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Computational Fluency, Algorithms, and Mathematical Proficiency
Elementary Mathematics Webinar November 2013 Computational Fluency, Algorithms, and Mathematical Proficiency

kitty.rutherford@dpi.nc.gov or
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Computational Fluency
Strategies vs. Algorithms Drill and Practice Memorization or Automaticity Timed Tests

Computational Fluency

What 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?

Computational 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

The 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

“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

How do we develop computational fluency in students?

Developing 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

Conceptual Understanding:
Important component of proficiency, along with factual knowledge and procedural facility Essential component of the knowledge needed to deal with novel problems and settings NCTM, Principles and Standards for School Mathematics, pg. 20

What Are the Expectations for Students?
Grade Level Common Core Standard Required Fluency K K.OA. 5 Add/subtract within 5 1 1.OA.6 Add/subtract within 10 2 2.OA.2 2.NBT.5 Add/subtract within 20 Add/subtract within 100 3 3.OA.7 3.NBT.2 Multiply/divide within 100 Add/subtract within 1000 4 4.NBT.4 Add/subtract within 1,000,000 5 5.NBT.5 Multi-digit multiplication 6 6.NS.2 6.NS.3 Multi-digit division Multi-digit decimal operations

Kindergarten Understand addition, and understand subtraction. K.OA.A.5 Fluently add and subtract within 5

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).

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.

Third 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.¹ 3.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.

Fourth Grade Use 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.

Fifth Grade Perform 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.

Strategies vs. Algorithms

Why should we spend time teaching strategies instead of teaching only the standard algorithm?

The 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 another Computation 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 correctly Progressions for the Common Core State Standards in Mathematics, K- 5 Number and Operations in Base Ten, pg. 3

Building 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

Instruction Should Focus On:
Strategies for computing with whole numbers so students develop flexibility and computational fluency Development and discussion of strategies, so various “standard” algorithms arise naturally or can be introduced by the teacher as appropriate NCTM, Principles and Standards for School Mathematics, pg. 35

One-More- Than/Two-More- Than Facts with zero Doubles Near Doubles Make 10 Think-Addition Build up through 10 Back down through 10

Multiplication/Division Strategies
Commutative Property Doubles Fives Facts Helping Facts Double and Double Again Double and one more set Near facts Looking for patterns Allow students to invent strategies that work for them to remember the facts.

Students 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

Common 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

“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

Kamii, Young Children Reinvent Arithmetic, pg 8

Standard 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

456 + 167 How would you solve this problem?

Fuson & Beckman, Standard Algorithms in the Common Core State Standards, NCSM Fall/Winter Journal, pgs

The 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

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

For 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

Drill and Practice

How does drill and practice impact a student’s ability to become proficient in math?

x 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
ag ah ai Think 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.

We 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

The 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

Meaningful 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

Do 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

Memorizing facts with flashcards or through drill and practice on worksheets will not develop important relationships: Double plus or minus Working with the structure of five Making tens Using compensations Using known facts Fosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 98

Drill 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

Memorization or Automaticity?

How should students develop automaticity
How should students develop automaticity? Through drill, practice, and memorization, or through a focus on relationships?

February 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 with the school year.

Teaching for Memorization: refers to committing the results of unrelated operations to memory so that thinking is unnecessary Teaching for Automaticity: refers to answering facts automatically, in only a few seconds without counting, but thinking about the relationships within facts is critical Fosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 98

There 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.

It’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

Memorization 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

When Relationships are the Focus:
Fewer facts to remember Big ideas: compensation, hierarchical inclusion, part/whole relationships Strategies for quickly finding answers when memory fails Fosnot & Dolk, Constructing Number Sense, Addition, and Subtraction, pg. 99

Is Memorization Faster?
A comparison of two first grade classrooms Classroom A focused on relationships and working toward automaticity Classroom B memorized facts with drill sheets and flashcards Students 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

Students 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

Timed Tests

Is it necessary to assign a time limit for students to demonstrate knowledge of math facts?

If 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

Faster Isn’t Smarter As 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

Computational 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

Those 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 skills Children who have difficulty with skills, or who work more slowly, run the risk of reinforcing wrong learning under pressure Children can become fearful and negative toward their math learning. Burns , About Teaching Mathematics: A K-8 Resource, pg.191

Overemphasizing 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

Timed tests do not measure children’s understanding.
An instructional emphasis on memorizing does not guarantee the needed attention to understanding Timed 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 answers Burns , About Teaching Mathematics: A K-8 Resource, pg.192

Timed Tests: Cannot promote reasoned approaches to fact mastery
Will produce few long-lasting results Reward few Punish many Should generally be avoided Van de Walle & Lovin, Teaching Student-Centered Mathematics Grades K-3, pg. 119

In Conclusion: After looking at research from experts in the field, reflect on your practices. Computational Fluency Strategies vs. Algorithms Drill and Practice Memorization or Automaticity Timed Tests Does the research affirm your current teaching practices? Are there practices you would like to revisit?

“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. 144 Important to note: algorithms are based on well-understood properties and number relationships…strategies have to be built first, then algorithms

What questions do you have?

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Contact Information Kitty Rutherford Denise Schulz Website: maccss.ncdpi.wikispaces.net

For all you do for our students!

Algorithms 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”

Students who understand the structure of numbers and the relationship among numbers can work with them flexibly NCTM, Principles and Standards for School Mathematics, pg. 149

Once 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)