Fluency
Number SenseGrades 3-5Math & Science Collaborative at the
Allegheny Intermediate UnitDay 3SharingShare the mental math
activity and game you did with your class.What successes did your
students have?What challenges did they face?
Math & Science Collaborative at the Allegheny Intermediate
Unit2Read and DiscussDiscuss: Why Children Have Difficulties
Mastering the Basic Number Combinations and how to Help Them. from
TCM, August 2006What 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?
Math & Science Collaborative at the Allegheny Intermediate
Unit
Corinnes Notes from the Article:Three phases of mastering basic
number combinationsCounting strategies (object counting or verbal
counting) starting from 1 or counting onReasoning Strategies (using
known facts to derive new ones)Mastery that is efficientAdding it
Up says attaining computational fluency (efficient, appropriate and
flexible) is essential aspect of math proficiencyVignettes (there
are 4 in this article) can confront some of the things teachers
might now emphasize or do in their classroomsHOW CHILDREN LEARN
BASIC COMBINATIONS:Conventional wisdom says they grow out of rote
memorization of facts through practice and reinforcement (e.g flash
cards and tables)Number sense view says that mastery that UNDERLIES
COMPUTATIONAL FLUENCY grows out of discovering the patterns and
relationships that interconnect the basic computations (part of
phases 1 and 2 above)People more easily learn a body of knowledge
by focusing on its structureWell-connected factual knowledge is
easier to retain in memory and to transfer (ex. Kids who understand
composition, are helped in understanding decomposition)Children
with a rich grasp of no. and arith.patterns are more likely to get
to phase 2Research shows that mental-arithmetic experts have a web
of interconnected ideasPhases 1 and 2 are NEEDED for laying
conceptual groundwork and providing reasoning strategies that
underlie computational fluencyREASONS FOR DIFFICULTIES Defects in
learners or inadequate instruction?Conventionally, many kids
labeled as LD have a heavy reliance on counting strategies; do not
invent reasoning strategies; do not retain basic no. combos greater
than a sum of 5; make mistakes recalling factsConventional view
makes learning facts unduly difficult and anxiety provoking;
discourages looking for patterns and relationships; deflects
reasoning things out; undermines a productive
dispositionInefficiency promoted by memorization of discrete facts
because it is much more challenging learning interrelated ideas. If
sts do not understand teacher-imposed rules, then prone to
associative confusionInappropriate application when memorizing
instead of connecting or making sense no effort or way to check.
(e.g 2+5 is 3 or 7*0=7Inflexibility because less likely to invent
their own strategies and rely on countingSO IN A NUTSHELL, KIDS
HAVE DIFFICULTIES BECAUSE OF ROTE MEMORIZATION TECHNIQUES FOR
DISCRETE FACTSHELPING KIDS MASTER BASIS COMBINATIONSShort-term vs
long-term viewConventional wisdom says lots and lots of practice
and drill with timed tests, etc. Updated version of this gives set
amount of time per fact and limits sets of facts to be learned, but
this still does away with phases 1 and 2Number sense view says that
purposeful, meaningful, inquiry-based instruction is the way to
go.Adding it Up says that computational fluecy should be
intertwined with conceptual understanding, strategic thinking and
productive disposition. (Note that CC is explicit about conceptual
and procedural being balanced using conceptual to build to
procedural)Encourage the invention of , refinement of and sharing
of informal strategies these will get more efficient as number
sense increasesPromote meaningful memorization by looking at
patterns and relationships (e.g. fact families, build on known
ideas, use different strategies)Use practice wisely to discover
patterns, to make reasoning strategies more automaticPut forth idea
that Number combination proficiency should be broadly defined as
any efficient strategies, not just fact retrieval
3Mastery of a basic fact means that a student can give a quick
response (in about 3 seconds) without resorting to inefficient
means, such as counting by ones (Van de Walle, Karp, Bay-Williams,
2013, page 171). Math & Science Collaborative at the Allegheny
Intermediate UnitThis is what John Van de Walle has to say about
basic fact mastery.
Lets see what the CCSSM have to say about it.4Math & Science
Collaborativeat the Allegheny Intermediate UnitModules are built on
the strands
Math & Science Collaborativeat the Allegheny Intermediate
Unitfoundational development occurs PreK- Grade 2
16%K-2=49%30%25%6Mental 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 Math &
Science Collaborative at the Allegheny Intermediate Unit4 x 7, 2 x
14Does this strategy always work?How might we show one of these
strategies with a concrete model? Array. Area 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.
7Video: Doubling and HalvingThe teacher chooses to use a context
as students consider the relationship between 4 x 7 and 2 x 14. How
does this context help support student thinking?How is the
commutative property addressed in the discussion?How do the
teachers questions help students begin to build an understanding of
doubling and halving?The teacher is continually assessing her
students during the number talk. What student understandings and
misconceptions will she use to guide her next instructional
steps?Math & Science Collaborative at the Allegheny
Intermediate Unit8 x 258Array DiscussionThe class had shared
strategies for solving 8 x 25 before the teacher introduced the
array model. Why do you think this instructional decision was made?
How did she link previous strategies to the array?Noel incorrectly
refers to columns as rows in the array. The teacher does not
correct her. Why do you think she choose to ignore this error?How
does the teacher connect the students additive thinking to
multiplication?How does the array support student understanding of
multiplication? The commutative property? The distributive
property?Math & Science Collaborative at the Allegheny
Intermediate Unit32 x 15932 X 15What examples support the strong
learning community that exists in the classroom?What strategy
closely resembles the standard algorithm? How are they similar?
Different?The distributive property is interwoven in most of the
students strategies. What examples do you notice of this property
being used?The topic of efficiency surfaces in this discussion.
Which strategies would you deem ot be most efficient and why?Choose
one of the strategies to model with an open array.Math &
Science Collaborative at the Allegheny Intermediate Unit496 810
496 8How does the succession of problems provide a scaffold for
students to solve 496 8?How does the scaffold provide multiple ways
for students to access the problem,?In what other ways could 496 8
be solved using the prior problems?Jillian had a lovely strategy
for solving this problem; however, she struggled to state what her
answer was. Where is Jillian's answer in the recording on the
board? What mathematical concepts are embraced in her
strategy?Students struggle to follow Jacksons strategy. What
components of division and multiplication does Jackson understand?
The teacher repeatedly asks students to explain where their answer
is in their strategy. Why is that an important focal point
throughout the discussion?
Math & Science Collaborative at the Allegheny Intermediate
Unit11Place Value StandardsPA Core Standards talk about strategies
based on place-value, properties of operations, and/or
relationships between operations.Look at the standards. What
strategies are they focusing on? What are strategies based on
place-value, properties of operations, and/or relationships between
operations.?
Math & Science Collaborative at the Allegheny Intermediate
UnitPA CoreMath & Science Collaborative at the Allegheny
Intermediate Unit
2.OA.2Math & Science Collaborative at the Allegheny
Intermediate Unit
1.OA.6Math & Science Collaborative at the Allegheny
Intermediate Unit
Precursors to FluencyThe sequence of number names, both starting
at 1 and not starting at 1How to count a set, keeping track of the
items they counted Understanding relationships of more, less , and
sameSkip counting starting from 1 and from other numbers
CardinalityConservation 1-to-1 correspondence Making tens link to
understanding place valueSubitizing Decomposing and composing
numbers Understanding part-part-wholeNumber sense
Math & Science Collaborative at the Allegheny Intermediate
UnitTake a few minutes and talk with your group about each one of
these. Pass out concepts cards, 1 or 2 to a table, and tell
participants their table will be responsible for reporting out on
that concept. What does it look like?How will you know if students
have understanding of the concept?How can you help students move
beyond counting by ones to other chunks?What does understanding
part-part-whole mean?
This activity is meant as more of a formative assessment
activity to get information for us.16Fluency demandsAn
understanding of the meaning of operations and their relationship
to each other.The knowledge of a large repertoire of number
relationships. (The patterns of our number system.)Math &
Science Collaborative at the Allegheny Intermediate UnitThis
includes inverse relationships means more than saying addition is
the opposite of subtraction. Students need to know why this is
so.This includes how 4 x 5 is related to 4 x 50, how 6 + 6 is
related to 6 + 7.10 behaves differently in 24 + 10 than it does in
24 x 1017Fluency demands A thorough understanding of the base-ten
number system, how numbers are structured in this system, and how
this system behaves in different operations.Knowing how a number
can be composed and decomposed and using that information to be
flexible and efficient with solving problems.
Involves three components. Math & Science Collaborative at
the Allegheny Intermediate UnitFluencyEfficiency 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.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.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.Math &
Science Collaborative at the Allegheny Intermediate UnitHow 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 Math
& Science Collaborative at the Allegheny Intermediate
Unithttps://www.teachingchannel.org/videos/multiplication-division-in-the-core?fd=1
mult anddiv 20Conceptual Understanding Understanding operations
Symbolic representations Relationship between parts and whole
Investigating the meaning of facts through hands-on activities and
thoughtful discussionsUnderstanding is gained through:Problem
posingHands-on exploration Classroom discussions Real-world
examples Math & Science Collaborative at the Allegheny
Intermediate Unitp.2 Our goal in todays math class has shifted from
memorizing facts and procedures to increased understanding of math
skills and concepts.21Meaningful 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 Math & Science Collaborative at the
Allegheny Intermediate UnitUnderstanding Multiplication and
DivisionAddress the big ideasGuide the types of questions that are
posed Explore symbolic representationsUse models to represent
multiplication and divisionNumber lines, manipulatives, area
models, arraysExplore concepts through problems and literature Math
& Science Collaborative at the Allegheny Intermediate
UnitUnderstanding of the operations is essential. Initial
understanding develops through a contextDeeper understanding can
develop through additional contexts while fluency with the
operation in play develops23Understanding Multiplication and
DivisionWhat does it mean to understand multiplication?Look at the
standards for the operations. Read the standards dealing with
multiplication and division.Then talk with your group. Develop a
chart, graphic, or model showing what students need to know to show
an understanding of multiplication.Math & Science Collaborative
at the Allegheny Intermediate UnitNote that many of these are in
OA, beginning at grade 2 with foundational work for multiplication.
However, some are in the NB standards as well.24Building
Understanding While Focusing on Fluency Use models to represent
multiplication and division Arrays, set models, area models, number
linesUse problem contexts/real-life situations Make sure all four
categories of problems are addressed Productive talkClassroom
environment Math & Science Collaborative at the Allegheny
Intermediate UnitBig Ideas Central to Understanding Multiplication
and DivisionOur number system is a system of patterns.Numbers can
count objects or groups.The order of factors does not change the
product.Addition and multiplication are related
operations.Multiplication and division are inverse
operations.Numbers are flexible.Math & Science Collaborative at
the Allegheny Intermediate UnitPatterns in facts can help students
make sense of, and remember, the facts. For example, noticing that
the products of x6 facts are twice the products of x3 facts or that
x 5 facts have a 0 or 5 in the ones place.From experience with
counting, addition and subtraction, students have developed an
understanding that 2 means 2 objects. In multiplication, a number
can represent the size of each group and the other number
represents the number of groups.Five groups of two has the same
product as 2 groups of five. The representation may look different,
but the total is the same.Both operations combine parts, but
multiplication deals with equal groups and a more efficient way of
combining the groups.This means more than just saying the
operations are related or inverse.Numbers can be broken apart in
different ways. The distributive property is the foundation for
understanding that splitting large factors and then finding the sum
of two products is a way to find unknown products.26Questions Posed
What do the numbers in the equation represent? What patterns do you
notice in the factors and products?Does the order of the factors
affect the product? Give examples to justify your thinking?Do you
notice a connection between this multiplication equation and this
division equation? Explain.Can you break apart one of the factors
to help you find the product?Math & Science Collaborative at
the Allegheny Intermediate Unit27Classroom Environment
DiscussionPartner WorkInteractive bulletin boardsWord wallsCenters
Math & Science Collaborative at the Allegheny Intermediate
UnitIntroducing Concepts of Multiplication and DivisionCompare
methods for solving problems.Provide multiple opportunities for
students to represent and solve multiplication problems.Set
modelsArraysArea ModelsNumber linesExplore divisionMath &
Science Collaborative at the Allegheny Intermediate UnitShow
students a word problem, such as Bailey made favor bags for each of
the 5 guests at her birthday party. She put 3 stickers in each
favor bag. How many stickers were in the 5 bags altogether? Have
students share strategies. Students need to represent
multiplication situations with concrete, pictorial, and symbolic
representations.
29Looking 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.Math &
Science Collaborative at the Allegheny Intermediate Unit30Possible
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?Math & Science Collaborative at the Allegheny
Intermediate Unit31Literature ConnectionTwo of Everything read and
discuss. 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?
Math & Science Collaborative at the Allegheny Intermediate
Unitpp. 35-37http://www.youtube.com/watch?v=TY_NP528ph4 32Word
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.Math & Science Collaborative at the 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?
Math & Science Collaborative at the 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?
Math & Science Collaborative at the 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? Math &
Science Collaborative at the Allegheny Intermediate Unit36Building
AutomaticityShort practice daily routineGames Rolling for
DoublesDouble UpFact Card JumpsDoubles Match UpConnect to
Division
Math & Science Collaborative at the Allegheny Intermediate
Unit37Other Groups of FactsFor the group of facts assigned to your
group, read the pages for your set of facts.Math & Science
Collaborative at the Allegheny Intermediate
UnitX10X5x1X0x3X4X6X9X8x7
Refer to chart on page 10 for rationale for this sequence.38Math
& Science Collaborative at the Allegheny Intermediate
UnitSharingGive a brief overview of you group of facts.Share: An
interesting activity for your fact
39Article DiscussionRead and discuss: Fluency: Simply Fast and
Accurate? I Think Not! by NCTM Past President Linda M. GojakHow do
the ideas in the article resonate with your ideas?What is the same?
What is different?What stood out for you?Math & Science
Collaborative at the Allegheny Intermediate UnitMake sure that it
comes out that fluency entails 3 main
parts:EfficiencyAccuracyFlexibility
My takeaways (Corinne)Entire 2nd and 3rd paragraph really stand
out to meMemorization is one of the least effective ways to reach
fluency (4th paragraph)Like the questions Linda G poses in the 7th
paragraph (Do we give students the opportunity to think about what
they know and understand and use it in ways that make sense to
them? .)8th paragraph mentions mastery of facts I think that as
soon as we see the work master we assume that means to memorize
WRONGAs we think about fluency, we should realize that it is more
than procedural.Last paragraphI am not sure our students actually
enter school thinking the goal is to do math fast and get it right.
I agree that we encourage it, sometimes inadvertantly, sometimes
purposefully.40FluencyLook back at your ideas about fluency. What
ideas, if any, do you want to add to your ideas?Math & Science
Collaborative at the Allegheny Intermediate UnitPut up their
post-its from day 2.41
