old computations, new representations lynn t. goldsmith nina shteingold [email protected] lynn t....
TRANSCRIPT
Old Computations,
New Representations
Old Computations,
New Representations
Lynn T. GoldsmithNina Shteingold [email protected]
Lynn T. GoldsmithNina Shteingold [email protected]
© EDC. Inc., ThinkMath! 2007© EDC. Inc., ThinkMath! 2007
http://www2.edc.org/thinkmath/
© EDC. Inc., ThinkMath! 2007© EDC. Inc., ThinkMath! 2007
Plan of the presentation:Plan of the presentation:• ThinkMath: examples of using different representations in teaching addition, subtraction, multiplication, and division.
• Discussion: - how does using a variety of representations help to build computational fluency?- how does using a variety of representations help to de-bug a concept?
• ThinkMath: examples of using different representations in teaching addition, subtraction, multiplication, and division.
• Discussion: - how does using a variety of representations help to build computational fluency?- how does using a variety of representations help to de-bug a concept?
© EDC. Inc., ThinkMath! 2007© EDC. Inc., ThinkMath! 2007
(Some of) The Problems that Teachers Experience:(Some of) The Problems
that Teachers Experience:
• Different students have different learning styles
• Different students learn with different pace• Without computational fluency students cannot progress to fully comprehend related concepts
• Flows in conceptual understanding are frequent
• There is just not enough time!
• Different students have different learning styles
• Different students learn with different pace• Without computational fluency students cannot progress to fully comprehend related concepts
• Flows in conceptual understanding are frequent
• There is just not enough time!
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One Way of Solving These Problems: Using Multiple Representations
One Way of Solving These Problems: Using Multiple Representations
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Example of Addition and Subtraction
Example of Addition and Subtraction
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From representing number as a quantityand as a position…
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… to representing addition and subtractionboth as a change in the position on the number line…
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… and as a changein quantity.
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What are someof characteristics of the number line representation of addition and subtraction?
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Observing patterns
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Numbers grow…Students do nothave to use the number lineto complete the task, but they can if they need.
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The level of abstractiongrows.
Students rely moreand more on theirinternal representation.
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Cross Number Puzzles
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6 small counters,4 large counters
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7 blue counters,3 gray counters
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Does not matter how you count counters,small and then large,or blue and then gray,you’ll always have the total of 10.
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Cross Number Puzzles
Underline “any order,any grouping” propertyof addition and subtraction
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Interplay of different representations:numbers are represented by “sticks” (each worth10) and “dots”(each worth 1); addition is represented by a partof a Cross NumberPuzzle.
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Moving towards addition algorithm
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Adding moneyis a very good concreterepresentation ofaddition
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Using place valueto add and subtract:1. Same amount on both sides of a thick line;2. Only multiples of 10 in one of the columns.
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It works with more than 2-digit Numbers too.And with more than 2 numbers
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Multiplication and Division Representation
Multiplication and Division Representation• Repeated jumps on a number line• Counting objects in equal groups• Counting North-South and East-West roads and intersections
• Counting lines in one direction, lines in another direction, and intersections
• Counting combinations• Counting dots in an array• Counting rows, columns, and blocks• Calculation “area”
• Repeated jumps on a number line• Counting objects in equal groups• Counting North-South and East-West roads and intersections
• Counting lines in one direction, lines in another direction, and intersections
• Counting combinations• Counting dots in an array• Counting rows, columns, and blocks• Calculation “area”
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Repeated jumps
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Groups ofthe same size
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Combinations
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Combinations of letters (and digits)
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Lines and intersections
This representationIs good forshowing commutativeproperty ofmultiplication as well as for showing what multiplying by 0 means.
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Underlying distributive property
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Underlying distributive property - on a more complexlevel
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Array representationof multiplication
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One cannot just count any more!
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And then to area.
This representation iswell expandable to include multiplication of fractions.
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Interplay of array and Cross Number Puzzle
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How multiplicationand division arerelated
Notice how standard notation for division is being introduced(lower part of the page).
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Connections:Multiplication and division sentencesare used to describe different situations (representations);earlier number sentences were introduced as theirRecords.
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How does using a variety of representations help to build computational fluency:
How does using a variety of representations help to build computational fluency:
• Allows for students’ different learning styles
• Allows for different pace• Helps to increase practice in computation yet to avoid boredom
• ?
• Allows for students’ different learning styles
• Allows for different pace• Helps to increase practice in computation yet to avoid boredom
• ?
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How does using a variety of representations help to de-bug a concept:
How does using a variety of representations help to de-bug a concept:
A representation underlines some properties of a concept but obscures others.
A representation underlines some properties of a concept but obscures others.