Get it right and get it fast! Building automaticity to strengthen mathematical proficiency.

Introduction

The important role mathematical fluency, or automaticity, plays in daily life is perhaps best appreciated when the fluency is absent. Have you ever spent time in a checkout line as the shopper in front of you slowly counted out the correct payment for groceries? In the classroom, have you ever felt concern during a lesson as the momentum ground to a halt, while students looked up facts and formulas they should have memorized? Incidents like these remind teachers that comprehension is necessary but insufficient for mathematical proficiency. Automaticity, the ability to perform a skill fluently with minimal conscious effort, is also necessary (Bloom, 1986; Schneider & Shiffrin, 1977). According to Hasselbring, Goin, and Bransford (1988), "[t]he ability to succeed in higher-order skills appears to be directly related to the efficiency at which lower-order processes are executed" (p. 1).

The purposes of this article are to: (a) discuss a rationale for moving beyond mathematical accuracy to automaticity, (b) offer a model illustrating the fluid relationship between comprehension and fluency training in the promotion of mathematical proficiency, and (c) provide recommendations for increasing mathematical automaticity. The prominent role that mathematical proficiency plays in today's global, information-driven economy supports the need for addressing this issue. Efficiency reduces the costs of achieving results in terms of both time and effort. The development of automaticity enables standard mathematical processes, such as facts about families of functions and formulas, to become useful tools for facilitating higher-order thinking. Underlying our discussion of automaticity is the assumption that teachers interested in building their students' mathematical fluency have first taught and confirmed students' comprehension of the material.

The Problem

Based on our experiences and discussions with students in middle and secondary school settings, many students are developing mathematical skills without concurrent development of automaticity. Teachers commonly relate that students do not know basic mathematical operations. Many educators discuss this problem when talking about elementary school students. We also observed similar deficiencies with older students. One high school teacher reported that her students do not know the basic arithmetic facts. A precalculus teacher said her students do not know how to graph the basic families of functions. Calculus students could not give the formulas for the derivatives of trigonometric functions without consulting a reference sheet.

Perhaps the availability of formula sheets and calculators in the classroom contributes to the lack of automatic recall by students. We observed students in higher-level high school mathematics classes (algebra and precalculus) using calculators to multiply single-digit numbers. There were even calculus students using graphing calculators to graph the function y = [x.sup.2].

Rationale for Building Automaticity

Comprehension

Although there is no universal agreement about the use of the term comprehension, we are using it to include the understanding of mathematical concepts, rules, principles, and generalizations. Comprehension is essential for the development of proficiency in mathematics (National Research Council, 2001) and should be developed along with fluency "in a coordinated, interactive fashion" (p. 11). There are various ways to teach students to understand and develop comprehension. According to Piaget (1977), logical thinking develops systematically through stages. During the concrete operational stage, typical from ages seven to eleven, students can think logically with the help of concrete materials. For instance, working with pie-shaped manipulatives can help students understand that one-fifth is equal to two-tenths. The formal operational stage follows and extends into adulthood, with students able to understand and solve abstract problems without consciously depending on models or other prompts.

Interestingly, when we attempt to learn something new in an area where we have no knowledge background, we often regress to the concrete operational stage, regardless of our age. To confirm this, simply reflect on how helpful it is to have someone "walking you through" a new skill, such as developing your own Internet web page. The advantage of working at the concrete level is that it allows us to build a strong foundation for new understandings, thus increasing the probability that our problem solving will be accurate when we proceed to thinking at the formal level. Prompts offer the mental scaffolding that leads to deeper and accurate understanding. The implication for middle and high school mathematics teachers is that they should allow students to manipulate concrete materials or draw pictures when beginning new areas of study.

The body of research underlying the constructivist movement in education supports our argument that students at all grade levels should be actively engaged in learning activities. Using appropriate manipulative materials, questioning techniques, and discussions can accomplish this. According to the philosophy of constructivism (Good & Brophy, 1986; Wittrock, 1998), (a) students construct their own understandings when presented with new material, (b) new understandings depend upon prior knowledge, (c) authentic tasks make learning more meaningful, (d) learning is facilitated by active engagement by the learner, and (e) social interaction facilitates learning.

Research about the Concrete/Semi-concrete/Abstract (CSA) approach (Miller, Butler, & Lee, 1998) illustrates both Piaget's theory of cognitive development and the learning benefits of constructivist teaching for students with learning difficulties. In the concrete learning stage, students used manipulatives. When tasks were completed at an 80% success rate (and not before), teachers moved students on to the semi-concrete stage, when the same problems were presented using drawings on paper instead of concrete objects. Again, at the 80%-or-higher level of success, the learning stimulus was changed. This time, students were asked to complete the problems using only abstract numbers, operational signs, and symbols written on paper. Overall, this approach created a 25-85% improvement in students' mathematics test scores. While all students may not progress through these stages at the same rate, this staging strategy for stimulus presentation offers a systematic way to structure learning that supports the development of students' understanding.

The practical value of the CSA approach can be easily illustrated. Using algebra tiles is a sound approach for teaching integer arithmetic. But it would be awkward to pull out algebra tiles each time a person needed to add integers. Once students have experience with algebra tiles, they should move to a semi-concrete experience of drawing those tiles. The final goal is to perform arithmetic operations using only symbols. Thus, the initial dependence on algebra tiles fades as a more efficient strategy develops (Howell & Lorson-Howell, 1990).

Teachers prone to lecturing need to question the effectiveness of habitually lecturing compared to a more constructivist approach. Peterson, Mercer, and O'Shea (1988) compared the CSA approach to an abstract-only approach. The abstract-only approach is the most frequently used method in middle grades through high school in general classrooms. The researchers found that the CSA sequence was more effective at each stage than the abstract-only approach. Statistically significant differences favoring the CSA group were noted on the post-test, maintenance, and retention of academic material.

Automaticity

Automaticity, or fluency, involves both accuracy and speed. For instance, when you suddenly realize that you have arrived at work but cannot recall passing any of the landmarks along the way, you have evidence of automaticity with your driving skills. According to Garnett (1992), mathematical fluency implies that attention is only minimally needed to coordinate the operations of a complex problem. Similarly, Glaser (1987) found that when performance on some tasks "becomes sufficiently automated through practice and requires little conscious attention, then effort can be devoted to other, frequently higher level, tasks" (p. 335).

What information should students be able to recall effortlessly? The National Council of Teachers of Mathematics [NCTM] (2000) reported: "Knowing the basic number combinations--the single-digit addition and multiplication pairs and their counterparts for subtraction and division--is essential. Equally essential is computational fluency--having and using efficient and accurate methods for computing" (p. 32).

We are not suggesting a "drill and kill" approach to mathematics instruction or that "either/or" thinking should drive choices about how to allocate class time. Suydam (1978) cautioned, "Drill-and-practice plays an important role in the mastery of computational skills, but strong reliance on drill-and-practice alone is not an effective approach to learning" (p. ix). Rather, both comprehension and fluency practice have a legitimate place in mathematics curricula (NCTM, 2000). Zentall and Ferkis (1993) found a significant relationship between fluency and performance on higher-level tasks. Students whose performance demonstrated both comprehension and fluency in computation received higher grades in mathematics and performed better on higher-level problem-solving skills than less fluent students. Zentall and Ferkis also concluded that computational speed is a significant predictor of word-problem performance.

Fluency is also related to other dimensions of mathematical proficiency. One example is estimation, a frequently used problem-solving strategy. "The ability to compute rapidly was related to the ability to estimate numerical computation" (Reys, Rybolt, Bestgen, & Wyatt, 1982, p. 185). Another dimension of mathematical proficiency is application. The link between fluency and application is provided by the test score intercorrelations for the Math Fluency and Applied Problems tests on the Woodcock-Johnson III (McGrew & Woodcock, 2001). The Applied Problems subtest "is a measure of quantitative reasoning, math achievement, and math knowledge. This test requires the ability to analyze and solve math problems" (Mather & Woodcock, 2001, p. 84). The Math Fluency subtest is a three-minute timed test of single-digit addition, subtraction, and multiplication problems. One third of the variance of the Applied Problems scores is related to the variance in the Math Fluency scores for 6-8 year olds. More than one-fourth of the variance of the Applied Problems scores is related to the variance in the Math Fluency for 20-39 year olds (McGrew & Woodcock, 2001, Appendix D).

We suggest there is a fluid relationship between the attainment of comprehension and the attainment of automaticity in the learning process. Figure 1 illustrates the interflow of these two dimensions of mathematical proficiency. The hot and cold faucets suggest that the amount of teaching focus on either comprehension or fluency will be decided by the teacher. For example, if the teacher decides to turn on the "comprehension faucet" much harder than the "fluency faucet," the students' mathematical proficiency will be affected.

[FIGURE 1 OMITTED]

Figure 2 suggests that teaching comprehension but neglecting fluency training, or vice versa, will not only impact students' levels of mathematical proficiency but also may leave them too uncomfortable to want to engage in problem solving. In terms descriptive of the model's images, students may find mathematics too hot or too cold (too frustrating).

Fluency and comprehension should be taught in a balanced manner, not in isolation, because each supports the development of the other. NCTM (2000) cautions against teachers sacrificing curriculum for drill. Even with students who are struggling, the curriculum should stay rich. According to Glaser (1987), it is critical to develop proficiency to a level where processes facilitate one another, rather than operate independently. One illustration is that an understanding of place value in the base-10 system supports the development of fluency in multidigit computation" (National Research Council, 2001, p. 121).

[FIGURE 2 OMITTED]

Well-intended but misplaced emphasis on teaching to a standardized test can contribute significantly to an imbalance between comprehension and fluency. Given that teachers are very concerned about improving standardized test scores, they may rely on strategies such as providing formula sheets to students to support that purpose rather than considering the appropriateness of such strategies for student learning. We found an example of this situation in a regular tenth-grade geometry class. Needing to find the area of a circle before he could solve the remainder of a word problem, a student sought help from a teaching intern. A diagram of a circle accompanied the word problem in question. When the student told the intern that the area of a circle is "base times height," the intern responded that he should look on his formula sheet. This same formula sheet is provided on the state-level examinations given to tenth graders. Later, several interns, experienced teachers, and the authors discussed this student's difficulty with knowing the area of a circle. Some explained that, since students could use the formula sheet on the standardized test, they saw no need for students to memorize the formula for a circle. In fact, without the formula sheet, some teachers said that students would be more likely to recall wrong formulas, thereby risking lower test scores!

Regular practice is a significant factor in the development of fluency. Teachers need to help students at all grade levels understand that many skills can become automatic if they practice them. According to the American Association for the Advancement of Science (1993), "[a]ttending closely to any one input of information usually reduces the ability to attend to others at the same time (p. 141). Teachers who neglect to provide students with sufficient practice of basic facts and operations risk seeing them increasingly struggle with mathematics as they not only fail to add to their automatic repertoires, but also lose fluency with what they had learned in previous years. We suspect that students who say they dislike math are reflecting, in part, their frustrations at having to manage new material without adequate support of a fluent knowledge base.

Building and Sustaining Automaticity

Building and sustaining automaticity requires a consistent way to measure progress. Mathematical fluency can be easily measured with timing exercises. Rate per minute is a more sensitive measure of changes in performance than an accuracy measure alone (Howell & Lorson-Howell, 1990; Zentall & Ferkis, 1993). Some researchers also advocate using timed trials because they motivate students to abandon the use of inefficient strategies (Hasselbring, Goin, & Bransford, 1988).

Some teachers might worry that timing academic task completion may be too stressful for students. This belief may stem from a sensitive and kind, but perhaps misguided perception that time pressure is somehow harmful. Let the doubters consider what Air Force Colonel Eileen Collins, the first female shuttle commander, said about stress when she first piloted the shuttle: "I'm a role model whether I like it or not, and that does add stress to my job. But it's a good stress because I will work harder and focus more" (Hoversten, 1995). Colonel Collins claimed that she always wanted to be a teacher and, in fact, taught mathematics at the United States Air Force Academy. Her comments serve as a reminder of the positive impact stress can have on motivation to perform well (Osipow & Spokane, 1984). An additional feature of positive stress, or eustress, is that it leads to sustained or increased selfesteem on the part of the individual experiencing it (Seyle, 1974).

Many students actually enjoy the challenge of being timed. Upon completion of a Florida Institute of Education PreCollegiate Program camp, 88% of the students reported on a written survey that they liked the fluency training provided by the authors. McDougall and Brady (1998) found that students who observed special education members of the class using self-monitoring techniques, including timing, asked to be included in these exercises. However, teachers need to avoid pushing students to achieve unrealistic rates (Howell & Lorson-Howell, 1990). The optimum rate of fluency for different skills varies, depending upon age and grade level (Woodcock, McGrew, & Mather, 2001).

In addition to increased motivation and self-esteem, the academic benefits resulting from self-monitoring of progress, including fluency, have been documented in educational research. McDougall and Brady (1998) found that self-monitoring increased both mathematical accuracy and fluency that maintained over time for students with and without disabilities. During independent practice sessions, self-management increased both academic productivity and engagement. McDougall and Brady also found evidence that individuals' productivity and engagement continued to increase when components of the full self-management treatment packages were faded. Not surprisingly, students who self-monitored least accurately improved least on the performance of mathematics tasks. The implication for instruction is that teachers need to closely supervise students to ensure that they performing self-monitoring procedures correctly.

In the PreCollegiate Program camp mentioned above, students completed fluency training including timed tests across several trials. The task for each trial was to write the answers to as many single-digit integer addition problems as possible in one minute. Each student self-monitored by graphing the number of correct answers for each trial. All students increased their speed over the course of the six trials. Figure 3 shows the graph that one of the students created, followed by his comments about his self-monitoring. Not included here are the student's graph and journal writing about his accuracy. His accuracy was at least 90% on every trial.

[FIGURE 3 OMITTED]

Speed

My graph shows the data about speed. Speed relating to the number of problems in sixty seconds or one minute. My data proves that every test my number correct improves. Most of my tests went up by 3 answers correct. On Trial 1 my speed was 21 answers per minute. By Trial 5 my speed was 34 answers per minute. If I say so myself, I improved.

All students graphed both their accuracy and speed and wrote about their performance in their journals. All of the students showed improvement, but not all reached fluency. In the case of this exercise, a goal of 30 problems per minute was established. This mark was established by:

* Reasonable alignment with the scoring used on the Woodcock-Johnson III (McGrew & Woodcock, 2001) for completing single-digit whole number facts;

* Student and teacher data collected from the same integer-operations timed trials by the authors; and

* The recommendation of Miller and Heward (1992): "The minimum correct rate for basic facts should be set at 30 to 40 problems per minute, since this rate has been shown to be an indicator of success with more complex tasks" (pp. 100-101).

Estimating from these points of reference, the authors decided that 30 problems per minute would be an age-appropriate goal for the participating middle school students. The point to keep in mind in deciding optimum fluency rates is that the goals should be both developmentally and age appropriate.

Those students who demonstrated fluency for adding single-digit integers, like the one in the above example, moved to subtraction exercises (and then to multiplication when their subtraction skills were automatized). For students whose accuracy was low on the first two trials, the remediation focused on strategies such as using learning tiles and the number line to find the sum of two integers. Once accuracy was achieved, they began speed training.

Applications of self-monitoring can be found in many aspects of life outside of school. For instance, adults in almost every profession attend time-management and goal-setting seminars. Students at one of the United States military academies record study-time data for individual purposes as well as to help faculty assess course goals. Joggers maintain logs of their daily runs to make certain they are putting enough stress on their system while avoiding harm caused by over-training. Given its benefits in and beyond mathematics class, it appears that the building and sustaining of automaticity through self-monitoring strategies is certainly a worthwhile, lifelong habit to foster in students.

Strategies for Promoting Automaticity

Following are five suggested strategies that, in combination, will support the development of automaticity in student learning.

Strategy 1: Getting It Right

Maintain a strong focus on helping students both understand and be able to perform skills accurately (Samuels, 1988). This might be an arduous task for some students. For this reason, incorporate novelty into problems, actively engage students, help them see the relevancy of mathematical problems in real life, and explain why and how the process was developed. Strengthen student motivation and academic self-esteem by presenting lessons that are challenging but, with effort, possible to master. According to Bandura (1995):

  Successes build a robust belief in one's personal efficacy. Failures
  undermine it, especially if failures occur before a sense of efficacy
  is firmly established.... If people experience only easy successes,
  they come to expect quick results and are easily discouraged by
  failure. A resilient sense of efficacy requires experience in
  overcoming obstacles through perseverant effort. Some difficulties and
  setbacks in human pursuits serve a useful purpose in teaching that
  success usually requires sustained effort. (p. 3)

Strategy 2: Introduce Fluency Training

As soon as a student demonstrates comprehension of a new skill, begin fluency training with aspects of the skill that need to become automatic for future lessons. For instance, the formulas a student uses for computing the amount of turf needed to sod a family's newly landscaped yard should be practiced until recall becomes automatic. However, avoid pushing students to speed up while they are still focused on grasping the concepts involved in the problem. When this happens, students either may trade accuracy for speed, thereby becoming better at executing the skill incorrectly, or simply give up from frustration.

Strategy 3: Promote Student Interest in Automaticity

Promote student interest in automaticity by talking about it and praising student efforts to become mathematically fluent. Discuss with students how automaticity is helpful in daily life. Invite them to ask significant adults in their lives about the skills they can perform to an automatic level and how this level of fluency benefits them. Share interesting examples from several fields, such as concert pianist, sculptors, tennis stars, Olympic swimmers, research mathematicians, and research neurologists.

Impress upon students that achieving expertise takes time. Bloom (1986), in a study of "geniuses" in a variety of fields, found that no individual "reached this high level of attainment in less than a dozen years, and it took the average person about 16 years" (p. 71). "By adolescence, most were spending about 25 hours a week on practice and learning in the talent field. They 'over-learned' particular aspects of their repertoire of skills" (p. 72). Seeing similarities, such as gender, age, ethnic origin, or beginning struggles between themselves and successful role models in the community increases the likelihood that students will believe that they, too, possess the capabilities to master comparable activities (Bandura, 1995).

Strategy 4: Provide Adequate Practice

The development of skill fluency requires frequent and correct practice. According to Rhymer, Dittmer, Skinner, and Jackson (2000), teachers can increase fluency through procedures such as providing immediate performance feedback, using time limits, and reinforcing higher rates of fluency. The nature of the practice should be focused, with a strong tempo and high interest for students, and within a short time span. Just as you would not push students to be speedy before they understand a process and can repeatedly perform it correctly, do not interrupt practice for fluency with long explanations about why and how the process was developed (Howell & Lorson-Howell, 1990). Based on our observations of class activities, one of the best times to have students practice fluency is just before the end of class. An additional benefit of scheduling practice at this time is that a focused, high-tempo practice exercise will extend instructional time for many students who otherwise may not use the last ten minutes of class well.

Formatting practice sessions as a game can promote motivation to participate, but attention to game design is important. In designing a game format, keep in mind that all students should have equal opportunity to participate. Students should not be pitted against one another so some students win and others end up feeling stupid. Finally, the game structure should not put class management at risk, as in the case of some students shouting out answers while others sit idly by or engage in unrelated distractions.

A learning game called Brain Box was developed by the authors and used for over fifteen years with middle and high school students as a way to promote automaticity in a high-interest, light-hearted, and systematic way. Materials and directions needed for utilizing Brain Box follow the illustration of the Brain Box board in Figure 4.

[FIGURE 4 OMITTED]

1. Materials needed to build a Brain Box are (a) a large poster board, (b) a marker for drawing the grid, (c) a library pocket glued in each of the blank squares, and (d) index cards (3"x5") to fit vertically into the pockets.

2. Write any information students need to learn to an automatic recall level on the index cards. Write only one question per card.

3. Place the cards randomly in the pockets. As you add more cards, you may put more than one card in a pocket.

4. To plan for Brain Box, set aside frequent and consistent time periods in your instructional schedule.

5. To play Brain Box, have students, in turn, pick a pocket (e.g., 1C, 4A, 2C). Pull the card from that pocket (if more than one card is in the pocket, pick the top card) and ask the question (e.g., "4/5 equals what percent?").

6. Students answer as quickly as possible. As fluency develops, answers will come within a second after the question. If a student cannot answer a question or responds slowly, simply return the card to its pocket. This will "recycle" the question. Put cards that are answered correctly in a pile. These questions are out of the game for that day.

7. When a student misses a question, refrain from making a lengthy explanation of the correct answer. Simply state the right answer (e.g., "It's 80%") and replace the card. Maintain a supportive tone of voice. Research has confirmed that the most effective feedback is immediate, specific, offers corrective information, and has a positive tone (Bransford, Brown, & Cocking, 1999; Good & Brophy, 1986).

8. The group goal is to see how many cards the class can answer within the game's time limit. Graphing each game's number of cards in the pile is one way of letting the class "raise the bar" by selecting a slightly higher number of questions to master for the next Brain Box session.

9. When a question card is being answered fluently by the entire class, you can "retire" it permanently.

Specific advantages of using Brain Box are that (a) all students participate, (b) students do not compete against one another but, rather, against their own previous performance, (c) students are not criticized when they miss a question, and (d) the fast pace of the game is motivating. In addition, the teacher can easily add material to the game and take out material that all students have mastered fluently. Although we are discussing Brain Box in the context of mathematics instruction, information that students should automatically know in any area of study or simply as general life knowledge can be included in the game. The author, for instance, mixed questions from all subject areas. One card asked for the capital of Florida. Before long, all students were answering Tallahassee rather than Miami, Jacksonville, or Disney World.

Strategy 5: Provide Timed Practice Exercises

We found Miller and Heward's procedures for fluency training to be very effective from elementary through college level. Using a sheet of basic multiplication facts as an example, steps for implementing the procedure follow (Miller & Heward, 1992):

1. Create a set of questions containing more than enough problems to last all students for at least one-minute work session.

2. Establish a purposeful climate: no clutter on desks, quiet in the room, and no disruptions.

3. Hand out a copy of the exercise to each student, instructing the class to leave the paper face down on their desks until told to "go."

4. Encourage students to have fun and do their best, confirm that everyone has a working pencil, and encourage the students to relax.

5. To lend a sense of fun and challenge to the exercise, cue students with "get ready, get set, go!" This is common in informal sports activities, and can help increase students' focus. As soon as you say "go," students should turn their papers over and write answers as rapidly as they can and still be accurate.

6. After one minute, say "Stop. Pencils down."

7. Project a transparency of the answers or provide an answer sheet to each student. Have students record the number of problems attempted and the number correct.

8. Assessing performance with a standard percentage grade will measure accuracy. The accuracy is the number of correct responses out of the number of problems attempted.

9. To measure fluency, record the number correctly answered items per minute. To measure improvement in fluency, consider a graphing strategy. To do this, first take a baseline measure, followed by regular and frequent rate measurements. Have students score and graph their own progress.

Strategy 6: Teach Self-Management

As mentioned previously, we taught students to self-monitor. We can not emphasize enough the importance of self-management for mathematics proficiency. Having students record and manage their own progress can facilitate academic improvement (McLaughlin & Skinner, 1996) and help to maintain student interest in learning mastery (Fuchs, Bahr, & Bahr, 1990). Be sure to have students correct their errors as to avoid reinforcing faulty thinking. Incorrect patterns can be difficult to unlearn. For example, the golfer who repeatedly practices the wrong form on the driving range will tee off the same way when playing 18 holes. Self-correction can be done as homework to avoid using valuable instructional time in class for independent tasks.

One simple self-management tool is graphing. McLaughlin and Skinner (1996) found that self-graphing increased initial improvements and maintained performance at high levels. Graphing provides a relevant, real-world link between academic material and students' own, measurable self-improvement. Graphing progress also permits students to practice goal setting, an important skill in the development of self-determination. Avoid posting any individuals graphed scores publicly.

Another self-management procedure for improving fluency and skill maintenance is the Cover, Copy, and Compare procedure (Ivarie, 1986; McLaughlin & Skinner, 1996; Skinner, Ford, & Yunker, 1991). To implement, the student would first look at the material, such as the formula for a circle, then cover it up, write it down, and finally, compare the written response with the model. If wrong, the student would correct his/her response and then try again. An important point for educators to remember is to applaud students for managing their own performance and for the corrective actions they take. Emphasizing these procedures can lead to lifelong habits that help students select behaviors that will contribute to their own improvement across many areas.

Challenge to Teachers

Our advocacy for a plausible approach to building mathematical proficiency has been persuaded by our experiences with students as well as a variety of research studies. We believe that overlearning mathematical skills is essential for mathematical proficiency. Without automaticity, students will expend too much time and energy focusing on basic skills rather than on processes such as, but not limited to, understanding, representing, interpreting, and selecting appropriate operations for problem solving.

We recommend that teachers be mindful of their instructional practices and closely monitor the balance between comprehension and automaticity. Questions they might ask are:

* How much time are they devoting to fluency training?

* Is fluency training balanced with training of other dimensions of mathematical proficiency?

* What can they do when accuracy during fluency training becomes problematic?

* How should they set expected rates of fluency for different grade levels and for various applications?

Teaching students to become fluent should be integrated with development of other components that contribute to mathematical proficiency. Teachers should not be lured into falsely believing that they have to focus on either comprehension or fluency. Instead, teachers must accept the challenge of balancing the instructional time (including practice) between comprehension and fluency. Our model in Figure 1 illustrates the instructional balance we are advocating. Consequently, we recommended techniques to increase automaticity that have been helpful to our students. It is important that teachers explicitly plan for specific fluency-building activities in order to maximize the probability that the time will be spent wisely toward facilitating automatization of students' basic skills.

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Tuiren A. Bratina and Kathryn M. Krudwig

University of North Florida