WK5: THE ROLE OF ALGORITHMS Childrens 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 Governments 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_pla ce_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 NCTMs 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 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. (b) Computational Proficiency Have skill with mathematics (computation) Mathematical proficiency includes: conceptual understanding,
procedural fluency, strategic competence, adaptive reasoning What is an algorithm? Some Definitions 1. A precise rule (or set of rules) specifying how to solve some problem wordnetweb.princeton.edu/perl/webwn 2. 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
The Addition Algorithm More than one Algorithm The Addition Algorithm An Alternative Point to Remember 1. Children construct knowledge all the time 2. They frequently construct algorithms that make sense to them 3. The constructed algorithms may not match the standard algorithms of teachers
Learning Algorithms with Understanding Concrete materials help Algorithms are abstractions of real life problems Follow Bruners enactive (concrete materials), iconic (pictorial) and symbolic (mathematical symbols) stages in helping children understand how algorithms work Place-value ideas are important 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 Childrens Addition Algorithms The Standard Addition Algorithm Building Ideas of Regrouping Writing it Down 11 4
+ 7 2 1 The Partial Sum Addition Algorithm A common occurrence Separate sums are written down for tens and ones The separate (or partial) sums are then added together Less chance of regrouping errors Writing it down 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
1 2 4 6 7 How would you normally show your pupils the process of decomposition? (Meaning show the borrowing) The Equal Additions Algorithm 3
9 11 2 4 6 7 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 1 12 4 6 7 No cancellations are required. Mentally, pupils think like this: (10-4)+1=7
9-2-1=6 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 Childrens Multiplication Algorithms Mental multiplication The Partitioning / Expanded Method Partial Products Multiplication Lattice Multiplication To compute 234 x 189 2
3 4 Draw the lines 2 3 4 1 1 8 8
9 9 Lattice Multiplication Add the totals beginning from bottom right cell Multiply each row & column 2 3 0 0 2 1
1 6 8 2 2 4 3 4 7 0 3 3 2 4
2 6 3 0 1 0 1 3 2 8
4 9 4 1 6 1 8 2 4 2 4 2
7 2 0 2 1 4 3 2 3 6 6 1 8
9 Lattice Multiplication To get the answer 1. Read the number from the left column down and to the right at the bottom of the grid as indicated by the arrows 2. Thus, the answer is given by 234 x 189 = 44226 2 3 0 0
1 3 2 4 4 1 6 1 8 2 4 2
4 2 7 2 0 2 1 4 3 2 3 6 6
1 8 9 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 x 0 3 8 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 Distributive Algorithm) (The 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 its 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 dont 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 1. Which ones can be considered the standard algorithms? 2. To what extent should algorithms be standardised? All of them? Some of them? 3. 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
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