Students’ Understandings of the Meaning of Letters
Incorrect Pre-Algebra Concepts:
Letter Evaluated
A letter is given a numerical value instead of being treated as an unknown or generalised number.
For example, in response to, ‘if e + f = 8, then e + f + g = …’, many students write 12 (from 4 + 4
+ 4) or 15 (g is the 7th letter of the alphabet).
Letter as Object
A letter is used to denote an object. For example, in 2a + 5b + a = … the expression can be
thought of as ‘2 apples, 5 bananas, and another apple’, or even as ‘2a’s, 5b’s, and another a which
makes 3a’s and 5b’s altogether’.
Although many items can be solved very efficiently in this way, the usage of letters in this way can
lead to severe confusion such as in examples connecting dollars and cents where it is common to
write d = 100c rather than 100d = c. The letters c and d represent numbers, not objects.
Letter Not Used
Here the letter is acknowledged without being given a meaning, or is ignored entirely. For
example, in Add 4 onto 3n, students sometimes combine the familiar elements, 4 and 3, to get 7n
or just 7. The category also applies to items such as a + b = 43, a + b + 2 = … which can be
solved successfully by just focusing on + 2, by which the left-hand sides differ.
Correct Algebra Concepts
Letter as Specific Unknown
Here the letter is thought of as a particular but unknown number that the student is willing to
manipulate despite the ‘lack of closure’ that this involves. For example, adding 4 onto 3n gives
3n + 4 rather than 7n or even 7.
Letter as Generalised Number
Here the letter is seen as being able to take several values rather than just one. For example,
‘What can you say about c if c + d = 10 and c is less than d?’
Letter as Variable
Here the letter is seen as representing a range of values, but in addition, the student is able to
describe the degree to which changes in one set relate to changes in another. For example, ‘Given
a = b + 3, what happens to a if b is increased by 2?’ A response indicating the notion of ‘letter as
variable’ is ‘a is always 3 bigger than b’.
Some Unstated Conventions in Algebra:
Italics
Textbooks show letters in algebra in italics.
Letters as Specified Numbers and Letters as Variables
Letters from the beginning of the alphabet are used for specific numbers - they are literal, and
letters towards the end of the alphabet are used as variables e.g. y  ax2  bx  c
x is the independent variable, y the dependent variable, and a, b, and c are constants.
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Page 1 of 10
The Length of Division Lines -Vinculums - Prioritise Bracketing
Vinculums are used in a number of ways in mathematics usually to indicate grouping. In algebra a
vinculum is effectively a bracketing device indicating division.
3
For example,
means 3 ÷ (x + 4).
x4
The different lengths of vinculums are used to order divisions.
3
For example,
means 3 ÷ {(x + 4) ÷ (y + 6)}
x4

y6
3
whereas x  4 means {3 ÷ (x + 4)} ÷ (y + 6).
y6

The usefulness of the vinculum notation in complex expression is evident: it is easy to read and
understand.

Correct Uses of Letters that are Not Algebra:
Letter as Shorthand for a Specific Number
Examples: the exponential e, , c as the speed of light being nearly 300 000 km/second.
Letters as Shorthand for Units
Examples: m for metres, mg for milligrams. A signal that this is the case is that the letters are not
italicised.
Page 2 of 10
Algebra Assessment
Name:
1
Stage 1
What can you say about a if a + 5 = 8?
Answer:
2
If a + b = 43 then what does a + b + 2 equal?
Answer:
3
What is that area of this shape?
6
Answer:
10
5
4
The perimeter of this shape is 5 + 4 + 6 + 2,
equals 17.
4
which
2
6
10
Work out the perimeter of this shape.
2
Answer:
1
9
Page 3 of 10
5
g
This square has sides of length g. So, for its perimeter, we can write
P = 4g.
g
g
g
What can we write for the perimeter of this triangle?
Answer:
e
e
P=
e
6
Write 2a + 5a more simply.
Answer:
2a + 5a =
Marker only: Stage criterion: 4/6.
Stage achieved is
or higher
(Award stage 0 for 3/6 or less)
Page 4 of 10
1
Stage 2
What is that area of this shape?
Answer:
n
A=
m
2
What is the perimeter of this shape?
h
Answer:
P=
h
h
h
t
3
What is the perimeter of this shape?
Answer:
u
u
5
5
P=
6
4
What can you say about u if u = v + 3 and v = 1?
Answer:
5
What can you say about m if m = 3n + 1 and n = 4?
Answer:
6
u=
m=
Simplify 2a + 5b + a.
Answer:
Marker only: Stage criterion: 4/6.
Stage achieved is
or higher
Page 5 of 10
1
Stage 3
Add 4 onto 3n.
Answer:
2
If e + f = 8 then what does e + f + g equal?
Answer:
3
e+f+g=
Part of this shape is not drawn. There are n sides altogether, each of
length 2. What is the perimeter?
Answer: P =
This information applies to questions 17 and 18.
Cabbages cost $2 each and pumpkins are $3 each. c stands for the number of cabbages bought and
p stands for the number of pumpkins bought.
4
What does 2c + 3p stand for?
Answer:
5
What is the total number of vegetables bought?
Answer:
6
Can 2a + 5b be written more simply?
Answer:
Yes
No
(Circle one.)
Page 6 of 10
7
Write 3a - b + a more simply.
Answer:
8
3a – b + a =
What can you say about r if r = s + t and r + s + t = 30?
Answer:
9
In a shape like this, you can work out the number of diagonals
by taking away 3 from the number of sides.
A shape with k sides has how many diagonals?
Answer:
10
What can you say about c if c + d = 10 and c is less than d?
Answer:
Marker only: Stage criterion: 7/10.
Stage achieved is
or higher
Page 7 of 10
1
Stage 4
Which is larger, 2n or n + 2? Explain.
Answer:
2
n multiplied by 4 can be written as 4n. Multiply n + 5 by 4.
Answer:
3
What is that area of this shape?
5
Answer:
e
4
2
Mary’s basic wage is $200 per week. She is also paid another $20 for each hour of overtime
she works.
h stands for the number of hours of overtime that she works.
w stands for her total wage (in dollars),
Write down an equation connecting w and h.
Answer:
5
Circle the correct answer:
Always
6
Never
e+f+g =e+h+g
Sometimes, when
Cakes cost $c each and pies cost $p dollars each. I buy 4 cakes and 3 pies,
What does 4c + 3p stand for?
Answer:
Page 8 of 10
7
(x + 1)3 + x = 349
This is true when x = 6.
If (3x + 1)3 + 3x = 349 what value of x will make this equation true?
Answer: x =
8
Blue stickers cost 5 cents each and red stickers cost 6 cents each. I buy some blue stickers
and some red stickers and altogether it costs me 90 cents. If b is the number of blue stickers
bought, and if r is the number of red stickers bought, what can you write down about b and
r?
Answer:
9
You can feed any number into this machine:
+ 10
x5
Fill in two numbers in this machine so it has the same overall effect as the first machine:
+
x
Marker only: Stage criterion: 7/9.
Stage achieved is
Overall highest stage achieved is
Adapted from Brown, M., Hart K., & Kuchemann, D., (1984). Chelsea Mathematics Diagnostic
Tests - Algebra. Windsor, England NFER-Nelson.
See also Hart, K.(1981). Children's understanding of mathematics: 11-16. London: John Murray.
Page 9 of 10
Assessing Understanding of Algebra
Description of Stage
Stage
0
Student fails at algebra conceptually and procedurally
1
Student solves problems without having to operate on letters representing numbers. Items are
numerical, or letters can be evaluated, regarded as objects, or not used.
Stage 2 items have increased complexity over items in Stage 1. Student understanding is
procedural. Student cannot utilise algebra effectively.
2
3
4
Student uses letters as specific unknowns or generalised numbers and in some cases, variables. Stage
4 items have increased complexity over items in Stage 3. Student understands and can utilise
algebra.
Percentage of Students at Each Algebra Stage
English Results
13 years
14 years
15 years
Stage 0
10%
6%
5%
Stage 1
50%
35%
30%
Stage 2
23%
24%
23%
Stage 3
15%
29%
31%
Stage 4
2%
6%
9%
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