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Teaching multiplication facts to students with learning problems.

Teaching Multiplication Facts to Students with Learning Problems

Proficiency in arithmetic contributes to students' academic and vocational advancement. As Silbert, Carnine, and Stein (1981) stated:

The mastery of basic facts is critical if students are to develop fluency in working mathematical problems. The negative atitudes many children have about math can be traced to not having mastered the basic facts. (p. 44)

Hasselbring, Goin, and Bransford (1987) indicated that the ability to succeed in higher level arithmetic skills is directly related to the efficiency with which lower level processes (including basic facts) are executed. Instructional techniques have been identified by which some students with handicaps may successfully practice and learn the basic arithmetic facts. These techniques include multisensory training (Lombardo & Drabmen, 1985), flash card drill (Schilling, 1985), instructional games (Beattie, 1981), manipulatives (Juliano, 1982), self-monitoring (Coble, 1982), peer tutoring (Bullard & McGee, 1983), and computer-assisted instruction (Walkins & Webb, 1981). Although there is evidence to support the effectiveness of these procedures for arithmetic instruction, most involve trial-and-error learning. In contrast, near-errorless learning procedures allow learners to quickly experience successful responding and avoid practicing errors. High levels of correct responding during initial instruction and practice--80% correct or higher--is one aspect of effective instruction (Rosenshine, 1983) and may be very important for learners with disabilities.

An errorless learning technique known as time delay transfer of stimulus control has been effectively used with students ranging in age from preschoolers to adults. This procedure initially was described by Touchette (1971), who implemented delayed prompting strategies to teach letter-form discrimination to adolescents with severe retardation. The time delay procedure consists of systematically increasing the interval of time between the teacher's presentation of task directions paired with a novel stimulus and a "controlling prompt," or the teacher's model of the correct response (Snell & Gast, 1981).

Time delay procedures have been used primarily with learners with moderate and severe handicaps to teach such skills as requesting lunch (Halle, Marshall, & Spradlin, 1979), object identifications (Godby, Gast, & Wolery, 1987), manual signing (Browder, Morris, & Snell, 1981; Kleinert & Gast, 1982; Oliver & Halle, 1982), and food preparation (Schuster, Gast, Wolery, & Guiltinan, 1988). Recent research has also shown time delay to be a successful instructional procedure for teaching reading and spelling skills to students with mild handicaps (Gast, Kleinert, Isaac, Eizenstat, & Bausch, 1983; Johnson, 1977; Jones, 1985; Precious, 1985; Stevens & Schuster, 1987; Yancey, 1987). The utility of time delay for teaching arithmetic skills has received little attention from researchers.

The purpose of the present study was to investigate the effects of a constant time delay procedure in teaching multiplication facts to elementary school students with mild learning and behavior disorders. The present investigation used the constant time delay procedure, which begins with zero-second delay trials in which the controlling prompt is delivered immediately after presentation of the novel stimulus, and the learner is to repeat the correct response. On subsequent trials, the prompt is delayed a specified number of seconds after the presentation of the novel stimulus. If the learner knows the answer, he or she is to respond following presentation of the stimulus; if the learner is unsure of the correct response, he or she is to wait for the teacher to give the prompt.


A multiple-probe design (Gast, Skouge, & Tawney, 1984) across behaviors was used to assess the effectiveness of the constant time delay procedure. The students' special education teacher (first author of the present article) conducted all sessions and collected the data. This individual was trained in the time delay procedure and had previously applied it in the classroom.


Four students, two enrolled in fifth grade and two enrolled in sixth grade, were involved in this study. All students were placed in special education and, in addition to their regular class placement, also received daily services in the same learning and behavior disorders (LBD) resource room. All students qualified for and received all of their ongoing mathematics instruction in the LBD resource room. Students spent from 3.5 to 4.5 hours per day in the resource room. Students were included in the study because they had not yet demonstrated mastery of all multiplication facts. In addition, students selected for the study demonstrated the following characteristics or behaviors required for the time delay procedure: (a) possess visual and auditory acuity (with correction if necessary) within normal limits, (b) attend to teacher and materials for at least 15 minutes (min), (c) wait 5 s for a response prompt, (d) verbally imitate spoken words, (e) respond correctly to verbal instructions used in time delay, (f) identify potential reinforcers, and (g) write correctly the digits 0 through 9. Additional descriptive information for the four students appears in Table 1.


Assessment and training sessions took place in the same special education resource room within a public, suburban elementary school in a moderate-sized city in central Kentucky. Sessions were conducted twice daily for each student, Monday through Friday, between 8:15 and 10:15 a.m. and between 12:15 and 1:45 p.m. Throughout the study, all daily mathematics instruction for all students took place in the LBD resource room setting.


A total of 100 multiplication facts, using combinations of the single digits 0 through 9, were used for screening of students. Written facts were presented on separate 3x5-inch cards and students were to verbally state the product. There were 6 separate screening sessions; each session consisted of exposure to 50 different multiplication facts. At the conclusion of screening sessions, each fact had been presented 3 times. Facts were considered unknown if the student could not verbally state the answer to a fact within 5 s after the flash card was presented for at least 2 of the 3 trials. For each student, a set of 30 facts identified as not known were selected for training. The unknown facts were then randomly divided into six groups of 5 facts each for training sessions. When the reverse of a fact was also unknown, the 2 were placed in the same group (e.g., 3x2 and 2x3). Three-by-five-inch white cards containing 1 multiplication fact per card written horizontally in black marker were used for probe, training, intermix, and maintenance sessions. Worksheets containing multiplication facts printed horizontally on one side and vertically on the reverse side were used for generalization probes. Token reinforcer stimuli consisted of colored discs, which were traded at the end of each session for stickers--which all students had previously selected as preferred reinforcers.


Response Definitions and Data Collection. During all phases, students' first oral and written responses were judged as either correct or incorrect. Self-corrections were not considered because the teacher wanted to promote immediate and accurate responding.

During the training condition, data were collected on the following types of student responses:

1. Anticipations, defined as the student correctly stating the fact answer within the 5-second delay interval, or before the teacher provided the prompt; for example:

Teacher: "Read the fact. Say the answer."

Student: "Two times two equals four."

2. Waits, defined as the student's not responding during the 5-second delay interval because of uncertainty about the answer; the teacher then provides the prompt, and the student immediately repeats the correct response; for example:

Teacher: "Read the fact. Say the answer."

Student: (Waits 5 seconds.)

Teacher: "Two times two equals four."

Student: "Two times two equals four."

3. Non-waits, defined as the student's stating an incorrect fact answer before the teacher provided the prompt; for example:

Teacher: "Read the fact. Say the answer."

Student: "Two times two equals three.

4. Wait errors, defined as the student's stating an incorrect answer following the prompt; for example:

Teacher: "Read the fact. Say the answer."

Student: (Waits 5 seconds.)

Teacher: "Two times two equals four."

Student: "Two times two equals seven."

5. No response, defined as the student's making no response within 5 seconds of the teacher's providing the prompt; for example:

Teacher: "Read the fact. Say the answer."

Student: (Waits 5 seconds.)

Teacher: "Two times two equals four."

Student: (Waits 5 seconds.)

During training, only correct anticipations were graphed as a percentage and counted toward criterion, but all responses were summarized.

Immediately after a student met criterion in training sessions, he or she completed three probe sessions during which students responded verbally and were not prompted; these responses were summarized as percent correct. Generalization sessions required a written response to printed math facts and were recorded as percent correct. Maintenance probes required verbal responses, and were summarized as percent correct. In addition, the amount of direct instruction time (in minutes), number of trials to criterion, number of sessions to criterion, and number of errors to criterion were recorded.

Reliability Measures. Reliability checks for procedures and scoring of students' responses were conducted by an independent observer once during baseline and once each week per student for the duration of the study. Reliability was assessed at least once in each of the six time delay conditions and at least once per student during probe conditions. The independent observer sat behind the students and unobtrusively collected the reliability data during designated sessions. These checks were counterbalanced across days of the week. Procedural and scoring reliability were calculated using the point-by-point method (Gast & Tawney, 1984) dividing the number of agreements by the number of agreements plus disagreements and multiplying by 100. These calculations resulted in 100% interobserver agreement for procedures and for scoring student responses for all sessions and all students, indicating a high degree of reliability.


The time delay intervention used in this study is actually a package including many features of effective instruction. The specific procedures described here incorporate token and social reinforcement, modeling of desired responses, intermixing of learned facts, multiple response opportunities, systematic prompting, and near-errorless learning.

General Procedures. For each student, one instructional session occurred each day in the morning, and one occurred in the afternoon. Each student was trained individually, and training sessions were counterbalanced across students. A counterbalanced schedule of training was used to avoid the possibility that the sequence in which students were called to work with the teacher had any confounding effect on the outcome of the intervention. Each training session included 25 trials. Facts were presented in 5 blocks, and each block consisted of 1 trial per target multiplication fact. Facts were randomized within blocks.

During all conditions, tokens used as reinforcers were delivered on the table in front of the students, contingent on good attending and correct responses. The tokens were paired with descriptive verbal praise and were later traded for a sticker if the student had earned sufficient tokens to indicate 90% or better correct responding during the session. In all conditions, reinforcement was withheld following incorrect responses.

Baseline and Probe Procedures. As part of the multiple probe design, a brief baseline phase consisting of at least 3 sessions was completed before training on each of the 6 sets of multiplication facts. This eliminated the need for a continuous baseline while contributing to experimental control. Baseline sessions consisted of 1 trial for each of the 30 target facts. Written multiplication facts were presented on separate 3x5-inch cards, and students were to state verbal responses. The teacher reinforced the student for attending on a variable ration (VR3) schedule with descriptive verbal praise and a token. Correct responding was reinforced with descriptive verbal praise and a token on a continuous schedule. Reinforcement and feedback were withheld following incorrect responses.

Constant Time Delay Procedures. Following baseline, the constant time delay procedure was implemented daily with each student. During Session 1, a zero-second delay was implemented, in which the teacher presented a card, said, "Look"; obtained an attending response; said, "Read the fact. Say the answer"; and immediately provided the controlling prompt, which the student then repeated (e.g., "Six times four equals 24"). The 25 trials at zero-second delay provided models of correct responses, which the student then imitated.

All remaining sessions for the set of multiplication facts used a 5-s delay between the student's reading the fact and the teacher's delivering the response prompt (if necessary). For example, during the 5-s delay trials, the teacher showed a fact card and said, "Read the fact. Say the answer." The student read the fact aloud and then had 5 s in which to state the correct product or, if unsure of the correct response, wait for the prompt.

When the student made a correct response, defined as an anticipation or wait, the teacher said, "Good," repeated the fact with the answer, and delivered one token. For errors defined as non-waits, the teacher said, "No, wait for me to tell you if you don't know." The teacher then removed the card from the table and dropped her head for 5 s before proceeding to the next fact as a brief

"time-out" from teacher attention.

Mastery criteria applied during the time delay training first required students to respond with 100% correct anticipations across three consecutive sessions with a CRF schedule of reinforcers. At this point, the reinforcement schedule was changed to VR3, and students were required to meet an additional mastery criterion, which was 100% correct anticipations for 2 consecutive sessions under the thinned schedule of reinforcers.

Intermix Procedures. Following the attainment of criterion on Fact Set 1 and Fact Set 2, the two fact sets were randomly intermixed across 25 trials to promote discrimination among learned facts. Intermixed trials were conducted exactly as probe trials. Following attainment of 100% mastery criterion for 2 consecutive sessions on a CRF schedule of reinforcement and 2 consecutive sessions on a VR3 schedule, the probe conditions were reimplemented across all facts. As each set of facts was mastered, it was intermixed with those previously mastered.

Generalization and Maintenance

Generalization data were collected concurrently with training. Following mastery of Fact Set 1, the student was presented daily with a printed worksheet containing the set of math facts mastered. The facts were presented horizontally, (as in training) on the front side of the worksheet and vertically on the reverse side. Graded worksheets were returned to the students following the second training session of the day. As each new set of facts was mastered, it was added to the daily worksheet. In addition, maintenance checks were conducted as oral probes at 2-week, 3-week, and 4-week intervals following completion of the training portion of the study. All facts were randomly intermixed and were probed for 3 trials each at each maintenance check for a total of 9 trials on each fact.


The individual performances of each student are depicted in Figures 1, 2, 3, and 4. All students consistently responded at a 0% correct level on all baseline probes of untrained fact sets throughout the duration of the study. Each introduction of the time delay intervention resulted in a marked accelerating trend in the number of correct anticipations, and each student achieved the mastery criterion 100% correct anticipations for each set of arithmetic facts under CRF, VR3, and intermix conditions. (The Training Phase depicted in all figures combines these conditions for the targeted set of multiplication facts.)

For all students, the range of number of sessions to criterion on a set of five facts was 6 to 19; the median number of sessions to criterion was 10. Note that the number of sessions to criterion includes the required 3 consecutive sessions at 100% correct responding on CRF and 2 consecutive sessions at 100% correct responding on VR3 schedules of reinforcement. The mean duration of a session across all students was 3 min, 22 s. The total number of minutes of direct instruction time required to train 30 facts to the criterion were 388 min (111 sessions) for Student 1; 293 min (92 sessions) for Student 2; 334 min (94 sessions) for Student 3; and 280 min (86 sessions) for Student 4.

All errors committed by all students were non-waits; that is, the students stated an incorrect response during the 5-s delay before the teacher delivered the controlling prompt. Total error responses ranged from 37 (1.5%) for Student 4, to 60 (1.7%) for Student 1.

During intermix sessions, the performances of Students 1 and 2 occasionally showed a slight decelerating trend; this was not the case for Students 3 and 4. On the final intermix probe, all students performed at 100% correct anticipations. Maintenance probe data indicated initially high levels of sustained responding for all students with a slight deterioration in accuracy over time. Mean correct responses across all maintenance probes were 83.5% for Student 1, 94.5% for Student 2, 79.8% for Student 3, and 85.3% for Student 4. All students scored 0% correct on the pencil-and-paper generalization pretest, and all students scored 100% correct on the generalization posttest. The mean scores on daily generalization probes ranged from 95% to 98% correct.


The constant time delay procedure used in this study was effective in teaching the targeted multiplication facts, and the findings indicated that all four students learned their sets of 30 facts to an accuracy criterion of 100% in a near errorless fashion. Experimental control was demonstrated when the introduction of the intervention resulted in each student meeting criterion on each fact set and by replication of intervention effects across sets of multiplication facts and across students.

The most outstanding difference among students in terms of efficiency was the performance of Student 1, who took more trials, sessions, and direct instruction time to reach criterion and also committed more errors than the other students. This performance may have been related to the impulsive oral language behavior of Student 1 observed during the experiment and in other instructional settings. This student frequently gave an immediate incorrect response upon presentation of the stimulus card but then made a spontaneous self-correction before the end of the 5-s delay period. According to scoring criterion, only the first response was considered in judging an answer as correct or incorrect. The differences between the performance of Student 1 and those of other students may be, in part, an artifact of the scoring criterion used in the study.

The mean rate of correct responding for all responses and all students was 98.3%; this error rate of 1.7% is similar to the findings of other studies using errorless learning procedures (Browder et al., 1981; Jones, 1985; Kleinert & Gast, 1982; Precious, 1985). Errorless learning procedures appear to be efficient and less frustrating when compared with many of the more traditional trial-and-error methods of instruction (Carr, Newsom, & Binkoff, 1980; Weeks & Gaylord-Ross, 1981). It is assumed that the avoidance of error responses when learning a skill will contribute to the learner's motivation toward the learning task and self-concept as a competent learner; further study of the affective aspects of errorless learning is needed.

In addition to low error rates, the time delay procedure appears to have advantages relative to other methods used in teaching arithmetic facts. Expensive instructional materials and equipment are not required, and the time delay procedure is relatively easy for teachers to learn. Instructional sessions are very brief, and time delay may be a helpful tool to remediate the gaps in skill acquisition frequently seen in upper elementary and secondary students with mild learning and behavior disorders.

It is important to note several limitations of the present study. First, the constant time delay procedure was used to promote accurate responding to basic multiplication facts only and was not designed to teach fast rates of responding or the underlying concepts involved in multiplication important for application and problem solving. Whereas accurate recall of basic facts is essential to acquisition of arithmetic skills, it should be considered only as part of the total mathematics program.

Second, the maintenance probes were conducted without reteaching or review; teaching procedures used in this study did not address maintenance. This may explain, in part, any deterioration in student performance. Also, as implemented in this study, time delay required the teacher to work with students individually. Adaptations of this procedure for small group instruction should be considered. Finally, due to the number of students included in this study, educators should use caution when applying these findings in planning for academic instruction. Teachers using the time delay procedure should continuously collect performance data and base instructional decisions on the student's performance.

The authors have several suggestions for future implementations and study of the time delay procedure with students experiencing mild learning problems. First, a shorter delay time (possibly 2 or 3 s) could serve to promote a more fluent performance. Also, the 5-s "time out" contingency for errors (replicated from earlier studies with learners experiencing more severe disabilities) may not serve as an instructive error-correction procedure for learners with mild learning and behavior problems. Preferred approaches may include consequating errors with relevant feedback statements or contingent error drills. Finally, accepting self-corrected responses as correct anticipations may encourage students to monitor and modify their own performance.

The results of this study indicated that the instructional package featuring the time delay procedure can result in mastery of arithmetic facts with a low rate of student errors during acquisition, generalization, and maintenance. Although this study is among the first to examine the effectiveness of the time delay procedure in teaching arithmetic skills, the positive results are similar to those obtained in other studies of time delay across a wide range of academic and self-help skills with a variety of populations. The time delay procedure is clearly a powerful instructional tool that should be considered by teachers, teacher educators, and researchers.


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Browder, D. M., Morris, W. W., & Snell, M. E. (1981). Using time delay to teach manual signs to a severely retarded student. Education and Training of the Mentally Retarded, 16(4), 252-258.

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Gast, D. L., & Tawney, J. W. (1984). Dependent measures and measurement systems. In J. W. Tawney & D. L. Gast (Eds.), Single subject research in special education (pp. 112-141). Columbus, OH: Charles Merrill.

Godby, S., Gast, D., & Wolery, M. (1987). A comparison of time delay and system of least prompts in teaching object identification. Research in Developmental Disabilities, 8, 283-306.

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Jones, M. R. (1985). Comparison of delay procedures in teaching functional word reading to moderately handicapped children. Unpublished master's thesis, University of Kentucky, Lexington.

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Kleinert, H. L., & Gast, D. L. (1982). Teaching a multihandicapped adult signs using a constant time delay procedure. Journal of the Association for the Severely Handicapped, 6, 25-32.

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Oliver, C. G., & Halle, J. W. (1982). Language training in the everyday environment: Teaching functional sign use to a retarded child. Journal of the Association for the Severly Handicapped, 7(3), 50-62.

Precious, C. J. (1985). Efficiency study of two procedures: Constant and progressive time delay in teaching oral sight word reading. Unpublished master's thesis, University of Kentucky, Lexington.

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Schilling, D. E. (1985). Drill work can be fun and effective--here's how! Academic Therapy, 20(5) 551-556.

Schuster, J. W., Gast, D. L., Wolery, M., & Guiltinan, S. (1988). The effectiveness of a constant time-delay procedure to teach chained responses to adolescents with mental retardation. Journal of Applied Behavior Analysis, 21(2), 169-178.

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Snell, M. E., & Gast, D. L. (1981). Applying time delay procedure to the instruction of the severely handicapped. Journal of the Association for the Severely Handicapped, 6(3), 3-14.

Stevens, K. B., & Schuster, J. W. (1987). Effects of a constant time delay procedure on the written spelling performance of a learning disabled student. Learning Disability Quarterly, 10(1), 9-16.

Touchette, P. E. (1971). Transfer of stimulus control: Measuring the moment of transfer. Journal of the Experimental Analysis of Behavior, 15(3), 347-354.

Watkins, M. W., & Webb, C. (1982). Computer-assisted instruction with learning disabled students. Educational Computer Magazine, 1(3), 24-27.

Weeks, M., & Gaylord-Ross, R. (1981). Task difficulty and aberrant behavior in severely handicapped students. Journal of applied Behavior Analysis, 14(4), 449-463.

Yancey, B. S. (1987). Instruction for oral reading of basic sight words with a constant time delay procedure. Unpublished master's thesis, University of Kentucky, Lexington.

JUDITH C. MATTINGLY is Special Education Teacher, Fayette County Schools, Lexington, Kentucky; and DEBORAH A. BOTT is Assistant Professor, Department of Special Education, University of Kentucky, Lexington.
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Author:Mattingly, Judith C.; Bott, Deborah A.
Publication:Exceptional Children
Date:Feb 1, 1990
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