Behavior Modification

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DOI: 10.1177/0145445510361331

2010; 34; 79 Behav Modif

Kelly K. Brady and Richard M. Kubina, Jr

Attention Deficit Hyperactivity Disorder

Endurance of Multiplication Fact Fluency for Students With

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Behavior Modification

34(2) 79 –93

DOI: 10.1177/0145445510361331

http://bmo.sagepub.com

Endurance of

Multiplication Fact

Fluency for Students

With Attention Deficit

Hyperactivity Disorder

Kelly K. Brady

and

Richard M. Kubina Jr.

Abstract

This study examines the relationship between a critical learning outcome of

behavioral fluency and endurance, by comparing the effects of two practice

procedures on multiplication facts two through nine. The first procedure,

called whole time practice trial, consisted of an uninterrupted 1 minute

practice time. The second procedure, endurance building practice trials, had

three 20 second practice trials. A total of 3 students with attention deficit

hyperactivity disorder participated. Results indicated that multiplication facts

with the endurance building practice trials produced more efficient learning

when compared to the whole time practice trial procedure for all 3 partici-

pants. Additionally, results show that even with the amount of practice time

being equal, 1 minute in both conditions, on average participants practiced

30% more problems with the endurance building practice trials procedure

than they did with the whole time practice trial procedure.

Keywords

endurance, behavioral fluency, precision teaching, Attention Deficit

Hyperactivity Disorder

Pennsylvania State University

Corresponding Author:

Richard M. Kubina Jr., Pennsylvania State University, University Park, PA 16802-3109

Email: [email protected]

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80 Behavior Modiﬁcation 34(2)

Precision Teaching, a system that includes “defining instructional targets, moni-

toring daily performance, and organizing and presenting performance data in a

uniform manner to facilitate timely and effective instructional decisions” (White,

2005, p. 1433), has a history in education that now spans over 40 years (e.g.,

Lindsley, 1964, 1968, 1971, 1990, 1997). Through the years and applications of

Precision Teaching with many learners in schools and other educational settings,

a concept called behavioral fluency came into existence. The theory of behavioral

fluency posits that when a learner reaches a performance standard, a frequency

range (e.g., typing 60-90 words per minute) three critical learning outcomes will

also occur (Binder, 1996, 2005). The three learning outcomes are long-term

retention, endurance or the ability to perform a behavior over a given period of

time and with environmental distraction, and application or behavioral elements

combining to form a behavioral compound.

A database has been steadily emerging that details the effects of attaining

performance standards and the occurrence of the three critical learning out-

comes associated with behavioral fluency (Kubina, 2010). Specifically, the

research tentatively supports the proposition that behavioral fluency, as quan-

titatively defined as a performance standard, does produce skill retention and

maintenance, skill endurance and subsequent resistance to distraction, and the

application or combining of behaviors (Binder, 1996, 2005; Kubina, 2010;

Kubina & Morrison, 2000). Although the bulk of behavioral fluency

research exists for retention and application, there are fewer research stud-

ies focusing on endurance (Kubina, 2010). Endurance, or the ability to

attend to a certain task for a given length of time and in the presence of envi-

ronmental distractions (Binder 1984, 1996; Binder, Haughton, & Van Eyk,

1990), offers unique opportunities to understand and positively affect learn-

ing and behavior change programs.

For example, a study by Binder et al. (1990) examined the effects of

endurance on writing fluency. A group of more than 75 students ranging

from Kindergarten through eighth grade served as participants. The

assignment had students writing digits from 0 to 9 as quickly as possible.

As keeping all other conditions constant, teachers changed the length of

task timings. Students wrote for 15 seconds, 30 seconds, 1 minute, 2

minutes, 4 minutes, 8 minutes, or 16 minutes intervals on different days.

The results demonstrated that the students who reached a rate of 70

responses per minute performed at the same level regardless of whether

the duration was 15 seconds or 16 minutes. Students who could not write

quickly declined in performance as they progressed through the intervals.

Some students, who wrote at about 20 digits per minute, stopped writing

before the end of the 16 minute period. The study suggests that students

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Brady and Kubina 81

who have not attained minimal fluency levels, or lack endurance, can-

not maintain their performance for any time period longer than brief

durations before slowing down or stopping completely.

Based on the results of their study Binder et al. (1990) suggested that

“attention span” may be thought of as endurance. In other words, before

invoking complex cognitive processes to explain attention span, direct behav-

ioral observations of endurance offer a parsimonious alternative. Another

study showing the relationship between task fluency and increasing attention

and endurance assessed a 9-year-old boy with attention deficit hyperactivity

disorder (ADHD) on his level of endurance (McDowell & Keenan, 2001).

The researchers assessed the student by measuring how many letter sounds

were produced correctly and incorrectly per minute and how much time was

spent on task. The results showed that as the participant became more fluent

with the academic task, his attention and time spent on task also became more

pronounced. By the end of the study, the student was performing at the flu-

ency level for identifying letter sounds and was consistently maintaining his

attention for 10 minute practice periods.

The studies on endurance have shown positive effects with attention and

seem particularly well suited for students with ADHD. Endurance interven-

tions directly target specific skills and attention. If endurance is a viable alter-

native for attention span then students with attention problems should greatly

benefit from endurance interventions. In the present experiment we chose to

examine endurance in applied setting and with a skill students needed for

their current educational goals. The 3 participants, students with ADHD could

not fluently multiply basic facts. To study endurance, we evaluated the differ-

ence in students’ performance in two equivalent conditions; a 1 minute multi-

plication practice trial and three 20 second practice trials. Both conditions

were followed by a 1 minute assessment. As students with ADHD generally

have difficulty paying attention for long time intervals we hypothesized that

the students should learn more in a condition that maximizes their endurance

as opposed to one that stretches it.

Method

Participants and Setting

There were three male students, Kurt who was in fourth grade and Miles and

Mitch in fifth grade, who participated in the study. Each participant had a

diagnosis of ADHD and received special education services as Other Health

Impaired. All of the ADHD diagnoses were given by a physician or licensed

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82 Behavior Modiﬁcation 34(2)

psychologist and were documented in the students educational files. Two of

the students had comorbid conditions, Learning Disabilities. The study took

place in a learning support classroom where students received instruction for

reading, writing, and math as specified by their individualized education

plans (IEPs). The classroom was located in a midsize northeastern university

city where 14% of the students were classified as low income.

Materials

Materials used in the study included pretest screening probes. Each probe

contained all of the basic multiplication facts 0-9. The probes were randomly

generated using the single skill multiplication option from the Curriculum-

Based Measurement Warehouse for Mathematics at http://Interventioncentral

.com. Other materials were the multiplication probe sheets used during the

intervention, pencils, and a countdown timer. The construction of the multi-

plication fact sheets is discussed in the following section.

Dependent Variables

The dependent variables were the number of correct and incorrect digits writ-

ten per minute. Individual digits were defined as correct when the digit was in

the proper position. For example, the answer 4 would count as one correct

digit in response to the problem 2 × 2 =__. Digits that were rotated or reversed,

as long as the reversal or rotation did not affect the value of the number, were

counted as correct digits. If the rotation or reversal did affect the value of the

number by making it appear to be another number then it was counted as an

incorrect digit. Individual digits were also defined as incorrect when the digit

was the incorrect number to answer the problem. For instance, the answer 6

would count as one incorrect digit in response to the problem 2 × 2 =__. If the

digit was in an incorrect position or place value, for example answering 01

instead of 10 to the problem 2 × 5 =__, the response would be counted as two

incorrect digits. If only one digit was incorrect in a two digit answer, such as

answering 23 to the problem 6 × 4 =__, the response would count as one cor-

rect digit (the 2) and one incorrect digit (the 3). Skipped problems did not

count for or against the total score. Sessions were conducted at approximately

the same time each day for each participant in the same physical setting

(i.e., their desk or a small table with the experimenter).

There were a total of × 8 multiplication fact sheets. To control for diffi-

culty, all problems containing a 0 or 1 were excluded during preassessment

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Brady and Kubina 83

and during the construction of the 8 multiplication fact sheets. Math facts were

generated to form a set of problems used with either the endurance building

practice trials or the whole time practice trials on an alternating schedule. All

problems from 2 × 2 to 9 × 9 were listed on a sheet of paper. Then, each

problem along with its associate problem was assigned to one of the practice

trial conditions. Associate problems were defined as a problem that would be

found in the same fact family. For example, the associate problem of 6 × 8

would be 8 × 6. Problems with identical numbers such as 2 × 2 have no asso-

ciate problems but these problems were distributed evenly by the alternating

schedule.

Once all problems were assigned, the endurance building practice trials set

contained 31 problems with the opportunity to write 58 correct digits per min-

ute. The whole time practice trials set had 31 problems with the opportunity to

write 60 correct digits per minute. Problems in both sets were of equal diffi-

culty because of the split distribution used to create each set. Furthermore, an

analysis of the students’ performance with each set before the intervention

began showed that all students performed similarly with each set. Four sheets

were created for each practice condition by taking the stock of problems for

the set and randomizing all of the problems. The randomization step helped

guard against the any serial memorization.

Independent Variables

The independent variables in this study were endurance building practice tri-

als and whole time practice trials. Both conditions provided an equivalent

amount of practice, 1 minute and then 30 seconds of corrective feedback. The

conditions differed only in the breakdown and sequencing of the practice tri-

als and corrective feedback. The endurance building practice trials condition

allowed for three 20 second practice trials with immediate 10 second correc-

tive feedback sessions following each practice trial. The whole time practice

trial condition provided 1 minute uninterrupted practice trial followed by

30 seconds of corrective feedback. Therefore, both conditions had the same

amount of practice time and the same amount of feedback with the only dif-

ference coming in the temporal segmentation of practice trials (i.e., three

20 second practice trials vs. one 60 second practice trial).

Experimental Design

An alternating treatments design (Cooper, Heron, & Heward, 2007) was used

to examine the effects of the two practice conditions. Each participant was

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84 Behavior Modiﬁcation 34(2)

evaluated using 1 minute assessment timing. During intervention, the two prac-

tice conditions were alternated daily over 4 weeks during the experimental

sessions. Additionally, the practice conditions were counterbalanced to mini-

mize multitreatment interference effects.

Procedure

Initial assessment. To determine current level of performance, each partici-

pant had to take five pretests containing math facts with numbers 0 through 9.

They were timed for 1 minute and were told to stop and mark where they

finished at the end of the minute. Then they were instructed to continue the

rest of the problems that they knew on the sheet. The pretest determined when

the students had acquired the requisite amount of fact to be considered in the

instructional stage. For Grades 4 and up the instructional level for multiplica-

tion facts are 20-39 correct digits per minute (Deno & Mirkin, 1977). In the

present study, two of the three students (i.e., Miles and Mitch) met the criteria

for instructional level before beginning the intervention. Kurt did not attain

instructional level before beginning the intervention but the experimenter

decided to implement the intervention because it could still benefit him.

Intervention. Two different interventions, endurance building practice trials

and a whole time practice trial, were introduced to the participants after the

initial assessment. The endurance building practice trials procedure consisted

of three 20 second practice trials interspersed with 10 second corrective feed-

back sessions, followed by a 1 minute multiplication math fact sheet assess-

ment. The whole time practice trial incorporated a one 1 minute practice trial

with a 30 second corrective feedback session, followed by a 1 minute multipli-

cation math fact sheet assessment. The students sat at a desk across from the

experimenter and were read standardized directions for each of the timings.

Directions included the instruction to perform the skill as quickly as possible.

For each timing, the participants wrote the answers to the multiplication

problems on each of the multiplication math fact sheets as quickly as they

could for the amount of time allotted. Corrective feedback was given in each

condition to correct errors and provide answers to skipped problems. During

each of the practice trials procedure two timers were used. One timer was

used to time the students’ performance on the multiplication math fact sheets

and the other timer was set for either 10 seconds or 30 seconds to time the

experimenter giving feedback. The two timers ensured equivalency of feed-

back between the conditions.

The error correction procedure used for both incorrect and skipped prob-

lems was a model-lead-check method (Stein, Kinder, Silbert, & Carnine, 2006).

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Brady and Kubina 85

For example, if the participant wrote the answer 5 to the problem 4 × 4 =__, the

examiner would restate the problem with the answer, ask the student to say the

problem, and then ask the student to answer the problem. In both conditions,

incorrect answers were corrected first and if time permitted the model-lead-

check method was used with answers to skipped.

Interobserver Agreement

Earlier to baseline, an independent observer was instructed how to identify

correct and incorrect digits for each probe. The independent observer was

shown how to determine correct and incorrect digits, as described in the depen-

dent variable section, and then asked to apply those rules and score a variety

of hypothetical multiplication math fact sheets for correct and incorrect digits.

This procedure was repeated until the experimenter and independent observer

reached 80% agreement on the identification of correct and incorrect digits.

Interobserver agreement checks were conducted for 100% of baseline ses-

sions and 100% of intervention sessions. Using total agreement (dividing the

smaller total by the larger total and multiplying by 100), interobserver agree-

ment for this study was 100%.

Procedural Integrity

Fidelity of treatment was observed using a checklist of procedures during

25% of intervention sessions. An independent observer was given the check-

list and observed the experimenter to determine her consistency in applying

the methods outlines in the procedure section. The results from the indepen-

dent observer indicated that the experimenter followed the steps with 100%

accuracy.

Results

Figures 1-3

show the results for Kurt, Miles, and Mitch. Each graph has an

alternating treatments design with solid dots showing the assessment of

the endurance building practice trials or the three 20 second practice trials.

The open circles represent the measurement of the whole time practice trial

or the one 60 second trial. Figure 1 shows the minute assessment frequen-

cies for Kurt. Kurt’s whole time practice trial responses ranged from 8 to 15

correct per minute. His performance in the endurance building practice tri-

als ranged from 6 to 22. As the two interventions are implemented a separa-

tion emerges in the endurance building practice trial data set with only one

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86 Behavior Modiﬁcation 34(2)

Figure 1. Kurt’s correct per minute frequency during endurance building practice

trials and whole time trial conditions

instance where the two conditions meet on the second session. Kurt’s data

show the endurance building practice trials produce a higher level of learn-

ing when compared to the other condition. Although not as rapid, Kurt does

show an upward trend of answering multiplication facts in the whole time prac-

tice trial condition.

Figure 2 shows the results for Miles. The range for the endurance building

practice trials was 26 to 45 per minute whereas Miles performance in the

whole time practice trial condition ranged from 18 to 36. In all instances,

Miles performed at a higher level in the endurance building practice trials

condition. There is a wide degree of separation between the two conditions.

Similar to Kurt, although Miles performed at a lower level in the whole time

practice trial condition he still showed an increase answering multiplication

facts per minute.

The last graph, Figure 3, displays the results for Mitch. Mitch has a range

of 28 to 32 correct answers per minute for the whole time practice trial con-

dition and a range of 22 to 36 for the endurance building practice trials con-

dition. Mitch begins by answering more problems in the whole time practice

trial condition with a crossover to a higher performance in the endurance

building condition during the second session. On the sixth session Mitch also

has a performance higher than his performance in the endurance building

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Brady and Kubina 87

Figure 2. Mitch’s correct per minute frequency during endurance building practice

trials and whole time trial conditions

Figure 3. Mile’s correct per minute frequency during endurance building practice

trials and whole time trial conditions

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88 Behavior Modiﬁcation 34(2)

practice trials condition. Mitch shows a modest increase in the number of

multiplication facts answered per minute during the whole time practice trial

condition. For the majority of data points Mitch performs at a higher level

during the endurance building practice trials condition.

Beyond the alternating treatments design performances, the number of

problems practiced within the multiplication fact sheets were calculated for

both conditions. On average, the students practiced 30% more problems with

the endurance building practice trials condition than they did in the whole

time practice trial condition. Specifically, Kurt practiced 127 problems during

the endurance building practice trials condition and only 81 in the whole time

practice trial condition. Similarly, Miles practiced 286 to 213 problems and

Mitch practiced 289 to 192 problems in the endurance building and whole

time trial conditions.

Discussion

Endurance refers the ability to attend to a certain task for a given length of

time and in the presence of environmental distractions (Binder 1984, 1996,

2005; Binder et al., 1990). In this study we hypothesized that students with

ADHD would benefit from an intervention that directly focused on building

endurance. The results of this study indicate the endurance building practice

trials did indeed produce greater learning. The three alternating treatments

design shows a separation of data paths favoring the endurance building

practice trials condition for all participants.

Each of the students responded similarly to the endurance practice trial

conditions but had slight differences. For example, Kurt’s first session shows

he performed better in the whole practice trial condition but by the third ses-

sion he answered significantly more multiplication problems on the 1 minute

multiplication fact sheet, 17 versus 11 per minute. At the end of the study Kurt

maintained a differential of seven more digits written correctly per minute

(i.e., 22 vs. 15) with the endurance practice trial conditions. Among the 3 par-

ticipants, Kurt started off with the lowest frequencies for both conditions but

still responded positively to both practice conditions by steadily answering

more multiplication problems correctly.

Miles data, Figure 2, shows the clearest separation of data paths within

the 3 participants. Miles first performance was higher during the endurance

building practice trials from the beginning and never overlapped with the

whole time practice condition thereby showing the clearest effect for an

endurance building condition. Even though Miles did not practice the prob-

lems in the endurance practice trial condition for 1 minute, his performance

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Brady and Kubina 89

for 1 minute on the multiplication fact sheet rapidly increased and was

clearly superior to the other condition. The differential between responding

on the first trial was eight more digits written correctly per minute, 26 versus

18 digits written correctly per minute. The last session data show a very

similar differential with nine more digits written correctly per minute, 45

versus 36 digits written correctly per minute. Developing endurance with the

brief practice trials was demonstrated very clearly by Miles’ data.

Mitch’s performance also supports the use of shorter practice trials for

building endurance though are not as clear as Miles. Mitch began the session

with a higher data point for the whole trial condition but for the majority of the

other sessions, aside from the sixth, Mitch performed better with the endur-

ance building practice trials. Mitch’s performance along with the two other

participants support the findings by McDowell and Keenan (2001); building

endurance produces a higher frequency of responding when compared to con-

ditions that tax endurance.

On a practical level, the students’ performance data points to an important

teaching practice; namely, using procedures that enhance endurance. All stu-

dents’ learned in the whole time trial condition but did so faster when the

practice method more readily augmented endurance. The endurance building

condition had students practice for a shorter period thereby facilitating atten-

tion or more accurately sharpening stimulus control. With the increased atten-

tion (i.e., endurance) the students also practiced more problems overall when

compared to the whole trial condition. In contrast, the whole trial practice

condition may have produced fatigue resulting in a decreased number of

problems the students could attend to and subsequently practice. Similar to

previous studies (i.e., Berens, Boyce, Berens, Doney, & Kenzer, 2003; Kim,

Carr, & Templeton, 2001; McDowell & Keenan, 2001) teachers examining

other academic behaviors may discover interventions directed specifically at

addressing and enhancing endurance.

The results from the present study not only demonstrate endurance function-

ing in a capacity of attending to a task for a given length of time, but also sug-

gest endurance may function as a sensitive indicator for behavioral persistence.

For instance, Berens et al. (2003) found how well students performed for 3 min-

utes predicted how well students would perform for 5 minutes. McDowell and

Keenan (2001) also noted the effects of 1 minute performance and its relation-

ship to performing for 10 minutes. In the present study, the shorter timing

intervals (i.e., 20 seconds) were also positively related to changes on the larger

timing interval (i.e., 1 minute). Future studies may further validate the use of

shorter timing intervals for reliably predicting performance with longer time

intervals. Such information would help assessment and intervention plans.

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90 Behavior Modiﬁcation 34(2)

One last implication of this study is the fact that all participants were diag-

nosed with ADHD. Children with ADHD have been described as having a

difficult time sustaining attention to relevant stimuli and typically have impul-

sive responses to nonrelevant stimuli (Barkley, 1991). Additionally, the

American Psychiatric Association (2000) has defined ADHD as a “persistent

pattern of inattention and/or hyperactivity-impulsivity that is more frequently

displayed and more severe than is typically observed in individuals at a com-

parable level of development” (p. 85). Attention, as a cognitive construct how-

ever, does not serve a behavioral analysis well because attention reifies attending

behavior (Dube, MacDonald, Mansfield, Holcomb, & Ahearn, 2004).

Attending behavior and the concomitant concept of attention span or how

much time a person can concentrate or attend to an activity, benefits from a

directly observable measure such as endurance. Binder et al. (1990) describes

the idea as follows:

Until students attain certain minimum levels of speed and accuracy . . .

they typically lack the ability to maintain steady performance levels for

extended periods of time. . . . when learners approach fluency . . . they

become able to work steadily for significant durations. . . . Endurance, or

attention span, thus follows or is a byproduct of behavioral fluency. (p. 25)

As demonstrated by Kurt, Miles, and Mitch, as their performance levels for

the three 20 second intervals improved so too did their ability to perform over

the longer time interval of 1 minute. Stated differently, the participants’ perfor-

mance with writing answers to multiplication problems improved most under

a condition of focused repetition (i.e., endurance building practice trials) and

improved modestly during a condition of extended repetition (i.e., whole time

practice trial condition). At best, the construct of attention in clinical definitions

of ADHD does little more than provide a broadly descriptive pattern of behav-

ior. At worse, attention may suggest interventions that target the actual problem

indirectly and perhaps incorrectly. Endurance interventions offer a direct, mea-

surable, and reliable alternative for addressing many learning problems.

Limitations

First, although the directions for each probe were to work across the page

and not skip to known problems, participants did not always follow the

directions despite redirecting prompts. Skipping problems could have

affected the students’ rate at which they were learning and developing flu-

ency. Students skipping problems and not attempting harder problems also

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Brady and Kubina 91

meant that students could have been responding to easiest and then slighter

harder problems rather than attempting all problems in the domain of prob-

lems selected for each multiplication facts sheet. Second, this study was

ended due to the academic year ending. Therefore, with extended time the

students would have been able to reach a fluency criterion and have

extended time with both conditions thereby more clearly establishing the

effects of each condition. And third, this study mainly focused on the first

part of the definition of endurance and did not assess students’ responses in

the face of environmental distractions.

Conclusion

The findings from the study support the behavioral concept of endurance and

its importance as an intervention. Both practice interventions helped the three

students with ADHD to improve their frequency of writing correct answers to

basic multiplication problems. The endurance building practice trials condi-

tion, three 20 second practice trials, produced more rapid learning and a greater

number of problems practiced when compared to the whole time practice trial

condition, an uninterrupted 1 minute practice time. The findings support

endurance, an outcome of behavioral fluency, suggesting the ability to attend

to a certain task for a given length of time and in the presence of environmental

distractions can be experimentally tested and verified. Furthermore, endurance

provides an alternative, behavioral description of attention span, and a possi-

ble means to help students with ADHD to improve multiplication skills.

Note

1. All data are available on Standard Celeration Charts. For copies please email the

second author at rmk1[email protected].

Declaration of Conflicting Interests

The authors declared no potential conflicts of interests with respect to the authorship

and/or publication of this article.

Funding

The authors received no financial support for the research and/or authorship of this

article.

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ematics instruction: A direct instruction approach. Reading, MA: Pearson.

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applications (Vol. 3, pp. 1433-1437). Thousand Oaks, CA: Sage.

Bios

Kelly K. Brady received her MA in Special Education at Pennsylvania State University.

She is currently working as a special educator in Alexandria, Virginia.

Richard M. Kubina, Jr., PhD, is an associate professor in the Special Education Pro-

gram at Pennsylvania State University. His current research interests include mea-

surably effective curricula and the effects of behavioral fluency within educational

and social settings.

at PENNSYLVANIA STATE UNIV on March 31, 2010 http://bmo.sagepub.comDownloaded from