Fluency

Supporting Students Who Struggle with MathGrades 3-5Day2

1Day 2: AgendaBuilding fluencyAn example with one set of factsStrategies for fact fluency

Allegheny Intermediate UnitBuilding Fluency Allegheny Intermediate UnitMemorizing MathematicsConceptual gaps force children to memorize mathematics, often producing the following difficulties:Incomplete or inaccurate memorization of informationLack of retentionLack of transferAllegheny Intermediate UnitStudents may memorize the information partially or incorrectly, for example, memorizing part of the traditional subtraction algorithmStudents may memorize information for a test, then forget most of what they learned, which necessitates constant review.Because children do not understand what they memorized, they cannot use it to assimilate new, even closely related material. For example, after memorizing the subtraction procedure for two-digit numbers, children often misapply it (12 8) or cannot apply it to three-digit numbers. It does not guarantee that they can apply it to a slightly different case such as 304 128.4Mastery of FactsNumber sense (meaningful connections) may affect how basic number combination knowledge is mentally represented in long-term memory and processed Facts are a network of interconnecting relations

Allegheny Intermediate UnitFor example, an understanding of commutatively may allow us to store 5 x 8 = 40 and 8 x 5 = 40 as a single triplet: 5, 8, 40.5Conceptual Understanding Understanding operations Symbolic representations Relationship between parts and whole Understanding is gained through:Problem posingHands-on exploration Classroom discussions Real-world examples Allegheny Intermediate UnitPossible Teaching Sequence X 2X10X5x1X0Allegheny Intermediate UnitX3X4X6X9X8x7

Understanding Multiplication and DivisionAddress the big ideasGuide the types of questions that are posed Explore symbolic representationsUse models to represent addition and subtraction Number lines, manipulatives, ten frames, number charts Explore concepts through problems and literature Allegheny Intermediate UnitBefore math fact practice begins, understanding fo the operations is essential. Students who understand the concepts of addition and subtraction are able to understand the connections between math facts and real situations are better equipped to effectively solve problems by choosing the operation that makes sense. They are better able to make reasonable judgments about sums and differences and are better prepared to begin to remember the math facts because they understand what they are being asked to recall/memorize.8Classroom Environment DiscussionPartner WorkInteractive bulletin boardsWord wallsCenters Allegheny Intermediate UnitLooking at One Example: x2What are the big ideas around multiplying by 2?Multiplication by 2 is same as doubling.Numbers stand for a variety of things. Operation symbols help us determine what the numbers represent.Our number system is a system of patterns.Order of factors does not change the product.Allegheny Intermediate Unit10Possible Questions to Support the Big Ideas X2What does it mean to have twice as much? What does it mean to double a quantity?What does it mean to have half as much?What do the numbers in the equation mean?What patterns do you notice in the products?Does the order of the factors affect the products? Give examples to justify your thinking.How are a sum and a product the same? How are they different?Allegheny Intermediate Unit11Literature ConnectionTwo of Everything read and discuss. http://www.youtube.com/watch?v=TY_NP528ph4 After story, discuss what doubled. Ask:Are doubling and twice as many the same? Explain.Can you find twice as many by adding? How?Can you find twice as many by multiplying? How?

Allegheny Intermediate Unitpp. 35-3712Doubling TaskPlace 1 to 10 counters in plastic bags and give each pair of students 3 or 4 bags containing different quantities. Have additional counters available for students who need to explore the problems by actually creating the doubles. Ask students to determine how many coins are in the bag now and how many would be in the bag after it fell into the pot. Have them record an addition and multiplication equation that would solve the problem and to explain their answer.Allegheny Intermediate UnitWord ProblemsStudents need to visualize the facts using a concrete model and move from concrete/visual experiences to symbolic representations. They need to use concrete items and draw pictures.Allegheny Intermediate UnitWord ProblemsPose problem such as the following:Mrs. Short baked some chocolate brownies. She placed 6 plates on the table and put 2 brownies on each plate. How many brownies did she put on plates?

Allegheny Intermediate UnitObserve Patterns with TwosHave students think about a series of brownie problems, and write a multiplication equation to solve each one.1 plate with 2 brownies on each plate. 2 plates with 2 brownies on each plate. ..10 plates with 2 brownies on each plate.

What patterns do you notice?

Allegheny Intermediate UnitCommutative PropertyProvide students with manipulatives and paper. The paper can represent the baskets. Have them determine the answer to the following:Colin had 2 baskets with 3 apples in each basket, how many apples did he have? Colin had 3 baskets with 2 apples in each basket. How many apples did he have? Allegheny Intermediate Unit17Writing Doubles Word ProblemsWork with the class to write a doubles word problem together..Record the story on chart paper and label it with the appropriate number sentence.Have students work with a partner to write a doubles word problem. Allegheny Intermediate UnitBuilding AutomaticityShort practice daily routineGames Rolling for DoublesDouble UpFact Card JumpsDoubles Match UpConnect to Division

Allegheny Intermediate Unit19GamesPlay the games with a partner.How is this targeted game practice the same or different than other multiplication games?

Allegheny Intermediate UnitMastery of FactsSlow Process: Three PhasesRelatively slow counting strategiesRelatively slow reasoning strategiesMastery relatively fast fact retrieval

Allegheny Intermediate UnitAllowing children to use counting or reasoning strategies:Using informal strategies can lead to discovering patterns and relations that underlie an efficient mental representation and processing of basic combinationsPractice using counting and reasoning strategies will, over time, lead to their efficient execution 21Fluency and AutomaticityLearning Progression stagesUnderstanding manipulatives and pictorial representationsRelationship making connections within and acrossFluency strategy development for accuracyAutomaticity practice to facilitate automaticity Fluency and Automaticity Basic FactsComputation Allegheny Intermediate UnitFluency and AutomaticityMass practice or Drill-and-Kill is NOT an efficient or targeted technique for many struggling students Most practice efforts are NOF FLUENCY based more accuracy based Allegheny Intermediate UnitSteps to Fluency and AutomaticitySpecific criterion for introducing new factsIntensive practice on newly introduced facts (more than 2 x)Systematic practice on previously introduced factsAdequate allotted time (5-10 min/day)Record keepingMotivational procedures

Allegheny Intermediate UnitSteps to Fluency and AutomaticitySpecific criterion for introducing new factsIntensive practice on newly introduced facts (more than 1x) each fact on 4 cardsSystematic practice on previously introduced factsAdequate allotted time (5-10 min/day)Record keepingMotivational procedures

Allegheny Intermediate Unit3-4 factsContinuous cycleRegular schedule Practice Math FactsPeer to peer or individual or small groupStudents must say the fact (Four times five is twenty)Error Correction Procedure The only correct response is the correct answer to the problem All other responses should be correctedFor example, hesitating, saying the incorrect fact, using a strategyAllegheny Intermediate UnitPractice Math FactsStop student and say correct answerSay correct answer with studentHave student say correct answerPartner says correct answerFact is placed three cards back to make sure student has opportunity to re-practice the fact while the correction is still in short-term memory Allegheny Intermediate UnitFacilitating Fluency and Automaticity Instruction for conceptual understanding must occur firstAutomaticity activities must be cumulativeNewly introduced facts receive intensive practice while previously introduced facts receive less intensive practice, but still systematically plannedFluency/automaticity activities should occur for no less than 10 minutes Automaticity practice must be purposeful and systematic as well as carefully controlled by the tecaher

Allegheny Intermediate UnitAllegheny Intermediate Unit

Allegheny Intermediate Unit

Role of TeachersEncourage children to use efficient strategies to retrieve facts, not just memorization Help children conceptually understand operations before drilling factsEncourage children to look for patterns and relations Encourage children to build on what they know Practice should focus on making reasoning strategies more automatic, not on drilling isolated facts Allegheny Intermediate Unit31How Can We Help Students with Facts?Ongoing practice and engagement with math facts tasks Hands-on activities and thoughtful discussions Conceptual understanding of operations Strategic thinking Allegheny Intermediate Unithttps://www.teachingchannel.org/videos/multiplication-division-in-the-core?fd=1 mult anddiv 32Strategic Thinking Math fact strategiesFocuses attention on number sense, operations, patterns, properties, number concepts Big ideas: concept of tens, knowing the order of addends will not affect the sum, various numbers can create the same product (e.g., 8 x 5 = 40 and 4 x 10 = 40)Allegheny Intermediate UnitMeaningful Practice Builds on understanding of operations and using strategic reasoning to explore math factsPractice 5 10 minutes daily throughout the school year Vary the practice activities - ensures that students are motivated and engaged Automaticity is achieved through brief, frequent, interactive activities Allegheny Intermediate UnitTime to PracticeUse the index cards to create a set of math fact cards you might use with a particular student or a small group of students.Practice with your partner.Students must say the fact (Four times five is twenty)Error Correction Procedure The only correct response is the correct answer to the problem All other responses should be correctedFor example, hesitating, saying the incorrect fact, using a strategyStop student and say correct answerSay correct answer with studentHave student say correct answerPartner says correct answerFact is placed three cards back to make sure student has opportunity to re-practice the fact while the correction is still in short-term memory

Allegheny Intermediate UnitRead and DiscussRead the article Why Children Have Difficulties Mastering the Basic Number Combinations and how to Help Them.What are some reasons why students have difficulty mastering basic combinations. What new ideas did you discuss in your group about the relationship between number sense and fluency after reading the article?

Allegheny Intermediate Unit

36Understanding Multiplication and Division Students who understand these concepts recognize the connection between math facts and real situations i.e., 2 vases of flowers with 9 flowers in each vase would be represented by 2 x 9)Are better equipped to effectively solve math problems by choosing the operation that makes sense Are better able to make reasonable judgments about products and quotients E.g., 2 x 9 cant be 11 because you have 2 groups of 9 flowers. That doesnt make sense.Allegheny Intermediate UnitStandards for Mathematical Practice Take time to read the Standards for Mathematical Practice.What is something that stood out/interested you? Something you have a question about?

As a table group, discuss: To what extent do you think these practices are embedded in the daily work of teachers and students?How do number talks support development of the Mathematical Practices?

Math & Science Collaborative at the Allegheny Intermediate UnitMental MathEncourages students to build on number relationships to solve problems instead of memorized procedures Using number relationships helps students develop efficient, flexible strategies with accuracyCauses students to be efficient to avoid holding numerous quantities in their heads Strengthens students understanding of place value Allegheny Intermediate Unit57 + 24 = 83 56 =How might we show one of these strategies with a concrete model? Number line? Story problem?Writing the problem horizontally encourages a student to think about and utilize the value fo the whole number. Number relationships provide the foundation for strategies that help students remember basic facts.

39Number Talks: Supporting FluencyEngage students in mental math through grappling with mathematical problemsA powerful tool for helping students develop computational fluency because the expectation is that they will use number relationships and the structures of numbers to add, subtract, multiply and divide.

Math & Science Collaborative at the Allegheny Intermediate Unit7 x 7407 x 7As you watch the video, consider the following questions:What evidence in the video supports student understanding of multiplication?How do students strategies exhibit number sense?How does the teacher connect math ideas throughout the number talk?How does fluency with smaller multiplication problems support the students strategies?What examples of properties can be observed in the strategies and discussion?Which strategies were most accessible to you? More challenging?How could the strategies for 7 x 7 be connected using an array?

Allegheny Intermediate Unit4 x 72 x 14

41Doubling and HalvingWatch the video and consider the following questions:How does using a context support student reasoning about the commutative property?How is a context used to support student thinking about the doubling and halving strategy?How does the context and the array bring student understandings and misconceptions to the forefront?What questions does the teacher pose to help students confront their misconceptions?Allegheny Intermediate Unit8 x 2542Arrays: 8 x 25As you watch the video, consider the following questions:How does the array model support the student strategies?How does breaking the factors into friendly numbers promote the goals of efficiency and flexibility?How do the teachers questions foster an understanding of multiplication?How could you represent other multiplication problems with an array?How does the array help connect additive thinking to multiplicative reasoning?Which math understandings and misconceptions are addressed in this model?Allegheny Intermediate Unit