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

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Presentation on theme: "Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 1."— Presentation transcript:

1 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 1 A White Paper on Computational Fluency (K-12) Presented by Mark Jewell, Ph.D. Chief Academic Officer Federal Way School District

2 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 3 Computational Fluency A look at what the research says and classroom implications.

4 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 6 Project Timeline Initial meeting Review of research literature Compile preliminary research and implications Nov. 20, 2006 Dec. 2006–Feb Jan. 2–4, 2007

7 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 8 Project Timeline 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

9 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 9 What is Computational Fluency? A concept with deep historical roots in the literature of mathematics instruction and assessment.

10 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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. What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

13 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 NCTM, 2000, p. 144

15 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 17 Adding it Up, National Research Council, p. 117

18 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 19 What is Computational Fluency? More Contemporary Thinking Susan Jo Russell on “Accuracy” 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 20 What is Computational Fluency? More Contemporary Thinking Susan Jo Russell on “Efficiency” 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 21 What is Computational Fluency? More Contemporary Thinking Susan Jo Russell on “Flexibility” 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 34 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 35 How Does Computational Fluency Develop? Examples of Addition Strategies Strategy Representative Use to Solve Counting All “1, 2…1, 2, 3, 4…1, 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 “2…3, 4, 5, 6” or “3, 4, 5, 6”

36 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 36 How Does Computational Fluency Develop? Examples of Addition Strategies Strategy Representative Use to Solve Counting on from the larger addend “4…5, 6, “ or “5, 6” Linking “2 + 2 = 4, + 2 more = 6” Retrieval“6”

37 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 as that for addition.” Adding it Up, National Research Council, p. 191

38 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 38 GradeGuessing Counting All Counting-OnDerivedFacts Known Facts K30%22%30%2%16% 18%1%38%9%44% 25%0%50%11%45% Percentage of Time of Students Use Various Addition Procedures (Siegler,1987)

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

51 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 51 How Does Computational Fluency Develop? Automaticity 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 52 How Does Computational Fluency Develop? Automaticity 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 53 How Does Computational Fluency Develop? Automaticity 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 57 How Does Computational Fluency Develop? Computational 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 58 How Does Computational Fluency Develop? Computational 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 differ from a conceptual deficit, but rather from some sort of disruption to normal development of their network of relationships between facts and answers.

59 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 59 How Does Computational Fluency Develop? Computational Fluency and Diverse Students These students often have well-developed number sense and procedural knowledge—they 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 their ability to recall basic math facts from memory (Hasselbring et al., 1988).

61 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 61 How Does Computational Fluency Develop? More About Math-Delayed Students In contrast to their skilled peers, 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 strategies—either developed naturally or through explicit instruction—invariably lead to automaticity.

63 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 generality— that 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 are efficient, accurate, and flexible. 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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). –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. –Identifying and separating fluent from non-fluent facts is important (Woodward, 2006).

71 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 71 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). What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

72 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 72 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). What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

73 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 73 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 student’s 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. What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

74 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 74 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 (Baroody & Coslick, 1998; Gersten & Chard, 1999). –Christensen (1991) found that fact practice, combined with fluency building, produced better effects than strategy instruction. What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

75 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 75 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, as well as others (Howell & Garcia, 1985; Reith, 1985), has shown that computer-based drill and practice results in little or no improvement on the part of handicapped students” (p. 1). What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

76 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 76 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 nothing but strengthen students’ use of counting strategies and does little to move the student toward a state of automaticity (Hasselbring, Goin & Sherwood, 1986). What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

77 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 77 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. What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

78 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 78 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. What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

79 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 79 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. What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

80 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 80 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. What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

81 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 81 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). What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

82 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 82 Integrating Strategy Instruction and Timed Practice Drills –Per Kameenui and Simmons (1990): The learning and retention of basic facts is 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). What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

83 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 83 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. What Learning Experiences are Most Conducive to the Attainment of Computational Fluency?

84 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 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 Frustration0-9 8 or more Instructional to 7 Mastery 20 or more 2 or fewer Grades 4 & Up Frustration or more Instructional to 7 Mastery 40 or more 2 or fewer

85 Mathematics Initiative Office of Superintendent of Public Instruction A White Paper on Computational Fluency (K-12) 2007 WERA Spring Conference Slide 85 Effective computational fluency programs provide students with knowledge of their progress by charting their improvement over practice sessions (Binder, 1993; Spence & Hively, 1993). What Are the Characteristics of Effective Computational Fluency Programs?


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