A White Paper on Computational Fluency (K-12)

1. A White Paper on Computational Fluency (K-12) Presented by Mark Jewell, Ph.D. Chief Academic Officer Federal Way School District

2. A View of Mathematics from OSPI • 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 mathematics, content must be taught and assessed in meaningful situations.

3. Computational Fluency A look at what the research says and classroom implications.

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. Initial meeting Review of research literature Compile preliminary research and implications Nov. 20, 2006 Dec. 2006–Feb. 2007 Jan. 2–4, 2007 Project Timeline

7. 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 Project Timeline

8. Review computational fluency programs Submit final recommendations to Superintendent Bergeson for review and approval Present recommendations during OSPI Summer Institutes March 26-30, 2007 May 2007 Summer 2007 Project Timeline

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.

11. 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) demonstrated 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 is already being used.

13. What is Computational Fluency?More Contemporary Thinking • NCTM’s 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).

14. What is Computational Fluency?More Contemporary Thinking • NCTM’s Principles and Standards for School Mathematics (2000) • “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

15. What is Computational Fluency?More Contemporary Thinking • NRC’s 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. 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 one’s own efficacy. U.S. National Research Council, 2001, p. 5

17. Adding it Up, National Research Council, p. 117

18. What is Computational Fluency?More Contemporary Thinking • NCTM’s (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.

19. 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)

21. 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)

22. 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 children’s awareness of the processing steps that are required to solve a problem.

24. How Does Computational Fluency Develop?Normal Development of Computational Fluency • Research into the study of children’s 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.

25. 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.

26. 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). Children’s Mathematics. Portsmouth, NH: Heinemann.

27. 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

28. 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 with little effort and some that need to be calculated using some counting strategy.

29. 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

31. 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.

33. 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.

34. 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 How Does Computational Fluency Develop?Strategies to Memorization of Basic Facts: Keys to Mastery Adding It Up National Research Council, p. 187, 190

35. How Does Computational Fluency Develop?Examples of Addition Strategies

36. How Does Computational Fluency Develop?Examples of Addition Strategies

37. Strategies to Memorization:Keys 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 asthatfor addition.” Adding it Up, National Research Council, p. 191

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

39. 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?Computational 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?Computational 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?Computational 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 do 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 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 • 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 student’s 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?Developmental 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?Automaticity as a Foundation for Traditional Algorithm Proficiency • Kirby and Becker (1988) indicated that lack of automaticity in basic operations and strategy use–either the use of an inefficient strategy or the use of the right strategy at the wrong time–were 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.”