by the pro-
gressive training sequence that required
demonstration of the target as well as all pre-
viously trained components. Mayfield and
Chase (2002), for example, found that the
rate and accuracy of algebra problem solving
and novel application were higher for college
students who received instruction that in-
corporated cumulative practice on a mix of
previously trained component skills (rules)
than for students who received instruction
that incorporated simple practice on one
component at a time or one session of extra
practice. The authors argued that the jux-
taposition of different kinds of problems in
the cumulative review enabled learning of

31
multiple discriminations for rule application
necessary for problem solving with novel
rule combinations.
Nevertheless, it is possible that training in
the present study may have required less
time without the known prompted and un-
known prompted phases. The relatively large
number of sessions needed to reach criterion
during the random unprompted phase sug-
gests that performance may not have been
facilitated by the two preceding training
phases, but the design did not allow that
determination. Similarly, it may be that
teaching all components simultaneously
rather than sequentially would be more ef-
ficient.
Our results are limited in several addi-
tional respects. First, because of time con-
straints, we were unable to examine the ex-
tent to which the effects were maintained
over an extended period of time. Second, a
more complete analysis of the function of
the behaviors as precurrents would require
a demonstration that correct solutions de-
crease when the precurrent behaviors were
prevented, or that those behaviors were not
maintained when they were not followed by
a correct solution. Parsons (1976) and Par-
sons et al. (1981), for example, showed dec-
rements in terminal symbol counting and
matching-to-sample behaviors, respectively,
when collateral precurrent behaviors were
prohibited. It would be difficult, however,
to use such preparations in the context of
the present study. For example, unlike in
the Parsons et al. investigation, the precur-
rent behaviors could not be prevented be-
cause, once learned, they may become co-
vert (Skinner, 1976); indeed, that would be
a desirable outcome. In addition, to teach
behaviors that are ineffective in producing
the current operant seemed counterproduc-
tive to the purpose of our applied investi-
gation.
Our purpose was to extend the concept
of precurrent operants examined in basic re-
search (Parsons, 1976; Parsons et al., 1981;
Polson & Parsons, 1994; Torgrud & Hol-
born, 1989) to our own problem-solving ef-
forts in developing effective procedures for
teaching an educationally significant skill
that has proven to be difficult for students
with developmental disabilities. The proce-
dure was efficient in that the identification
of the basic elements and operations in-
volved in a problem, regardless of their po-
sition or sequence, enabled solution of all
types of problems within that class. Thus,
the procedure enabled students to respond
to problems that require a novel synthesis of
responses in the presence of a novel stimu-
lus, consistent with Becker, Engelmann, and
Thomas’ (1975) definition of problem solv-
ing.
Our results were limited to simple addi-
tion and subtraction story problems and did
not encompass all the categories and sub-
types of problems represented in various tax-
onomies (e.g., Riley & Greeno, 1988).
However, the procedures could easily be ap-
plied to other types of problems and oper-
ations. Efficiency might be improved by ap-
plying a general case strategy that systemat-
ically samples the range of stimulus and re-
sponse variations in the universe of possible
problems (Endo, 2001).
Because story problem-solving skills have
been infrequently addressed in behavioral re-
search, some have suggested that they are be-
yond the reach of a behavioral analysis. But-
ler et al. (2001), for example, recently char-
acterized the current problem-solving stan-
dards for mathematics curricula as
demonstrating ‘‘a shift from a behaviorist ap-
proach of teaching rote learning of facts and
procedures to a constructivist approach’’ (p.
20). Extensions of the current approach in
subsequent research may make more prom-
inent the potential contributions of behavior
analysis to the acquisition of complex prob-
lem-solving skills of all types.

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Received September 12, 2001
Final acceptance November 4, 2002
Action Editor, Iser DeLeon
STUDY QUESTIONS
1. What are the differences between precurrent- and current-operant behaviors? Provide an
original example that illustrates the interaction of these behaviors.
2. Explain the importance of a precurrent analysis to the understanding and improvement of
problem-solving behavior.
3. Describe the five components into which word problems were divided.
4. How were correct responses defined?
5. What independent variables were included in the training procedures used to establish the
precurrent responses?
6. Summarize the results of the study with respect to acquisition of (a) precurrent and (b)
current responses.
7. How were the results of the current study consistent with Skinner’s conceptualization of
problem solving?
8. What useful information might be gained by including a condition in which participants
received reinforcement following correct solutions in the absence of any precurrent training?
Questions prepared by Stephen North and Jessica Thomason, The University of Florida
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