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

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 http://mathnet.preprints.org/EMIS/journals/ZDM/zdm003r3.pdf 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 http://www.dest.gov.au/NR/rdonlyres/5D4B0095-AA36-4ED9-BD50-07A931B22C91/1645/understanding_place_value.pdf 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. 233-244). Osnabrueck: Forschungsinstitut fuer Mathematikdidaktik. Available http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1-proceedings-1-v1-0-2.pdf Howe, R. (1999). [Review of the book L. P. Ma(1999). Knowing and Teaching Mathematics. London: Lawrence Erlbaum Associates]. Available http://www.ams.org/notices/199908/rev-howe.pdf

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.

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

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

More than one Algorithm The Addition Algorithm An Alternative

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

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

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

Yr 2 Children’s Addition Algorithms

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

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

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

Higher Decade Addition Sometimes a child may know 2 + 7 but does not add correctly 52 + 7 give the correct answer to 6 + 9 but errs in 36 + 9. Help children transfer from simple combinations to the higher decade combinations.  6 1 2 3 4 + 9 5 7 8

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?

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?

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

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

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

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

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

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

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

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

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?

Meaning of Division Equal Distribution Repeated Subtraction

Developing the Division Algorithm Using base-ten blocks

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.

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.

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

The Subtractive Division Algorithm Subtracting a Fixed Amount Subtracting Unequal Amounts

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?

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

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!

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

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?

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