Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 1

Mathematics InitiativeMathematics Initiative

Office of Superintendent

of Public Instruction

A White Paper on Computational Fluency (K-12)A White Paper on Computational Fluency (K-12)

2007 WERA Spring Conference2007 WERA Spring ConferenceSlide Slide 11

A White Paper on A White Paper on Computational Fluency (K-12)Computational Fluency (K-12)

Presented byPresented by

Mark Jewell, Ph.D.Mark Jewell, Ph.D.

Chief Academic OfficerChief Academic Officer

Federal Way School DistrictFederal Way School District






A View of Mathematics from OSPIA View of Mathematics from OSPI

Mathematics is a language and science of Mathematics is a language and science of patterns.patterns.

Mathematical content (EALR 1) must be Mathematical content (EALR 1) must be embedded in the mathematical processes embedded in the mathematical processes (EALRs 2-5).(EALRs 2-5).

For all students to learn significant For all students to learn significant mathematics, content must be taught and mathematics, content must be taught and assessed in meaningful situations.assessed in meaningful situations.






Computational FluencyComputational Fluency

A look at what the research saysA look at what the research says

and and

classroom implications.classroom implications.






Computational Fluency: Computational Fluency: Research and Implications for PracticeResearch and Implications for Practice

Six Focus Questions Six Focus Questions What is computational fluency?What is computational fluency?

How does computational fluency develop?How does computational fluency develop?

How does computational fluency differ from How does computational fluency differ from simply being able to add, subtract, multiply, simply being able to add, subtract, multiply, and divide?and divide?






Computational Fluency: Computational Fluency: Research and Implications for PracticeResearch and Implications for Practice

How is computational fluency related to How is computational fluency related to automaticity?automaticity?

What learning experiences are most What learning experiences are most conducive to the attainment of computational conducive to the attainment of computational proficiency?proficiency?

What are the characteristics of effective What are the characteristics of effective computational fluency programs?computational fluency programs?






Project TimelineProject Timeline

Initial meetingInitial meeting

Review of research Review of research literatureliterature

Compile preliminary Compile preliminary research and research and implications implications

Nov. 20, 2006Nov. 20, 2006

Dec. 2006–Feb. 2007Dec. 2006–Feb. 2007

Jan. 2–4, 2007Jan. 2–4, 2007







Present status report at Present status report at OSPI January OSPI January Conference Conference

Develop preliminary Develop preliminary recommendations and recommendations and obtain feedback from obtain feedback from practitioners across the practitioners across the state and national state and national expertsexperts

Jan. 10, 2007Jan. 10, 2007

Jan.–Feb. 2007Jan.–Feb. 2007







Review computational Review computational fluency programsfluency programs

Submit final Submit final recommendations to recommendations to Superintendent Bergeson Superintendent Bergeson for review and approvalfor review and approval

Present recommendations Present recommendations during OSPI Summer during OSPI Summer InstitutesInstitutes

March 26-30, 2007March 26-30, 2007

May 2007May 2007

Summer 2007Summer 2007






What is Computational Fluency?What is Computational Fluency?

A concept with deep historical roots in the A concept with deep historical roots in the literature of mathematics instruction and literature of mathematics instruction and assessment.assessment.







– William Brownell (1935; 1956)William Brownell (1935; 1956)Described “meaningful habituation,” in many ways Described “meaningful habituation,” in many ways a historical precursor to computational fluency.a historical precursor to computational fluency.

Advocated an instructional approach that balanced Advocated an instructional approach that balanced meaningmeaning and and skillskill..

Maintained that “meaning” and “skill” are mutually Maintained that “meaning” and “skill” are mutually dependent, even though some people attempt to dependent, even though some people attempt to portray them as distinct.portray them as distinct.







– Stuart Appleton Courtis (1906; 1942)Stuart Appleton Courtis (1906; 1942)Developed one of the first published arithmetic Developed one of the first published arithmetic tests in the U.S.tests in the U.S.

Believed that rate tests represented “an avenue of Believed that rate tests represented “an avenue of development largely unexplored” (p. 9).development largely unexplored” (p. 9).






1978 NCTM Year Book1978 NCTM Year Book– Drill has long been recognized as an essential Drill has long been recognized as an essential

component of instruction in the basic facts. Practice component of instruction in the basic facts. Practice is necessary to develop immediate recall. Brownell is necessary to develop immediate recall. Brownell and Chazai (1935) demonstrated quite convincingly and Chazai (1935) demonstrated quite convincingly that drill increases the speed and accuracy of that drill increases the speed and accuracy of responses to basic-fact problems. Those are the responses to basic-fact problems. Those are the purposes for which drill should be used. Drill alone purposes for which drill should be used. Drill alone will not change the thinking that a child uses; it will will not change the thinking that a child uses; it will only tend to speed up the thinking that is already only tend to speed up the thinking that is already being used.being used.

What Learning Experiences are Most Conducive What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?to the Attainment of Computational Fluency?






What is Computational Fluency?What is Computational Fluency?More Contemporary ThinkingMore Contemporary Thinking

NCTM’sNCTM’s Curriculum and Evaluation Standards for Curriculum and Evaluation Standards for School Mathematics (1989)School Mathematics (1989)– ““Children should master the basic facts of arithmetic that are Children should master the basic facts of arithmetic that are

essential components of fluency with paper-and-pencil and essential components of fluency with paper-and-pencil and mental computation and with estimation” (p. 47).mental computation and with estimation” (p. 47).

– ““Practice designed to improve speed and accuracy should be Practice designed to improve speed and accuracy should be used, but only under the right conditions: that is, practice with a used, but only under the right conditions: that is, practice with a cluster of facts should be used only after children have cluster of facts should be used only after children have developed an efficient way to derive the answers to those facts” developed an efficient way to derive the answers to those facts” (p. 47).(p. 47).

– ““It is important for children to learn the sequence of steps, and It is important for children to learn the sequence of steps, and the reasons for them, in the paper-and-pencil algorithms used the reasons for them, in the paper-and-pencil algorithms used widely in our culture. Thus instruction should emphasize the widely in our culture. Thus instruction should emphasize the meaningful development of these procedures, not the speed of meaningful development of these procedures, not the speed of processing” (p. 47).processing” (p. 47).






What is Computational Fluency?What is Computational Fluency?More Contemporary ThinkingMore Contemporary Thinking

NCTM’s NCTM’s Principles and Standards for School Principles and Standards for School Mathematics (2000)Mathematics (2000)

– ““Fluency refers to having efficient, accurate, and Fluency refers to having efficient, accurate, and generalizable methods (algorithms) for computing that generalizable methods (algorithms) for computing that are based on well-understood properties and number are based on well-understood properties and number relationships.”relationships.”

NCTM, 2000, p. 144NCTM, 2000, p. 144






What is Computational Fluency?What is Computational Fluency? More Contemporary ThinkingMore Contemporary Thinking

NRC’s NRC’s Adding it UpAdding it Up

– Conceptual Understanding:Conceptual Understanding: Comprehension of Comprehension of mathematical concepts, operations, and relations.mathematical concepts, operations, and relations.

– Procedural Fluency: Procedural Fluency: Skill in carrying out procedures Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.flexibly, accurately, efficiently, and appropriately.

– Strategic Competence: Strategic Competence: Ability to formulate, Ability to formulate, represent, and solve mathematical problems.represent, and solve mathematical problems.







– Adaptive Reasoning: Adaptive Reasoning: Capacity for logical thought, Capacity for logical thought, reflection, explanation, and justification.reflection, explanation, and justification.

– Productive Disposition: Productive Disposition: Habitual inclination to see Habitual inclination to see mathematics as sensible, useful, and worthwhile, mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own coupled with a belief in diligence and one’s own efficacy.efficacy.

U.S. National Research Council, 2001, p. 5U.S. National Research Council, 2001, p. 5






Adding it Up, National Research Council, p. 117







NCTM’sNCTM’s (2006) Curriculum Focal Points for (2006) Curriculum Focal Points for Prekindergarten through Grade 8 MathematicsPrekindergarten through Grade 8 Mathematics– Grade 2:Grade 2: Developing “quick recall” of addition and Developing “quick recall” of addition and

subtraction facts and fluency with supporting algorithms is subtraction facts and fluency with supporting algorithms is a focus.a focus.

– Grade 4:Grade 4: Developing “quick recall” of the basic Developing “quick recall” of the basic multiplication facts and related division facts and fluency multiplication facts and related division facts and fluency with whole number multiplication.with whole number multiplication.

– Grade 5:Grade 5: Developing an understanding of and fluency with Developing an understanding of and fluency with division of whole numbers.division of whole numbers.

– Grade 5/6:Grade 5/6: Developing an understanding of and fluency Developing an understanding of and fluency with addition and subtraction of fractions and decimals.with addition and subtraction of fractions and decimals.







Susan Jo Russell on “Accuracy”Susan Jo Russell on “Accuracy”

Accuracy depends on several aspects of the problem Accuracy depends on several aspects of the problem solving process, among them, careful recording, the solving process, among them, careful recording, the knowledge of basic number combinations and other knowledge of basic number combinations and other important number relationships, and concern for important number relationships, and concern for double-checking results.double-checking results.

(2000, p. 154)(2000, p. 154)







Susan Jo Russell on “Efficiency”Susan Jo Russell on “Efficiency”

Efficiency implies that the student does not get bogged Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student strategy. An efficient strategy is one that the student can carry out easily, keeping track of sub-problems can carry out easily, keeping track of sub-problems and making use of intermediate results to solve the and making use of intermediate results to solve the problem.problem.

(2000, p. 154)(2000, p. 154)







Susan Jo Russell on “Flexibility”Susan Jo Russell on “Flexibility”

Flexibility requires the knowledge of more than one Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. approach to solving a particular kind of problem. Students need to be flexible to be able to choose Students need to be flexible to be able to choose an appropriate strategy for the problem at hand an appropriate strategy for the problem at hand and also to use one method to solve a problem and also to use one method to solve a problem and another method to double-check the results.and another method to double-check the results.

(2000, p. 154)(2000, p. 154)







Is there more to computational fluency Is there more to computational fluency than identified by Russell (2000)?than identified by Russell (2000)?

– Accuracy: Accuracy: Being careful and keeping good Being careful and keeping good records.records.

– Efficiency: Efficiency: Not getting lost or being bogged Not getting lost or being bogged down.down.

– Flexibility: Flexibility: Able to use multiple approaches.Able to use multiple approaches.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Types of Mathematical KnowledgeTypes of Mathematical Knowledge

According to cognitive psychologists, learning is a According to cognitive psychologists, learning is a process in which the learner actively builds mental process in which the learner actively builds mental structures, or schemata. These structures consist of:structures, or schemata. These structures consist of:– Conceptual Knowledge: Conceptual Knowledge: This is a highly structured This is a highly structured

and interrelated body of knowledge of schemata.and interrelated body of knowledge of schemata.– Declarative Knowledge: Declarative Knowledge: This type of knowledge This type of knowledge

refers to memorized facts involving arithmetical refers to memorized facts involving arithmetical relations among numbers.relations among numbers.

– Procedural Knowledge: Procedural Knowledge: This type of knowledge This type of knowledge involves children’s awareness of the processing steps involves children’s awareness of the processing steps that are required to solve a problem.that are required to solve a problem.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Normal Development of Computational FluencyNormal Development of Computational Fluency

Research into the study of children’s Research into the study of children’s mathematical thinking tells us there is a mathematical thinking tells us there is a continuum of strategies through which continuum of strategies through which students develop computational fluency with students develop computational fluency with basic facts and multi-digit numbers in all four basic facts and multi-digit numbers in all four operations.operations.

For basic facts, there are three stages before For basic facts, there are three stages before recall, or memorization in each operation.recall, or memorization in each operation.







For computation with multi-digit numbers, For computation with multi-digit numbers, there are four stages before the student can there are four stages before the student can use the traditional algorithm with use the traditional algorithm with understanding.understanding.If a student has only memorized without the If a student has only memorized without the opportunity to develop through the opportunity to develop through the continuum, and then forgets the fact, he or continuum, and then forgets the fact, he or she will have no way to solve the problem.she will have no way to solve the problem.







Experience along the continuum enables the Experience along the continuum enables the student to better determine the reasonableness of student to better determine the reasonableness of an answer.an answer.

Students move along the continuum at individual Students move along the continuum at individual rates.rates.

Often it is the difficulty of the problem that Often it is the difficulty of the problem that determines the strategies the student will use.determines the strategies the student will use.

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children’s Mathematics. Children’s Mathematics. Portsmouth, NH: Heinemann. Portsmouth, NH: Heinemann.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?The Acquisition of Basic Math FactsThe Acquisition of Basic Math Facts

The acquisition of math facts generally The acquisition of math facts generally progresses from a deliberate, procedural, progresses from a deliberate, procedural, and error-prone calculation to one that is and error-prone calculation to one that is fast, efficient, and accurate.fast, efficient, and accurate.

Ashcraft, 1992; Fuson, 1982, 1988; Siegler, 1988Ashcraft, 1992; Fuson, 1982, 1988; Siegler, 1988






How Does Computational Fluency How Does Computational Fluency Develop?Develop?

The Acquisition of Basic Math FactsThe Acquisition of Basic Math Facts

For many students, at any point in time For many students, at any point in time from preschool through at least the fourth from preschool through at least the fourth grade, they will have some facts that can grade, they will have some facts that can be retrieved from memory with little effort be retrieved from memory with little effort and some that need to be calculated using and some that need to be calculated using some counting strategy.some counting strategy.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?The Acquisition of Basic Math FactsThe Acquisition of Basic Math Facts

From the fourth grade through adulthood, From the fourth grade through adulthood, answers to basic math facts are recalled answers to basic math facts are recalled from memory with a continued from memory with a continued strengthening of relationships between strengthening of relationships between problems and answers that results in problems and answers that results in further increases in fluency.further increases in fluency.

Ashcraft, 1985Ashcraft, 1985






How Does Computational Fluency Develop?How Does Computational Fluency Develop?The Acquisition of Addition and Subtraction FactsThe Acquisition of Addition and Subtraction Facts

In a typical developmental path in addition, In a typical developmental path in addition, students begin adding using a strategy students begin adding using a strategy called “counting on” strategy, which in turn called “counting on” strategy, which in turn gives ways to linking new facts to known gives ways to linking new facts to known facts.facts.

Garnett, 1992Garnett, 1992






How Does Computational Fluency Develop?How Does Computational Fluency Develop?The Acquisition of Addition and Subtraction FactsThe Acquisition of Addition and Subtraction Facts

The most frequently used and most The most frequently used and most efficient counting strategy among efficient counting strategy among kindergarten, first, and second grade kindergarten, first, and second grade students was a minimum addend counting.students was a minimum addend counting.

Siegler 1987; Siegler & Shrager, 1984Siegler 1987; Siegler & Shrager, 1984







The Acquisition of Addition and Subtraction FactsThe Acquisition of Addition and Subtraction Facts

The acquisition of minimum addend The acquisition of minimum addend counting strategy is an essential predictor counting strategy is an essential predictor of success in early mathematics (Siegler of success in early mathematics (Siegler 1988). Although most children learn or 1988). Although most children learn or deduce this strategy readily, LD and other deduce this strategy readily, LD and other struggling math students do not.struggling math students do not.







The Acquisition of Addition and Subtraction FactsThe Acquisition of Addition and Subtraction Facts

The finding that students with learning The finding that students with learning disabilities do not spontaneously produce disabilities do not spontaneously produce task-appropriate strategies necessary for task-appropriate strategies necessary for adequate performance leads to the need adequate performance leads to the need for direct and explicit instruction before for direct and explicit instruction before they show signs of performing they show signs of performing strategically.strategically.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Strategies to Memorization of Basic Facts: Keys to MasteryStrategies to Memorization of Basic Facts: Keys to Mastery

AdditionAddition– Count AllCount All– Just One MoreJust One More– Count OnCount On– Small DoublesSmall Doubles– -Doubles +/--Doubles +/-– Makes a 10Makes a 10– Related FactsRelated Facts

SubtractionSubtraction– Count BackCount Back– Just One LessJust One Less– Count UpCount Up– Related FactsRelated Facts– Subtraction NeighborsSubtraction Neighbors– Finding DoublesFinding Doubles– Over the HillOver the Hill

Adding It UpNational Research Council, p. 187, 190






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Examples of Addition StrategiesExamples of Addition Strategies

StrategyStrategy Representative Use to Representative Use to Solve 2 + 4Solve 2 + 4

Counting AllCounting All ““1, 2…1, 2, 3, 4…1, 2, 3, 4, 1, 2…1, 2, 3, 4…1, 2, 3, 4, 5, 6”5, 6”

Shortcut SumShortcut Sum ““1, 2, 3, 4, 5, 6”1, 2, 3, 4, 5, 6”

Finger DisplayFinger Display ““Displays 2 fingers, then 4 Displays 2 fingers, then 4 fingers; says 6”fingers; says 6”

Counting on from the first Counting on from the first addendaddend ““2…3, 4, 5, 6” or “3, 4, 5, 6”2…3, 4, 5, 6” or “3, 4, 5, 6”






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Examples of Addition StrategiesExamples of Addition Strategies

StrategyStrategy Representative Use to Representative Use to Solve 2 + 4Solve 2 + 4

Counting on from the larger Counting on from the larger addendaddend ““4…5, 6, “ or “5, 6”4…5, 6, “ or “5, 6”

LinkingLinking ““2 + 2 = 4, + 2 more = 6”2 + 2 = 4, + 2 more = 6”

RetrievalRetrieval ““6”6”






Strategies to Memorization:Strategies to Memorization:Keys to MasteryKeys to Mastery

““When counting up is not introduced, many When counting up is not introduced, many children may not invent it until the second or children may not invent it until the second or third grade, if at all. Intervention studies with third grade, if at all. Intervention studies with U.S. first graders that helped them see U.S. first graders that helped them see subtraction situations as taking away the first x subtraction situations as taking away the first x objects enabled them to learn and understand objects enabled them to learn and understand counting-up-to procedures for subtraction. Their counting-up-to procedures for subtraction. Their subtraction accuracy became as high assubtraction accuracy became as high as thatthat for for

addition.”addition.” Adding it Up, Adding it Up, National Research Council, p. 191National Research Council, p. 191






GradeGrade GuessingGuessing Counting AllCounting All Counting-OnCounting-OnDerivedDerived

FactsFactsKnown FactsKnown Facts

KK 30%30% 22%22% 30%30% 2%2% 16%16%

11 8%8% 1%1% 38%38% 9%9% 44%44%

22 5%5% 0%0% 50%50% 11%11% 45%45%

Percentage of Time of Students Use Various Addition Procedures (Siegler,1987)






How Does Computational Fluency Develop?How Does Computational Fluency Develop?The Acquisition of Multiplication and Division FactsThe Acquisition of Multiplication and Division Facts

In multiplication, a student might employ a In multiplication, a student might employ a repeated addition or skip counting as initial repeated addition or skip counting as initial procedures for calculating the facts (Siegler, procedures for calculating the facts (Siegler, 1988). With repeated exposures, most 1988). With repeated exposures, most normally developing students establish a normally developing students establish a memory relationship with each fact. Instead memory relationship with each fact. Instead of calculating it, they recall it automatically.of calculating it, they recall it automatically.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Computational Fluency and Brain ScienceComputational Fluency and Brain Science

Recent research in cognitive science using Recent research in cognitive science using functional magnetic resonance imaging functional magnetic resonance imaging (FMRI), has revealed the actual shift in brain (FMRI), has revealed the actual shift in brain activation patterns as untrained math facts activation patterns as untrained math facts are learned.are learned.

Delazer et al., 2003Delazer et al., 2003







Instruction and practice cause math fact Instruction and practice cause math fact processing to move from a quantitative area processing to move from a quantitative area of the brain to one related to automatic of the brain to one related to automatic retrieval.retrieval.

Dehaene, 1997; 1999; 2003Dehaene, 1997; 1999; 2003







Delazer and her colleagues suggest that this Delazer and her colleagues suggest that this shift aids the solving of complex shift aids the solving of complex computations that require “the selection of an computations that require “the selection of an appropriate resolution algorithm, retrieval of appropriate resolution algorithm, retrieval of intermediate results, storage and updating in intermediate results, storage and updating in working memory” by substituting some of the working memory” by substituting some of the intermediate steps with automatic retrieval.intermediate steps with automatic retrieval.

Delazer et al., 2004Delazer et al., 2004






How Does Computational Fluency Develop?How Does Computational Fluency Develop?The Importance of Automaticity in MathematicsThe Importance of Automaticity in Mathematics

All human beings have a limited information-All human beings have a limited information-processing capacity. That is, an individual simply processing capacity. That is, an individual simply cannot attend do too many things at once.cannot attend do too many things at once.

Some of the sub-processes, particularly basic facts, Some of the sub-processes, particularly basic facts, need to be developed to the point that they are done need to be developed to the point that they are done automatically. If this fluent retrieval does not automatically. If this fluent retrieval does not develop, then the development of higher-order develop, then the development of higher-order mathematical skills, such as multiple digit addition mathematical skills, such as multiple digit addition and subtraction, and fractions--may be severely and subtraction, and fractions--may be severely impaired (Resnick, 1983).impaired (Resnick, 1983).







Studies have found that lack of math fact Studies have found that lack of math fact retrieval can impede math class retrieval can impede math class discussions (Woodward & Baxter, 1997), discussions (Woodward & Baxter, 1997), successful mathematics problem solving successful mathematics problem solving (Pelligrino & Goldman, 1987), and even (Pelligrino & Goldman, 1987), and even the development of everyday life skills the development of everyday life skills (Loveless, 2003). (Loveless, 2003).







Rapid math fact retrieval has been shown Rapid math fact retrieval has been shown to be a strong predictor of performance on to be a strong predictor of performance on mathematics achievement tests (Royer, mathematics achievement tests (Royer, Tronsky, Chan, Jackson, & Marchant, Tronsky, Chan, Jackson, & Marchant, 1999).1999).







““Once procedures are automatized, they require Once procedures are automatized, they require little conscious effort to use, which, in turn, frees little conscious effort to use, which, in turn, frees attentional and working memory resources for attentional and working memory resources for use on other more important features of the use on other more important features of the problem” (Geary, 1995).problem” (Geary, 1995).

When a basic fact is executed without conscious When a basic fact is executed without conscious monitoring and attention, it is considered to have monitoring and attention, it is considered to have become automatic (Goldman & Pellegrino, become automatic (Goldman & Pellegrino, 1987).1987).







Automaticity is useful both in and out of Automaticity is useful both in and out of the classroom (Isaacs & Carroll, 1999).the classroom (Isaacs & Carroll, 1999).

Counting strategies and the use of Counting strategies and the use of electronic calculators interfere with electronic calculators interfere with learning higher level math skills such as learning higher level math skills such as multiple-digit addition and subtraction, long multiple-digit addition and subtraction, long division, and fractions (Resnick, 1983). division, and fractions (Resnick, 1983).







If a student is constantly having to compute the answers If a student is constantly having to compute the answers to simple addition and subtraction facts, part of the to simple addition and subtraction facts, part of the student’s thinking capacity is reduced and less is left for student’s thinking capacity is reduced and less is left for interrelating higher-order concepts that the student has interrelating higher-order concepts that the student has to learn. For example, a child who is performing a long to learn. For example, a child who is performing a long division must monitor constantly where he or she is in division must monitor constantly where he or she is in that procedure, requiring a certain amount of attention that procedure, requiring a certain amount of attention resources. If the students must use counting strategies resources. If the students must use counting strategies to subtract or multiply during the division process, these to subtract or multiply during the division process, these procedures also must be monitored. This draws upon procedures also must be monitored. This draws upon the limited attention resources, and the student often the limited attention resources, and the student often fails to grasp the concepts involved in multiple-digit fails to grasp the concepts involved in multiple-digit division.division.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Developmental Perspective of AutomaticityDevelopmental Perspective of Automaticity

Early counting strategies are replaced with more Early counting strategies are replaced with more efficient rule-based strategies (Hopkins & efficient rule-based strategies (Hopkins & Lawson, 2002). Lawson, 2002). At the automatic stage, learners quickly At the automatic stage, learners quickly recognize the problem pattern (e.g., division recognize the problem pattern (e.g., division problem, square root problem) and implement problem, square root problem) and implement the procedure without much conscious the procedure without much conscious deliberation. deliberation. As a skill develops, learners are able to execute As a skill develops, learners are able to execute it rapidly and achieve greater accuracy in their it rapidly and achieve greater accuracy in their answers.answers.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Automaticity as a Foundation for Traditional Algorithm ProficiencyAutomaticity as a Foundation for Traditional Algorithm Proficiency

Kirby and Becker (1988) indicated that lack of Kirby and Becker (1988) indicated that lack of automaticity in basic operations and strategy use–either automaticity in basic operations and strategy use–either the use of an inefficient strategy or the use of the right the use of an inefficient strategy or the use of the right strategy at the wrong time–were responsible for the strategy at the wrong time–were responsible for the majority of math problems that children experience. majority of math problems that children experience.

Based on the results of their research, Kirby and Becker Based on the results of their research, Kirby and Becker concluded that “children with learning problems in concluded that “children with learning problems in arithmetic do not have any major structural defect in their arithmetic do not have any major structural defect in their information processing systems or that they are information processing systems or that they are qualitatively different from normally achieving students in qualitatively different from normally achieving students in any enduring sense.”any enduring sense.”







““Instead, the results are consistent with the interpretation Instead, the results are consistent with the interpretation that such children may not be carrying out even simple that such children may not be carrying out even simple arithmetic in the correct manner, and that they require arithmetic in the correct manner, and that they require extensive practice in the correct strategies” (p. 15). extensive practice in the correct strategies” (p. 15). Speed of mathematical fact retrieval from memory Speed of mathematical fact retrieval from memory relates directly to overall mathematical achievement in relates directly to overall mathematical achievement in students from elementary school through college (Royer, students from elementary school through college (Royer, Tronsky, Chan, Jackson, & Marchant, 1999).Tronsky, Chan, Jackson, & Marchant, 1999).







Students have achieved behavioral fluency when they can Students have achieved behavioral fluency when they can perform a skill quickly and with minimal or no errors perform a skill quickly and with minimal or no errors (Spence & Hively, 1993). Information-processing theorists (Spence & Hively, 1993). Information-processing theorists refer to behavioral fluency as automaticity. Although there refer to behavioral fluency as automaticity. Although there certainly is some controversy about the need to build certainly is some controversy about the need to build behavioral fluency, there are data to suggest that fluency behavioral fluency, there are data to suggest that fluency with basic skills can help students with later learning and with basic skills can help students with later learning and application of those skills (Binder, 1993; Spence & Hively, application of those skills (Binder, 1993; Spence & Hively, 1993). For example, Haughton (1972) found that children 1993). For example, Haughton (1972) found that children who could solve single-digit arithmetic problems at a who could solve single-digit arithmetic problems at a minimum of fifty to sixty correct per minute were more minimum of fifty to sixty correct per minute were more successful at later parts of a math curriculum. successful at later parts of a math curriculum.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Automaticity as a Means for Developing Automaticity as a Means for Developing

Number SenseNumber Sense

Isaacs and Carroll (1999) note that automaticity Isaacs and Carroll (1999) note that automaticity in math facts is essential to estimation and in math facts is essential to estimation and mental computations.mental computations.

These skills, particularly the ability to perform These skills, particularly the ability to perform mental computations (e.g., make approximations mental computations (e.g., make approximations based on rounded numbers such as 10s and based on rounded numbers such as 10s and 100s), are central to the ongoing development of 100s), are central to the ongoing development of number sense.number sense.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Why Speed of Recall MattersWhy Speed of Recall Matters

One of the indications of whether a “fact” is learned One of the indications of whether a “fact” is learned to the point of automaticity is speed of recall. to the point of automaticity is speed of recall. When attention must be divided between the task at When attention must be divided between the task at hand and the search for a calculation answer, the hand and the search for a calculation answer, the student may not have enough working memory to student may not have enough working memory to search for an algorithm, translate the problem, and search for an algorithm, translate the problem, and so forth.so forth.A strong argument for teaching mathematics facts is A strong argument for teaching mathematics facts is that if facts are learned to the point of automaticity, that if facts are learned to the point of automaticity, then the limited resources of working memory are then the limited resources of working memory are available for problem solving. available for problem solving.







Zentall and Ferkis (1993) stated that slow and Zentall and Ferkis (1993) stated that slow and inaccurate computational skill may place further inaccurate computational skill may place further attention load on the problem solving process.attention load on the problem solving process.

Zawaiza and Gerber (1993) noted that many Zawaiza and Gerber (1993) noted that many researchers believe that automaticity can “free researchers believe that automaticity can “free attentional resources necessary for more complex attentional resources necessary for more complex and abstract aspects of some problem solving” (p. and abstract aspects of some problem solving” (p. 65).65).

High rates of accurate responding have been called High rates of accurate responding have been called fluent (Haring & Eaton, 1978; Marston, 1989) or fluent (Haring & Eaton, 1978; Marston, 1989) or automatic responding (Gagne, 1983). automatic responding (Gagne, 1983).







Gagne (1983) suggested that automatic responding Gagne (1983) suggested that automatic responding to basic mathematics problems allows students to basic mathematics problems allows students more cognitive energy to focus on higher level skills.more cognitive energy to focus on higher level skills.

Haring and Eaton (1978) suggested that students Haring and Eaton (1978) suggested that students who can accurately perform basic skills at higher who can accurately perform basic skills at higher rates have been exposed to over learning and, rates have been exposed to over learning and, therefore, are more likely to maintain those skills.therefore, are more likely to maintain those skills.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Computational Fluency and Diverse StudentsComputational Fluency and Diverse Students

Cognitive research on mathematical difficulties Cognitive research on mathematical difficulties reveals that students with learning disabilities reveals that students with learning disabilities have deficits in fact retrieval (Garnett & have deficits in fact retrieval (Garnett & Fleischner, 1983; Geary, 1994; Geary, Hoard, & Fleischner, 1983; Geary, 1994; Geary, Hoard, & Hamson, 1999). They make more mistakes in Hamson, 1999). They make more mistakes in giving simple answers in various areas of giving simple answers in various areas of arithmetic and sometimes recall facts more arithmetic and sometimes recall facts more slowly than their peers. Such fact retrieval slowly than their peers. Such fact retrieval problems are probably related to deficits in problems are probably related to deficits in working memory.working memory.







Most math-delayed children, along with those who have Most math-delayed children, along with those who have never received systematic math fact instruction, show a never received systematic math fact instruction, show a serious problem with respect to the retrieval of basic serious problem with respect to the retrieval of basic math facts. math facts. Learning-disabled children are substantially less Learning-disabled children are substantially less proficient than their non-disabled peers in retrieving the proficient than their non-disabled peers in retrieving the answers to basic math facts in addition and subtraction. answers to basic math facts in addition and subtraction. Although information is still emerging about the particular Although information is still emerging about the particular difficulties experienced by these children in the retrieval difficulties experienced by these children in the retrieval of this information, the evidence that does exist suggests of this information, the evidence that does exist suggests that these children do not differ from a conceptual deficit, that these children do not differ from a conceptual deficit, but rather from some sort of disruption to normal but rather from some sort of disruption to normal development of their network of relationships between development of their network of relationships between facts and answers. facts and answers.







These students often have well-developed number These students often have well-developed number sense and procedural knowledge—they can figure out sense and procedural knowledge—they can figure out the answer to any fact given enough time. But because the answer to any fact given enough time. But because they have poorly developed declarative knowledge, they they have poorly developed declarative knowledge, they have minimal ability to recall anything buy the most basic have minimal ability to recall anything buy the most basic facts from memory.facts from memory.






How Does Computational Fluency Develop?How Does Computational Fluency Develop?More About Math-Delayed StudentsMore About Math-Delayed Students

What this suggests is that there are huge What this suggests is that there are huge differences in the amount of instruction individual differences in the amount of instruction individual children need to become fluent at retrieving answers children need to become fluent at retrieving answers to basic math facts.to basic math facts.

By age seven, non math-delayed students can recall By age seven, non math-delayed students can recall more facts from memory than their math-delayed more facts from memory than their math-delayed peers, and this discrepancy increases as age peers, and this discrepancy increases as age increases. increases.

As math-delayed students get older, they fall farther As math-delayed students get older, they fall farther and farther behind their non math-delayed peers in and farther behind their non math-delayed peers in their ability to recall basic math facts from memory their ability to recall basic math facts from memory (Hasselbring et al., 1988).(Hasselbring et al., 1988).







In contrast to their skilled peers, struggling math In contrast to their skilled peers, struggling math students have a serious problem with respect to the students have a serious problem with respect to the retrieval of basic number facts.retrieval of basic number facts.

Fleischner, Garnett, and Ginsburg (1984) have Fleischner, Garnett, and Ginsburg (1984) have found that students with learning disabilities are found that students with learning disabilities are substantially less proficient than students without substantially less proficient than students without learning disabilities in retrieving basic math facts in learning disabilities in retrieving basic math facts in addition and subtraction.addition and subtraction.







Cumming and Elkins (1999) point out that many Cumming and Elkins (1999) point out that many educators and researchers make the unwarranted educators and researchers make the unwarranted assumption that strategies—either developed assumption that strategies—either developed naturally or through explicit instruction—invariably naturally or through explicit instruction—invariably lead to automaticity.lead to automaticity.







Research indicates that students with LD do not Research indicates that students with LD do not develop sophisticated fact strategies naturally (e.g., develop sophisticated fact strategies naturally (e.g., Geary, 1993; Goldman et al., 1988) . Geary, 1993; Goldman et al., 1988) .

Empirical research on strategy instruction in math Empirical research on strategy instruction in math facts for students with LD is limited, and the results facts for students with LD is limited, and the results are mixed in terms of the effective development of are mixed in terms of the effective development of automaticity (see Putnam, deBettencourt & automaticity (see Putnam, deBettencourt & Leinhardt, 1990; Tournaki, 2003).Leinhardt, 1990; Tournaki, 2003).






How Does Computational Fluency Develop?How Does Computational Fluency Develop?Add, Subtract, Multiply, and DivideAdd, Subtract, Multiply, and Divide

Although there is some controversy about the need Although there is some controversy about the need to build computational fluency, there are data to to build computational fluency, there are data to suggest that fluency with basic skills can help suggest that fluency with basic skills can help students with later learning and application of those students with later learning and application of those skills (Binder, 1993; Spence & Hively, 1993).skills (Binder, 1993; Spence & Hively, 1993).

Torbeyns, Verschaffel, and Ghesiquiere (2005) Torbeyns, Verschaffel, and Ghesiquiere (2005) investigated the fluency with which first graders of investigated the fluency with which first graders of different mathematical achievement levels applied different mathematical achievement levels applied multiple, school-taught strategies for finding multiple, school-taught strategies for finding arithmetic sums over 10. High-achieving students arithmetic sums over 10. High-achieving students applied the strategies more efficiently but not more applied the strategies more efficiently but not more adaptively than did their lower achieving peers.adaptively than did their lower achieving peers.







At any point in time from preschool through at least fourth grade, At any point in time from preschool through at least fourth grade, most students will have some facts that they can retrieve from most students will have some facts that they can retrieve from memory automatically and some that have to be reconstructed memory automatically and some that have to be reconstructed using procedural knowledge. From the fourth grade through using procedural knowledge. From the fourth grade through adulthood, simple addition and subtraction problems are solved with adulthood, simple addition and subtraction problems are solved with a continued strengthening of relationships between problems and a continued strengthening of relationships between problems and answers, which results in further increases in the speed of retrieving answers, which results in further increases in the speed of retrieving all facts (Ashcraft, 1985). all facts (Ashcraft, 1985).

Hung-Hsi Wu (2001), professor of mathematics at the University of Hung-Hsi Wu (2001), professor of mathematics at the University of California at Berkeley, has argued that computational fluency is a California at Berkeley, has argued that computational fluency is a prerequisite for success in algebra. According to Wu, “if students prerequisite for success in algebra. According to Wu, “if students are not sufficiently fluent with the basic skills to take the numerical are not sufficiently fluent with the basic skills to take the numerical computations for granted, either because they lack practice or rely computations for granted, either because they lack practice or rely too frequently on technology, then their mental disposition toward too frequently on technology, then their mental disposition toward computations of any kind would soon be one of apprehension and computations of any kind would soon be one of apprehension and ultimately instinctive evasion” (p. 3).ultimately instinctive evasion” (p. 3).







Differing Perspectives on Standard AlgorithmsDiffering Perspectives on Standard Algorithms

The term “algorithm” sometimes provokes disdain among The term “algorithm” sometimes provokes disdain among educators because of the oppressive ways in which traditional educators because of the oppressive ways in which traditional algorithms often are taught. In fact, algorithms are algorithms often are taught. In fact, algorithms are remarkable tools in mathematics and computer science. They remarkable tools in mathematics and computer science. They have great practical and theoretical importance.have great practical and theoretical importance.

Standard algorithms were gradually developed many Standard algorithms were gradually developed many centuries ago for their efficiency, accuracy, and generality—centuries ago for their efficiency, accuracy, and generality—that is, they work in all situations. They are theoretically and that is, they work in all situations. They are theoretically and practically important methods for computing. They contain in practically important methods for computing. They contain in their very structure all the basic properties of the base-ten their very structure all the basic properties of the base-ten place-value system, set forth in as efficient a manner as place-value system, set forth in as efficient a manner as possible. An understanding of how and why they work, as possible. An understanding of how and why they work, as well as the ability to use them fluently, provides the foundation well as the ability to use them fluently, provides the foundation for mathematical competence. for mathematical competence.







Differing Perspectives on Standard AlgorithmsDiffering Perspectives on Standard AlgorithmsAs children acquire knowledge of the underlying structure of a As children acquire knowledge of the underlying structure of a particular operation and explore different ways to perform it, they particular operation and explore different ways to perform it, they should also learn how to use the standard algorithm for the should also learn how to use the standard algorithm for the operation. After they learn a standard algorithm for an operation, operation. After they learn a standard algorithm for an operation, whatever they then choose to use routinely should be judged on the whatever they then choose to use routinely should be judged on the basis of efficiency and accuracy. Children should be able to explain basis of efficiency and accuracy. Children should be able to explain whatever method they use and see the usefulness of methods that whatever method they use and see the usefulness of methods that are efficient, accurate, and flexible.are efficient, accurate, and flexible.

A 15-member group of mathematicians, appointed by the A 15-member group of mathematicians, appointed by the Mathematical Association of America to respond to a set of Mathematical Association of America to respond to a set of questions about algorithms and algorithmic thinking posed by the questions about algorithms and algorithmic thinking posed by the National Council of Teachers of Mathematics Commission on the National Council of Teachers of Mathematics Commission on the Future of the Standards, stated that “standard mathematical Future of the Standards, stated that “standard mathematical definitions and algorithms serve as a vehicle of human definitions and algorithms serve as a vehicle of human communication” and that they should be taught to all children (Ross, communication” and that they should be taught to all children (Ross, 1997).1997).







Differing Perspectives on Standard AlgorithmsDiffering Perspectives on Standard Algorithms

Notices of the American Mathematical Society states Notices of the American Mathematical Society states that: “all the algorithms of arithmetic are preparatory for that: “all the algorithms of arithmetic are preparatory for algebra . . . The division algorithm is also significant for algebra . . . The division algorithm is also significant for later understanding of real numbers” (American later understanding of real numbers” (American Mathematical Society Association Resource Group for Mathematical Society Association Resource Group for the NCTM Standards, 1988).the NCTM Standards, 1988).






What Learning Experiences are Most Conducive to What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?the Attainment of Computational Fluency?

A Preliminary List of RecommendationsA Preliminary List of Recommendations– Early Numeracy ProgramsEarly Numeracy Programs

Griffin (2005) recommends that early numeracy Griffin (2005) recommends that early numeracy programs include activities that “provide opportunities programs include activities that “provide opportunities for children to acquire computational fluency as well as for children to acquire computational fluency as well as conceptual understanding” (p. 283).conceptual understanding” (p. 283).

– Drill and Practice versus Strategy InstructionDrill and Practice versus Strategy InstructionTeaching students the use of effective strategies to Teaching students the use of effective strategies to solve basic math fact problems enhances learning, solve basic math fact problems enhances learning, leading to automaticity (e.g., Morin & Miller, 1998; leading to automaticity (e.g., Morin & Miller, 1998; Thornton, 1978).Thornton, 1978).







Drill & Practice ProgramsDrill & Practice Programs– Drill and practice programs have demonstrated a Drill and practice programs have demonstrated a

positive effect on improving the retrieval speed for positive effect on improving the retrieval speed for facts already being recalled from memory facts already being recalled from memory (Woodward, 2006). (Woodward, 2006).

– Drill and practice had no effect on developing Drill and practice had no effect on developing automaticity for non-recalled facts (Hasselbring, automaticity for non-recalled facts (Hasselbring, Goinn, & Sherwood, 1986). Goinn, & Sherwood, 1986).

– To facilitate the automatic recall of all facts, instruction To facilitate the automatic recall of all facts, instruction must be focused on non-automatized facts while must be focused on non-automatized facts while practice and review are given on facts that are practice and review are given on facts that are already being recalled from memory. already being recalled from memory.

– Identifying and separating fluent from non-fluent facts Identifying and separating fluent from non-fluent facts is important (Woodward, 2006). is important (Woodward, 2006).






Strategy-Based FluencyStrategy-Based Fluency– Issacs and Carroll (1999) emphasize that students Issacs and Carroll (1999) emphasize that students

naturally develop strategies for learning math facts if naturally develop strategies for learning math facts if given the opportunity. given the opportunity.

– Research supporting the natural development of Research supporting the natural development of strategies may be found for addition and subtraction strategies may be found for addition and subtraction (Baroody & Ginsburg, 1986; Carpenter & Moser, (Baroody & Ginsburg, 1986; Carpenter & Moser, 1984; Resnick, 1983; Siegler & Jenkins, 1989) as well 1984; Resnick, 1983; Siegler & Jenkins, 1989) as well as more recent work in the area of multiplication as more recent work in the area of multiplication (Angghileri, 1989; Baroody, 1997; Clark & Kamii, (Angghileri, 1989; Baroody, 1997; Clark & Kamii, 1996; Mulligan & Mitchelmore, 1997; Sherin & Fuson, 1996; Mulligan & Mitchelmore, 1997; Sherin & Fuson, 2005).2005).







Strategy-Based FluencyStrategy-Based Fluency– A number of educators emphasize the use of explicit A number of educators emphasize the use of explicit

strategy instruction over traditional rote learning when strategy instruction over traditional rote learning when teaching math facts. Methods vary from the use of teaching math facts. Methods vary from the use of visual displays such as ten frames and number lines visual displays such as ten frames and number lines (Thompson & Van de Walle, 1984; Van de Walle, (Thompson & Van de Walle, 1984; Van de Walle, 2003) to more general techniques such as classroom 2003) to more general techniques such as classroom discussion where students share fact strategies discussion where students share fact strategies (Steinberg, 1985; Thornton, 1990; Thornton & Smith, (Steinberg, 1985; Thornton, 1990; Thornton & Smith, 1988).1988).







Integrating Strategy Instruction and Timed Integrating Strategy Instruction and Timed Practice DrillsPractice Drills– Cumming and Elkins’ research (1999) suggests that a Cumming and Elkins’ research (1999) suggests that a

middle-ground position for teaching facts to middle-ground position for teaching facts to academically low-achieving students and students academically low-achieving students and students with LD consists of integrating strategy instruction with LD consists of integrating strategy instruction with frequent timed practice drills. Results of their with frequent timed practice drills. Results of their research indicate that instruction in strategies does research indicate that instruction in strategies does not necessarily lead to automaticity. Frequent timed not necessarily lead to automaticity. Frequent timed practice is essential. However, strategies help practice is essential. However, strategies help increase a student’s flexible use of numbers, and for increase a student’s flexible use of numbers, and for that reason, Cumming and Elkins advocate the use of that reason, Cumming and Elkins advocate the use of strategy instruction for all students through the end of strategy instruction for all students through the end of elementary school.elementary school.







Integrating Strategy Instruction and Timed Integrating Strategy Instruction and Timed Practice DrillsPractice Drills– Strategy instruction can benefit the development of Strategy instruction can benefit the development of

estimation and mental calculations. In this respect, estimation and mental calculations. In this respect, strategy instruction helps develop number sense strategy instruction helps develop number sense (Baroody & Coslick, 1998; Gersten & Chard, 1999).(Baroody & Coslick, 1998; Gersten & Chard, 1999).

– Christensen (1991) found that fact practice, combined Christensen (1991) found that fact practice, combined with fluency building, produced better effects than with fluency building, produced better effects than strategy instruction. strategy instruction.







Integrating Strategy Instruction and Timed Integrating Strategy Instruction and Timed Practice DrillsPractice Drills– Hasselbring, Goin, & Bransford (1988) concluded that Hasselbring, Goin, & Bransford (1988) concluded that

“computer-based drill and practice can be used to “computer-based drill and practice can be used to develop automaticity, but only when specific develop automaticity, but only when specific prerequisite conditions are met. If these prerequisite prerequisite conditions are met. If these prerequisite conditions are not met, our research, as well as conditions are not met, our research, as well as others (Howell & Garcia, 1985; Reith, 1985), has others (Howell & Garcia, 1985; Reith, 1985), has shown that computer-based drill and practice results shown that computer-based drill and practice results in little or no improvement on the part of handicapped in little or no improvement on the part of handicapped students” (p. 1).students” (p. 1).







Integrating Strategy Instruction and Timed Integrating Strategy Instruction and Timed Practice DrillsPractice Drills– According to Hasselbring, Goin, and Bransford According to Hasselbring, Goin, and Bransford

(1988), “Neither paper and pencil drill and practice nor (1988), “Neither paper and pencil drill and practice nor computer-based drill and practice seems to be computer-based drill and practice seems to be sufficiently powerful in itself for developing sufficiently powerful in itself for developing automaticity in learning handicapped students.” automaticity in learning handicapped students.” Additional work on developing a declarative Additional work on developing a declarative knowledge network is needed before drill and practice knowledge network is needed before drill and practice is effective. Practice that allows students to use is effective. Practice that allows students to use counting strategies does nothing but strengthen counting strategies does nothing but strengthen students’ use of counting strategies and does little to students’ use of counting strategies and does little to move the student toward a state of automaticity move the student toward a state of automaticity (Hasselbring, Goin & Sherwood, 1986).(Hasselbring, Goin & Sherwood, 1986).







Integrating Strategy Instruction and Timed Integrating Strategy Instruction and Timed Practice DrillsPractice Drills– Computational Fluency and Curriculum-Based Computational Fluency and Curriculum-Based

MeasurementMeasurementDeno and Mirkin (1977) suggested that in order to Deno and Mirkin (1977) suggested that in order to demonstrate mastery in mathematics, students should demonstrate mastery in mathematics, students should complete mathematics computation problems at a rate complete mathematics computation problems at a rate of 20 digits correct per minute in first through third of 20 digits correct per minute in first through third grades, and 40 digits correct per minute in subsequent grades, and 40 digits correct per minute in subsequent grade levels.grade levels.







Integrating Strategy Instruction and Timed Integrating Strategy Instruction and Timed Practice DrillsPractice Drills– Time Needed for PracticeTime Needed for Practice

The learning of mathematical procedures, or The learning of mathematical procedures, or algorithms, is a long, often tedious process (Cooper & algorithms, is a long, often tedious process (Cooper & Sweller, 1987). To remember mathematical Sweller, 1987). To remember mathematical procedures, student must practice using them. procedures, student must practice using them. Students should also practice using the procedure on Students should also practice using the procedure on all the different types of problems for which the all the different types of problems for which the procedure is typically used. Practice, however, is not procedure is typically used. Practice, however, is not simply solving the same problem or type of problem simply solving the same problem or type of problem over and over again. Practice should be provided in over and over again. Practice should be provided in small doses (about 20 minutes per day) and should small doses (about 20 minutes per day) and should include a variety of problems.include a variety of problems.








These recommendations are based on studies of These recommendations are based on studies of human memory and learning that indicate that most of human memory and learning that indicate that most of the learning occurs during the early phase of a the learning occurs during the early phase of a particular practice session (Delaney et al., 1998). In particular practice session (Delaney et al., 1998). In other words, for any single practice session, 60 minutes other words, for any single practice session, 60 minutes of practice is not three times as beneficial as 20 of practice is not three times as beneficial as 20 minutes. In fact, 60 minutes of practice over three minutes. In fact, 60 minutes of practice over three nights is much more beneficial than 60 minutes of nights is much more beneficial than 60 minutes of practice in a single night.practice in a single night.








Moreover, it is important that the students not simply Moreover, it is important that the students not simply solve one type of problem over and over again as part solve one type of problem over and over again as part of a single practice session (e.g., simple subtraction of a single practice session (e.g., simple subtraction problems, such as 6-3, 7-2). This type of practice problems, such as 6-3, 7-2). This type of practice seems to produce only a rote use of the associated seems to produce only a rote use of the associated procedure. One result is that when students attempt to procedure. One result is that when students attempt to solve a somewhat different type of problem, they tend solve a somewhat different type of problem, they tend to use in a rote manner, the procedure they have to use in a rote manner, the procedure they have practiced the most, whether or not it is applicable. practiced the most, whether or not it is applicable.








Per Geary (1995): “Procedural learning requires Per Geary (1995): “Procedural learning requires extensive practice on the whole range of problems on extensive practice on the whole range of problems on which the procedure might eventually be used” (p. 33).which the procedure might eventually be used” (p. 33).Effective behavioral fluency programs should also Effective behavioral fluency programs should also provide students with knowledge of their progress by provide students with knowledge of their progress by charting their improvement over practice sessions charting their improvement over practice sessions (Binder, 1993; Spence & Hively, 1993).(Binder, 1993; Spence & Hively, 1993).







Integrating Strategy Instruction and Timed Integrating Strategy Instruction and Timed Practice DrillsPractice Drills– Per Kameenui and Simmons (1990)Per Kameenui and Simmons (1990): The learning and : The learning and

retention of basic facts is facilitated by teaching retention of basic facts is facilitated by teaching computations according to their relationships to each computations according to their relationships to each other, instead of according to the sizes of other factors other, instead of according to the sizes of other factors (Cook & Dossey, 1982, Steinberg, 1985; Thorton, 1978).(Cook & Dossey, 1982, Steinberg, 1985; Thorton, 1978).

– Sequencing facts according to their relationships to each Sequencing facts according to their relationships to each other reduces the number of facts that must be learned other reduces the number of facts that must be learned through sheer memorization. Thus, sequencing the through sheer memorization. Thus, sequencing the instruction of basic facts by relationships (e.g., for addition: instruction of basic facts by relationships (e.g., for addition: doubles series 2 + 2, 3 + 3, 4 + 4; plus one facts 4 + 1; 5 + doubles series 2 + 2, 3 + 3, 4 + 4; plus one facts 4 + 1; 5 + 1; doubles plus one 6 + 7, 4 + 5; and reciprocals) is 1; doubles plus one 6 + 7, 4 + 5; and reciprocals) is superior to factor size sequences (e.g., plus one facts; plus superior to factor size sequences (e.g., plus one facts; plus two facts; plus three facts).two facts; plus three facts).







Integrating Strategy Instruction and Timed Integrating Strategy Instruction and Timed Practice DrillsPractice Drills– Teaching rules, principles and relationships for basic fact Teaching rules, principles and relationships for basic fact

mastery will result in greater efficiency of learning, and is thus mastery will result in greater efficiency of learning, and is thus worth the extra attention for instructional design (Baroody, worth the extra attention for instructional design (Baroody, 1984).1984).

– Speed of mathematical fact retrieval from memory relates Speed of mathematical fact retrieval from memory relates directly to overall mathematical achievement in students from directly to overall mathematical achievement in students from elementary school through college (Royer, Tronsky, Chan, elementary school through college (Royer, Tronsky, Chan, Jackson, & Marchant, 1999).Jackson, & Marchant, 1999).

– Haughton (1972) found that children who could solve single-Haughton (1972) found that children who could solve single-digit arithmetic problems at a minimum of fifty to sixty correct digit arithmetic problems at a minimum of fifty to sixty correct per minute were more successful at later parts of a math per minute were more successful at later parts of a math curriculum. As a teacher, you have to determine if you want curriculum. As a teacher, you have to determine if you want students to develop behavioral fluency for some skills, and students to develop behavioral fluency for some skills, and how much time this goal merits in your classroom.how much time this goal merits in your classroom.







What Levels of Computational Fluency Are Desirable?What Levels of Computational Fluency Are Desirable?Curriculum-Based Assessment Research Norms for Math Curriculum-Based Assessment Research Norms for Math

Computational Fluency (Shapiro, 1996)Computational Fluency (Shapiro, 1996)

Digits CorrectDigits Correct

Per MinutePer Minute

Digits IncorrectDigits Incorrect

Per MinutePer Minute

Grades 1-3Grades 1-3

FrustrationFrustration 0-90-9 8 or more8 or more

InstructionalInstructional 10-1910-19 3 to 73 to 7

MasteryMastery 20 or more20 or more 2 or fewer2 or fewer

Grades 4 & UpGrades 4 & Up

FrustrationFrustration 0-190-19 8 or more8 or more

InstructionalInstructional 20-3920-39 3 to 73 to 7

MasteryMastery 40 or more40 or more 2 or fewer2 or fewer






Effective computational fluency programs Effective computational fluency programs provide students with knowledge of their provide students with knowledge of their progress by charting their improvement over progress by charting their improvement over practice sessions (Binder, 1993; Spence & practice sessions (Binder, 1993; Spence & Hively, 1993).Hively, 1993).

What Are the Characteristics of Effective What Are the Characteristics of Effective Computational Fluency Programs?Computational Fluency Programs?

Documents

Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 1