Old Computations, New Representations

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/


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?


(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!


One Way of Solving These Problems: Using Multiple Representations

One Way of Solving These Problems: Using Multiple Representations


Example of Addition and Subtraction

Example of Addition and Subtraction


QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

From representing number as a quantityand as a position…




… to representing addition and subtractionboth as a change in the position on the number line…




… and as a changein quantity.




What are someof characteristics of the number line representation of addition and subtraction?




Observing patterns







Numbers grow…Students do nothave to use the number lineto complete the task, but they can if they need.




The level of abstractiongrows.

Students rely moreand more on theirinternal representation.


Cross Number Puzzles


6 small counters,4 large counters


7 blue counters,3 gray counters


Does not matter how you count counters,small and then large,or blue and then gray,you’ll always have the total of 10.


Cross Number Puzzles

Underline “any order,any grouping” propertyof addition and subtraction




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.




Moving towards addition algorithm




Adding moneyis a very good concreterepresentation ofaddition




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.




It works with more than 2-digit Numbers too.And with more than 2 numbers


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”




Repeated jumps


Groups ofthe same size




Combinations






Combinations of letters (and digits)




Lines and intersections

This representationIs good forshowing commutativeproperty ofmultiplication as well as for showing what multiplying by 0 means.




Underlying distributive property


Underlying distributive property - on a more complexlevel






Array representationof multiplication




One cannot just count any more!


And then to area.

This representation iswell expandable to include multiplication of fractions.






Interplay of array and Cross Number Puzzle




How multiplicationand division arerelated

Notice how standard notation for division is being introduced(lower part of the page).







Connections:Multiplication and division sentencesare used to describe different situations (representations);earlier number sentences were introduced as theirRecords.








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

• ?


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.

Documents

Old Computations, New Representations