math coaches and the struggling learner valerie faulkner coordinating teacher sped services wake...
TRANSCRIPT
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Math Coaches and the
Struggling Learner
Valerie Faulkner
Coordinating Teacher SpEd Services
Wake County Public School System
858-1599
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Agenda
The Math – Working on our own skills: How can I improve my own mathematical thinking?
The Culture – How is it different to work with a population of students versus a population of teachers?
The Instruction – What specific changes can I help teachers implement?
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The Questions
Best practices – How do I get teachers to use them?
Number Sense – How do I get teachers to embed it in their instruction?
Differentiation – How do I support stronger differentiation in classrooms?
Assessment – How do we assess and use it for instruction?
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The Math
Elementary Math is Hard!
Are you seeing the
STRUCTURES of the arithmetic
or still looking for Answers?
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The Culture
Remember – We have a Cultural problem.
We tend to think about getting the answers in this lesson, rather than SYSTEMATICALLY working on mathematical structures.
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Analyzing Student Work:Question posed to me by a colleague
names changed to protect the excellent!
I was in a 3rd grade math class and the students were working on this problem:
Jack had picked 12 apples.
Syreeta picked 5 fewer apples than Jack.
How many apples did Jack and Syreeta pick altogether?
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BRAINSTORM:
How is this problem difficult for students?
SMALL GROUP:
How would you support teacher?
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The word fewer confuses them
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Research Based Approach # 1Use number line to ground student in
magnitude and quantity. Process the vocabulary through the
number line.Develop “diagram literacy” grounded in
magnitudeMath Talk!
Subtext taught: Vocabulary grounded in the number line.Numbers ‘live’ on the number line.
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Mathematical Problem
At Food Lion, butter costs 65 cents per stick.This amount is 2 cents less per stick, than butter at
Lowes.If you need to buy 4 sticks of butter, how much will
you pay at Lowes?
(adapted from Hegarty, Mayer, Green, 1992)
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Converting a sentence into a mental representation
At Food Lion, butter costs 65 cents per stick.
So --- Food Lion = 0.65
65
Assignment Sentence: Easy - peasy
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Relational Sentences:Uh – oh…
This amount is 2 cents less per stick, than butter at Lowes.
Say What?
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Cultural Problem
The problem is that almost all of the strategies we learned in math pretend that life is filled with
nothing but assignment sentences!
The only thing that can really help is to Deeply Develop and Understand
Arithmetic Relationships –VISUALLY AND STRUCTURALLY.
This allows us to make sense of the relational sentences.
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• Building a mental model of the problem situation– The hard part of this problem is NOT the
multiplication. It is figuring out the cost of butter at the two stores!
FOOD LION LOWES?
65 LOWES?
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Younger than, fewer than, fewer, less, less than, more than, help!
“More than means add”
“Less than means subtract”
“For less than take the two numbers and subtract but switch the two numbers”
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Number Worlds
Vocabulary Development with Meaning - Not “disembodied”
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Develop Vocabulary in CONTEXT
The number line IS the context
The Otter lesson
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Integrating
• Yolanda is four years younger than Zelda
Y ZZ?
Z – 4 = YY + 4 = ZZ – Y = 4
?
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Comparison bars and math mountains don't work very well.
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Develop diagram literacy grounded in magnitudeDevelop diagram literacy grounded in the
structures of the arithmetic operationsDevelop student understanding that numbers are
made of other numbers
Subtext taught: Math takes timeRespect the depth of Arithmetic: “Profound Understanding of Fundamental
Mathematics”
Research Based Approach # 2
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Doug Clements, Julie Sarama
Subitizing
This is a critical skill and may lay underneath early math number sense difficulties with addition and subtraction.
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Number Sense and Instructional Choices
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Number Sense and Instructional Choices
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Math Mountains
Total/Sum
Jack14
Syreeta 5
Syreeta
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Jack Syreeta
?12
Jack 12
Syreeta5
Jack – 5 = Syreeta12 – 5 = Syreeta12 – 5 = _____
Jack + Syreeta = _____12 + ______ = ______
Comparison Bars
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Structure addition
Join and Part-Part Whole
– There is something, and you get more of it?
– There are two kinds, how many all together?
Start Unknown
– Some are given away, some are left, how many were there to start?
Compare--total unknown
- I know one amount and I have some amount more than that--how many do I have?
?
?
?
Addition types Adapted from Carpenter, Fennema, Franke, Levi and Empson, 1999, p. 12 in Adding it Up, NRC 2001.
Taken Left
What didI start with?
How many altogether?
How many do I have?
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Structure3 types of subtraction
Ask yourself if a problem is a subtraction problem—
Does it fit one of these three types?:
– The Classic “Take away”
(how many left?)
– Comparison
(difference between? who has more?)
– Deficit/Missing amount
(what’s missing?)
You’ve got some amount and “take away” from it: What’s left?
You compare to see: Who has more or less?
You need some more to get where you want to be:What is the missing amount?
?
?
?
Subtraction types adapted from Ma and Kessel, Knowing Mathematics, 2003
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If they have had Number Worlds maybe they would know to count backward from 12 five times.
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Research Based Approach # 3
Teach through Base-TenMath Facts ARE important
- but, perhaps, for a different reason than we think!
Conceptual Repetitions
Subtext taught: Our number system is based on tenI can derive answers using ten as my cornerstone
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Math Facts
8 + 7
Fact Families?Flash Cards?
Rewards?
Problem? We see this as a one-step building block
Cognitively, it is not.
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Makes 10 Left over
________ + _________ = __________
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Makes 10 Left over
____8____ + ____5_____ = _________
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Makes 10 Left over
9 + 7
9 + ( ____ + ____ )
( 9 + ____ ) + ____
(____) + _____
_________
Demonstrate the Associative Property of Addition
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Makes 10 Left over
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The Instruction
• How might The Math affect how you will support instruction?
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NumerationNumeration
Quantity/Quantity/
MagnitudeMagnitude
Base TenBase Ten
EqualityEqualityForm of aForm of aNumberNumber
ProportionalProportional
ReasoningReasoning
Algebraic andGeometricThinking
The Big Ideas© 2007 Cain/Doggett/Faulkner/Hale/NCDPI
Language
The Components of Number Sense
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Embedding Number Sense in Instruction
How many Therapists does it take to change a lightbulb?
How many educators does it take to change a lesson plan?
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So how do you deal with the CULTURAL issues to get to the INSTRUCTIONAL
issues?
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Staff Development Mini-Modules
Create
Objective
and Measurable ‘challenges’
for your staff to grapple with as colleagues
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Equality
8 + 5 = ___ + 3
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When the entire class blows it…
The Math Coach Swoops In!
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Two trucks = one elephant
Are two trucks “the same thing” as one elephant?
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How might Subitizing support instruction in Equality?
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Video: Does 6 = 6?
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Remember!
Part of Assessing is the
Victorious Post-Test!
Give the problems again!
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fluency
Can you use a fluency challenge to sneak base-ten into the school?
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Is it possible that ASSESSMENT
can help us with this?
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Makes 10 Left over
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Makes 10 Left over
9 + 7
9 + ( ____ + ____ )
( 9 + ____ ) + ____
(____) + _____
_________
Demonstrate the Associative Property of Addition
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Stations, Interventions, Activities to support Meaningful Fluency?
Discuss in Small Group
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Group Discussion
Discuss One target issue in your small (2-4 people) group
Develop a plan to:
AssessIntervene
Affect
So you are working Together with your teachers!
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Assess Intervene Affect!
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Knowledge is the Engine
Suggested Books and Articles for Group Study
Ma, Liping. Knowing and Teaching Elementary Mathematics
Carpenter, Franke, Levi. Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School
Burns, Marilyn. About Teaching Mathematics: A K-8 Resource
Carpenter, Hiebert, Fennema, Fuson. Making Sense: Teaching and Learning Mathematics with Understanding
Ball, Deborah: Select journal articles by topic (late elem) at: http://www-personal.umich.edu/~dball/articles/inex.html (articles posted on-line!)
Griffin, Sharon Select journal articles by topic (late elem) at: http://www.clark.edu/academicCatalog/facultybio.cfm?id=39 (you can also
email me—here articles are not posted on-line, just the titles) Clements, Doug Subitizing: What is it? Pdf on Blackboard
Faulkner, Valerie and Cain, Chris Components of Number Sense and How it affected one teacher. Pdf on blackboard