1. WK5: THE ROLE OF ALGORITHMS
Children’s use & understanding of algorithms in whole number operations

2. ADDITIONAL READING LIST
Anderson,A. (2000). [Review of the book Thompson, I. (1999). Issues in teaching numeracy in primary schools. Buckingham: Open University Press]. Available Commonwealth of Australia. (2004). Understanding place value: A case study of the Base Ten Game. [A project funded under the Australian Government’s Numeracy Research and Development Initiative and conducted by the Association of Independent Schools of South Australia]. Available Hedren, R. (1999). The teaching of traditional standard algorithms for the four arithmetic operations vs the use of pupils’ own methods. In I. Schwank (Ed.), Proceedings of the First Conference of the European Society for Research in Mathematics Education (pp ). Osnabrueck: Forschungsinstitut fuer Mathematikdidaktik. Available Howe, R. (1999). [Review of the book L. P. Ma(1999). Knowing and Teaching Mathematics. London: Lawrence Erlbaum Associates]. Available

3. NCTM’s Vision for School Mathematics
Imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction.
Knowledgeable teachers have adequate resources to support their work and are continually growing as professionals.
The curriculum is mathematically rich, offering students opportunities to learn important mathematical concepts and procedures with understanding.

4. Computational Fluency
(a) Conceptual Understanding
(b) Computational Proficiency
understand various meanings of operations of whole numbers and the relationship between the operations;
understand the effects of operations on whole numbers;
understand situations that require different operations, such as equal groupings of objects and sharing equally.
Have skill with mathematics (computation)
Mathematical proficiency includes:
conceptual understanding,
procedural fluency,
strategic competence,
adaptive reasoning

5. What is an algorithm? Some Definitions The Addition Algorithm
 A precise rule (or set of rules) specifying how to solve some problem wordnetweb.princeton.edu/perl/webwn
An effective method for solving a problem using a finite sequence of instructions. Algorithms are used for calculation, data processing …. en.wikipedia.org/wiki/Algorithm

6. More than one Algorithm
The Addition Algorithm
An Alternative

7. Point to Remember Children construct knowledge all the time
They frequently construct algorithms that make sense to them
The constructed algorithms may not match the standard algorithms of teachers

8. Learning Algorithms with Understanding
Concrete materials help
Place-value ideas are important
Algorithms are abstractions of real life problems
Follow Bruner’s enactive (concrete materials), iconic (pictorial) and symbolic (mathematical symbols) stages in helping children understand how algorithms work
When children master place-value ideas, regrouping and renaming, they can construct algorithms that work and understand why they work
Without these ideas, they learn algorithms by rote, become prone to errors

9. Working with Materials
Concrete materials allow pupils to experiment with different ways of representing numbers, try out ideas in regrouping and renaming a number

10. Yr 2 Children’s Addition Algorithms

11. The Standard Addition Algorithm
Building Ideas of Regrouping
Writing it Down 1 1 4 + 7 2

12. The Partial Sum Addition Algorithm
A common occurrence
Writing it down
Separate sums are written down for tens and ones
The separate (or partial) sums are then added together
Less chance of regrouping errors

13. Column Addition: 3 or more numbers
Involves an unseen addend
Different ways of doing column addition
Adding down
Adding up
Grouping numbers to 10

14. Higher Decade Addition
Sometimes a child may
know but does not add correctly
give the correct answer to but errs in
Help children transfer from simple combinations to the higher decade combinations.  6 1 2 3 4 + 9 5 7 8

15. Standard Subtraction Algorithms
The Decomposition Algorithm
The Equal Additions Algorithm 9 1 – 2 4 6 7 9 1 – 2 4 6 7 1 3
How would you normally show your pupils the process of decomposition?
(Meaning show the “borrowing”)
Do you recall an algorithm similar to this that was commonly used in Chinese primary schools (1970s)? Is it still used today?

16. Other Possible Algorithms
Have you come across other algorithms used by your pupils? 9 1 – 2 4 6 7 1
No cancellations are required.
Mentally, pupils think like this:
(10-4)+1=7
9-2-1=6 1
This was commonly used in Chinese primary schools in the 1970s.
Can you identify other algorithms that are used by your pupils today?

17. More Difficult Subtractions
Why are these subtraction problems difficult for young children?
How can you help them deal with these problems?

18. More Difficult Subtractions
Manipulative materials or a place value chart can be used to help children understand the subtraction algorithm involving zeroes

19. Meanings of Multiplication
Multiplication represented as repeated addition
Multiplication represented as an array
3 x 5 means 3 rows of 5 apples

20. Standard Multiplication Algorithm
The Standard Written Method: Efficient and elegant
The Multiplication Grid: Reducing the abstractness x 50 7 40
2000
280
2280 6
300 42
342
2622
Easy for teachers, abstract for pupils!
Easy to understand for multiplication with two digit multipliers

21. Children’s Multiplication Algorithms
Mental multiplication – The Partitioning / Expanded Method
Partial Products Multiplication

22. Lattice Multiplication
To compute 234 x 189
Draw the lines 2 3 4 1 8 9 2 3 4 1 8 9

23. Lattice Multiplication
Multiply each row & column
Add the totals beginning from bottom right cell 2 3 4 1 8 9 2 3 4 1 8 9 6 1 2 1 2 3 4 2 3 4 1 2 3 1 2 3 4 6 4 2 6 4 2 1 2 3 1 2 3 8 7 6 4 8 7 6 2 2

24. Lattice Multiplication
To get the answer ……
Read the number from the left column down and to the right at the bottom of the grid as indicated by the arrows
Thus, the answer is given by 234 x 189 = 44226 2 3 4 1 8 9 6 1 2 1 2 3 4 1 2 3 4 6 4 2 1 2 3 4 8 7 6 2 2

25. Dealing with zeroes In order to avoid children ignoring the zeroes
Encourage children to estimate the answer
Use a place value chart (see p. 267)
Use the expanded multiplication technique (the distributive property of multiplication)
Think about your classroom experiences 2 8 x 3 4
What possible answers can children give to this problem?

26. Meaning of Division
Equal Distribution
Repeated Subtraction

27. Developing the Division Algorithm
Using base-ten blocks

28. Developing the Division Algorithm
Regroup the hundred as 10 tens.
Now there are 17 tens.
Put an equal number of tens in each of the three groups.
There are now 2 tens and 4 ones left.

29. Developing the Division Algorithm
Regroup the 2 tens as 20 ones.
Divide the 24 ones blocks into the three groups, putting an equal number in each group
8 ones blocks will be added to each group .
Each of the 3 groups has 5 tens and 8 ones.

30. Writing the Std Division Algorithm
The Standard Algorithm (The Distributive Algorithm)

31. The Subtractive Division Algorithm
Subtracting a Fixed Amount
Subtracting Unequal Amounts

32. How do kids understand remainders?
About remainders
Is it good to leave division with remainders till the very last?
Can kids understand the concept of remainders?
How do kids make sense of remainders?

33. How did the kids think about 54 ÷ 4?
It is not important to think about what to do with the remainder ie the 2 cubes left
It is up to us to decide what to do with the remainder ie the 2 cubes left

34. How did the kids think about 54 ÷ 4?
The remainder ie the 2 cubes left is a very mysterious thing but mathematics is always full of surprises so it’s Ok!
We know exactly what to do with the remainder ie the 2 cubes left but we may not know how to write it down the way adults do!

35. Points to Note
New developments in relation to standard written algorithms
Arrival of calculators
Research shows kids don’t use teacher-taught algorithms much
Need to go beyond rote learning of algorithms to understanding them

36. Points to Note 3 computational techniques Standard written algorithms
Mental computation
Calculators
Standard written algorithms
Quick, efficient, neat computations
Good for mental agility
May help concept formation
THE DEBATE ON ALGORITHMS
Which ones can be considered the standard algorithms?
To what extent should algorithms be standardised? All of them? Some of them?
When or in which years should the standards ones be taught?

37. The Use of Calculators
First recommended for use in classrooms since 1990s
In practice, still not widely used
Evidence that it is potentially useful as a
counting device
teaching aid to enhance understanding
computational tool for complicated sums
play, explorations and investigations