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2Computational Fluency, Algorithms, and Mathematical Proficiency Elementary Mathematics WebinarNovember 2013Computational Fluency, Algorithms, and Mathematical Proficiency
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4Computational Fluency Strategies vs. AlgorithmsDrill and PracticeMemorization or AutomaticityTimed Tests
6What does it mean to have computational fluency? Think about the terms “computational fluency”. What do you think this means? How does this translate to instruction?
7Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently.NCTM, Principles and Standards for School Mathematics, pg. 152
8The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships.NCTM, Principles and Standards for School Mathematics, pg. 152
9“Computational fluency entails bringing problem solving skills and understanding to computational problems.”Bass, Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective, Teaching Children Mathematics, pgs
10How do we develop computational fluency in students?
11Developing fluency requires a balance and connection between conceptual understanding and computation proficiency.Computational methods that are over-practiced without understanding are forgotten or remembered incorrectly.Understanding without fluency can inhibit the problem solving process.NCTM, Principles and Standards for School Mathematics, pg. 35
12Conceptual Understanding: Important component of proficiency, along with factual knowledge and procedural facilityEssential component of the knowledge needed to deal with novel problems and settingsNCTM, Principles and Standards for School Mathematics, pg. 20
13What Are the Expectations for Students? Grade LevelCommon Core StandardRequired FluencyKK.OA. 5Add/subtract within 511.OA.6Add/subtract within 1022.OA.22.NBT.5Add/subtract within 20Add/subtract within 10033.OA.73.NBT.2Multiply/divide within 100Add/subtract within 100044.NBT.4Add/subtract within 1,000,00055.NBT.5Multi-digit multiplication66.NS.26.NS.3Multi-digit divisionMulti-digit decimal operations
14KindergartenUnderstand addition, and understand subtraction. K.OA.A.5 Fluently add and subtract within 5
15First Grade Add and subtract within 20. 1.OA.C.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).1.OA.C.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., = = = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding by creating the known equivalent = = 13).
16Second Grade Add and subtract within 20. 2.OA.B.2 Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.Use place value understanding and properties of operations to add and subtract.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
17Third Grade Multiply and divide within 100. 3.OA.C.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.Use place value understanding and properties of operations to perform multi-digit arithmetic.¹3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
18Fourth GradeUse place value understanding and properties of operations to perform multi-digit arithmetic.4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
19Fifth GradePerform operations with multi-digit whole numbers and with decimals to hundredths.5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
21Why should we spend time teaching strategies instead of teaching only the standard algorithm?
22The CCSSM distinguish strategies from algorithms: Computation strategy: purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into anotherComputation algorithm: a set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctlyProgressions for the Common Core State Standards in Mathematics, K- 5 Number and Operations in Base Ten, pg. 3
23Building Strategies“Strategy” emphasizes that computation is being approached thoughtfully with an emphasis on student sense making.Fuson & Beckman, Standard Algorithms in the Common Core State Standards, NCSM Fall/Winter Journal, pgs
24Instruction Should Focus On: Strategies for computing with whole numbers so students develop flexibility and computational fluencyDevelopment and discussion of strategies, so various “standard” algorithms arise naturally or can be introduced by the teacher as appropriateNCTM, Principles and Standards for School Mathematics, pg. 35
25Addition/Subtraction Strategies One-More- Than/Two-More- ThanFacts with zeroDoublesNear DoublesMake 10Think-AdditionBuild up through 10Back down through 10
26Multiplication/Division Strategies Commutative PropertyDoublesFives FactsHelping FactsDouble and Double AgainDouble and one more setNear factsLooking for patternsAllow students to invent strategies that work for them to remember the facts.
27Students who used invented strategies before they learned standard algorithms demonstrated a better knowledge of base-ten concepts and could better extend their knowledge to new situations.When students compute with strategies they invent or choose because they are meaningful, their learning tends to be robust—they are able to remember and apply their knowledge.NCTM, Principles and Standards for School Mathematics, pg. 86
28Common school practice has been to present a single algorithm for each operation. However, more than one efficient and accurate computational algorithm exists for each arithmetic operation. If given the opportunity, students naturally invent methods to compute that make sense to them.NCTM, Principles and Standards for School Mathematics, pg. 153
29“In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable.”Progressions for the Common Core State Standards in Mathematics, K- 5 Number and Operations in Base Ten, pg. 13
31Standard algorithms for base-ten computations with the four operations rely on decomposing numbers written in base-ten notation into base-ten units. The properties of operations then allow any multi-digit computation to be reduced to a collection of single- digit computations. These single-digit computations sometimes require the composition or decomposition of a base-ten unit.Fuson & Beckman, Standard Algorithms in the Common Core State Standards, NCSM Journal, pgs
33Fuson & Beckman, Standard Algorithms in the Common Core State Standards, NCSM Fall/Winter Journal, pgs
34The standard algorithms are especially powerful because they make essential use of the uniformity of the base-ten structure.Fuson & Beckman, Standard Algorithms in the Common Core State Standards, NCSM Journal, pgs
35Students use strategies for addition and subtraction in grades K-3. Students are expected to fluently add and subtract whole numbers using the standard algorithm by the end of grade 4.Progressions for the Common Core State Standards in Mathematics, K- 5 Number and Operations in Base Ten, pg. 3
36For students to become fluent in arithmetic computation, they must have efficient and accurate methods that are supported by an understanding of numbers and operations. “Standard” algorithms for arithmetic computation are one means of achieving this fluency.NCTM, Principles and Standards for School Mathematics, pg. 35
38How does drill and practice impact a student’s ability to become proficient in math?
39x a b c d e f g h i j k l m n o p q r s t u v w y z aa ab ac ad ae af agahaiThink about a student who is lacking in number sense and doesn’t understand the relationships between numbers…For that student, multiplication (or addition drills) can be as meaningless as this chart.
40We know quite a bit about helping students develop fact mastery, and it has little to do with quantity of drill or drill techniques. If appropriate development is undertaken in the primary grades, there is no reason that all children cannot master their facts by the end of grade 3.Van de Walle & Lovin, Teaching Student-Centered Mathematics Grades K-3, pg. 94
41The kinds of experiences teachers provide clearly play a major role in determining the extent and quality of students’ learning.Students’ understanding can be built by actively engaging in tasks and experiences designed to deepen and connect their knowledge.Procedural fluency and conceptual understanding can be developed through problem solving, reasoning, and argumentation.NCTM, Principles and Standards for School Mathematics, pg. 21
42Meaningful practice is necessary to develop fluency with basic number combinations and strategies with multi-digit numbers.Practice should be purposeful and should focus on developing thinking strategies and a knowledge of number relationships rather than drill isolated facts.NCTM, Principles and Standards for School Mathematics, pg. 87
43Do not subject any student to fact drills unless the student has developed an efficient strategy for the facts included in the drill.Van de Walle & Lovin, Teaching Student-Centered Mathematics Grades K-3, pg. 117
44Memorizing facts with flashcards or through drill and practice on worksheets will not develop important relationships:Double plus or minusWorking with the structure of fiveMaking tensUsing compensationsUsing known factsFosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 98
45Drill can strengthen strategies with which students feel comfortable—ones they “own”—and will help to make these strategies increasingly automatic. Therefore, drill of strategies will allow students to use them with increased efficiency, even to the point of recalling the fact without being conscious of using a strategy. Drill without an efficient strategy present offers no assistance.Van de Walle & Lovin, Teaching Student-Centered Mathematics Grades K-3, pg. 117
47How should students develop automaticity How should students develop automaticity? Through drill, practice, and memorization, or through a focus on relationships?
48February 25, 2013*H146-v-1*A BILL TO BE ENTITLED AN ACT TO REQUIRE THE STATE BOARD OF EDUCATION TO ENSURE INSTRUCTION IN CURSIVE WRITING AND MEMORIZATION OF MULTIPLICATION TABLES AS A PART OF THE BASIC EDUCATION PROGRAM.The General Assembly of North Carolina enacts:SECTION 1. G.S. 115C-81 is amended by adding new subsections to read:(l) Multiplication Tables. – The standard course of study shall include the requirement that students enrolled in public schools memorize multiplication tables to demonstrate competency in efficiently multiplying numbers."SECTION 2. This act is effective when it becomes law and applies beginning withthe school year.
49Teaching for Memorization: refers to committing the results of unrelated operations to memory so that thinking is unnecessaryTeaching for Automaticity: refers to answering facts automatically, in only a few seconds without counting, but thinking about the relationships within facts is criticalFosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 98
50There are no “tricks” in math. Understanding math makes it easier Setting up opportunities for students to discover rules or generalizations allows them to exercise reasoning skills as they are making sense of math concepts.O’Connell & SanGiovanni, Putting the Practices Into Action: Implementing the Common Core Standards for Mathematical Practice K-8, pg 124.
51It’s not wise to focus on learning basic facts at the same time children are initially studying an operation. A premature focus gives weight to rote memorization, instead of keeping the emphasis on developing understanding of a new idea.When learning facts, children should build on what they already know and focus on strategies for computing.Burns , About Teaching Mathematics: A K-8 Resource, pg.191
52Memorization of basic facts usually refers to committing the results of unrelated operations to memory so that thinking is unnecessary.Isolated additions and subtractions are practiced one after another as if there were no relationships among them.The emphasis is on recalling the answers.Children who struggle to commit basic facts to memory often believe that there are hundreds to be memorized because they have little or no understanding of the relationships among them.Fosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 98
53When Relationships are the Focus: Fewer facts to rememberBig ideas: compensation, hierarchical inclusion, part/whole relationshipsStrategies for quickly finding answers when memory failsFosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 99
54Is Memorization Faster? A comparison of two first grade classroomsClassroom A focused on relationships and working toward automaticityClassroom B memorized facts with drill sheets and flashcardsStudents in Classroom A significantly outperformed the traditionally taught students in being able to produce correct answers to basic addition facts within three seconds (76% vs 55%)Fosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 99
55Students who memorize facts or procedures without understanding often are not sure when or how to use what they know, and such learning is often quite fragile.NCTM, Principles and Standards for School Mathematics, pg. 20
57Is it necessary to assign a time limit for students to demonstrate knowledge of math facts?
58If teachers highly value speed in mathematics, what are the potential gains for student learning? The potential barriers?Seeley, Faster Isn’t Smarter: Messages about Math, Teaching, and Learning in the 21st Century, pg. 95
59Faster Isn’t SmarterAs part of a complete and balanced mathematics program it is useful to be able to add, subtract, multiply, and divide quickly.It is important to know basic addition and multiplication facts without having to figure them out or count on your fingers.Asking students to demonstrate this knowledge within an arbitrary time limit may actually interfere with their learning.Seeley, Faster Isn’t Smarter: Messages about Math, Teaching, and Learning in the 21st Century, pg. 93
60Computational recall is important, but it is only part of a comprehensive mathematical background that includes more complex computation, and understanding of mathematical concepts, and the ability to think and reason to solve problems.Measuring one aspect of mathematics—fact recall—using timed tests is both flawed as an assessment approach and damaging to many students’ confidence and willingness to tackle new problems.Seeley, Faster Isn’t Smarter: Messages about Math, Teaching, and Learning in the 21st Century, pg. 93
61Those who use timed tests believe that the tests help children learn basic facts. This makes NO instructional sense.Children who perform well under time pressure display their skillsChildren who have difficulty with skills, or who work more slowly, run the risk of reinforcing wrong learning under pressureChildren can become fearful and negative toward their math learning.Burns , About Teaching Mathematics: A K-8 Resource, pg.191
62Overemphasizing fast fact recall at the expense of problem solving and conceptual experiences gives students a distorted idea of the nature of mathematics and of their ability to do mathematics.Seeley, Faster Isn’t Smarter: Messages about Math, Teaching, and Learning in the 21st Century, pg. 95
63Timed tests do not measure children’s understanding. An instructional emphasis on memorizing does not guarantee the needed attention to understandingTimed tests do not ensure that students will be able to use the facts in problem-solving situations.Timed tests convey to children that memorizing is the way to mathematical power, rather than learning to think and reason to figure out answersBurns , About Teaching Mathematics: A K-8 Resource, pg.192
64Timed Tests: Cannot promote reasoned approaches to fact mastery Will produce few long-lasting resultsReward fewPunish manyShould generally be avoidedVan de Walle & Lovin, Teaching Student-Centered Mathematics Grades K-3, pg. 119
65In Conclusion:After looking at research from experts in the field, reflect on your practices.Computational FluencyStrategies vs. AlgorithmsDrill and PracticeMemorization or AutomaticityTimed TestsDoes the research affirm your current teaching practices?Are there practices you would like to revisit?
66“Fluency refers to having efficient, accurate, and generalizable methods (algorithms) for computing that are based on well- understood properties and number relationships.”NCTM, Principles and Standards for School Mathematics, pg. 144Important to note: algorithms are based on well-understood properties and number relationships…strategies have to be built first, then algorithms
71Algorithms Standard algorithm: “for each operation, there is a particular mathematical approach that is based on place value and properties of operations”“an implementation of the particular mathematical approach is called the standard algorithm for that operation”
72Students who understand the structure of numbers and the relationship among numbers can work with them flexiblyNCTM, Principles and Standards for School Mathematics, pg. 149
73Once children understand the process of multiplication and can represent multiplication situations with symbols, they are ready to focus on the number patterns and relationships that will help them internalize the basic multiplication facts. They should spend much of their time exploring and recording multiplication patterns. The search of patterns and relationships will help children learn multiplication facts in a much more powerful way than they would by simply memorizing the times table.(Richardson)