© University of Reading 2007 www.reading.ac.uk

The 'algebra as object' analogy: a view from schoolDr Geoff Tennant and Kate Colloff

g.d.tennant@reading.ac.uk

Plan of session• Background to the issue;• Original research;• Some reflections on the process;• Questions?

The issueTeaching of early simplification of algebra, need to

communicate that:

3a + 2a = 5aAlgebra as object: ‘a’ is the object ‘apple’, so can

read this as ‘3 apples plus 2 apples is 5 apples’

From: CRANK, V et al. (1998). Key Maths 7. Cheltenham: Stanley Thornes.

From: CRANK, V et al. (1998). Key Maths 7. Cheltenham: Stanley Thornes.

From: CRANK, V et al. (1998). Key Maths 7. Cheltenham: Stanley Thornes.

Questions from Oxford GCSE Maths for Edexcel, 2010, p41.

Number: Ian, Chris and Tim Statistics: Helen, Fiona, Chris, Mairead, and Tim.

Number: Ian, Chris and Tim Statistics: Helen, Fiona, Chris, Mairead, and Tim.

Three objections (Tennant 2009)

1. It’s wrong: this is abbreviation, not algebra!

Tennant, G. D. (2009). 2a+3a=5a without apples: the algebra as object analogy. Mathematics in School, 38(1), pp. 4-6.

2. It ‘works’ only in a very narrow band of examples

eg. 3a – 2a = a would work

but -2a + 3a doesn’t, nor does 3a – 4a

…..

Three objections (Tennant 2009)

And what about

abor

a2 ?

Three objections (Tennant 2009)

3. It leads to problems further on (Clement 1982);Eg 1: in a University there are six students to each

professor. If S is the number of students and P is the number of professors, write down a formula connecting S and P;

For one professor there are 6 students:

P = 6S

Three objections (Tennant 2009)

Clement, J. (1982). Algebra word problem solutions: thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13(1), pp. 16-30.

Eg 2: 5 miles is (approximately) the same as 8 kilometres. If m is the number of miles and k is the number of kilometres, write down a formula connecting m and k.

5 miles = 8 kilometres

5m = 8k

Three objections (Tennant 2009)

Textbook approaches to an introduction to simplification

• ‘Algebra as object’ already noted;• Ambiguous;• ‘Algebra as object’ merges into algebra as

variable;• Algebra as variable.

Textbook approach: ambiguous

From GREER, A (1993). A complete GCSE mathematics basic course: third edition. Cheltenham: Stanley Thornes.

a + a + a + a + a is 5 x a. In algebra it is written as 5a (ie. with the multiplication sign left out). 5a is called an algebraic term.In the same way3y = 3 x y = y + y + y

Exercise: write down each of the following as an algebraic term:1. m + m + m….

Textbook approach: algebra as object merges into variable

From MILLER et al. (1992). MSM Mathematics 2w. Walton-on-Thames: Nelson.

Textbook approach (3): algebra as object merges into variable

From MILLER et al. (1992). MSM Mathematics 2w. Walton-on-Thames: Nelson.

Textbook approach: simplification with algebra as variable

From VICKERS, K M et al. (1995). National Curriculum Mathematics 5. Cheltenham: Stanley Thornes (Publishers) Ltd.

DISCUSSION EXERICSEIs a + b the same as b + a? Is 5a + 3a = 8a?Is 5a – 3a = 2a? Is 3a + 4b = 7ab?Is 5a + 4b + 3a + 7b = 8a + 11b? Is 5a + 4b – 3a – 7b = 2a -3b?Discuss. As part of your discussion replace a and b with many different numbers.

Continues with ‘ambiguous’ approach.

Textbook approach (4d): simplification with algebra as variable

From LEDSHAM, A & WARDLE, M E (1996). Foundation GCSE mathematics: revision and practice. Oxford: Oxford University Press.

The perimeter of an equilateral triangle is given by the formula p = a + a + a or p = 3aFind the perimeter of an equilateral triangle whose side length a is 5cm.The perimeter, p = (3 x 5) cm = 15 cm.

a a

a

Textbook approach (4d): simplification with algebra as variable

From LEDSHAM, A & WARDLE, M E (1996). Foundation GCSE mathematics: revision and practice. Oxford: Oxford University Press.

The perimeter of an isosceles triangle is given by the formula p = a + a + b or p = 2a + bFind the perimeter of an isosceles triangle with a = 3cm and b = 5cm.The perimeter, p = (2 x 3 + 5) cm = (6 + 5) cm = 11 cm.

a a

b

Textbook approach (4d): simplification with algebra as variable

From LEDSHAM, A & WARDLE, M E (1996). Foundation GCSE mathematics: revision and practice. Oxford: Oxford University Press.

The perimeter of a square is given by the formula p = l + l + l + l or p = 4lFind the perimeter of a square whose side length is 7cm. The perimeter, p = (4 x 7) cm = 28 cm.

l

l

l

l

Textbook approach (4d): simplification with algebra as variable

From LEDSHAM, A & WARDLE, M E (1996). Foundation GCSE mathematics: revision and practice. Oxford: Oxford University Press.

The perimeter of a rectangle is given by the formula p = l + w + l + w or p = 2l + 2w or p = 2(l + w)Find the perimeter of a rectangle with l = 5 cm and w = 3 cm. The perimeter, p = 2(5 + 3) cm = (2 x 8) cm = 16 cm.

w

l

w

l

Textbook approach (4e): simplification with algebra as variable

From ATKINSON, C et al. (1998). Nelson GCSE mathematics: intermediate 1. Walton-on-Thames: Thomas Nelson & Sons Ltd.

The function is n → 3n + 1. Copy and complete this box.

n →3n + 1 6 →? 23 →? ? →100 ? →1

Textbook approach (4e): simplification with algebra as variable

From ATKINSON, C et al. (1998). Nelson GCSE mathematics: intermediate 1. Walton-on-Thames: Thomas Nelson & Sons Ltd.

The function is n → 3n + 2 + 4.

(a) Copy and complete this box.

(b) What is a simpler way of writing the function?

n → 3n + 2 + 4 8 → ? 23 → ? 0 → ? n → ?

Textbook approach (4e): simplification with algebra as variable

From ATKINSON, C et al. (1998). Nelson GCSE mathematics: intermediate 1. Walton-on-Thames: Thomas Nelson & Sons Ltd.

Copy and complete these boxes, and write each function in a simpler way:

n → 5n - 3n 6 → ? 10 → ? 13 → ? n → ?

Further suggestions (1):consecutive numbers

• Ask volunteer to give a number (eg. 23), offer a prize for any pupil who can get add up that number and the next 4 consecutive numbers (ie. 23+24+25+26+27) faster than you.

• Repeat until it becomes clear to the pupils that you have a method which enables you to get the answer quickly.

Further suggestions (1):consecutive numbers

Suggest that they gather information systematically:

First no. Sum Total

0 0+1+2+3+4 10

1 1+2+3+4+5 15

2 2+3+4+5+6 20

3 3+4+5+6+7 25

10 10+11+12+13+14 60

Formula

5x0 + 10

5x1 + 10

5x2 + 10

5x3 + 10

5x10 + 10

n n+(n+1)+(n+2)+(n+3)+(n+4) 5n+10 5n+10

Further suggestions (2):number pyramids

Volunteer gives 5 numbers (eg. 1,2,3,4,5) which go into the bottom of the pyramid. Then each box in the rows above is the sum of the two boxes below.

48

20 28

8 12 16

3 5 7 9

1 2 3 4 5

Further suggestions (2):number pyramids

Possible questions to ask:

(a) How would you arrange the numbers in the bottom row to get the biggest total on the top row?

(b) As (a), the lowest total?(c) What happens to the number in the box on top if you add 1 to each of

the boxes in the bottom row, one at a time? Can you explainyour answer?

……(d) If you put a, b, c, d, e, in the bottom row, can you write a formula

for the number in the top row?(e) Look at the answers different people have got. What do you notice?

Further suggestions (2):number pyramids

a b c d ea+b b+c c+d d+e

a+2b+c b+2c+d c+2d+ea+3b+3c

+eb+3c+3d

+e

a+4b+6c+4d+e

Further suggestions (3):matching exerciseMake up sets of cards as a matching activity. Have students explain why they think any set of cards do or don’t match.

3n +

2n

5 lots of n

4n

n + n + n + n

n + (n+1) +

(n+2)

3n + 3

3(n+1)

25

2n + 9 + n + 7

5n + 10 + 6 –

2n

n + m + n +

m

2n +

2m

2(n+m)3n + 8+ 2m + 12

5n + 20 5m + 20

3n + 16

BEWARE: not all cards have partners!

New fieldwork• Review of up to date textbooks;• Interviews with colleagues;• Focus group of year 7 pupils;• Focus group of year 10 pupils.

Textbook analysis• 10 texts and 3 electronic resources in current

usage within school. Of the textbooks:• 4 straight to algebra;• 3 generalised arithmetic;• 2 symbols (implying algebra as object);• 1 kinaesthetic object (length of rod).

No clear patterns by publication date, with a range from 1955-2010.

Teacher interviews (1)Aware that ‘algebra as object’ not recommended:

• You cannot multiply apples.• The letter represents a number, not an object.• You cannot get cubic or squared bananas.• It means that students struggle to access other

things later.• It is then difficult when you introduce a

negative number of apples.

One teacher knew ‘not allowed’ but couldn’t explain why.

Teacher interviews (2)• Teachers using eg. faces of pupils to illustrate

simplification, rationalising the approach as not being ‘algebra as object’ ;

• Differences of opinion as to how to teach basic simplification;

• Differences of opinion as to ‘good’ approaches for high and low attaining pupils – latter more prone to ‘algebra as object’ through frustration and pressure for exam passes.

Year 7 interviews (1)- 35 pupils interviewed;- In general, pupils who have previously achieved

higher at maths able to answer simplification questions more accurately;

- When explaining simplification, answers included:- “2+3 then put 'a' back on” (74%);- “Collecting like terms” (9%);- Wrong, including 5, 5aa and 7 (9%);- “Guess / don’t know” (9%).

Year 7 interviews (2)- No year 7 pupil explicitly appealed to an

‘algebra as object’ approach (eg. “3 apples plus 2 apples is 5 apples”).

Year 10 interviews (1)30 top set pupils in total of 8 groups, asked how they would explain introductory simplification of algebra to year 7 pupils.

6 of 8 groups appealed to ‘algebra as object’

Remaining 2 went straight to algebra and appeals to ‘rules’: “It’s just law”.

Year 10 interviews (2)Answers confused when dealing with simplification not easily undertaken with ‘algebra as object’:

“a is blue, b is red, so ab is purple”“a2 is two apples”“a x b = apple x banana = apple and banana.”

ImplicationsNeed to ensure approaches to algebra, making

links with number and generalisations in words, covered thoroughly in ITT and CPD courses;

Ensure plenty of examples of other approaches readily available.

Questions arisingWhere does the idea of ‘algebra as object’ arise

from between year 7 and 10?

Do teaching approaches which avoid ‘algebra as object’ work in practice?

What happens in other countries?

Reflections on the processKeep research interests close to professional

interests;Explore something that is a burning interest;Can you get students to work with you? So their assignments feed into your larger project?

General principle: data is all around you – even your annual reports!

And finally….

Thank you!Dr Geoff Tennant

g.d.tennant@reading.ac.ukPersonal website: www.geofftennant.name