The 1-minute explicit timing intervention: the influence of mathematics problem difficulty.

This study examined the effects of explicit timing with varying levels of mathematics tasks. Fifty-four students in the sixth-grade completed a one-step addition task (1st grade level), a three-step subtraction task (3.5 grade level), and a complex multiplication task (6th grade level) during a no timing condition and a timing condition. During baseline, students were told to correctly complete as many problems as possible. Students were not informed of a time deadline; however, 3 minutes were allowed per mathematics task (i.e., researcher covertly timed). During intervention, students were told that they had 3 minutes to correctly complete as many problems as possible. A 2 (timing) by 3 (assignment) within-subjects analysis of variance indicated that students completed more problems correct per minute during the explicit timing condition than during the no timing condition while maintaining accuracy. Furthermore, on the one-step addition task and the three-step subtraction task, students completed more problems correct per minute during the explicit timing condition when compared to the no timing condition. In conclusion, explicit timing appears to have differential effects based on the complexity of the academic task. This intervention is effective in the classroom for basic skills review but not for more complex mathematics tasks.

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Explicit timing is a procedure that alerts students to a time limit while they are completing an academic assignment. Van Houten and Thompson (1976) compared one-minute explicit timing intervals to a 30-minute interval without the explicit timing. Second grade students with poor school performance completed 1 digit plus 1 digit addition problems and 1 digit minus 1 digit subtraction problems. Van Houten and Thompson (1976) used an ABAB reversal design for this study. The results indicated that students completed more problems correct per minute during the explicit timing conditions. During the second baseline, the number of problems completed correct per minute decreased but then increased again when the second explicit timing condition was implemented. Accuracy rates remained between 90 and 100% for all experimental conditions. Therefore, explicit timing was effective with second grade students for single digit addition and subtraction problems.

Miller, Hall, and Heward (1995) replicated Van Houten and Thompson (1976) by comparing one-minute explicit timing intervals to a 10-minute interval without the explicit timing. Two classrooms of students participated in this experiment, 23 students in a first grade classroom and 11 students between the ages of 9 and 12 years old in a special education classroom. A pre-experimental assessment was conducted to identify the type of math facts that most students could answer correctly. Four types of math facts were assessed: addition problems with sums less than 10, addition problems with sums between 10 and 18, subtraction problems with minuends less than 10, and subtraction problems with minuends between 10 and 18. The pre-experimental assessment indicated that most students could not answer correctly the subtraction problems with minuends between 10 and 18; therefore, these math problems were not included in the math packet. For the experiment, mixed math sheets were created which included addition problems with sums less than 10, addition problems with sums between 10 and 18, and subtraction problems with minuends less than 10. The results of the Miller, Hall, and Heward (1995) study replicated the Van Houten and Thompson (1976) study. Students completed more math problems correct during the one-minute timing intervals than during the 10 minute work interval without the explicit timing. Furthermore, accuracy rates remained between 82-89% for the explicit timing condition and the 10 minute condition. Therefore, explicit timing appears to be effective for children in regular education and special education when completing simple addition and subtraction problems.

Furthermore, explicit timing has been shown to be effective for African-American children and Caucasian children when comparing four 1-minute timing intervals to a four-minute interval without the explicit timing. Rhymer, Henington, Skinner, and Looby (1999) implemented the explicit timing procedure with African-American and Caucasian second-grade students. The mathematics sheet consisted of 1 digit plus 1 digit and 1 digit minus 1 digit problems. Results indicated that the children completed more problems during the explicit timing intervention than during the no timing condition. Furthermore, accuracy rates remained consistent in the low to middle 80's from the no timing condition to the timing condition. Therefore, explicit timing is effective at increasing the number of problems completed while maintaining accuracy for second-grade students on 1 digit plus 1 digit and 1 digit minus 1 digit mathematics problems. However, some research with explicit timing has shown a decrease in accuracy rates.

Rhymer, Skinner, Henington, D'Reaux, and Sims (1998) implemented the explicit timing procedure with third-grade students using a multiple baseline design and accuracy rates decreased. The mathematics sheet consisted of mixed problems with 2 digits plus 1 digit, 2 digits plus 2 digits, 2 digits minus 1 digit, and 1 digit times 1 digit problems. None of the problems involved carrying or borrowing. Students completed more problems during the explicit timing intervention. However, accuracy levels decreased slightly which was contrary to previous research (Rhymer et al., 1999; Miller, Hall, & Heward, 1995; Van Houten & Thompson, 1976). The researchers hypothesized that the decrease in accuracy was related to students' pre-intervention accuracy levels. Therefore, the researchers ranked the students based on accuracy levels and then divided the students into three equal groups (low, medium, and high accuracy levels). The ranked accuracy data suggested that decreases in accuracy were due to the students' pre-intervention level of accuracy. Specifically, the groups with low and middle accuracy levels during baseline had the greatest decrease in accuracy levels daring the timing condition. Therefore, Rhymer, et al. (1998) suggested that the differential results regarding the explicit timing procedure may be contingent on students' skill profiency development levels (i.e., acquisition, fluency, adaptation, and generalization). However, the researchers did not examine the type of mathematics problems being studied. Maybe the students were fluent in 2 digits plus 1 digit, but not fluent on 1 digit times 1 digit. Therefore, explicit timing may have differential results based on the type of mathematics problem. Therefore, the purpose of the current study was to examine explicit timing with older children using a variety of math problems ranging from easy (1 digit plus 1 digit addition) to difficult (3 digits times 3 digits multiplication).

Method

Participants and Setting

Two 6th grade teachers who taught mathematics volunteered three classrooms each for this study. The classrooms consisted of mixed ability students. All students were enrolled in the 6th grade in a Mid-South public district. A total of 54 students participated in this study. The average age of the students was 11.5 years with a range of 10 to 13 years. Twenty-eight of the students were male and 26 were female. Nineteen students were Caucasian, 34 were African-American, and one was of Indian descent. All sessions were conducted in the students' mathematics classroom in their assigned seats.

Materials

Experimenters constructed the mathematics assignment sheets used in this experiment. An assignment sheet consisted of 98 total problems printed on both sides of an 8.5 by 11 inch sheet of paper. There were three types of sheets: addition, subtraction, and multiplication problems. The addition sheet consisted of one-digit plus one-digit addition problems using the numbers zero through nine. The addition sheet represented a mathematics task taught in the beginning of 1st grade and only involved one-step addition (i.e., grade level 1.0). The subtraction sheet consisted of three-digits minus three-digits subtraction problems using the numbers zero through nine with no borrowing from adjacent columns. The subtraction sheet represented a mathematics skill taught in the middle of 3rd grade and involved a three-step solution (i.e., grade level 3.5). The multiplication sheet consisted of three-digits times three-digits multiplication problems using the numbers zero through nine. The multiplication sheet represented a mathematics skill taught in the beginning of 6th grade and involved a complex multiplication and addition solution (i.e., grade level 6.0). The numbers were randomly chosen using a random numbers table; however, if a particular number was not appropriate for a problem (i.e., 213-107) then the next number on the table was selected (i.e., 213-101). These random numbers were used to create two forms for each type of sheet (i.e., two addition sheets, two subtraction sheets, and two multiplication sheets). These mathematics sheets were stapled together to form a packet. Each packet consisted of an addition sheet, a subtraction sheet, and a multiplication sheet. These sheets were presented in counterbalanced order across packets. Wrist watches were used for covert timing and stopwatches were used for explicit timing.

Design

A 2 by 3 within-subjects design was utilized. The within subjects variables were the timing condition (untimed and timed) and the type of assignment (grade 1.0, grade 3.5, and 6.0). A Tukey HSD test was used to follow-up all significant effects. Stevens (1990) indicated that repeated measures is a powerful design because each subject is compared to themselves; therefore, fewer subjects are needed. However, possible disadvantages of this design include sequence effects and carryover effects (Stevens, 1990). In this study, counterbalancing of assignment controlled for some sequence effects (i.e., which assignment was first-addition, subtraction, or multiplication). Furthermore, sessions were separated by 24 hours which may have decreased carryover effects (Barlow & Hersen, 1984). An alpha of .05 was used for statistical significance. Power and effect size was also obtained.

Dependent variables

The dependent variables in this study were problems completed per minute and percent of completed problems that were correct. A problem was scored as complete if it contained the minimum number of digits in the answer. The number of problems completed per minute was a measure of response rate. Percent of completed problems correct was a measure of accuracy. It was calculated by dividing the number of problems correct by the number of problems complete and multiplying by 100.

Procedures

Days one, two, and three consisted of the untimed conditions. Packets were distributed to all participants and students were told to: (a) work as many mathematics problems correct as they could on the first sheet, (b) try to work all of the problems, (c) not skip around, (d) work on the back side of the sheet if they completed the front side, (e) put their pencil in the air and be quiet if they completed both sides of the sheets, and (f) not to go to the next sheet. The researcher timed covertly for 3 minutes. This procedure continued for two more trials, so that each student completed an addition, subtraction, and multiplication sheet.

Days four, five, and six consisted of the timed conditions. These conditions were identical to the untimed conditions with two exceptions: students were given information regarding the amount of time to complete the task and 1 minute timing intervals. Students were told: (a) they would have 3 minutes to work as many mathematics problems correct as they could on the first sheet, (b) during these 3 minutes they would be timed for 1 minute intervals with a stopwatch, (c) try to work all of the problems, (d) not skip around, (e) work on the back side of the sheet if they completed the front side, (f) put their pencil in the air and be quiet if they completed both sides of the sheets, and (g) not to go to the next sheet. The stopwatch was shown to students. After 1 minute had expired, students were told to stop, underline the last number written, and put their pencils in the air. Researchers then instructed students to begin (i.e., continue working on the same problem) for the next 1 minute timing. After all 3 intervals of 1 minute each had expired on the first sheet, students were instructed to turn to the next sheet and the same procedures were repeated for the next two sheets.

Treatment Integrity

A trained graduate student collected treatment integrity data on 33% of the sessions. Treatment integrity was calculated by dividing the number of steps completed correctly by the total number of steps completed correctly and incorrectly. Treatment integrity was 95% for the observed sessions.

Interscorer Agreement

Thirty percent of the assignment sheets were randomly selected for interscorer agreement on digits correct. Total interscorer agreement was calculated by dividing the number of agreements of digits correct by the number of agreements of digits correct plus disagreements of digits correct and multiplying by 100. Total interscorer agreement of digits correct was 91% for the scored assignments.

Results

Problems Completed per Minute

The interaction effect for timing by assignment was statistically significant, F (2, 106) = 10.05, MSE = 5.76. Students completed significantly more problems per minute on the addition assignment during the timing condition than on the no timing condition. Students also completed significantly more problems per minute on the subtraction assignment during the timing condition than on the no timing condition. See Table 1 for means and standard deviations for problems completed per minute.

The effect for timing was statistically significant, F (1, 53) = 34.39, MSE = 6.60. Students completed significantly more problems per minute in the timing condition than in the no timing condition. The effect for assignment was statistically significant, F (2, 106) = 808.33, MSE = 26.20. Students completed significantly more problems per minute on the addition assignment than both the subtraction assignment and the multiplication assignment. Students also completed significantly more problems per minute on the subtraction assignment than on the multiplication assignment.

Percent of Completed Problems Correct

The effect for timing was not statistically significant, F (1, 53) = 0.254, MSE = 190.5. The effect size was .005. The observed power for this effect was 0.079. The effect for assignment was statistically significant, F (2, 106) = 76.57, MSE= 618.28. The percent of completed problems correct was significantly higher on the addition assignment than the multiplication assignment. The percent of completed problems correct was also significantly higher on the subtraction assignment than the multiplication assignment. The interaction effect for timing by assignment was not statistically significant, F (2, 106) = 0.201, MSE = 191.99, p = .82. The observed power for the interaction effect was 0.081.

Discussion

Overall, explicit timing increased the number of mathematics problems completed per minute in the 6th grade students. These results replicate previous research with 2nd and 3rd grade students (e.g., Rhymer et al., 1998; Van Houten & Thompson, 1976). Therefore, explicit timing procedures appear to be effective for older students as well as for younger students.

The current study extended the literature on timing procedures by examining the effects of explicit timing along with mathematics problem difficulty. Previous research on explicit timing has not accounted for problem difficulty. Data analysis indicated that the assignment sheets were statistically significantly different from each other on problems completed per minute. Students performed better on the one-step addition assignment than on both the three-step subtraction and complex multiplication assignment. Furthermore, students performed better on the three-step subtraction assignment than on the complex multiplication assignment. Students also had higher accuracy rates on the one-step addition assignment as compared to the complex multiplication assignment and had higher accuracy rates on the three-step subtraction assignment as compared to the complex multiplication assignment. Therefore, the 1 digit plus 1 digit addition problems were the easiest, while the 3 digits times 3 digits multiplication problems were the hardest.

On problems completed per minute, students performed significantly better on the one-step addition assignment during the timing condition than during the no timing condition. Therefore, explicit timing was effective on the easy 1 digit plus 1 digit addition problems. Also, students completed significantly more problems per minute on the three-step subtraction assignment during the timing condition than during the no timing condition. Therefore, explicit timing was also effective on the 3 digits minus 3 digits subtraction problems. However, students did not perform statistically significantly better on the number of problems complete per minute on the complex multiplication assignment during the timing condition. Therefore, explicit timing was not effective on the difficult 3 digits times 3 digits multiplication problems. The results suggest that explicit timing should only be utilized when the academic assignment involves simple steps versus complex steps. When the student is completing difficult problems such as 3 digits times 3 digits multiplication, the data indicate that explicit timing may not be effective at increasing the number of problems completed per minute.

Percent of completed problems correct remained constant from the no timing condition to the timing condition. Constant accuracy levels from the no timing condition to the timing condition during explicit timing procedures replicates the accuracy findings in Rhymer, et al. (1999). However, this finding is in contrast to Rhymer et al. (1998) findings of lower accuracy rates during the timing condition and Van Houten and Thompson's (1976) findings of higher accuracy rates during the timing condition.

Limitations

The current study may be limited due to sequence effects. Students received three days of the no timing condition and then three days of the timing condition; therefore, it is possible that the increase during the timing condition was due to the practice on the three days of the no timing condition. However, no corrective feedback was provided during the no timing condition; therefore, learning would be minimal. Furthermore, Rhymer et al. (1998) conducted a multiple baseline with three different groups of students and the number of problems completed increased for each group when the explicit timing condition was implemented. Therefore, it is unlikely that sequence effects accounted for the results of this study.

Furthermore, the current study may be limited due to issues of external validity. For example, researchers implemented the procedures rather than teachers. Different results may have been obtained if the students' general education teacher implemented the protocol.

The results may also be due to the novelty effects of the explicit timing procedures. Researchers have not examined the effectiveness of long-term implementation of explicit timing (i.e., 10 minutes daily for the entire school year).

Future research

This was the first study to investigate the effects of explicit timing and mathematics problem difficulty. Therefore, replication of these results with different aged students along with varying levels of problem difficulty is needed. Future research in explicit timing also needs to examine the longitudinal effects of explicit timing. The increased rates of accurate responding occasioned by explicit timing may decrease over repeated sessions without reinforcement for high rates of responding. Therefore, researchers should investigate the effects of the explicit timing intervention on a daily basis in a classroom for a school year.

Rates of learning are influenced by the quantity of trials during the time allotted for the academic task (Skinner, Fletcher and Henington., 1996). During explicit timing, students respond to more academic problems (i.e., more trials) in the same amount of time as that given for the no timing condition. However, researchers should examine why explicit timing increases rates of responding. Determining the causal mechanism will allow educators to identify when explicit timing will be effective, for whom it will be effective, and develop other interventions that may be effective.

The explicit timing procedure should also be compared to other interventions that increase learning rates during independent seat work. The amount of time allotted for each intervention should also be examined. For example, one intervention may require 30 minutes to learn 15 problems and another intervention may require 15 minutes to learn 10 problems. The intervention that requires the least amount of classroom time and results in the higher rates of learning is the more effective intervention (Skinner et al., 1996).

Lastly, research should be conducted on students' and teachers' perception of the explicit timing procedure. Anadodal reports suggest that students enjoy the explicit timing procedure. However, researchers should ask students if they prefer the explicit timing condition over the no timing condition. Furthermore, teachers should be surveyed regarding treatment acceptability for this intervention. It is hypothesized that students will prefer the explicit timing condition over the no timing condition and that teachers will rate this intervention as acceptable.

Table 1
Problems Completed per Minute
for Timing Condition and
Level of Assignments

                  No Timing     Timing
Level             M      SD    M     SD

Addition         22.7   6.7   30.9   7.7
Subtraction      10.6   6.5   12.1   4.2
Multiplication    1.5   1.7    1.9   2.2
(n=54)

Table 2
Percent of Completed Problems
Correct for Timing Condition and
Level of Assignment

                  No Timing      Timing
Level              M     SD     M      SD

Addition         98.4    1.8   98.2    2.9
Subtraction      96.6    4.6   96.7    6.7
Multiplication   62.3   36.3   60.2   34.5
(n=54)

References

Miller, A. D., Hall, S. W., & Heward, W. L. (1995). Effects of sequential 1-minute time trials with and without inter-trial feedback and self-correction on general and special education students' fluency with math facts. Journal of Behavioral Education, 5, 319-345.

Rhymer, K. N., Henington, C., Skinner, C. H., & Looby, E. J. (1999). The effects of explicit timing on mathematics performance in second-grade Caucasian and African-American students. School Psychology Quarterly, 14, 397-407.

Rhymer, K. N., Skinner, C. H., Henington, C., D'Reaux, R. A., & Sims, S. (1998). Effects of explicit timing on mathematics problem completion rates in African-American third-grade elementary students. Journal of Applied Behavior Analysis, 31, 673-677.

Skinner, C. H., Fletcher, P. A., & Henington, C. (1996). Increasing learning rates by increasing student response rates: A summary of research. School Psychology Quarterly, 11, 313-325.

Stevens, J. P. (1990). Intermediate statistics: A modern approach. Hillsdale, NJ: Lawrence Erlbaum.

Van Houten, R., & Thompson, C. (1976). The effects of explicit timing on math performance. Journal of Applied Behavior Analysis, 9, 227-230.

Katrina N. Rhymer, Central Michigan University. Christopher H. Skinner, University of Tennessee--Knoxville. Shantwania Jackson, Stephanie McNeill, Tawnya Smith, and Bertha Jackson, Mississippi State University.

Correspondence concerning this article should be addressed to Dr. Katrina N. Rhymer, Central Michigan University, Department of Psychology, Sloan 101, Mt. Pleasant, MI 48859; Email: [e-mail]