A White Paper on Computational Fluency (K-12)

1
A White Paper on Computational Fluency (K-12)

• Presented by
• Mark Jewell, PhD
• Chief Academic Officer
• Federal Way School District

2
Computational Fluency

• A look at what the research says
• and
• classroom implications.

3
Mathematics

• Mathematics is a language and science of
  patterns.
• Mathematical content (EALR 1) must be embedded in
  the mathematical processes (EALRs 2-5).
• For all students to learn significant mathematic,
  content must be taught and assessed in meaningful
  situations.

4
Computational Fluency Research and
Implications for Practice

• Six Focus Questions
• What is computational fluency?
• How does computational fluency develop?
• How does computational fluency differ from simply
  being able to add, subtract, multiply, and divide?

5
Computational Fluency Research and Implications
for Practice

• How is computational fluency related to
  automaticity?
• What learning experiences are most conducive to
  the attainment of computational proficiency?
• What are the characteristics of effective
  computational fluency programs?

6
Project Timeline

• Initial meeting
• Review of research literature
• Compile preliminary research and implications

• Nov. 20, 2006
• Dec. 2006Feb. 2007
• Jan. 24, 2007

7
Project Timeline

• Present status report at OSPI January Conference
• Develop preliminary recommendations and obtain
  feedback from practitioners across the state and
  national experts

• Jan. 10, 2007
• Jan.Feb. 2007

8
Project Timeline

• Review computational fluency programs
• Submit final recommendations to Superintendent
  Bergeson for review and approval
• Present recommendations during OSPI Summer
  Institutes

• MarchApril 2007
• May 2007
• Summer 2007

9
What is Computational Fluency?

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

10
What is Computational Fluency?

• William Brownell (1935 1956)
• Described meaningful habituation, in many ways
  a historical precursor to computational fluency.
• Advocated an instructional approach that balanced
  meaning and skill.
• Maintained that meaning and skill are
  mutually dependent, even though some people
  attempt to portray them as distinct.

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What is Computational Fluency?

• Stuart Appleton Courtis (1906-1942)
• Developed one of the first published arithmetic
  tests in the U.S.
• Believed that rate tests represented an avenue
  of development largely unexplored (p. 9).

12
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• 1978 NCTM Year Book
• Drill has long been recognized as an essential
  component of instruction in the basic facts.
  Practice is necessary to develop immediate
  recall. Brownell and Chazai (1935) have shown
  quite convincingly that drill increases the speed
  and accuracy of responses to basic-fact problems.
  Those are the purposes for which drill should be
  used. Drill alone will not change the thinking
  that a child uses it will only tend to speed up
  the thinking that a child uses it will only tend
  to speed up the thinking that is already being
  used.

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What is Computational Fluency?More Contemporary
Thinking

• NCTMs Curriculum and Evaluation Standards for
  School Mathematics (1989)
• Children should master the basic facts of
  arithmetic that are essential components of
  fluency with paper-and-pencil and mental
  computation and with estimation (p. 47).
• Practice designed to improve speed and accuracy
  should be used, but only under the right
  conditions that is, practice with a cluster of
  facts should be used only after children have
  developed an efficient way to derive the answers
  to those facts (p. 47).
• It is important for children to learn the
  sequence of steps, and the reasons for them, in
  the paper-and-pencil algorithms used widely in
  our culture. Thus instruction should emphasize
  the meaningful development of these procedures,
  not the speed of processing (p. 47).

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What is Computational Fluency?More Contemporary
Thinking

• NCTMs Principles and Standards for School
  Mathematics (2001)
• Fluency refers to having efficient, accurate,
  and generalizable methods (algorithms) for
  computing that are based on well-understood
  properties and number relationships.
• NCTM, 2000, p. 144

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What is Computational Fluency? More Contemporary
Thinking

• NRCs Adding it Up
• Conceptual Understanding Comprehension of
  mathematical concepts, operations, and relations.
• Procedural Fluency Skill in carrying out
  procedures flexibly, accurately, efficiently, and
  appropriately.
• Strategic Competence Ability to formulate,
  represent, and solve mathematical problems.

16
Adding it Up, National Research Council, p. 117
17
What is Computational Fluency? More Contemporary
Thinking

• Adaptive Reasoning Capacity for logical thought,
  reflection, explanation, and justification.
• Productive Disposition Habitual inclination to
  see mathematics as sensible, useful, and
  worthwhile, coupled with a belief in diligence
  and ones own efficacy.
• U.S. National Research Council, 2001, p. 5

18
What is Computational Fluency? More Contemporary
Thinking

• NCTMs (2006) Curriculum Focal Points for
  Prekindergarten through Grade 8 Mathematics
• Grade 2 Developing quick recall of addition
  and subtraction facts and fluency with supporting
  algorithms is a focus.
• Grade 4 Developing quick recall of the basic
  multiplication facts and related division facts
  and fluency with whole number multiplication.
• Grade 5 Developing an understanding of and
  fluency with division of whole numbers.
• Grade 5/6 Developing an understanding of and
  fluency with addition and subtraction of
  fractions and decimals.

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What is Computational Fluency? More Contemporary
Thinking

• Susan Jo Russell on Accuracy
• Accuracy depends on several aspects of the
  problem solving process, among them, careful
  recording, the knowledge of basic number
  combinations and other important number
  relationships, and concern for double-checking
  results.
• (2000, p. 154)

20
What is Computational Fluency? More Contemporary
Thinking

• Susan Jo Russell on Efficiency
• Efficiency implies that the student does not get
  bogged down in many steps or lose track of the
  logic of the strategy. An efficient strategy is
  one that the student can carry out easily,
  keeping track of sub-problems and making use of
  intermediate results to solve the problem.
• (2000, p. 154)

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What is Computational Fluency? More Contemporary
Thinking

• Susan Jo Russell on Flexibility
• Flexibility requires the knowledge of more than
  one approach to solving a particular kind of
  problem. Students need to be flexible to be able
  to choose an appropriate strategy for the problem
  at hand and also to use one method to solve a
  problem and another method to double-check the
  results.
• (2000, p. 154)

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What is Computational Fluency?

• Is there more to computational fluency than
  identified by Russell (2000)?
• Accuracy Being careful and keeping good records.
• Efficiency Not getting lost or being bogged
  down.
• Flexibility Able to use multiple approaches.

23
How Does Computational Fluency Develop?Types of
Mathematical Knowledge

• According to cognitive psychologists, learning is
  a process in which the learner actively builds
  mental structures, or schemata. These structures
  consist of
• Conceptual Knowledge This is a highly structured
  and interrelated body of knowledge of schemata.
• Declarative Knowledge This type of knowledge
  refers to memorized facts involving arithmetical
  relations among numbers.
• Procedural Knowledge This type of knowledge
  involves childrens awareness of the processing
  steps that are required to solve a problem.

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How Does Computational Fluency Develop?Normal
Development of Computational Fluency

• Research into the study of childrens
  mathematical thinking tells us there is a
  continuum of strategies through which students
  develop computational fluency with basic facts
  and multi-digit numbers in all four operations.
• For basic facts, there are three stages before
  recall, or memorization in each operation.

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How Does Computational Fluency Develop?Normal
Development of Computational Fluency

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

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How Does Computational Fluency Develop?Normal
Development of Computational Fluency

• Experience along the continuum enables the
  student to better determine the reasonableness of
  an answer.
• Students move along the continuum at individual
  rates.
• Often it is the difficulty of the problem that
  determines the strategies the student will use.
• Carpenter, T., Fennema, E., Franke, M., Levi, L.,
  Empson, S. (1999). Childrens Mathematics.
  Portsmouth, NH Heinemann.

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

• The acquisition of math facts generally
  progresses from a deliberate, procedural, and
  error-prone calculation to one that is fast,
  efficient, and accurate.
• Ashcraft, 1992 Fuson, 1982, 1988 Siegler, 1988

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

• For many students, at any point in time from
  preschool through at least the fourth grade, they
  will have some facts that can be retrieved from
  memory without little effort and some that need
  to be calculated using some counting strategy.

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

• From the fourth grade through adulthood, answers
  to basic math facts are recalled from memory with
  a continued strengthening of relationships
  between problems and answers that results in
  further increases in fluency.
• Ashcraft, 1985

30
How Does Computational Fluency Develop?The
Acquisition of Addition and Subtraction Facts

• In a typical developmental path in addition,
  students begin adding using a strategy called
  counting on strategy, which in turn gives ways
  to linking new facts to known facts.
• Garnett, 1992

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

• The most frequently used and most efficient
  counting strategy among kindergarten, first, and
  second grade students was a minimum addend
  counting.
• Siegler 1987 Siegler Shrager, 1984

32
How Does Computational Fluency Develop?The
Acquisition of Addition and Subtraction Facts

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

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

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

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

• Addition
• Count All
• Just One More
• Count On
• Small Doubles
• -Doubles /-
• Makes a 10
• Related Facts

• Subtraction
• Count Back
• Just One Less
• Count Up
• Related Facts
• Subtraction Neighbors
• Finding Doubles
• Over the Hill

Adding It Up National Research Council, p. 187,
190
35
How Does Computational Fluency Develop?Examples
of Addition Strategies
Strategy Representative Use to Solve 2 4
Counting All 1, 21, 2, 3, 41, 2, 3, 4, 5, 6
Shortcut Sum 1, 2, 3, 4, 5, 6
Finger Display Displays 2 fingers, then 4 fingers says 6
Counting on from the first addend 23, 4, 5, 6 or 3, 4, 5, 6
36
How Does Computational Fluency Develop?Examples
of Addition Strategies
Strategy Representative Use to Solve 2 4
Counting on from the larger addend 45, 6, or 5, 6
Linking 2 2 4, 2 more 6
Retrieval 6
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Strategies to MemorizationKeys to Mastery

• When counting up is not introduced, many
  children may not invent it until the second or
  third grade, if at all. Intervention studies
  with U.S. first graders that helped them see
  subtraction situations as taking away the first x
  objects enabled them to learn and understand
  counting-up-to procedures for subtraction. Their
  subtraction accuracy became as high as that for
  addition.
• Adding it Up, National Research Council, p. 191

38
Percentage of Time of Students Use Various
Addition Procedures (Siegler,1987)
Grade Guessing Counting All Counting-On Derived Facts Known Facts
K 30 22 30 2 16
1 8 1 38 9 44
2 5 0 50 11 45
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How Does Computational Fluency Develop?The
Acquisition of Multiplication and Division Facts

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

40
How Does Computational Fluency Develop?Computatio
nal Fluency and Brain Science

• Recent research in cognitive science using
  functional magnetic resonance imaging (FMRI), has
  revealed the actual shift in brain activation
  patterns as untrained math facts are learned.
• Delazer et al., 2003

41
How Does Computational Fluency Develop?Computatio
nal Fluency and Brain Science

• Instruction and practice cause math fact
  processing to move from a quantitative area of
  the brain to one related to automatic retrieval.
• Dehaene, 1997 1999 2003

42
How Does Computational Fluency Develop?Computatio
nal Fluency and Brain Science

• Delazer and her colleagues suggest that this
  shift aids the solving of complex computations
  that require the selection of an appropriate
  resolution algorithm, retrieval of intermediate
  results, storage and updating in working memory
  by substituting some of the intermediate steps
  with automatic retrieval.
• Delazer et al., 2004

43
How Does Computational Fluency Develop?The
Importance of Automaticity in Mathematics

• All human beings have a limited
  information-processing capacity. That is an
  individual simply cannot attend to too many
  things at once.
• Some of the sub-processes, particularly basic
  facts, need to be developed to the point that
  they are done automatically. If this fluent
  retrieval does not develop, then the development
  of higher-order mathematical skills, such as
  multiple digit addition and subtraction, and
  fractions-may be severely impaired. (Resnick,
  1983).

44
How Does Computational Fluency Develop?The
Importance of Automaticity in Mathematics

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

45
How Does Computational Fluency Develop?The
Importance of Automaticity in Mathematics

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

46
How Does Computational Fluency Develop?The
Importance of Automaticity in Mathematics

• Once procedures are automatized, they require
  little conscious effort to use, which, in turn,
  frees attentional and working memory resources
  for use on other more important features of the
  problem (Geary, 1995).
• When a basic fact is executed without conscious
  monitoring and attention, it is considered to
  have become automatic (Goldman Pellegrino,
  1987).

47
How Does Computational Fluency Develop?The
Importance of Automaticity in Mathematics

• Automaticity is useful both in and out of the
  classroom (Isaacs Carroll, 1999).
• Counting strategies and the use of electronic
  calculators interfere with learning higher level
  math skills such as multiple-digit addition and
  subtraction, long division, and fractions
  (Resnick, 1983).

48
How Does Computational Fluency Develop?The
Importance of Automaticity in Mathematics

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

49
How Does Computational Fluency Develop?Developmen
tal Perspective of Automaticity

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

50
How Does Computational Fluency Develop?Automatici
ty as a Foundation for Traditional Algorithm
Proficiency

• Kirby and Becker (1988) indicated that lack of
  automaticity in basic operations and strategy
  useeither the use of an inefficient strategy or
  the use of the right strategy at the wrong
  timewere responsible for the majority of math
  problems that children experience.
• Based on the results of their research, Kirby and
  Becker concluded that children with learning
  problems in arithmetic do not have any major
  structural defect in their information processing
  systems or that they are qualitatively different
  from normally achieving students in any enduring
  sense.

51
How Does Computational Fluency Develop?Automatici
ty as a Foundation for Traditional Algorithm
Proficiency

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

52
How Does Computational Fluency Develop?Automatici
ty as a Foundation for Traditional Algorithm
Proficiency

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

53
How Does Computational Fluency Develop?Automatici
ty as a Means for Developing Number Sense

• Isaacs and Carroll (1999) note that automaticity
  in math facts is essential to estimation and
  mental computations.
• These skills, particularly the ability to perform
  mental computations (e.g., make approximations
  based on rounded numbers such as 10s and 100s),
  are central to the ongoing development of number
  sense.

54
How Does Computational Fluency Develop?Why Speed
of Recall Matters

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

55
How Does Computational Fluency Develop?Why Speed
of Recall Matters

• Zentall and Ferkis (1993) stated that slow and
  inaccurate computational skill may place further
  attention load on the problem solving process.
• Zawaiza and Gerber (1993) noted that many
  researchers believe that automaticity can free
  attentional resources necessary for more complex
  and abstract aspects of some problem solving (p.
  65).
• High rates of accurate responding have been
  called fluent (Haring Eaton, 1978 Marston,
  1989) or automatic responding (Gagne, 1983).

56
How Does Computational Fluency Develop?Why Speed
of Recall Matters

• Gagne (1983) suggested that automatic responding
  to basic mathematics problems allows students
  more cognitive energy to focus on higher level
  skills.
• Haring and Eaton (1978) suggested that students
  who can accurately perform basic skills at higher
  rates have been exposed to over learning and,
  therefore, are more likely to maintain those
  skills.

57
How Does Computational Fluency Develop?Computatio
nal Fluency and Diverse Students

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

58
How Does Computational Fluency Develop?Computatio
nal Fluency and Diverse Students

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

59
How Does Computational Fluency Develop?Computatio
nal Fluency and Diverse Students

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

60
How Does Computational Fluency Develop?More
About Math-Delayed Students

• What this suggests is that there are huge
  differences in the amount of instruction
  individual children need to become fluent at
  retrieving answers to basic math facts.
• By age seven, non math-delayed students can
  recall more facts from memory than their
  math-delayed peers, and this discrepancy
  increases as age increases.
• As math-delayed students get older, they fall
  farther and farther behind their non math-delayed
  peers in the ability to recall basic math facts
  from memory (Hasselbring et al., 1988).

61
How Does Computational Fluency Develop?More
About Math-Delayed Students

• In contrast to their skilled peers, students with
  struggling math students have a serious problem
  with respect to the retrieval of basic number
  facts.
• Fleischner, Garnett, and Ginsburg (1984) have
  found that students with learning disabilities
  are substantially less proficient than students
  without learning disabilities in retrieving basic
  math facts in addition and subtraction.

62
How Does Computational Fluency Develop?More
About Math-Delayed Students

• Cumming and Elkins (1999) point out that many
  educators and researchers make the unwarranted
  assumption that strategieseither developed
  naturally or through explicit instructioninvariab
  ly lead to automaticity.

63
How Does Computational Fluency Develop?More
About Math-Delayed Students

• Research indicates that students with LD do not
  develop sophisticated fact strategies naturally
  (e.g., Geary, 1993 Goldman et al., 1988) .
• Empirical research on strategy instruction in
  math facts for students with LD is limited, and
  the results are mixed in terms of the effective
  development of automaticity (see Putnam,
  deBettencourt Leinhardt, 1990 Tournaki, 2003).

64
How Does Computational Fluency Develop?Add,
Subtract, Multiply, and Divide?

• Although there is some controversy about the need
  to build computational fluency, there are data to
  suggest that fluency with basic skills can help
  students with later learning and application of
  those skills (Binder, 1993 Spence Hively,
  1993).
• Torbeyns, Verschaffel, and Ghesiquiere (2005)
  investigated the fluency with which first graders
  of different mathematical achievement levels
  applied multiple, school-taught strategies for
  finding arithmetic sums over 10. High-achieving
  students applied the strategies more efficiently
  but not more adaptively than did their lower
  achieving peers.

65
How Does Computational Fluency Develop?Add,
Subtract, Multiply, and Divide?

• At any point in time from preschool through at
  least fourth grade, most students will have some
  facts that they can retrieve from memory
  automatically and some that have to be
  reconstructed using procedural knowledge. From
  the fourth grade through adulthood, simple
  addition and subtraction problems are solved with
  a continued strengthening of relationships
  between problems and answers, which results in
  further increases in the speed of retrieving all
  facts (Ashcraft, 1985).
• Hung-Hsi Wu (2001), professor of mathematics at
  the University of California at Berkeley, has
  argued that computational fluency is a
  prerequisite for success in algebra. According
  to Wu, if students are not sufficiently fluent
  with the basic skills to take the numerical
  computations for granted, either because they
  lack practice or rely too frequently on
  technology, then their mental disposition toward
  computations of any kind would soon be one of
  apprehension and ultimately instinctive evasion
  (p. 3).

66
How Does Computational Fluency Develop?Add,
Subtract, Multiply, and Divide?Differing
Perspectives on Standard Algorithms

• The term algorithm sometimes provokes disdain
  among educators because of the oppressive ways in
  which traditional algorithms often are taught.
  In fact, algorithms are remarkable tools in
  mathematics and computer science. They have
  great practical and theoretical importance.
• Standard algorithms were gradually developed many
  centuries ago for their efficiency, accuracy, and
  generalitythat is, they work in all situations.
  They are theoretically and practically important
  methods for computing. They contain in their
  very structure all the basic properties of the
  base-ten place-value system, set forth in as
  efficient a manner as possible. An understanding
  of how and why they work, as well as the ability
  to use them fluently, provides the foundation for
  mathematical competence.

67
How Does Computational Fluency Develop?Add,
Subtract, Multiply, and Divide?Differing
Perspectives on Standard Algorithms

• As children acquire knowledge of the underlying
  structure of a particular operation and explore
  different ways to perform it, they should also
  learn how to use the standard algorithm for the
  operation. After they learn a standard algorithm
  for an operation, whatever they then choose to
  use routinely should be judged on the basis of
  efficiency and accuracy. Children should be able
  to explain whatever method they use and see the
  usefulness of methods that efficient, accurate,
  and general.
• A 15-member group of mathematicians, appointed by
  the Mathematical Association of America to
  respond to a set of questions about algorithms
  and algorithmic thinking posed by the National
  Council of Teachers of Mathematics Commission on
  the Future of the Standards, stated that
  standard mathematical definitions and algorithms
  serve as a vehicle of human communication and
  that they should be taught to all children (Ross,
  1997).

68
How Does Computational Fluency Develop?Add,
Subtract, Multiply, and Divide?Differing
Perspectives on Standard Algorithms

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

69
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• A Preliminary List of Recommendations
• Early Numeracy Programs
• Griffin (2005) recommends that early numeracy
  programs include activities that provide
  opportunities for children to acquire
  computational fluency as well as conceptual
  understanding (p. 283).
• Drill and Practice versus Strategy Instruction
• Teaching students the use of effective strategies
  to solve basic math fact problems enhances
  learning, leading to automaticity (e.g., Morin
  Miller, 1998 Thornton, 1978).

70
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Drill Practice Programs
• Drill and practice programs have demonstrated a
  positive effect on improving the retrieval speed
  for facts already being recalled from memory
  (Woodward, 2006).
• However, drill and practice had no effect on
  developing automaticity for non-recalled facts
  (Hasselbring, Goinn, Sherwood, 1986).
• To facilitate the automatic recall of all facts,
  instruction must be focused on non-automatized
  facts while practice and review are given on
  facts that are already being recalled from
  memory.
• Thus identifying and separating fluent from
  non-fluent facts is important (Woodward, 2006). .

71
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Strategy-Based Fluency
• Issacs and Carroll (1999) emphasize that students
  naturally develop strategies for learning math
  facts if given the opportunity.
• Research supporting the natural development of
  strategies may be found for addition and
  subtraction (Baroody Ginsburg, 1986 Carpenter
  Moser, 1984 Resnick, 1983 Siegler Jenkins,
  1989) as well as more recent work in the area of
  multiplication (Angghileri, 1989 Baroody, 1997
  Clark Kamii, 1996 Mulligan Mitchelmore,
  1997 Sherin Fuson, 2005).

72
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

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

73
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Integrating Strategy Instruction and Timed
  Practice Drills
• Cumming and Elkins research (1999) suggests that
  a middle-ground position for teaching facts to
  academically low-achieving students and students
  with LD consists of integrating strategy
  instruction with frequent timed practice drills.
  Results of their research indicate that
  instruction in strategies does not necessarily
  lead to automaticity. Frequent timed practice is
  essential. However, strategies help increase a
  students flexible use of numbers, and for that
  reason, Cumming and Elkins advocate the use of
  strategy instruction for all students through the
  end of elementary school.

74
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Integrating Strategy Instruction and Timed
  Practice Drills
• Strategy instruction can benefit the development
  of estimation and mental calculations. In this
  respect, strategy instruction helps develop
  number sense, a topic of emerging interest in the
  special education literature (author Baroody
  Coslick, 1998 Gersten Chard, 1999).
• Christensen (1991) found that fact practice,
  combined with fluency building, produced better
  effects than strategy instruction.

75
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Integrating Strategy Instruction and Timed
  Practice Drills
• Hasselbring, Goin, Bransford (1988) concluded
  that computer-based drill and practice can be
  used to develop automaticity, but only when
  specific prerequisite conditions are met. If
  these prerequisite conditions are not met, our
  research, we well as others (Howell Garcia,
  9185, Reith, 1985), has shown that computer-based
  drill and practice results in little or not
  improvement on the part of handicapped students
  (p. 1).

76
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Integrating Strategy Instruction and Timed
  Practice Drills
• According to Hasselbring, Goin, and Bransford
  (1988), Neither paper and pencil drill and
  practice nor computer-based drill and practice
  seems to be sufficiently powerful in itself for
  developing automaticity in learning handicapped
  students. Additional work on developing a
  declarative knowledge network is needed before
  drill and practice is effective. Practice that
  allows students to use counting strategies does
  noting but strengthen students use of counting
  strategies and does little to move the student
  toward a state of automaticity (Hasselbring, Goin
  Sherwood, 1986).

77
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Integrating Strategy Instruction and Timed
  Practice Drills
• Computational Fluency and Curriculum-Based
  Measurement
• Deno and Mirkin (1977) suggested that in order to
  demonstrate mastery in mathematics, students
  should complete mathematics computation problems
  at a rate of 20 digits correct per minute in
  first through third grades, and 40 digits correct
  per minute in subsequent grade levels.

78
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

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

79
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

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

80
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

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

81
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Integrating Strategy Instruction and Timed
  Practice Drills
• Time Needed for Practice
• Per Geary (1995) Procedural learning requires
  extensive practice on the whole range of problems
  on which the procedure might eventually be used
  (p. 33).
• Effective behavioral fluency programs should also
  provide students with knowledge of their progress
  by charting their improvement over practice
  sessions (Binder, 1993 Spence Hively, 1993).

82
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Integrating Strategy Instruction and Timed
  Practice Drills
• Per Kameenui and Simmons (1990) The learning and
  retention of basic facts us facilitated by
  teaching computations according to their
  relationships to each other, instead of according
  to the sizes of other factors (Cook Dossey,
  1982, Steinberg, 1985 Thorton, 1978).
• Sequencing facts according to their relationships
  to each other reduces the number of facts that
  must be learned through sheer memorization.
  Thus, sequencing the instruction of basic facts
  by relationships (e.g., for addition 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 superior to factor size sequences
  (e.g., plus one facts plus two facts plus three
  facts).

83
What Learning Experiences are Most Conducive to
the Attainment of Computational Fluency?

• Integrating Strategy Instruction and Timed
  Practice Drills
• Teaching rules, principles and relationships for
  basic fact mastery will result in greater
  efficiency of learning, and is thus worth the
  extra attention for instructional design
  (Baroody, 1984).
• Speed of mathematical fact retrieval from memory
  relates directly to overall mathematical
  achievement in students from elementary school
  through college (Royer, Tronsky, Chan, Jackson,
  Marchant, 1999).
• Haughton (1972) found that children who could
  solve single-digit arithmetic problems at a
  minimum of fifty to sixty correct per minute were
  more successful at later parts of a math
  curriculum. As a teacher, you have to determine
  if you want students to develop behavioral
  fluency for some skills, and how much time this
  goal merits in your classroom.

84
What Levels of Computational Fluency Are
Desirable?Curriculum-Based Assessment Research
Norms for Math Computational Fluency (Shapiro,
1996)
Digits Correct Per Minute Digits Incorrect Per Minute
Grades 1-3
Frustration 0-9 8 or more
Instructional 10-19 3 to 7
Mastery 20 or more 2 or fewer
Grades 4 Up
Frustration 0-19 8 or more
Instructional 20-39 3 to 7
Mastery 40 or more 2 or fewer
85
What Are the Characteristics of Effective
Computational Fluency Programs?

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