do your students really know their math facts?

DO YOUR STUDENTS REALLY KNOW THEIR MATH FACTS? Using Daily Time M a i s to Build Fluency Guidelines for conducting an effective method of building fluency—the time thai

By April D. Miller and William L. Heward

E

ach day, in thousands of classrooms throughout the country, millions of students practice their math facts. For the vast majority of those students, the feedback they receive for their efforts is in terms of the percentage of problems they attempted that they answered correctly. The concern for accuracy is understandable,- students who cannot add and subtract correctly face major difficulties in school and in life. But how do we know when a student has mastered his or her math facts? Is he or she prepared to learn other more complex math skills once he or she has met that almost universal classroom standard of 90% accuracy? Tonya knows all the right answers to every math fact, but is that enough? Does 100% accuracy ensure that she is capable of adding and subtracting at the speed required on the playground or in the grocery store? Many of the everyday skills we use at home, in the community, or on the job must be performed at a certain rate or speed in order to be functional. 98

Accuracy measures alone do not tell the whole story. A student's level of proficiency, or fluency of responding, is a critical component of mastery. There is a need to move beyond simply assessing how accurately math facts are answered to include a measure of the speed at which those answers are forthcoming. All students who perform at the same level of accuracy are not equally skilled. While two students might each complete the same page of math problems with 100% accuracy, the one who finishes in 2 minutes is more accomplished than the one who needed 5 minutes to answer the same problems.

Fluency as a Measure ef Student Learning

There are several important reasons why a measure of fluency—number of correct responses per minute—should be part of assessing student progress. First, fluency provides a more complete picture of learning and performance. Accuracy measures provide information only on the correctness of

performance, whereas rate of response gives a precise indication of the accuracy of performance in relation to the amount of time required for response. This combination of the two dimensions of responses—accuracy and number of responses over time—gives a more complete picture of learning (West, Young, & Spooner, 1990). Second, rate per minute is a more sensitive measure of changes in performance than an accuracy measure alone (Howell & Lorson-Howell, 1990). An artificial ceiling of 100% correct is imposed when only accuracy is measured, making it impossible to detect performance differences from one measurement to the next once a student has learned to perform with accuracy. When correct responses per minute are measured and recorded, however, the teacher can detect and reward small improvements between repeated measures of the same student's performance. Third,fluencyhas critical functional implications both in and out of school. Many math skills and tasks need to be performed quickly. The required rate

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of response is determined by the environment in which the skill is performed. Some standardized tests, for example, require that correct answers be given within 3 to 5 seconds. A student may be able to calculate the correct answer, but not quickly enough to meet the test requirements. Fourthgrade students may be given 5 minutes to complete a 100-problem math fact test. A teacher of seventh graders may require a long division problem, with all of its basic fact components, to be completed in 2 minutes. A long line at the checkout counter means the cashier may not wait very long for a customer to pay or count his change. In each of these situations, both speed and accuracy are required. There are also some data suggesting a positive correlation between fluency and improved maintenance and generalization of skills. Ivarie (1986) found that fourth-grade students who were taught to write Roman numerals to corresponding Arabic numerals at a rate of 70 numbers per minute performed with fewer errors on a test of maintenance than did students whose fluency on the number writing task was only 35 digits per minute. This finding was most significant for students of below-average achievement in mathematics. Haughton (1972) found that students who could do math computations at a rate of 30 or more digits per minute progressed easily to more complex mathematical skills. Van Houten (1980) cited an unpublished study by Van Houten and Sharma in which daily measures of students' correct and incorrect rates on long division and complex multiplication problems were measured. Intervention consisted of fluency drills on basic, single-digit multiplication facts. During the phases when the fluency drills were implemented, students increased the number of correct answers per minute and decreased their error rate on long division and multiple-digit multiplication problems, even though no direct instruction or intervention was applied for these more complex problems.

Two Stages off Learning

Although numerous conceptualizations have been proposed describing the different stages through which a student progresses while learning and mastering a new skill, two basic

stages—the acquisition stage and the practice stage—are common to most of these theoretical arrangements. Acquisition Stage

In the first stage of learning a new skill—the acquisition stage of learning—the goal is for the student to learn how to do the skill. Instructional activities should be directed toward the goal of teaching the student to perform the skill correctly and accurately. Time trials and other fluency-building activities should not be used during the acquisition stage. Feedback during the acquisition stage should emphasize the qualitative aspects of performance and should be delivered following each attempt by the student. Although immediate feedback is ideal during the acquisition stage, the actual amount of clock time

by which feedback is delayed is not as critical as making sure that feedback is provided before the student attempts the new skill again. The biggest problem with delayed feedback during the acquisition stage is that it provides "an opportunity for children to practice mistakes and learn poor habits that are often difficult to replace7' (Van Houten, 1984, p. 115).

If asked whether or not students should practice newly acquired math skills, virtually every teacher in the world would answer "Yes, of course they should." Evidence of this belief is found in the almost daily scheduling of practice activities of one form or another. However, if teachers were asked "What is the purpose of practice, and how should it be conducted?" a

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wide range of answers would be given, not all of which would be compatible with one another. After the initial acquisition stage of learning, when a student has learned how to perform the skill correctly, comes the practice stage of learning, in which the focus should shift to fluency building. Although there is no standard event across skills and students that signals the end of the acquisition stage and the beginning of the practice stage of learning, when a student's correct responses are occurring significantly more often than errors, transition to the practice stage should begin. The teacher does not push fluency when the student cannot yet work the problems correctly. Similarly, when teaching a student to be fluent, techniques used to promote accuracy are not used. During fluency instruction, elaborate explanations and corrections are not needed; in fact, they might even slow the student down. Instead, the teacher talks about and rewards fluency. (Howell & Lorson-Howell, 1990, p. 21) Feedback in the practice stage should emphasize the quantitative aspects of performance, such as the number of problems completed and the rate or speed of performance. A series of responses can now be completed before feedback is delivered. In fact, feedback following each response or attempt in the practice stage will most often impede the student's effort to increase fluency.

Setting Fluency Aims and Objectives

A specific goal, or aim, should be determined before a fluency-building program is started. But how does a teacher determine "how fast is fast enough?" Setting aim rates should not be determined arbitrarily. Aims should be based on criteria relevant to your students, such as the skill's importance for both immediate and long-range goals. Eaton (1978) described five methods for determining fluency aims: (1) the student's previous performance, (2) an adult-to-child proportional formula, (3) peer comparison, (4) rates found functional in math research, and (5) normative data guidelines.

A student's previous performance is a good place to start in developing individual aims. Aims for the next or a more complex skill can be based on the student's current level of functioning on a math skill. For example, if the student can correctly answer 40 addition problems per minute, his or her aim for subtraction facts may also be set at 40.

"Accuracy measures alone do not tell the whole story. A student's level of proficiency, or fluency of responding, is a critical component of mastery/'

If a student's current performance is not considered a proficient rate, however, a new aim can be set by increasing the expected correct rate per minute and/or by reducing the number of errors allowed per minute. Fluency aims equal to a student's performance on a previous math skill may be inappropriate for a more difficult or complex skill. Most students, for example, can answer single-digit addition and subtraction facts at a higher rate than multiplication and division facts. It is important to set aims that are sufficiently high, however, and to give students opportunities to attain new levels of proficiency. If aims are not set high enough, students' fluency may actually decrease as they lower their productivity to match aims set below their level of proficiency (Koorland, Keel, & Ueberhorst, 1990). An adult-to-child proportional formula can also be helpful in setting performance aims. Measures are first taken of the student's tool skills rate and the rates at which a competent adult performs both the tool skill and the target math skill. The tool skill for answering math facts is writing random numbers without solving any problems. This fast-as-you-can number writing rate provides a ceiling for the fastest rate at which written answers

to math facts can be produced. The proportion obtained by dividing the adult's performance rate on the target math skill by his or her tool skill rate is then multiplied by the student's tool skill rate. The resulting figure is the fluency aim for the student. For example, if the adult solves math facts at a rate of 60 correct answers per minute and can write 120 random numbers per minute, his or her target skill to tool skill proportion is .5 (60 divided by 120). Based on the adult-child proportional formula, the fluency aim for a student whose tool skill rate is 80 would be set at 40 correct answers per minute (80 times .5). Providing direct and repeated practice on the relevant tool skills may be an effective way of improving the overall fluency of some children. Alternative modes of response, such as answering orally, should also be considered for children who exhibit very slow writing or poor fine motor control. Performance aims can also be based on what the student's peers are able to do. Time a student who is considered proficient at the skill in question. Measures can be taken on several "good" students at the end of the school year to use as aims for the next year. A student may also be selected from the next grade level as an example of expected rates for future performance. Criteria used by the teacher your students will have next year might also be considered as a longrange goal. Fluency aims can also be set in accordance with rates of response found to be prerequisite for student success in learning more complex skills. Howell and Morehead (1987) have suggested that single-digit math facts should be answered with 100% accuracy at a rate of 40 per minute. Data gathered by Haughton (1972) indicate that students who are able to compute basic math facts at a rate of 30 to 40 problems correct per minute (or about 70 to 80 digits correct per minute) continue to accelerate their rates as tasks in the math curriculum become more complex. Haughton also found that students whose correct rates were lower than 30 per minute showed progressively decelerating trends when more complex skills were introduced. The minimum correct rate for basic facts should be set at 30 to 40 problems per minute, since this rate has been shown

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to be an indicator of success with more complex tasks. This has important implications for teachers of students in the primary grades. Teachers of students with disabilities should also be aware of these minimum fluency rates and incorporate fluency-building activities designed to help their students attain them. Normative data may also be used as a guideline for performance aims. This method requires testing a large number of students and computing measures of central tendency (mean, median, and mode) on their scores. Although some types of useful normative data can be obtained from standardized tests, rate of response data are seldom available. Normative data represent the typical case and should not be considered appropriate for all students. Normative data can, however, provide basic information on how a student compares to other students of similar age and grade level. How fast is fast enough? You should set aims that are sufficiently high, yet reasonable. Students should not be required to reach excessive aims. The rates at which competent adults perform can be used as realistic indicators of functional fluency rates. Table 1 shows a range of fluency rates for various math skills exhibited by competent adults. While practicing to respond at a fluent rate, students should continue to make progress through the curriculum.

Measuring Fluency

Once fluency aims are set, direct and frequent measurement of each student's performance of the math skill should begin. Measurement is direct when it objectively records the learner's performance of the behavior of interest in the natural environment for that skill. Measurement is frequent when it occurs on a regular, ongoing basis; ideally, measurement should take place as often as instruction occurs. Two errors of judgement are common for [teachers] who do not collect direct and frequent measurements of their students' performance. First, many ineffective intervention programs are continued . .. Second, many effective programs are discon-

T a b l e 1 • Fluency Ranges of Competent Adults and Grade Levels at Which Students Should Approach Those Fluency Rates

Selected mathematics skills

Rates per minute for fluent adults"

Grades

60-150 60-125

2-3

60-90

2-3

70-90 60-90 50-90 50-90 50-90

3-4 4-5 4-6 5-6 6

Write numbers from 1 to N Addition facts with sums 0 to 9 Subtraction facts with minuends from 1 to 9 Addition facts with sums 0 to 18 and subtraction facts with minuends 1 to 18 2-digit addition with regrouping 2-digit subtraction with borrowing Multiplication facts through 9 Division facts through 9

i-2

a

Adult fluency rates and grade-level recommendations have been compiled from Mercer and Mercer (1985); Wood, Burke, Kunzelmann, and Koenig (1978); and the authors' experience and research.

tinued prematurely because subjective judgements find no improvement. For example, teachers who do not use direct and frequent measures might discern little difference between a student's reading 40 words per minute with 60% accuracy and 48 words per minute with 73% accuracy. However, direct and frequent data collected on the rate and accuracy of oral reading would show an improved performance. Decision making in education must be based upon performance data; the individual's behavior must dictate the course of action. (Cooper, Heron, & Heward, 1987, p. 60) Daily time trials are an excellent way of providing students with opportunities to improve their fluency while at the same time providing the teacher with direct, frequent measures of each student's progress.

Time Trials

In a time trial, students answer correctly as many problems as they can in a short, timed period. For most math skills, a 1-minute time trial is sufficient. Time trials help students improve their fluency by providing many opportunities to respond at a fast rate (Greenwood, Delquadri, & Hall, 1984), and they give teachers a direct and frequent measure of students' learning. Focusing attention and effort on timing students also increases their aware-

ness and rate of performance. Using daily timed trials in the practice stage of learning is an excellent way to build fluency (Lovitt, 1978; Van Houten, 1980). Research Findings

Understandably, some teachers fear that time trials may impair accuracy and that students may become frustrated and feel pressured when they are timed. Results from classroom studies in which time trials have been evaluated show just the opposite: Accuracy does not suffer, but usually improves, and students enjoy being timed (e.g., Allyon, Garber, Si Pisor, 1976; Van Houten Si Thompson, 1976). For example, Miller, Hall, and Heward (1992) evaluated the effects of 1-minute time trials on the rate and accuracy of answering single-digit math facts in a regular first-grade classroom and an intermediate-level classroom for students with developmental handicaps. The first graders answered correctly an average of 4.8 problems per minute when they were told to complete as many problems as they could during an untimed 10-minute work period. During the next phase, a series of seven 1-minute time trials was conducted with 20-second rest periods between each time trial (equaling a total of 10 minutes as in the baseline condition), and the students' correct rate increased to 9.2 per minute. Fluency improved to 12.5 problems per minute during a

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final phase when immediate feedback and self-correction was provided immediately after two consecutive time trials. Similar results were obtained in the special education classroom for the three experimental conditions: 8.4 problems per minute during the untimed 10-minute work period compared to 13.2 and 17.3 during the two time-trial conditions. Overall accuracy during the untimed work period was 83% for the first graders and 85% for the special education students. During the two time-trial conditions, mean accuracy increased slightly, to 88%, for both classes. When asked which method they preferred, 26 of the 34 students indicated they liked time trials better than the untimed work period. Although most of the students in both classes were still in the practice stage at the conclusion of this brief study— only a few of the students in either class met Haughton's suggested correct rate of 30 math facts per minute—after just 5 weeks of daily 10-minute time time-trial sessions, some students were answering the mixed math facts at correct rates four and five times higher than their initial rates, while maintaining high levels of accuracy. How to Do Time Trials

Time trials are easily implemented in the classroom. Some suggestions for planning and using time trials follow. Student Assessment. Before beginning a time trial program, measure the present level and rate of performance for each student. Begin by giving an informal test of skills to everyone in the class. The test should be very similar to the worksheets that will be used for time trials. Give the test as a time trial so information about both accuracy and proficiency can be obtained. Focus on skills that have already been taught, but that students need to practice in order to become proficient. Test several different skills on different occasions to find out which skills should be targeted for instruction at the acquisition stage and which students are at the practice stage for specific skills. Time trials can be individualized with specific target skills for each student. Use one or more of the methods described earlier to set proficiency aims for each student or, if appropriate, for the entire class.

Materials. Prepare worksheets with the appropriate skills. Each sheet should contain more problems than any student can finish during a 1minute time trial; about 50 to 60 problems are sufficient for single-digit math facts; fewer problems are needed for more difficult skills. Make at least five different forms of each type of worksheet, each form consisting of the

"There is a need to move beyond simply assessing how accurately math facts are answered to include a measure of the speed at which those answers are forthcoming/'

same or similar problems in various random orders. Using a different form of the worksheet from day to day eliminates the possibility of students learning to write the same answers in the same order each day without giving attention to the problems. Students should be ready with two sharpened pencils for time trials. Instead of using their erasers, instruct students to cross out any answers they wish to change while they work and write the new answer next to the X. A wristwatch with a count-down timer or a digital kitchen timer that signals when time has elapsed is recommended. Stopwatches, wristwatches, and wall clocks with a second hand may also be used, but require you to look at the watch to call out the end of the time trial. Standard kitchen timers that have a dial to turn to set the minutes are not accurate enough for time trials. Conducting the Time Trials. Time trials should be done every day. Time trials can occur either at the beginning or the end of each day's math period. They can be used as a transition into a different subject, or to fill effectively

a few minutes while your class waits to leave the room for another activity. Although time trials can be administered any time during the school day, it is best to conduct them at a regularly scheduled time initially so that students get used to the new routine. Introduce time trials to your students as a game. Tell them they will be racing against the clock, working as accurately and as fast as they can for 1 minute. Pass out the worksheets and tell students to place them face down on their desks. When everyone has their sheet and you are ready with the timer, say "Pencils up." This lets you see that all students are ready and prohibits anyone from starting too soon. When all pencils are up, begin the time trial with a consistent signal such as, "On your mark. Get set. Go!" Say the starting phrases in a rhythmic way, one phrase per second, to allow less room for error at the start and to synchronize your start of the timer. Have students hold the corner of their worksheet at "On your mark," turn the paper over on "Get set," and immediately begin working on "Go!" At the end of exactly 1 minute say, "Pencils up!" This niinimizes students writing answers after time is up and you can quickly make a visual check around the room for any students who are still writing. A single time trial or a series of consecutive time trials can be conducted each day. For example, with a 20second rest between each trial, a total of three 1-minute trials can be accomplished in less than 5 minutes. If more than one time trial is to be conducted, pass out a stapled set of worksheets, one for each trial. After completing the first time trial, students should turn to the next worksheet, place their packets face down, and wait for the "Pencils up!" signal for the next time trial. Scoring and Feedback. Students can be taught to grade their own papers. Self-scoring is not only a cost-effective method for giving feedback immediately after each time trial (Hundert & Batstone, 1978), it also provides students with additional opportunities to make academic responses. Use an overhead projector and a transparency of the time trial worksheet to guide students through the correct answers. Point to one problem at a time and say the problem and answer aloud. Students

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cross out their errors with a coloredink pen or marker and write the correct answer next to it. Additional student practice can be build into this teacher-led feedback procedure with choral responding (Heward, Courson, & Narayan, 1989). Have students repeat out loud in unison each problem and its answer along with you as they check their papers. Continue with checking on the overhead until all of the problems attempted by anyone in the class have been checked. Have every student chorally respond to and check all problems for which feedback is given. Students can write in their colored pen the correct answers to problems they did not attempt. When this method of teacher-led selfcorrection was conducted in the Miller, Hall, and Heward study (1992), each of the first graders and the intermediate-level students with developmental handicaps averaged approximately 100 math responses during each day's 10-minute session. Students should keep track of their own progress by recording or graphing their daily time-trial performances. If more than one time trial is conducted, allow each student to graph his or her highest score for that day. This gets students involved in their learning, allows them to see the progress they have made each day, and can be very motivating. Posting a classroom chart of timetrial scores is another simple, yet effective, technique for delivering positive and specific feedback to students (Van Houten, 1984). Before beginning the time trials each day, review the scores on the chart, congratulating specific students for reaching new personal highs. When goals are set for the entire class to meet (e.g., class party when 10,000 problems have been answered correctly), students are more likely to support and encourage one another rather than to compete in a negative fashion. Soma Additional Guidelines for Conducting Time Trials

1. Keep time short. A 1-minute time trial is long enough for math facts and for most other math skills. If necessary, time trails can be extended to 2 to 3 minutes for skills involving several steps. 2. Do time trials every day. Do at least one time trial every day.

Time trials should take place at about the same time each day, but timings of different skills can be interspersed throughout the day. Try using time trials for other academic skills, such as oral reading, spelling, written composition, and classifying animals or plants by species. Don't limit time trials to math skills; fluency is important for most skills.

"If asked whether or not students should practice newly acquired math skills, virtually every teacher in the world would answer 'Yes, of course they should/ Evidence of this belief is found in the almost daily scheduling of practice activities of one form or another/'

3. Use time trials only during the practice stage of learning. Students must first be able to perform the target skill with some degree of accuracy before time trials will help them become fluent. Time trials are not a replacement for initial instruction and skill acquisition. 4. Make time trials fun. Time trials should be conducted as a game, not as a test. Encourage students to "beat the clock" by using the starter's call for racers, "On your mark, get set, go!" The manner in which you present and conduct time trials will determine how your students respond to the new procedure. 5. Encourage each student to beat his or her own score. Give each student feedback relative to his or her own performance after each timed trial. Praise students for small improvements in their individual scores. Do not emphasize a student's scores in relation to others, instead, compare his present score to his previous scores.

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6. Tell students not to be afraid to make a mistake. Fluency building allows room for error. When only accuracy is stressed, some students become too fearful of making mistakes and work too slowly. But when students are encouraged to "go fast" in an effort to improve their fluency and are provided with positive feedback on their correct rate—which encompasses both accuracy and speed—they are more likely to take the risk of working faster without fear of hurting their "average." 7. Give more problems than anyone can do. Scores for students who finish all of the problems before time is up will not represent their true level of proficiency. Tell students that the worksheets include more problems than anyone could complete during a single time trial, and that they should not feel bad because they cannot answer all the problems. In this way, each student can be challenged by working as fast as he or she can throughout the time trial, and his or her score will reflect the actual level of proficiency rather than a ceiling rate imposed by the number of problems on the worksheet. 8. Follow time trials with a more relaxed activity. Students work hard during time trials, so do not move immediately into another activity or lesson that requires a high concentration level. Instead, change the pace by presenting a more relaxed activity. 9. Keep records of student progress. Keep a record of each student's time trial performance. These data are excellent indicators of student progress and also are useful for parent-teacher conferences and report card time. 10. Reward students for improvements in fluency. Measuring and recording correct rate per minute allows both teacher and student to detect improved performance. Positive feedback and recognition of some form should follow even the smallest improvements in fluency (e.g., "Wow, Andy, 31 problems correct in 1 minute! That's 1 more than yesterday and the best score you've ever had! Way to go!"). 11. Evaluate the effectiveness of your time-trial program. Look at stuVOLUME 28

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dents' scores and chart them. Better yet, have your students make their own graphs or charts! Use this information to make informed instructional decisions. Ask students how they feel about time trials, too. Listen to their suggestions for making time trials more fun and effective.

Conclusion Mastery of math facts requires more than accurate performance. Rate of performance, or fluency, is an important aspect of skill mastery that is often overlooked as a target for instruction in the classroom. Proficient responding is required in many situations both in and out of school. Conducting time trials can be an effective method for building fluency. Time trials provide students with many opportunities to respond, are easy to implement, and are enjoyed by students. So, get a timer and a chart and encourage each student in your class to beat his or her best score! B

A p r i l D . M i l l e r , PhD, is assistant professor in the Department of Special Education, University of Southern Mississippi. W i l l i a m L. H o w a r d , EdD, is professor in the Department of educational Services and Research, The Ohio State University. Address: April D. Miller, Department of Special Education, University of Southern Mississippi, Southern Station Box 5115, Hattiesburg, MS 39406-5115.

Authors' Note Preparation of this paper was supported by a Leadership Training Grant (No. G008715568) from the Office of Special Education Programs, U.S. Department of Education. Opinions expressed are those of the authors and no endorsement by the Department of Education is implied.

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