ERIC EJ853187: The Effects of Teaching Precurrent Behaviors on Children's Solution of Multiplication and Division Word Problems

2009·0.19 MB·English

by  ERIC

JOURNALOFAPPLIEDBEHAVIORANALYSIS 2009, 42, 361–367 NUMBER2 (SUMMER2009)
THE EFFECTS OF TEACHING PRECURRENT BEHAVIORS ON
CHILDREN’S SOLUTION OF MULTIPLICATION AND DIVISION
WORD PROBLEMS
HEATHER B. LEVINGSTON AND NANCY A. NEEF
THEOHIOSTATEUNIVERSITY
AND
TRACI M. CIHON
THECHICAGOSCHOOLOFPROFESSIONALPSYCHOLOGY
Weexaminedtheeffectsofteachingovertprecurrentbehaviorsonthecurrentoperantofsolving
multiplicationanddivisionwordproblems.Twostudentsweretaughtfourprecurrentbehaviors
(identificationoflabel,operation,largernumbers,andsmallernumbers)inadifferentorder,in
thecontextofamultiplebaselinedesign.Aftermeetingcriteriononthreeofthefourprecurrent
skills,thestudents demonstrated thecurrent operantof correctproblem solutions. Theseskills
generalized to novel problems. Correct current operant responses (solutions that matched
answers revealed by coloring over the space with a special marker) maintained the precurrent
behaviors inthe absence ofany other programmed reinforcement.
DESCRIPTORS: mathematics,precurrent behaviors, problem solving, word problems
_______________________________________________________________________________
The National Council of Teachers of tional Progress, 1992) indicate a need for
Mathematics (1989, 2000, 2005) has for many methods of instruction that can enhance
years recommended an increased focus on performance of all students, particularly given
problem-solving tasks, typically taught through the emphasis on educating students with
theuseofwordproblems.TheNationalCenter disabilities in inclusive classrooms (Owen &
for Educational Statistics reported, however, Fuchs, 2002). Nevertheless, relatively little
that only 36% of fourth graders in the United research has been reported on procedures for
States were performing at or above the teaching word-problem solving compared to
proficient level in identifying and appropriately teaching basic foundation and computation
using information to solve word problems skills, and most of the research has targeted
(Perie, Grigg, & Dion, 2005). Students with students with learning disabilities (see Xin &
developmentaldisabilities(e.g.,autism)lageven Jitendra, 1999).
further behind (Cawley, Parmar, Foley, Salm- From an operant perspective, the behaviors
on, & Roy, 2001; Parmar, Cawley, & Frazita, necessary to solve mathematics word problems
1996). The difficulties with word problem can be classified as precurrents. Precurrent
solving demonstrated by students across ability behaviors are those that increase the effective-
and age levels (National Assessment of Educa- ness of a subsequent (current) behavior in
obtainingareinforcer(Polson&Parsons,1994;
Skinner, 1968). Once established, precurrent
This study was part of a master’s thesis conducted by
thefirstauthor.WethankRonDeMuesy,AmandaGuld, behaviors can be reinforced and maintained by
and Kathleen Heron for their assistance with assessments the correct performance of the current behavior
ofinterobserver agreement andprocedural integrity.
(Parsons, 1976; Skinner). For example, the
Address correspondence to Nancy A. Neef, School of
PAES, College of Education and Human Ecology, The current behavior of getting to a particular
OhioStateUniversity,PAESBldg.,17thAve.,Columbus, destination can reinforce and maintain the
Ohio43210(e-mail: [email protected]).
precurrent behavior of reading a map. Neef,
doi:10.1901/jaba.2009.42-361
361
362 HEATHER B. LEVINGSTON et al.
Nelles, Iwata, and Page (2003) taught four old typically developing girl who was enrolled
precurrent behaviors for addition and subtrac- in a regular education summer school program.
tion word problems (identifying the initial During a preexperimental skills assessment,
value, change value, operation, and resulting both participants (a) calculated correct answers
value) in a sequential manner to 2 young adults for at least 80% of multiplication problems
with developmental disabilities. Once the pre- from 1 3 10 to 10 3 10, (b) calculated correct
current behaviors were established, the number answers for at least 80% of division problems
of correct problem solutions increased. These with a maximum dividend of 100, and (c)
behaviors generalized to untaught problems. discriminated larger and smaller numbers from
The current investigation replicated the Neef amongatleast80%ofpairsbetween1and100.
etal.(2003)studyandextendedtheresearchon Nevertheless, both participants demonstrated
instruction in mathematical problem solving in difficulty in solving word problems and were
several ways. First, we systematically replicated nominated by their parents and a classroom
theproceduresusedbyNeefetal.withyounger teacher for participation in the study. Teaching
(elementaryaged)students,1withautismand1 sessionswereconductedindividuallybyanother
without disabilities. Very few, if any, studies on teacher(firstauthor)atatableinthebackofthe
teaching mathematical problem solving have participants’ classrooms.
targeted students with autism, who often have
Stimulus Materials
intact computational skills but difficulty in
discriminating the type of operation or ap- A bank of 300 multiplication and division
proach to use to solve word problems (Min- word problems was created using the equations
shew,Goldstein,Taylor,&Siegel,1994;Siegel, A 3 B 5 C and A 4 B 5 C derived from a
Goldstein, & Minshew, 1996). Second, we review of third- and fourth-grade mathematics
assessed the relation between precurrent and curricula (e.g., ‘‘Nathan’s wall is 12 feet wide.
current behaviors with word problems that How many posters can he hang on his wall if
involvedmultiplicationanddivisionoperations. each poster is 2 feet wide?’’). Variations were
Almostalloftheinvestigationsofinstructionon created by substituting proper names, nouns,
word-problem solving with students with verbs, and numbers. Multiplication problems
disabilities have targeted addition and subtrac- had products of 100 or less, and division
tion problems. Third, we implemented a self- problems had quotients of 10 or less without
checking procedure to reduce the instructional remainders. Each worksheet had 10 problems
demands on the teacher and to determine the drawn randomly from the bank. Each problem
extent to which a matching answer (i.e., the was used only once during the study. Correct
current operant of correct problem solution) numbers for the word-problem equations on
was sufficient to maintain precurrent behaviors the training worksheets were written below the
in the absence of other sources of reinforce- corresponding spaces with a color-changing
ment. Finally, we assessed generalization to marker and were invisible until colored over
novelwordproblemsintheabsenceofspacesas with a developing marker.
stimulus prompts for the parts of the equation.
Experimental Design
A multiple baseline design across behaviors
METHOD
was used to examine theeffects of teaching four
Participants and Setting precurrent behaviors (identification of label,
Matt was a 10-year-old boy with a diagnosis operation, larger number, and smaller number)
of autism who was enrolled in an inclusive on the current behavior (correct solution) of
fourth-grade classroom. Maddix was a 10-year- word problems. Precurrent behaviors were
PRECURRENT BEHAVIORS AND WORD PROBLEMS 363
Table 1
Prompts andCorrect andIncorrect Responses for Prompted Teaching Trials
Identificationoflabel
Prompt ‘‘Whatdotheywantyoutofind?Howshouldyoulabelyouranswer?Writeyouransweronthe
line[whilepointingtotheline].’’
Correct Labelwrittenonthefinalline
Incorrect Incorrectword,incorrectplacementofword,noresponsewithin10s
Identificationofoperation
Prompt ‘‘Doyouneedtomultiplytofind[howmuchorhowmany]altogetherordoyouneedto
dividebybreakingthewholeintoparts?Writeyouranswerinthebox.’’
Correct Correctsymbolinbox
Incorrect Incorrectsymbol;incorrectplacementofsymbol;noresponsewithin10s
Identificationoflargernumber
Prompt ‘‘Whichnumberisthelargest?Writeyouranswerontheline.’’
Correct Largestnumberonfirstline
Incorrect Incorrectnumber;incorrectplacementofnumber;noresponsewithin10s
Identificationofsmallernumber
Prompt ‘‘Whichnumberisthesmallest?Writeyouranswerontheline.’’
Correct Smallestnumberonsecondline
Incorrect Incorrectnumber;incorrectplacementofnumber;noresponsewithin10s
trained in a different order for the 2 partici- except for the target precurrent behavior and
pants. Continued baseline measurement of the the current behavior. The solution was pre-
current operant permitted assessment of the written in invisible ink. The teacher read the
effects of sequential acquisition of each pre- problem aloud and delivered the corresponding
current behavior on the current operant. verbal prompt (Table 1). She then identified
and modeled the correct response. This con-
Teaching Procedure sisted of underlining and stating either the
Typically,onetotwo10-to15-minteaching specificwordsintheproblemthatindicatedthe
sessions were conducted 5 days per week solution’s label (identification of label), the
throughouttheintervention.Teachingincluded operation (identification of operation), or the
up to four phases, each corresponding to number (identification of larger number and
instruction in a precurrent behavior that had smaller number) depending on the specific
notyetbeenacquired.Instructionineachofthe precurrent behavior being taught. Next, the
four phases (identification of label, operation, teacher wrote the correct response in its
larger number, and smaller number, respective- corresponding space. She then solved the
ly) occurred as needed and followed the same resulting number sentence and entered the
procedure (described below). For each precur- current operant on the final answer space.
rent behavior, participants were taught to write Finally, the teacher modeled a self-checking
the response in the corresponding space of an procedure by coloring under the answer space
equation below the word problem. Prompts for with a developing marker that revealed the
eachprecurrentbehavior,alongwithdefinitions current operant (solution), and read the
of correctand incorrect responses, are described resulting number sentence aloud.
in Table 1. Following modeling, the teacher presented
At the beginning of each phase, the teacher the participant with a worksheet that contained
demonstrated the target precurrent behavior on 10 different word problems and corresponding
a worksheet with five practice problems. equations with blank spaces for the precurrent
Practice worksheets contained a word problem beingtaught,anypreviouslytaughtprecurrents,
and an equation with all elements filled in and the current operant (i.e., the solution). The
364 HEATHER B. LEVINGSTON et al.
participant was instructed to complete the plication or division) that would lead to the
equation with a written response in the solution, the larger number referenced in the
corresponding spaces for (a) the precurrent problem, the smaller number referenced in the
behavior being taught, (b) all previously taught problem, and the solution to the equation.
precurrent behaviors (e.g., identification of During baseline and intervention probes, par-
label, operation, larger number, or smaller ticipants wrote their answers below the word
number), and (c) the solution, and to then problem on lines and boxes that corresponded
checkthesolutionusingthedevelopingmarker. toeachofthefourprecurrentbehaviorsandthe
Teaching trials began with guided instruction current behavior. No consequences were deliv-
during which the teacher presented the prompt ered for correct or incorrect responses. After all
(Table 1) for the precurrent behavior being precurrent behaviors had met mastery criterion,
trained. Previously taught precurrent behaviors a generalization probe was conducted in the
were not prompted. After completion of the same manner, except that the worksheets did
self-check procedure, the teacher prompted not contain lines for numbers and labels or a
correction of any inaccurate responses, if box for the operation.
necessary, by underlining the appropriate part
of the word problem. Data Collection
When the participant achieved criterion of at Data were collected on the written responses
least 90% correct for the target precurrent during teaching trials and probes. Correct
behavior, the subsequent session used un- responses were defined, respectively, as having
prompted trials. The participant was given the thecorrectnumbersintheappropriateorderfor
worksheet to complete and check independent- larger number and smaller number, the correct
ly. When the participant met a criterion of at operation symbol separating the larger and
least 90% correct for two consecutive sessions smaller numbers, and the correctly labeled
with unpromptedtrials on the targetprecurrent solution following the equal sign. Labels were
behavior, a probe was conducted (described countedascorrectiftheycorrespondedwiththe
below). Teaching was then initiated on another noun represented in the math problem. Any
precurrent behavior (unless probe data indicat- other response (or no response) was scored as
ed an increasing trend, in which case probes incorrect.
continued until performance was stable). If the
participant did not meet criterion during a Interobserver Agreement
session with unprompted trials, the teacher Two graduate students in special education
returned to guided instruction with prompted independentlyscoredstudentresponseson31%
trials until the participant again met criterion. of the training and probe sessions across all
conditions for each participant. Point-by-point
Probes interobserver agreement was calculated by
Probes were conducted before training (base- dividing the number of agreements on each
line), each time the student met training component of a problem by the number of
criterion on a target precurrent behavior agreements plus disagreements and multiplying
(intervention),andafterallprecurrentbehaviors by 100%. Mean agreement scores were 98.7%
had been taught (generalization). During each or above (range, 80% to 100%) for both
20-min probe session, the teacher gave the trainingandprobesessionsforeachparticipant.
participant a worksheet that contained 10 word
problems. Completion of each word problem Procedural Integrity
consisted of writing the object of the word Procedural integrity was assessed on 36% of
problem (label), the correct operation (multi- alltrainingandprobesessionsbythreegraduate
PRECURRENT BEHAVIORS AND WORD PROBLEMS 365
Figure 1. The percentage of correct responses during mathematics probes across baseline, intervention, and
generalizationconditions for Matt (left)andMaddix (right).
students in special education who had been RESULTS AND DISCUSSION
trained to a criterion of at least 90% Figure 1 shows the percentage of correct
agreement in scoring a sample videotaped precurrent and current responses during base-
session using a procedural checklist of the line, intervention, and generalization probes
behaviors outlined in the experimental meth- (dataforteachingsessionsareavailablefromthe
ods. Procedural integrity was calculated by second author). For both participants, correct
dividing the total number of steps performed responses for label, operation, and larger
bytheteacherbythetotalnumberofstepsand number increased following training. Matt’s
multiplying by 100%. The mean score was meanpercentagesofcorrectprecurrentrespons-
99.3% (range, 93% to 100%). es during baseline and intervention, respective-
366 HEATHER B. LEVINGSTON et al.
ly,were13%and96%forlabel,52%and96% your work?’’; Naglieri & Gottling, 1995) that
for operation, and 53% and 96% for larger havecharacterizedotherapproaches.Second,by
number. Maddix’s mean percentages of correct teaching precurrent behaviors sequentially and
precurrent responses during baseline and inter- probing performance as each was mastered,
vention, respectively, were 60% and 90% for instruction was limited to those behaviors that
operation, 15% and 98% for larger number, did not emerge as a result of training previous
and 0% and 100% for label. Despite her low precurrents. Thus, once identification and
baseline for larger and smaller numbers, she placement of the larger number in the equation
often obtained a correct solution for multipli- wasmastered,studentswereabletodiscriminate
cation problems that, unlike division problems, the smaller number, obviating the need for
could be derived irrespective of the order of the training on that step, or on the solution. The
two sets of numbers in the equation. For both self-checkingprocedure(coloringoverthespace
participants, correct responding for the smaller with a disclosing marker to reveal the answers)
number emerged following training on the enabled students to receive immediate feedback
larger number and therefore was not trained. and reinforcement for correct responses while
Following training on the first three precur- they worked independently, thereby further
rents, correct solutions increased. Both partic- reducing demands on teacher time. Finally,
ipants maintained high percentages of correct instruction resulted in generative problem
responses on generalization probes (range, 90% solving (i.e., responding to novel stimuli in a
to 100%). way that enabled solution of multiple types of
The results replicate the findings of Neef et multiplication and division word problems).
al. (2003) in that systematic instruction on Efficiency might be further enhanced by
precurrent behaviors that represented the com- teaching the strategies in the context of group
ponent parts of a problem was an effective instruction.
method for establishing generative skills in the
Problem solving involves an interresponse
solution of arithmetic word problems. These
relation in which the occurrence of precurrent
results extend the generality of Neef et al. by
behaviors makes the solution (current operant)
demonstratingtheeffectivenessofthisapproach
morelikely(Skinner,1953,1966,1968,1969);
with a wider range of students (a student with
the solution, in turn, may reinforce the
autism and a typically developing student) and
precurrent behaviors. Parsons (1976) and
withwordproblemsthatinvolvedothertypesof Parsons, Taylor, and Joyce (1981) showed that
operations (multiplication and division), which precurrent collateral behavior was maintained
rarely have been targeted in research on when reinforcement was made contingent on a
problem solving. Given the move toward subsequent current operant, and that the
inclusion, there is a need for procedures that current operant decreased when collateral pre-
are effective in addressing the goals of mathe- current behaviors were prohibited. In the
maticsinstructionacross aspectrumof learners. present study, it was not possible (or desirable)
In addition to being effective, instruction to reverse or prevent the precurrent behaviors
should be efficient. The procedures used in the once they were acquired. However, precurrents
current study were efficient in several respects. maintained at a high level of accuracy in the
First, instruction was limited to the identifica- absence of any programmed reinforcement
tion of the basic elements and operations other than producing the correct response
involved in a problem without the use of (revealed by coloring over the space with a
cognitive strategies that prompt self-reflection developing marker). This lends support to the
(e.g., ‘‘What did you notice about how you did findings of Parsons and colleagues in demon-
PRECURRENT BEHAVIORS AND WORD PROBLEMS 367
strating the relation between precurrent and Owen, R. L., & Fuchs, L. S. (2002). Mathematical
problem-solving strategy instruction for third-grade
currentbehaviors.Thecurrentstudysuggestsan
students with learning disabilities. Remedial and
approach that might be applied in future Special Education,23,268–278.
research to teaching more complex discrimina- Parmar, R., Cawley, J., & Frazita, R. (1996). Word
problem solving by students with and without mild
tions (e.g., word problems that contain irrele-
disabilities. Exceptional Children,62, 415–429.
vant information) or advanced mathematical Parsons, J. A. (1976). Conditioning precurrent (problem
skills (e.g., multistep problems or those with solving) behavior of children. Mexican Journal of
Behavior Analysis,2,190–206.
remainders) to many students.
Parsons, J. A., Taylor, D. C., & Joyce, T. M. (1981).
Precurrent self-prompting operants in children:
‘‘Remembering.’’ Journal of the Experimental Analysis
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