Pedagogical Content Knowledge in Early Mathematics: What ...
Early Mathematics: What’s a Big Idea€¦ · Understanding the Big Ideas of early math develops...
Transcript of Early Mathematics: What’s a Big Idea€¦ · Understanding the Big Ideas of early math develops...
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Mary Hynes-BerryLisa Ginet
Erikson Institute, Chicago4 November 2011
NAEYC Annual Conference
Early Mathematics:What’s a Big Idea ?
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Welcome! My Grandpa is a funny guy.
He always tells people,When I look around my house,
I can count14 feet and 2 tails.
Turn & Talk with a few partners:Who’s Grandpa counting?
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If we want children to learn,we must teach MATHEMATICS.
We must teach for meaning,not test for mastery.
We must guide children toexplore the Big Ideas
that inform skills.
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THE ESSENTIAL ABCS
ALWAYS BE CONVERSING
ALWAYS BE CONNECTING
ALWAYS BUILD COMPETENCE
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Number is Complex!
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Quantity (numerosity) is anattribute of a set of objects;
we use numbers to namespecific quantities.
A Big Idea About Number
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There is no such thing as 3 –or any other number!
You can’t find 3 in the world like a ball;you have to construct the idea of 3 in your head.
Number is an ATTRIBUTE of sets –used to describe the group, not an object in the group.
In math, this attribute is called NUMEROSITY.
Here’s a Big Idea Problem:Naked Numbers look like Nouns
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Children need many opportunities to develop the
understanding that no matter how they are arranged or how sizes compare,
3 things are always 3 things.
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Rote Counting Skillsdon’t count for much
Rational Counting calls for
UNDERSTANDING
Counting has plenty of its own complexities.
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Each number represents a quantity one more than the number
before it and one less than the number
after it.
Rational Counting:Stable Order Principle (Big Idea)
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• Mastery of the number name sequence used by culture• Can count up from given number• Can count down from given number
Stable Order Principle:What learning looks like (skills)
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Video Analysis:Oral Counting
What do these children know about counting?What counting skills have these children
mastered?How can you tell?
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Each item in a collection must be counted
once and only once.
Rational Counting:1-to-1 Correspondence Principle
(Big Idea)
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One number is named for each object pointed at.
1-to-1 Correspondence Principle:What learning looks like (skill)
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It doesn’t matter in which order items are counted.
Rational Counting:Order Irrelevance Principle (Big Idea)
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To assure accuracy of counting, some system is used
such as lining up, pushing away
or somehow noting each item as it is counted.
Order Irrelevance Principle:What learning looks like (skill)
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The last number name used names the quantity of objects in the set.
Rational Counting:Cardinality Principle (Big Idea)
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• When asked, “How many altogether?” names the last number (without re-counting). • When given a story problem, can model the count, using manipulatives, drawings & words.
Cardinality Principle:What learning looks like (skills)
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Video Analysis:Finding an Unknown
What skills has this child mastered?What beliefs does this child seem to have
about doing math?What Big Ideas does this child seem to
understand?How can you tell?
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The C‐P‐S principle:Understanding…
Starts with the Concrete (hands‐on experience) ‐putting one cup with one plate for each person at a table, touching each item as we count, or stacking two piles of blocks to make one “bigger” ‐ taller ‐ than the other.
Moves into the Pictorial ‐ the child can look at (or create) pictures or tally marks and know how to count or compare sizes visually, without actually having hands‐on proof.
And finally progresses to theSymbolic ‐ the child knows that number word fiveand numeral 5 stand for 1,2,3, 4, 5 items.
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SymbolicUnderstanding
& Representation
ConcreteUnderstanding
& Representation
PictorialUnderstanding
& Representation
C-P-S Principle is dynamic
Deep Mathematical Understanding
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Basic Skills
Big Idea Concepts
Teach
ing Strateg
ies
State Standards
Common Core
Adequate Yearly
Progress
Differentiation
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WHOKnowledge of
Children
WHATContent
Knowledge
HOWInstructional
Methods
Early Mathematics Teaching
PCKPedagogical
ContentKnowledge
Shulman, 1986, 1987
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From Common Core Intro:•These Standards define what students should understand and be able to do in their study of mathematics. •Asking a student to understand something means asking a teacher to assess whether the student has understood it. •But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.
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High Impact Teaching
Strategies
Big IdeasSkills &
Procedures
Teaching for Understanding is Dynamic
Focal Points
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Big Ideas & Skills: Number SenseTopic Big Ideas Skills and Procedures
Numerosity Quantity (numerosity) is an attribute of a set of objects; we use numbers to name specific quantities
•Pre-Emergent Number Sense: Confuses the different uses of number; can sense quantities of 3-5 things but considers larger collections as “many,”•Emerging Number Sense: Can compose and decompose sets of 10 and less and up to 20: Understands hierarchical inclusion in these quantities. •Developing Number Sense: Can compose and decompose larger numbers - up to 100 by end of first grade.
Counting Rational counting, that is counting with meaning rather than rote recitation of numbers, involves 4 principles
Stable Order One to One
Correspondence Order Irrelevance Cardinality
The 4 principles of rational counting tend to emerge in the order given.The 4 principles are first mastered for smaller amounts (1-10) and with experience and cognitive development extended to increasingly larger numbers.
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How do the Big Ideas help teachers?Understanding the Big Ideas of early math develops teachers’ adaptive expertise in teaching & learning foundational mathematicswith their young students.
Big Ideas help teachers focus & clarify their goals for children’s learning.
Big Ideas help teachers be more flexible & responsive concerning how children are actually thinking about & doing math in their classrooms.