training emr children to solve missing minuend problems errorlessly: acquisition, generalization,...

Analysis and Intervention in Developraental Disabilities. Vol, 4. pp, 379-402. 1984 0270-4684/8.4 $3.00 + .00 Printed in the USA. All rights reserved. Copyright ~ 1985 Pergamon Press Ltd.

Training EMR Children To Solve Missing Minuend Problems Errorlessly:

Acquisition, Generalization, And Maintenance

Paul M. Smeets

University of Leiden

Giulio E. Lancioni

University of Nijmegen

Sebastian Striefel

Utah State University

Rob. I. Willemsen

University of Leiden

The present study evaluated an errorless procedure for teaching EMR students to solve missing minuend problems (i.e., missing number problems starting with a minus sign). The stud>' consisted of two experiments, with two phases of training in each. The first phase was directed at establishing a nonnumerical, differentiated response to a prompt, the shape of which was gradually transformed to make the discriminative

Requests for reprints should be addressed to Paul M. Smeets, Department of Developmental Psy- chology, University of Leiden, Hooigracht 15, 2312KM Leiden, The Netherlands.

379

380 P. M. Smeet,s, G. E. l.anc'ioni, S. Stri~7[~'l, and R. J. Wilh'msen

stimulus. 7he second phase was designed (a) to extend the control oJ thi.s stimulus to the required numerical operations, and (b) to gradually ~'liminate the.[irst trained nonnumerieal components of the re.~7~onse chain and the experimental eondition~ (presence of the e.~perimenter and immediate feedback) used fi~r acquisition training. Five subjet'ts participated in Experiment I, and fi~ur in Experiment II. The results indicated that seven subjects ac'quired the target skill in a nearly errorless J~shion in 7.5 to 172 minutes ~]" individual training time. The two other subjects required several programmatic alteration.~ beJore ~'ompleting the training. Their total training time was much longer, that ix, 212 and 318 minutes, respe~'tivel 3. Moreover, the obtained findings revealed that whenever measured, the acquired skill transJ~'rred to simih~r, more advun~'ed problems, and was maintained over periods ranging .#'onz several weeks to several months. The technic'al aspects are discussed in terms of the literature on stimulus control b.v prompts. Attention i~ also given to the educational relevance e?]" the aequired skill.

Studies on the acquisition of basic mathematic skills (e.g., Case, 1978a, b; De Corte & Verschaffel, 1981; Thompson & Babcock, 1978) indicate that most young (normal) students havc great difficulties in solving unfamiliar addition and subtraction problems, that is, problems of which the stimulus configurations (stories, sets of objects, numericals) or the response requirements (verbal ex- planations, manipulation of objects) are different from those initially trained ( c . g . , 5 + 3 = , 5 - 3 = ). On some tasks (e.g.. = 5 + 3 : = 5 - 3 ) , these differences are mathematically irrelevant; yet, they may prevent the child from performing an arithmetic operation (De Corte & Verschaffel, 1981). On other tasks, such as missing number problems, the child may fail to identify the relevant stimuli and arrive at the correct (missing subtrahend) or incorrect (missing addend and missing minuend) number by following the operation sign. Although these problems have been associated with the young child's limited capacities to respond to complex and abstract stimuli (Gold, 1976; Goodstein, 1973; lbarra & Lundvall, 1982; O'Hara, 1975), most reports (Case, 1978a, 1978b; De Corte & Verschaffel. 1981; Dunlap & Brennen, 1981; King, 1982: Leutzinger & Nelson. 1979; Peck & Jencks, 1976; Thompson & Babcock, 1978) indicate that thesc errors result from inadequate teaching methods. These authors, most of whom use the neo-Piagetian or East European (Davydov, 1972; Gal'perin, 1970) frame- work of referencc, maintain that in addition to gross programmatic inadequacies (Gold. 1976), traditional methods frequently fail to establish the prerequisite skills for understanding the logic underlying the concepts of addition and subtraction. These skills include the abilities to (a) partition cardinal numbers and quantities into subsets of other numbers and quantities (e.g., 7 may be composed of 1 and 6, 2 and 5, 3 and 4), and (b) understand that the " = ' " sign signifies that the given or derived numbers on either side must balance. The acquisition

This research was supported by in part by the "'Stichting vtx~r Onderzoek van het Onderwijs'" (SVO). Appreciation is expressed to the teachers of the Pancrasschcx~l for their ct~.~peration and to Hugo Veldkamp for his valuable assistance in conducting this study.

Missing Minuend Problems 381

of these skills, first with objects and then with abstract (numerical) problems should provide the child with solution methods that are transferable to other, unfamiliar tasks and situations.

However, the studies allowing a detailed analysis of the results suggest that the neo-Piagetian or East European approach for establishing and transferring stimulus control may be less effective than has been claimed (De Corte & Verschaffel, 1981; Peck & Jencks, 1976; Resnick & Ford, 1981). First, the data reported by Case (1978b) and Gold (1976) illustrate that their programs for teaching missing addend problems were not very beneficial for less intelligent, though nonretarded children. Second, the data reported in these and other studies (Thompson & Babcock, 1978) observed that after being trained on how to solve missing addend problems presented by combinations of objects and mathematical signs (+ and = ), the children switched immediately to the incorrect operation (addition) when, on the following step, the objects were replaced by numbers. Furthermore, in his extensive study, Gold (1976) reported that after being trained on missing second addend problems (3 + = 5), very few children were capable of solving these problems when the stimulus array was reversed (5 = 3 + ).

The present study was an effort to assess the feasibility of using a discrimination training paradigm for teaching educably mentally retarded (EMR) students to solve missing minuend problems (i.e., missing number problems starting with a minus sign). While these problems are more complex than the missing subtrahend problems (requiring the child to follow the sign) though not more complex than the missing addend problems (requiring a mathematical operation opposite from the one indicated by the sign), the teachers reported that many students continued to have great difficulties with only the missing minuend problems. Specifically, the program was designed to (a) discriminate the missing minuend from other missing number (as well as standard addition and subtraction) problems, and (b) respond to the discriminative stimuli with the required operation. The study consisted of two experiments. Both experiments assessed the efficacy of the training procedures in terms of acquisition (error rate and individual training time), generalization (across similar problems of different complexity), and re- tention (over several weeks and months). Although the procedures are rather complex, they were deemed appropriate as a first study to deal with a complex problem.

EXPERIMENT 1: METHOD

Subjects, Experimenter, and Setting

Two boys (Erik and Rachid) and three girls (Miranda, Monique, and Petra) from a school for EMR students participated. Their CAs ranged from 12.9 to 13.3 years (M = 13.0). Their IQs (WISC, Stanford-Binet) varied from 66 to

382 P. M. Smeets, G. E. Lancioni, S. Striefel, and R. J. Willemsen

85 (M = 74.5). They were selected for their inability to solve missing minuend problems, while making no or very few errors on other (missing number or standard addition and subtraction) problems (see pretest).

An adult male served as experimenter. The study was conducted in a quiet room of the school building. Except during the probes and the final step of the training program, the experimenter was always seated next to the subject. The materials included worksheets, pens of different colors and ink (red and green, respectively), record forms (to be used by the subjects), and stickers and special certificates to be used as back-up reinforcers. All sessions were conducted in- dividually. Sessions were scheduled once a day, 5 days a week, and lasted 7 to 30 minutes. During all sessions, subjects were allowed to use any material (objects, extra paper) or system (e.g., finger counting) that would help them to arrive at the correct numerical solution.

Classification of Tasks

The tasks used for subject selection and subsequent testing and training were divided into six categories and three levels. The categories represented critical differences in stimulus configuration. They consisted of: (a) standard addition problems, (b) standard subtraction problems, (c) missing first addend problems, (d) missing second addend problems, (e) missing subtrahend problems, and (f) missing minuend problems. The categories were presented at three levels. The levels represented degrees of more advanced computational (addition and subtraction) skills. They consisted of problems which, when solved correctly, resulted in equations showing 3 numbers equal to, or lower than 10 (Level I); or 1 or 2 numbers between 11 and 20 (Levels II and 1II, respectively). None of the problems involved zero as a given number or as the result. Examples of the categories and levels are presented in Table 1.

Probing

General. Three types of probes were used, that is, pretraining, posttraining, and follow-up probes. During each probe session, subjects received a green pen and

T A B L E 1

Examples of Problems of Each Category and L e v e l

C a t e g o r i e s L e v e l s

I II 111

S t a n d a r d a d d i t i o n 1 + 3 = 9 + 8 ~ 13 + 7 =

S t a n d a r d s u b t r a c t i o n 7 - 4 = 12 - 7 = 18 - 5 =

M i s s i n g f i r s t a d d e n d + 2 = 4 + 9 = 14 ~- 5 - 16

M i s s i n g s e c o n d a d d e n d 6 + = 9 3 + = 12 2 ~- = 18

M i s s i n g s u b t r a h e n d 7 - = 6 18 - = 9 17 = 14

M i s s i n g m i n u e n d - 4 = 5 - 9 = 3 4 = 15


one to three worksheets. Each sheet comprised 20 problems of a same level, that is, 10 target (missing minuend) problems randomly interspersed with 10 other problems (2 exemplars of each of the other 5 categories). The subjects never received more than one sheet on the same level problems in a session. For each category, the specific problems used for probing were different from those used for training. The subjects were encouraged to take sufficient time to consider their answers, while being allowed to omit any problems they could not solve. During all probe sessions, the experimenter was sitting at another table at the opposite end of the room and refrained from giving any feedback on the subjects' performance during or at the end of the session.

Response scoring procedures. The solutions could be classified as completely correct, operationally correct, or operationally incorrect. Completely correct solutions were recorded when the correct number recorded by the subject resulted from the accurate use of the appropriate operation. Operationally correct solutions were recorded when the incorrect number recorded by the subject suggested the inaccurate use of the correct operation. Specifically, this classification was used for any incorrect number higher than the highest of both given numbers of any problem requiring addition (standard addition and missing minuend problems) or lower than that number for problems requiring subtraction (standard subtraction, missing addend, and missing subtrahend problems). Operationally incorrect solutions were recorded whenever the incorrect number recorded by the subject did not meet any of the above criteria or when a subject recorded no solution to a problem.

Pretraining probes. The first pretraining probe was given to assess each subject's ability to solve the target and nontarget problems at all three levels. This probe was conducted during three consecutive daily sessions. During each session, the subject received three sheets (one at a time) on problems of all three levels. Each sheet contained problems from a same level. At the end of the third session, the (mean) percentages of completely correct and operationally correct solutions were calculated for the (a) target (N = 30) and (b) nontarget problems (N = 30) of each level. A level was scheduled for training if the percentage of operationally correct solutions on the target problems was below 60, and the percentages of completely and operationally correct solutions on the nontarget problems were not below 75 and 92, respectively. If these mean percentages were higher, that level was scheduled for further testing only, provided that one or more levels were scheduled for training. Criterion on a level was met when the completely and operationally correct solutions on the (a) target and (b) nontarget problems were 90% (N=27) and 93% (N=28), respectively. The conditions (setting, instructions, problems, and feedback) for the subsequent pretraining probes were the same.

384 P. M. Smeets, G. E. Lancioni, S. StrieJk, l. and R. J. Willemsen

Posttraining probes. These probes were conducted immediately after completing the training on any given level. The problems and procedures were the same as those for the pretraining probe(s) on that level. All previously trained levels were probed.

Follow-up probes. These probes were scheduled once a week. They were conducted after a subject met criterion on the post- and/or pretraining probes on all levels scheduled for training. Each probe lasted only one session. In each session, the subjects received one sheet (i.e., 10 target and 10 nontarget problems) for each mastered level.

Training

General. The program consisted of two phases containing seven and four steps, respectively. Except for the final step of Phase 2, the experimenter was always seated next to the subject, thereby allowing him to provide the instructions and immediate feedback on each problem. During all sessions, the subjects were given several worksheets (1 at a time) each with 10 problems, that is, 5 missing minuend and 5 other problems (one exemplar of each of the other five categories), and one or two (red and green) pens. For each problem, the pens were always placed above the worksheet with the green pen on the left. There were two purposes for the colored pens. The first purpose was to provide an immediate discriminative stimulus for the experimenter on whether the subject correctly differentiated the type of problem to be solved. The second purpose was to provide the subject a discriminative cue for differentiating between missing minuend and other problems. The red pen was to be used for the missing minuend problems, and the green pen for all other problems.

Phase I. This phase was directed at teaching the subjects to discriminate the missing minuend problems from all other problems, particularly those with a missing number. This was accomplished by (a) guiding the subjects' attention to location of the discriminative stimulus (missing number followed by the minus sign) through the use of a salient prompt which was gradually eliminated through stimulus shaping (Bijou, 1968; Etzel & LeBlanc, 1979), and then (b) requiring the subjects to make a differentiated response resulting in the appearance of the full prompt.

Step 1 - 1 . The subjects were given only one pen (red). At first, the experimenter presented two introduction trials in which he showed an enlarged version


of the full prompt. He made the subjects notice that it had the shape of a pan (see Figure 1), and requested them to use the red pen for tracing it from left to right. Next he gave them sheets with five target problems with the outline of the normal sized prompt, and five other problems. The subjects were required to use the red pen for tracing the prompt on all "panproblems," and bypass all other problems. Discrimination of the full prompt was determined when no errors were made on two consecutive sheets.

Step I -2 . The stimulus conditions were the same as for Step 1-1, except that the subjects were now given a second pen (green). They were instructed to (a) use the red pen for tracing the prompt (pan) of the target problems, and (b) use the green pen for solving the other problems. Criterion was met when on three consecutive sheets, a subject (a) responded correctly on all (N = 15) target problems, and (b) gave at least 14 completely correct solutions (with green pen) on the other 15 problems.

Steps 1-3 to 1-7. The same as for Step 1-2, except that from Step i - 3 on, the shape of the prompt was gradually reduced (see Figure 1) to that of the minus sign or panhandle (Step 1-7). During these steps, the subjects were required to draw the pan in all target problems, irrespective of the reduced shape of the prompt.

PHASE I

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FIGURE 1. Example of the target (pan) problems and correct solutions on each step. Except for Steps 2 - 3 and 2 -4 , subjects were to select the red pen from among the red and green pens for drawing the prompt and writing the number. During Steps 2 - 3 and 2 -4 , they were given only the green pen.

386 P. M. Smeets. G. E. Lancioni, S. Striefel. and R. J. Willemsen

Phase 2. This phase was designed to (a) establish the control of the discriminative stimulus (panhandle) for determining the appropriate numerical operation (ad- ditionl to be carried out; and (b) eliminate the initial components of the differentiated response, as well as the setting events and contingencies used for training. During all steps of this phase, the stimulus composition of the target and nontarget problems was the same as for Step I -7 .

Step 2 - 1 . The procedures were the same as for Step I - 7 , except that the subjects were now instructed to extend their responses on target problems, that is. to add the given numbers and write the sum in the pan. At first, the subjects were trained to do so on sheets showing only target problems. This training continued until they used the red pen for drawing the pan and for writing the correct missing minuend on five consecutive problems. At that point, they again received sheets with target and nontarget problems (see Step 1-7). Criterion was reached when on three consecutive sheets a subject (a) made no operation errors, and (b) his solutions on at least 14 target and 14 nontarget problems (93% for each) were completely correct.

Step 2 - 2 . The subjects were no longer allowed to draw the prompt in the target problems. In essence, they were now required to use the red pen for writing only the missing minuend. The green pen was still used for completing nontarget problems.

Step 2 - 3 . The subjects no longer received the red pen. They were required to use the green pen for both the target and nontarget problems. Moreover, the requirements for meeting criterion performance were increased. Training on this step continued until on five consecutive sheets (50 problems), a subject (a) made no operation errors, and (b) scored at least 23 target and 23 nontarget problems (92% for each) completely correct.

Step 2 - 4 . During this step, the experimenter was seated at a table at the other end of the room. The subjects were required to call the experimenter each time they had completed all problems of a sheet. The experimenter then examined the solutions in the subject's presence and provided praise or corrective feedback on each of them, before giving the next sheet.

8

Response Scoring Procedures. Completely correct solutions were recorded when a subject (a) demonstrated all required nonnumerical responses (selecting the correct pen and drawing the prompt) and/or (b) wrote the correct number. For the missing minuend problems, the subjects were required to write the correct number from Step 2-1 on. For the other problems, this applied from Step ! - 2 on.

Operationally correct solutions were recorded when a subject (a) demonstrated the required nonnumerical responses (Steps 1-1 through 1-7 for the target problems), and/or (b) wrote an incorrect number resulting from the inaccurate execution of the appropriate numerical operation (from Step 2 - 1 for the target


problems and from Step 1-2 for the other problems). Operationally incorrect solutions were recorded when a subject (a) failed to demonstrate the required nonnumerical responses, and/or (b) wrote an incorrect number resulting from an inappropriate numerical operation. For example, on Step 2 -1 , an incorrect operation on a missing minuend problem was recorded when a subject selected the green pen, did not draw the pan, or wrote a number equal to or lower than the highest of both given numbers.

Reinforcement and Correction Procedures. The subjects were praised and allowed to write a mark on their record form for each correct solution, that is, (a) for each correct operation solution on a problem not requiring a numerical response (Steps I - 1 to 1-7 for the target problems), and (b) completely correct solutions (from Step 2 -1 for the target problems, and from Step 1 -2 for the other problems). Incorrect solutions (i.e., those in which one or more required response components were incorrect) were followed by the experimenter (a) preventing the subject from completing his response (Duker, 1981), and (b) drawing the subject's attention to the relevant stimuli. For example, when a subject selected the red pen for a missing addend problem, the experimenter stopped him immediately by saying, "Stop," and asked, "Is this a pan(handle) problem? So, what pen should you use?" For Steps 1-1 through 2 - 3 , these procedures were used immediately, while on Step 2 - 4 they were implemented in a delayed fashion (i.e., after completing each sheet.)

In addition, the subjects received an attractive sticker each time they met criterion on a step. The stickers were attached on a card designed like a certificate. They could earn one certificate for each level successfully trained.

Experimental Design

A modified version of a multiple probe technique (Homer & Baer, 1978) within and across subjects was used. For each subject, the experiment started with a pretraining probe on target and nontarget problems at all three levels. This probe was followed by the implementation of the full program on the lowest level (Level I) problems scheduled for training. The subjects were then probed again (posttraining probe) on the level just trained. If they did not reach probe criteria on both the target and nontarget problems, they were trained again from Step 2 - 1 on. Otherwise, they were probed (pretraining probe) on all other levels (Levels II and III) scheduled for training and testing. If they did not reach the probe criterion on the next higher level (Level II) problems scheduled for training, Phase 2 of the program would be implemented; otherwise, that level was considered learned. After the training of each level, posttraining probes were conducted on the same and all previously trained levels. The probes on levels scheduled for testing only, served to assess any possible changes occurring

388 P. M. Smeets, G. E. Lancioni, S. Stri(]k,I, arid R. J. Willenlsen

concomitantly with the training of other level problems. The follow-up probes were implemented after the posttraining criteria on all mastered levels were met. They were scheduled on a weekly basis until the end of the schoolyear.

RESULTS

Interobserver Agreement

The experiment was comprised of a total of 7,692 recordings, 4,550 of which were made during probe sessions (pretraining, posttraining, and follow-up probes), and 3,142 of which were made during training sessions. All recordings were checked by a second observer. Agreement was defined as agreement on the classification of the solution (completely correct, operationally correct and operationally incorrect solutions) of each problem. The second observer agreed on all but two recordings, on both of which the experimenter acknowledged that he had failed to make the correct classification.

Probes

Considering the purpose of the study, much of the data focus on the correct operations rather than on the accuracy of the missing minuend problems. The percentages of operationally correct solutions on the missing minuend problems during probing are presented in Figure 2. Based on their performance on the target and nontarget problems, three subjects (Erik, Rachid, and Petra) were scheduled to be trained on all three levels, and two subjects (Monique and Miranda) on two levels. Level III was excluded for Monique because of her low performance on the nontarget problems. Miranda's perlbrmance was different from that of all other subjects. Instead of following the minus sign consistently. she seemed to respond in an unpredictable fashion on the target problems of Levels I and I1, while using the correct operation consistently for those of Level III. Hence, this level was scheduled for testing only.'

Furthermore, the obtained findings revealed that (a) none of the subjects learned to solve (Level 1) the target problems before the implementation of the program, and (b) the training was highly effective with regard to the acquisition, generalization (across levels), and maintenance of the target skill in all subjects

kit was not discovered until the end of the study, that Miranda's solutions on all levels were very systematic and that they were controlled by an irrelevant stimulus, that is the relative position of the higher and the lower number. She used the incorrect operation (subtraction) whenever the lower number was on the right of the equal sign (e.g., - 6 = 2). and used the correct operation when that number was on the left of that sign (e.g., - 2 = 6). For Levels I, II, and III, the proportions of target problems (used for probingJ in which the higher number was on the right of the equal sign were 50%. 60% and 100%. respectively.


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FIGURE 2. Percentages of operationally correct solutions on target problems of the trained Level I (open circles), and untrained Levels II (dots) and 111 (asterisks) for each subject during pretraining, posttraining and follow-up probes.

(see Figure 2). For example, the figure shows that Monique was scheduled to be trained on target problems of Levels I and II; performed the incorrect operation on all target problems of each level during (a) the first pretraining probe, and (b) the following pretraining probes conducted when Miranda and Erik reached posttraining probe criterion; after training on the Level I problems, reached probe criterion on the target problems of Levels 1 and II (untrained); and continued to demonstrate criterion performance on the target problems of all three levels during six of the seven follow-up probes.

Table 2 shows the mean percentages of operationally correct and completely

390 P. M. Smeets. (;. E. Lancioni. S. Stri~fel. and R. J. Willemsen

"IABLE 2 Mean Percentages of Operationally Correct (OC) and Completely Correct (CC) Solutions on

the Target and Nontarget Problems across Levels for the Pretraining, Posttraining, and Follow-Up Probes.

Subjects Target problems Nontarget problems

Pro Post F-U Pre Post F-U

Miranda OC 66.7 97.8 100.0 98.9 100.0 99.0 CC 58.9 97.8 100.0 96.7 98.9 98.7

/:ilk OC 0.6 97.8 99.7 98.9 100.0 99.0 CC 0.0 88.9 94.7 93.3 95.5 95.0

Rachid OC 0.4 98.9 100.0 99.3 100.0 99.0 CC 0.4 98.9 100.0 99.2 98.9 96.7

Monique OC 0.0 100.0 97.9 99.4 95.0 99.3 CC 0.0 100.0 ~ .7 92.8 93.3 86.4

Petra OC 0.4 100.0 I00.0 I00.0 CC 0.4 I00.0 98.9 I00.0

correct solutions on the target and nontarget problems for the pretraining, posttraining, and follow-up probes. These data indicate that (a) the percentages of operationally and completely correct solutions on the nontarget problems were consistently high (above 90%) and did not deteriorate from training, while (b) those on the target problems were very similar after training.

Training

The results of the training are presented in two sections, that is, one on tour subjects (Miranda, Erik, Rachid, and Monique) who completed the program as planned, and one on Petra, who required several changes in experimental conditions.

Miranda, Erik, Rachid, and Monique. These subjects completed the program without any difficulties. While some needed more than the minimum number of sheets to reach criterion on some steps (notably those of Phase 2), the results (see Table 3) indicate that they completed the training on the Level I target problems in a near error-free fashion. The total training time for Miranda, Erik, Rachid, and Monique was 93, 172, 75, and 128 minutes, respectively.

Petra. Figure 3 shows Petra's operationally correct solutions on the Level I target and nontarget problems on each step of the original program until it was stopped (upper graph), and of the revised program (lower graph). The data plotted in the upper graph indicate that Petra made very few incorrect operations on any problems until she was no longer allowed (Step 2 - 2 ) to draw the pan in the target problems. All errors consisted of her making the incorrect operation, while using the correct (red or green) pen. In order to establish control again, Step 2 - 1 was re-introduced. Initially, the drawing of the pan resulted in her responding correctly on all problems. But, from the third sheet on, her performance dete- riorated again. Moreover, there was a remarkable drift from her making errors on the final component (incorrect numerical operation on sheets 3 and 4) to the

Missing Minuend Problems

TABLE 3 Training Results on Miranda, Erik, Rachid, and Monique

391

Phases Sheets (N) and to Steps criterion

Mean percentages of operationally correct (OC) and completely correct (CC) solutions on target (TP) and nontarget (NTP)

problems

Mi Er Ra Mo Miranda Erik Rachid Monique

TP NTP TP NTP TP NTP TP NTP

1-1 2 2 2 2 OC 100 100 100 100 I - 2 3 3 3 3 o c 100 100 100 100 100 100 100 100

CC 100 I00 I00 100 1-3 3 3 3 3 OC I00 100 loll 100 100 100 100 I00

CC 100 93 93 100 I - 4 8 3 3 3 OC 96 98 1130 100 100 100 100 100

CC 98 100 100 93 1-5 3 3 3 3 OC 100 100 100 100 100 100 100 100

CC 100 100 100 100 1-6 3 5 3 3 OC 100 100 92 100 100 100 100 100

CC 100 96 100 100 1-7 3 5 3 4 OC 100 1130 96 100 100 100 100 95

CC 100 1130 100 95 2-1 3 3 3 9 0(2 1130 100 100 100 100 100 100 93

CC 100 100 93 100 100 100 100 93 2-2 3 3 3 3 OC 100 100 1130 100 100 100 100 100

CC 100 100 100 100 100 100 100 100 2-3 9 5 5 12 OC 98 98 100 1130 1130 100 93 100

CC 98 98 92 1130 100 96 93 98 2-4 9 10 5 5 OC 98 98 96 100 100 100 100 100

CC 98 98 96 100 100 100 100 100

first component (select ion o f the wrong pen on sheets 5 and 6) as training

continued. At this point it was concluded that (a) the program was inadequate for con-

troll ing Petra 's numerical operations, and (b) that training on previous steps

would not prevent the same difficult ies from re-occurring. Therefore , it was

decided to start all over again, but with a revised program.

Petra's revised program: Procedures and results. The revised program was

essentially the same as the original one, except that it a l lowed the exper imenter

additional control (Johnston, Whi tman, & Johnson, 1980) on the numerical

operations. This was achieved by requiring Petra to make an additional non-

numerical response immedia te ly before carrying out the numerical operat ion.

The response consis ted of encircl ing one of two operat ions words ( i .e . , the Dutch

equivalents for "add" and "subtract") printed next to each problem. This additional response was required for all steps (except for the final step of Phase 2)

in which she was to write a number. Initially, the two phases compr ised a total

of 13 steps, that is, Steps 1 - 0 through 1 - 7 (Phase 1), and Steps 2 - 1 through

2 - 5 (Phase 2). But, as will be described below, additional steps (Steps 2 - 0 ,

2 - 2 a , 2 - 2 b , and 2 - 2 c ) were added in order to re-establish or maintain st imulus

control . An outl ine o f the revised program is presented in Table 4.

392 P. M. Smeets, G. E. 12tncioni S. Striefel, and R. J. Willemsen

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F I G U R E 3. P e t r a ' s p e r c e n t a g e s o f o p e r a t i o n a l l y c o r r e c t so lu t ions on the t a r g e t (closed circles) a n d n o n t a r g e t p r o b l e m s (open circles) p e r sheet on each s tep o f the o r ig ina l a n d revised p r o g r a m s .


TABLE 4 Problems, Materials, and Required Responses for the Target (TP) and Nontarget (NTP)

Problems of Each Step of the Revised Program

Steps Problems and materials Required responds

1-0 Pens: I green pen. Problems: NTP with operation words. Pens: Red and green pen. Problems: TP and NTP with operation

words. Same as 1-7.

I - I to I - 7 2 -0

2-1 Same as 1-7.

2 -2 Same as 1-7.

2-2a Same as I -7 .

2-2b Same as 1-7.

2-2c Pens: 2 green pens. Problems: Same as 1-7.

2 -3 Pens: 1 green pen. Problems: Same as I -7 .

2 -4 Pens: Same as 2-3. Problems: Same as 2-3, but no operation

words. 2 -5 Pens: Same as 2-3.

Problems: Same as 2-4. Experimenter absent--delayed feedback.

NTP: Encircling operation word and writing number.

TP: Drawing pan (red pen). NTP: Same as I -0 .

TP: Drawing pan and encircling operation words (red pen).

NTP: Same as I - 0 TP: Drawing pan, encircling operation

word, and writing number. NTP: Same as 1-0.

TP: Encircling operation word and writing number (red pen).

NTP: Same as 1-0 TP: Encircling operation word (red pen)

and writing number (green pen). NTP: Same as 1-0.

TP: Touching red pen before encircling operation word and writing number (green pen).

NTP: Same as 1-0. TP: Encircling operation word and

writing number. N'FP: Same as I -0 .

TP: Same as NTP. NTP: Same as 1-0.

TP: Writing number. NTP: Same as TP.

TP: Same as 2-4. NTP: Same as TI~

The first step of the revised program was designed to re-establish stimulus control over the nontarget problems. The worksheets contained only nontarget problems with operation words next to them. Petra was required to use the only available green pen for (a) encircling the appropriate operation word, and (b) writing the sum or missing number. Following the completion of this step, she progressed to Step 1-7 without making any operation errors. However, her error rate increased again soon after the introduction of Step 2 -1 , that is, when in addition to using the red pen for drawing the prompt, she was also to (a) encircle the appropriate operation word, and (b) write the missing number in the pan. All errors consisted of her encircling operation words matching the sign ( + or

- ) of missing addend and missing minuend problems, rather than the required numerical operations. Since it could be that these inappropriate discriminations were caused by the abrupt extension of the behavior chain, she was first trained

394 P. M. Smeets, G. E. Lancioni, S. Striefel, and R. J. Willemsen

on step 1-7 again, and then on a new step (Step 2-0) inserted between Steps 1-7 and 2 - 1. This step was identical to Step 2 - I, except that the requirement to carry out the numerical operations on the target problems was dropped. She completed this step while making no more than one incorrect operation (standard addition problem) on a total of 110 problems, and made no errors when Step 2 - 1 was reintroduced. She also had no difficulties with the following step (Step 2-2) in which she was no longer allowed to draw the prompt in the target problems, but failed again (i.e., encircled the incorrect operation words of target problems) when the red pen was no longer available (Step 2-3) . Therefore, it was decided to re-establish control and to gradually eliminate the use of the red pen. This was accomplished by first re-introducing Step 2 -2 , followed by Steps 2 -2a (in which she was to use the red pen for encircling the operation word and the green pen for writing the missing minuend), 2 -2b (first touching the red pen and then using the green pen for encircling the operation word and writing the missing minuend), and 2 -2c (the red pen was replaced by a second green pen). These steps were scheduled to be followed by two others, that is, Steps 2 - 3 (only one green pen available), and 2 - 4 (elimination of operation words), before introducing the final step (Step 2-5) . However, since the school year was coming to an end, it was decided to take the risk of omitting Steps 2 - 3 and 2 -4 , and to proceed directly from Step 2 -2c to Step 2 -5 . The results indicate that, except for two incorrect operations (target problems) on the first sheet, she had no difficulties in meeting the step criterion.

During the entire training, Petra's accuracy was very high. On the 1015 (target and nontarget) problems requiring a numerical response, she made a total of eight errors due to inaccurate use of the correct operation. Her total training time was 318 minutes.

EXPERIMENT 2

The purpose of this experiment was (a) to replicate the findings of the original program (Experiment 1), and to assess the impact of the revised program piloted with Petra, in essence, the requirement to encircle the operation words. It was realized (Johnston et al., 1980; Lovitt & Curtis, 1968) that such a requirement (a) could help the subjects to reconsider their numerical responses, and (b) would provide the experimenter increased opportunity for guiding the subjects' responses. Conversely, it was possible that, as with Petra, the presence of these stimuli would induce subjects making incorrect discriminations, that is, to encircle the word semantically corresponding with the sign rather than with the required numerical operation.

Missing Minuend Problems

METHOD

395

Subjects

Four boys (Rene, Patrick, Marco, and Gerard) of the same school participated. Their CAs ranged from 8.4 to 15.4 years (M = 11.9). Their IQs (WISC, Stanford-Binet) varied from 53 to 83 (M = 69.7).

Experimental Procedures, Conditions, and Design

The experimenter, classification of problems and solutions, scoring and reinforcement procedures, reliability measures, and probes (pretraining, posttraining and follow-up probes) were the same as in Experiment I. Based on random assignment, Rene and Marco were trained on the original program (Experiment I), while Patrick and Gerard were scheduled to follow the revised program (i.e., Steps 1 -0 to 1-7 of Phase l, and Steps 2 -0 , 2 - l, 2 -2 , 2 - 3 , 2 - 4 , and 2 - 5 of Phase 2) used with Petra (see Table 4). In both programs, criterion on a step was reached when a subject (a) made no operation errors, and (b) scored at least 90% of the target and nontarget problems completely correct on two (Step l - l), three (all other but the final two steps), or five consecutive sheets (2 final steps). The two programs were each used concurrently with two subjects. A modified multiple probe technique was employed to assess the effects across pairs of subjects. The experiment continued until the final day of the school year.

RESULTS

lnterobserver Agreement

The experiment was comprised of a total of 3,550 recordings, 1,680 of which were made during probe sessions, and 1,870 during training sessions. All recordings were checked by another observer. Agreement was present on all but one recording.

Probes

The percentages of operationally correct solutions on the target problems during pretraining, posttraining, and follow-up probes are presented in Figure 4. Rene, Patrick, and Marco were scheduled for training on all three levels. Gerard had considerable difficulties in solving any Level II and III problems. Therefore, he was scheduled to be trained on only the Level I problems. The results revealed that all subjects (a) consistently used the incorrect operation on

396 P. M. Smeets. G. E. l,ancioni. S. Strie[el, anti R. J. Wilh,m.~en

the target problems prior to the training, (b) acquired the target skill through training, and (c) continued to respond correctly each time when given different (target) problems of the same and higher levels. Moreover, the data presented in Table 5 indicate that the (mean) percentages of operationally and completely correct solutions were (very) high across pretraining (nontarget problems), and posttraining and follow-up probes (target and nontargct problems).

Training

Rene and Marco, who were trained with the original program, and Patrick, who was trained with the revised program, acquired the target skill in a nearly errorless fashion. The percentages of operationally and completely correct solutions were never below 99 and 97, respectively. The total training time for Rene, Marco, and Patrick was 79, 150, and 90 minutes, respectively. Marco 's training time was much longer because of his slow responding.

In contrast to Patrick, Gerard did not progress through the revised program without major difficulties (see Figure 5). After he had learned to demonstrate all required responses on the nontarget (Step 1 -0 ) and target problems (Step 1 - 1) separately, he failed to perform the correct operations on the nontarget problems during the next step (Step 1 -2 ) , that is, when the target and nontarget problems were presented together. All errors consisted of encircling the incorrect operation word of missing addend problems. In order to assess if these incorrect discriminations resulted from the presence of the operation words or from other sources, Step 1 - 2 of the original program (no operation words) was introduced. Now his only operation error consisted of selecting the incorrect pen for solving a missing subtrahend problem. However, since this effect could be attributed to more training, Step I - 2 of the revised program was reintroduced. When at that point he encircled the incorrect operation words of missing addend problems again, it was decided that (a) for Gerard these stimuli were detrimental for the acquisition training, and hence (b) that he should proceed with the original

TABLE 5 Mean Percentages of Operationally if)C) and Completely Correct (CC) Solutions on the

Target and Nontarget Problems across Levels for the Pretraining, Posttraining, and Follow-Up Probes.

Subjects Target Problems Nontarget Problems

Pre Post F- U Pre Post F- U

Rene OC 0.0 I00.0 I00.0 100.0 I(X).O C(" 0.0 100.0 98.3 98.9 100.0

Patrick OC 0.0 10~).0 100.0 98.9 100.0 CC 0.0 100.0 98.3 97.8 100.0

Marco OC 0.0 98.9 95.6 I(X).0 CC 0.0 97.8 85.6 88.9

Gerard OC 0.0 100.0 96.7 96.7 CC (}.0 I(X).O 81.6 93.3

100.0 98.3

100.0 98.3


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FIGURE 4. Percentages of operationally correct solutions on target problems of the trained Level ! (open circles), and the untrained Levels I1 (dots) and I!1 (asterisks) for each subject during pretraining, posttraining, and follow-up probes.

398 P. M. Smeets. G. E. Lancioni, S. Striefel. and R. J. Willemsen

U. e bz ~ o r~ 90 /

" .~ i • i GERARD w ~ 80

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FIGURE 5. Gerard's percentages of operationally correct solutions on the target problems (closed circles) and nontarget problems (open circles) per sheet of each step of the revised (R) and original (O) programs.

program. Although his performance on the following steps was more erratic than that of the other three subjects, he completed this program without making systematic errors. His overall percentages of operationally and completely correct solutions on the target problems were 98.4 and 95.4, respectively. Those on the nontarget problems were about the same, that is, 94.8 and 93.9. His total training time was 212 minutes.

DISCUSSION

The results of both experiments indicate that, albeit with some modifications, the program was effective in teaching EMR students to calculate missing minuend problems. Moreover, the findings revealed that whenever measured, the acquired skill generalized to similar and more advanced problems and was maintained for several weeks or months depending upon the length of time available for

follow-up probing for the individual subjects. The plesent two-phase program was modeled after the theoretical notion that

learning to respond in discrimination learning is a two-stage process, that is, one in which the subject learns which stimuli are relevant to the discrimination (identifying the type of problem), and another one in which he learns to make the correct responses (performing the appropriate operations) to these stimuli (Trabasso & Bower, 1968; Zeaman & House, 1963). Present findings pertaining to the first phase of the (original and revised) program revealed that the procedures were very effective for training the subjects to identify the type of problem to be solved. For all subjects, the error rate on using the red pen for drawing the prompt (pan) was extremely low, unless other problems occurred (e.g., Petra). Although the design of this study does not verify which mechanisms account


for the establishment of the discrimination, it is possible (if not plausible) that the acquisition of these discriminations was facilitated by the manipulations of the prompt. If so, this would support previous findings (Bijou, Etzel, & Domash, 1978; Schmillmoeler, Schmillmoeler, Etzel, & LeBlanc, 1979; Etzel & LeBlanc, 1979; Lancioni, Ceccarani, & Oliva, 1981; LeBlanc, 1979; Schreibman, 1975; Smeets, Lancioni, & Hoogeveen, 1984; Wolfe & Cuvo, 1978) on the efficacy of stimulus shaping procedures for guiding attention to the relevant stimuli.

The planned procedures were successful with seven of the nine subjects on the two experiments for extending the discriminative control of the target stimulus (i.e., the problem starting with a minus sign) to the numerical operation. The procedures needed to be modified for two subjects, Petra and Gerard. Petra's data indicated that her calculations were not always guided by the target stimulus, but by the color of the pen and her drawing of the prompt. During the implementation of the original program, she made several incorrect operations when, at Step 2 - 2 , she was no longer allowed to draw the pan. Likewise, stimulus control was lost again on Step 2 - 3 of the revised program, when she was given only one green pen for all target and nontarget problems. Different, though related, difficulties were observed on the use of the operation words with Gerard. Thus, the results suggested that, instead of strengthening, planned procedures prevented the establishment of stimulus control for these two subjects. In ret- rospect, these or similar difficulties might have been expected. In contrast to the stimulus manipulations used during the first phase, the second phase did not include procedures to ensure the control of the numerical responses by the critical stimulus. Thus, it is possible that during the course of the program, the initial components of the response chain (color of the pen, drawing of the pan) and supplementary stimuli (operation words) mistakenly came to serve as inappro- priam discriminative stimuli (Koegel & Rincover, 1976; Schreibman, 1975; Sulzer- Azaroff & Mayer, 1977; Wolfe & Cuvo, 1978).

Furthermore, it was observed that there was near perfect generalization from the Level I target problems to those of both other levels (Levels II and III) and that this performance was maintained for the duration of the follow-up probes. While it can only be speculated as to why the generalization across levels occurred, it is plausible that it resulted from the training of sufficient different exemplars (Solnick & Baer, 1984; Stokes & Baer, 1977) of Level I problems during the course of which the subjects learned to discriminate the relevant (problems starting with a minus sign) from the irrelevant stimulus components (numbers higher or lower than lO on either side of the minus sign) common in all target problems (Prokasy & Hall, 1963; Striefel & Owens, 1980). On the other hand, it is also possible that the extension of stimulus control was facilitated by irrelevant variables (Rincover & Koegel, 1975) such as the identity of the experimenter (same adult across conditions), the location in which the generalization tests were conducted (same classroom in which the training took place), and even the worksheets (similar to those used during training). These and

400 P. M. Smeets, (;. E. Lancioni, S. Striefel, and R. J. Wilh'mscn

possibly other stimulus conditions (e.g., the weekly follow-up sessions) may also account for the continued high maintenance of the acquired skill. Therefore, the present data do not allow any conclusions as to whether the same findings would be obtained under different stimulus conditions (different adults, locations, sheets, or time intervals).

Since the present study was not directed at increasing the students" understanding of the logic underlying the mathematical concepts of addition and subtraction, questions may be raised about the educational validity of the acquired skill. Evidently, such questions cannot be answered without reference to the degrees and forms of restricted stimulus control to be tolerated when assuming conceptual behavior (Whaley & Malott, 1971). Several of the aforementioned authors (De Cone & Verschaffel, 1981; King, 1982: Peck & Jencks, 1976) assumed lack or inadequate mathematical understanding in children who, when tested, (a) could not solve unfamiliar problems, or (b) could solve these problems, but were not capable of giving adequate verbal justifications for their solutions (e.g., "'The teacher told us to do them this way"), or carrying out the same problems when different stimulus dimensions (e.g., three dimensional objects) were inw~lved. These shortcomings have been ascribed (De Corte & Verschaffel, 1981; O'Hara, 1975: Peck & Jencks, 1976) to inadequate or "symptomatic" teaching methods. But, as was pointed out before, the results of several empirical studies (Case, 1978b; Gold, 1976; Thompson & Babcock, 1978) clearly indicate that their "fundamental" approach for teaching mathematical skills does not prevent restricted stimulus control from occurring. Yet, none of the findings of these previous studies were given much weight or considered as evidence for the children's failure to understand the basic concepts. These inconsistencies suggest that unless the criteria 1or measuring conceptual behavior are firmly established, criticisms of conceptual skill can be relevant only to the personal orientation of the reader. The present study operationalized procedures for successfully solving missing minuend problems with EMR children.

The implications of the present procedures are that EMR children can be trained to solve missing minuend problems. Further research, however, is nec- essary to refine and simplify the procedures to the point where they can and will readily be used by teachers. One simple method for classroom use would be to develop a microcomputer program.

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