Module 5 - Whole Number Computation
Typical Progression
Common Difficulties
Manipulatives, Games & Technology
Pedagogy/Strategies for Teaching & Assessment
Mathematical Representation
Concepts initially arise out of real world situations familiar to children
- Division arise out of activities where children share things
- Model concept with manipulative materials and later with diff pictorial reps
- Initially actions and results are described in simple language, later symbolic form of operation and formal language introduced
- Final Stage of Dev - Application of concept to real world
Commutative Property of Addition
a + b = b + a
Associative Property of Addition
(a + b) + c = a + (b + c)
Partioning into part-part-whole shows how addition and subtraction are related (subtraction is inverse of addition)
Inverse Operations - Subtraction can be used to 'undo' an addition and vice versa
Subtraction Structures
Partitioning
Comparison
Mental Computation
Should encourage to use as first resort - using written methods too early can stifle dev of mental computation skills
Strategies
Basic Facts
Table - Only have to learn half of table due to commutative properties pg98
Step 1 - Understanding - Variety of concrete and pictorial aids - Concept dev and appropriate language and symbols
Step 2 - Networking - Dev efficient ways of recalling basic facts, take emphasis off speed and onto discussion and sharing of strategies
Basic Facts
Break a Tower
Number Windows
Cover Ups
Step 3 - Automatic Recalling
"today's number is..." - Write number on board, and students come up with an operation to match
"How did you do it?" Provide mental computation and discuss strategies for how students mentally calculated it
Early Year 1 - Mostly Counting Strategies for mental computation
Later year 1, year 2, students introduced to notion of partioning and splitting numbers
Addition strategies
Relate to a known fact
Identification of pattern
Bridging Tens (how many to ten, how many after ten)
Subtraction Strategies
Partioning
Counting Down
Complementary addition
Bridging Tens
Adding a constant - ie. 15-19 is same as 16-10
Complementary addition "What number must we add to 6 in order to get 10"? (Inverse of addition)
Flexible mental strategies = Facts + Understandings (properties of number system) + Skills (labour saving techniques) + Attitudes (have a go)
Make a Number (ie 3, 5, 8, 9 (3 + 5 + 8 -9 = 7)
Written Computation
Strategies
Informal Written Methods
The Empty Number Line pg 108 ie. 25 (+5) (+20) (+3) = 53
Student invented algorithms (pg 108) -should be efficient, valid, generalisable
Should not be introduced until children can mentally add and subtract two digit numbers.
Written Methods
MAB blocks
Put all the ones together, can you make ten? Trade
Number sense - Fluence with basic number facts, extending number facts, four basic operations, properties and connections between, efficient strategies
Addition
Aggregration structure (two or more quantities are combined)
Augmentation Structure (quantity is increased by some amount)
- number line
- "start at and count on" "Increase by"
- Context of Monday, Temp, Age
Models
Counters
Connecting Cubes
Hundred Square
Number Line
Subtraction - non-commutative!
Take Away - How many does he have left?
Reduction
How many left?
Difference between
523-385 = 262... take smaller number from the bigger number