The transition from arithmetic thinking
to algebraic thinking
Kaye Stacey
University of Melbourne, Australia
k.stacey@unimelb.edu.au
Introduction
At the invitation of the conference organisers, this talk addresses the transition from arithmetic
thinking to algebraic thinking. This has been a topic of substantial interest in mathematics education
research for over 20 years. The importance of algebra, especially as a gateway to high mathematics,
is appreciated by education systems around the world, but at the same time, in an era when an
increasingly large proportion of students have the opportunity to attend algebra classes in secondary
school, algebra is increasing recognised as a subject that is hard to learn and hard to teach well
(Stacey, Chick & Kendal, 2004). Designing instruction that can maximise the learning opportunities
of all students requires an in-depth understanding of the new challenges that students meet as they
approach algebra. Otherwise algebra becomes a wall that blocks their paths to progress.
In this presentation, I will give several examples that illustrate the differences that confront students
between the ways of solving problems that they have learned in arithmetic and the new ways of
solving problems that are inherent in using algebra. By looking at students’ mathematical thinking,
we can see the transitions that they have to make as they move from arithmetic to algebra. Teachers
who are aware of these transitions can make students aware of them too. Algebra leads to
arithmetic, but the path is not straightforward. In the second section, I will give some brief
examples of the way in which the teaching of arithmetic can be transformed in order to better
prepare students for algebra. In this sense, algebra need not follow arithmetic, but aspects of
algebraic thinking can be developed from the beginnings of school by teachers with the ‘algebra
eyes and ears” (Blanton & Kaput, 2003) to capitalise on opportunities.
Problem 1.
There are 100 buns to be shared by 100 monks. The senior monks get 3 buns each. Every 3 junior
monks have to share 1 bun. How many senior and junior monks are there?
Problem 2.
Mark and Jan share $47, but Mark gets $5 more than Jan. How much do they each get?
Problem 3,
Tiger Company rents cars for $100 for initial rental and 20 cents per km. How far can I drive for
$240?
Problem 4.
Tiger Company rents cars for $100 for initial rental and 20 cents per km. Kangaroo Company rents
cars for $120 for initial rental and 15 cents per km. Which company is cheaper?
Problem 5.
Our class is collecting items to sell. If we collect an average of 10 items every school day, how
many weeks will it take us to collect 500 items?
Problem 6.
Our class is collecting items to sell. If we collect an average of d items every school day, how many
weeks will it take us to collect N items?
Figure 1. Problems used in this article.
Solving problems in algebra or arithmetic.
Students’ preference for numerical methods
Historically, and in the education of nearly all children, algebra grows out of arithmetic. However,
there are significant transitions that students need to make from the approaches adopted in solving
problems arithmetically, to the new methods that open up with algebra. In Figure 2, there are three
solutions to Problem 1 in Figure 1, which was given to 124 Singapore Grade 6 to 8 students (Wong,
2008, & personal communication). Singapore is one of the world’s highest achieving countries in
mathematics.
Averaged across 12 similar problems, approximately 10% of students’ solutions used equations,
and approximately 50% of these solutions were correct. All other methods, such as the guess-checkimprove solution in Figure 2 had considerable better success rates (around 75%). We will return to
this observation later.
The equations solution in Figure 2 is an interesting example of a common phenomenon. Even
though the answer is correct, the solution is wrong. Although the equations look like algebra, the
meanings given to them by the student are not algebraic. In this solution x stands for a senior monk
(or something associated) and y stands for a junior monk (or something associated). The equation x
= 3 buns means a senior monk eats 3 buns, the equations 3y = 1 means that 3 junior monks eat 1
bun and the equation x+ 3y = 4 means that a group of 1 senior monk and 3 junior monks eat 4 buns.
This correct observation leads to the correct answer, but the algebra here is only abbreviated words.
This is hard for a teacher to detect – students often write what looks like algebra, and follow some
algebra rules, but intend a quite different meaning.
Figure 2. A guess-check-improve solution and an equations solution to Problem 1.
This solution shows two characteristics of students’ equations – that they see algebra as a shorthand
for writing abbreviated mathematical sentences, rather than a strictly formal system which has great
power provided everything is done correctly and according to the rules (see MacGregor & Stacey,
1997). Students often also use algebraic letters to stand for objects (such as people) rather than
numbers. In contrast, in school algebra, algebraic letters always stand for numbers. This example
reveals some of the new ways of thinking that students need to adopt to solve problems using
algebra.
Some of the other transitions that students have to make to use algebra come from Australian Years
9 – 11 students’ solutions to Problem 2 of Figure 2 (Stacey & MacGregor, 2000). These students
were requested to solve the problem using algebra. Some sample solutions are shown in Figure 3.
In general, once again, students were generally more successful without algebra than with it. The
solution methods (heuristics) were broadly the same as those used by the Singapore students.
Guess-check-improve solutions such as Sara’s were very common, and successful in this simple
problem with a whole number answer. Solutions such as William’s were rare. Many students used
logical arithmetic reasoning, and the two main solutions are illustrated by Brenda and Wylie.
Brenda, observing the request to solve the problem by algebra has added an x and a y at the end of
her solution. She has done the hard work of solving the problem (in this case realising that it is
possible to begin by halving the money, and then give some of Jan’s half to Mark) but has added the
‘algebra’ at the end. Wylie has realised that the problem can be solved by first giving Mark $5 and
then sharing the remainder, and has written a ‘formula; that expresses this. In both cases, these
clever students do not understand that algebra could help in solving the problem. Instead they have
adopted an arithmetic approach, working from the known numbers in stages towards the goal. This
is very different to the algebraic method, where the first step is simply to describe the relationships
in the problem, not to try to “solve” it. Then, following rules of transformation, this description of
the problem is worked upon, through a chain of equivalent descriptions until the solution is evident.
Students such as Brenda and Wylie believe they have to solve the problem in a more ‘active’ way.
This is an important difference between solving problems by arithmetic and by algebra.
Brenda
47  2 = 23.5 - 2.5 = x
47  2 = 23.5 + 2.5 = y
Sara
15 + 32
16 + 31
..
21 + 26
= 47, difference 17 too big
= 47, difference 15 too big,
.
= 47, difference 5 - solution!
Wylie
y = (47 - 5)  2 + 5 = 42/2 + 5 = 26
x = (47-5)  2 = 42/2 = 21
William
x
2
2
x
+ (x + 5 ) = 47
. x + 5 = 47
. x = 42
= 21
Figure 3. Four students’ solutions to Problem 2.
The unknown in arithmetic and algebra
Many students were interviewed while they were solving Problem 2 (Stacey & MacGregor, 1997).
In Figure 4, there are extracts from 4 students’ interviews. In Joel’s interview, it is evident that he
has 2 different meanings for x in his solution, both “the amount they both get” ($42) and also x is
Jan’s amount. Les uses x for any unknown quantity, in this extract for three unknowns (Jan’s
amount, Mark’s amount and the amount left after taking $5 from $47). Les is thinking
arithmetically, moving from knowns to unknowns and labeling the successive unknowns by x. Tim
also uses x as a general label for unknown quantities, not appreciating that an expert in algebra
believes this problem has only one “unknowns”. To Tim, thinking arithmetically, there are many
unknown quantities. Leonie’s interview puzzled us for some time. She appeared to be good at
algebra, but refused to write the equation x + (x+5) = 47, insisting that x and y were not the same
and so she could not substitute one for the other. Like the Singapore student in Figure 2, Leonie was
thinking of x and y not as numbers (the value of the money each person had), but as objects (the
actual money that each person had in their hand). Since the notes and coins are not physically the
same objects (although with the same total numerical value), she could not agree to substitute x for
y.
Joel
Joel writes x for Jan’s money and x+5 for Mark’s money, then x + 5 = 47
Interviewer: Points to x + 5 = 47. “What does this say?”
Joel: “(it’s) the amount they both get. The amount that Jan gets. I just like to keep the three of
them, 47 dollars, x and 5 dollars and make something out of them.”
Les
Les begins by writing 5 + x = 47
Les: “x is what is left out of $47 if you take 5 off it.”
Interviewer: “What might the x be?”
Les: “Say she gets $22 and he gets $27. They are sharing two x’s.”
Int: “What are the two x’s?”
Les: “Unknowns…they are two different numbers, 22 and 27.”
Int: “So what is this x?” (pointing to 5 + x = 47)
Les: “That was what was left over from $47, so its $42.”
Tim
Tim writes x + 5 for Mark’s amount and then writes x+5 = x, saying the x after the equals sign is
Jan’s x
Tim: (pointing to first x in x+5 = x) “That’s Mark’s x”.
Int: “And why do we add 5 to it?”
Tim: “Because Mark has 5 more dollars than Jan. No, that’s not right; it should be Jan’s x plus
5 equals Mark’s x.”
Int: “Could you write an equation to say that Mark and Jan have $47 in total? You don’t have
to work out the answer first.”
Tim: “x divided by a half equals x”(writes x 1/2 = x)
Leonie
Leonie writes (x + 5) + y = 47, and cannot progress; nothing interviewer say helps her.
Leonie explains to the interviewer that
 y is the money that Jan has
 (x+5) is the money that Mark has and that this= says Mark’s money is $5 more than Jan’s
money
 together they have $47
Leonie believes that the equation (x+5) + x = 47 is wrong because y is not the same as x.
Figure 4. Extracts from interviews with four students working on Problem 2.
Figure 5 summarises the differences in the methods of solving problems in arithmetic and in
algebra. These have been illustrated above. Making the transition from the arithmetic method to the
algebraic method requires deep new learning.
Arithmetic Problem Solving
Work from knowns to unknowns
Algebraic Problem Solving
Working with and on unknowns throughout
Unknowns change through problem
Unknown fixed
Equation as formula to produce an answer
Equation as description of relationship
Chains of successive calculations
Chains of successive equalities
Guess and check equation solving
“Do the same to both sides” equation solving
Intermediate results can be interpreted in
problem situation
Intermediate results are not interpreted in
problem situation
Undoing operations one by one
Undoing operations one by one
Figure 5. Characteristics of arithmetic problem solving and algebraic problem solving.
Developing new algebra concepts through arithmetic thinking
Dettori et al (2001) suggested that the transition from arithmetic to algebra requires a change in the
nature of problem resolution (which has been discussed above) and a change in the nature of the
objects of study (i.e. from numbers to symbols, variables, expressions, equations etc). Beginning
students need to make both of these transitions. In the discussion above, we have seen that the
making the change in the nature of problem resolution is very substantial. My position is that this
does not need special activities in class, but it does need a teacher who is aware of the transitions
that need to be made, and therefore can highlight them in instruction. Teachers who understand the
transitions that students need to make, can pick up clues to students’ thinking in the real-time
environment of the classroom interaction, and can build classroom discussion around the
differences that students need to come to see.
Given the substantial requirements of this transition, it seems likely to be successful to establish at
least part of the change in the nature of the objects of study in the context of arithmetic problem
solving, rather than in the context of algebraic problem solving. This is where the methods that we
have seen above are easier for students can be useful. Take, for example, the guess-check-improve
solution to Problem 1. This is not a naïve solution. Other students simply listed all of (or some of)
the possible numbers of senior and junior monks, and the numbers of buns they would eat, and
looked amongst this mass of information for the lucky combination. However, experience shows
that this strategy is reasonably easy to teach and students can reasonably easily make such
approaches more effective. In Wong’s study (2008), the number of students selecting this method
increased after a short intervention, and the success rate was high. The difference between this
guess-check-improve strategy and the listing strategy is the way in which the constraint that there
are 100 monks altogether is used. Identifying constraints (which become equations in the algebraic
method) is an essential first step to the algebraic method, and can well be taught in this nonalgebraic context.
Other new concepts for algebra can also be taught whilst the problem solving is essentially nonalgebraic. Dettori et al (2001) explored how a spreadsheet could be used for teaching in this way,
whilst pointing out its limitations for teaching about the algebraic problem solving method.
Building tables of values (including with a spreadsheet if available) in the context of problems such
as Problem 3 and Problem 4 of Figure 1 can assist in developing the concept of variable and
function and substitution. Tables of values or graphs of cost against rental time can be drawn to
solve problem 4, developing the concept of the solution of an equation. My suggestion is that there
is considerable space to develop algebraic concepts whilst nevertheless retaining arithmetic
(numerical and graphical) problem solving methods. Algebraic problem solving methods can be
developed later. Whilst students can use logical reasoning to solve Problem 3 arithmetically,
Problem 4, which is algebraically represented by an equation with unknowns on both sides is
considerably more challenging and requires working with and on unknowns in the algebraic sense.
Building algebraic thinking into arithmetic
An important thread in international research and thinking on mathematics education curriculum is
to consider ways in which the transition from arithmetic to algebra can be made more smooth. In
particular, the ‘early algebra’ movement has examined how to teach arithmetic in a way that
prepares student for algebra, and which emphasises the thinking processes which underlie algebra.
The intention is not to introduce algebraic symbols at an earlier age, but to change the emphasis of
arithmetic teaching. Given the advances in calculation aids of the twentieth century, it is no longer
appropriate to have an arithmetic curriculum which focuses exclusively on computation, so that
there is opportunity to include experiences of generalisation, mathematical structure and properties
of operations that underpin algebra (MacGregor & Stacey, 1999).
Figure 6. Some solutions to Problem 5.
The meaning of operations
Algebra requires a stronger understanding of the meaning of operations than arithmetic, especially
whole number arithmetic. Consider how students might solve Problems 5 and 6 in Figure 1. Figure
6 shows some solutions from Australian students in Grades 5 – 8. An interesting observation to
make about these solutions is that students have solved the problem using addition, subtraction,
multiplication and division. The division solution is the only one that is effective for all numbers
(e.g. including decimals and fractions), so it is important for their progress in arithmetic that
students should all be able to recognise that division is applicable here. However, this is also an
essential understanding for algebra. To solve Problem 6, the algebra variant of Problem 5, the
addition, subtraction and trial multiplication solutions cannot be used: only students who recognise
this as a problem where division is applicable can solve the algebraic version. In our research on
Australian students, we have found that many students, even in secondary school when they should
be learning algebra, have immature understandings of the four arithmetic operations, which hinders
their progress in algebra. The integration of ideas about division and fractions is another key
understanding.
The properties of operations
Algebra requires a stronger understanding of the properties of operations than does arithmetic.
Algebra studies these properties (e.g. the commutative property, the associative property) and it also
uses them to change algebraic form through the rules of algebraic manipulation. We have found that
many students have difficulty learning about algebra because they are unsure of the arithmetic
properties which algebra generalises (Bell et al, 1993). One of the emphases of early algebra is
therefore to build intuitive awareness of general properties of number operations throughout
schooling. Consider, for example, the algebraic relationship a - (b -c) = a - b + c. Students often
learn this as a rule about changing sign, but it is also a property which quite young students can use
intuitively and discuss together. The intention is not to introduce students to the formal expression,
but rather to have students experience the algebraic thinking that this is a type of number sentence
that is true for all numbers and that such relationships can help them in working with numbers.
Figure 7, which is adapted from Fujii & Stephens (2008), reports how one student, Peter (based on
a real student in Grade 3) uses the thinking underlying the algebraic expression above to simplify
subtraction. Fujii and Stephens asked other students to react to Peter’s method. Some students,
(such as Thomas in Figure 7) only check that Peter’s method produces correct answers in a few
cases. We know that many students much older than the research group also do this. Other students,
such as Alan in Figure 7, have been taught to look at Peter’s method as something that can be
generalised (it works for other starting numbers, and can be modified to work for subtracting other
numbers) and also as something that can be explained. Alan’s explanation is not yet very clear, but
it shows an orientation to key processes of algebraic thinking including generalising and explaining.
The key for producing more students like Alan, and fewer students like Thomas, is to engage
students in deep classroom discussion about operations and how they work. Students’ interest in
‘clever calculating’ is a key motivator. Figure 8 shows how students might also explain this
relationship with manipulative materials.
Alan’s work also shows the use of what we call an ‘unclosed’ or ‘unexecuted’ expression, namely
32 + 4 – 10 = 32 – 6. This is how Alan demonstrates the structure. The Japanese curriculum has
great strength in the way in which students use these expressions to highlight structure. Students
who immediately execute all calculations never get an opportunity to observe this. Alan’s
expression 32 + 4 – 10 = 32 – 6 also shows that Alan is equipped to understand the way that
equality is used in algebra. It is very well established in the research literature that many students
learn from their studies of arithmetic that the = sign is a command to compute. Teachers, for
example, will write 13 + 45 + 76 – 44 = ___ indicating by the sign = that a computation is required
and where to write the answer. This view of = needs to be expanded for algebra, so that the idea that
the two sides of the equality are equal is the dominant meaning. It is these statements of equality
that are operated upon when equations are solved by ‘doing the same to both sides’. A student
without this view may well complete the number sentence 32+4-10 = ? – 6 with the number 26
(because the ‘answer’ to the left hand side is written after the equality) and believe that the ‘answer’
to the complete expression is 20 (32+4 – 10 = 26 – 6 = 20).
Peter’s method
“It is easy to work out
37 – 6 = 31
59 – 6 = 53
86 – 6 = 80
but it is harder to work out questions like
32 – 6 , 53 – 6 or 84 - 6, so I do it this way:
First add 4, then subtract 10
32 + 4 = 36, then take off 10 gives 26 (the answer)
41 + 4 = 45, then take off 10 gives35”
Thomas’s comment on Peter’s method
“It works because Peter got the right answers.”
Alan’s comment on Peter’s method
“If you add 4 you need to take away 10 to equal it out to 6.”
32 + 4 = 36
36 – 10 = 26
32 + 4 – 10 = 32 – 6
It doesn’t matter which number you start with.”
“For any number you take away, you have to add the other number, which is between 1 and 10, that
equals 10; like 7 and 3 or 6 and 4. You take away 10 and that gives you the answer.”
Figure 7. Peter’s method for difficult subtractions (adapted from Fujii & Stephens, 2008).
Alan’s teacher has taken the opportunities that naturally arise in a curriculum that teaches about
number and number operations, to engage students in testing ideas about number and operations,
and explaining them to their fellow students in rich classroom discussion. The teacher has
emphasised whether methods always work, testing numerically but also looking for general reasons.
Because of this, Alan has been able to demonstrate an orientation to generalise mathematical
properties, to look for reasons, to show structure, and to understand equality. We can see that Alan
is further along the road to algebra than Thomas, long before either of these boys have been
introduced to algebraic symbols. The new book by Kaput, Carraher & Blanton (2008) explores
these ideas and other ideas of early algebra thoroughly.
A
A
A-5
A-6
10
A+4
Figure 8. Students can show why relationships are true using manipulative materials.
Conclusion
This paper has selected for examination only a few aspects of the transition from arithmetic
thinking to algebraic thinking. There is much more that can be said on the matter, especially as the
mathematics education research community has put a lot of effort into identifying the types of
activity and the patterns of thinking characterise algebra and arithmetic. Faced with the difficulties
which students around the world have with algebra, there have been many experiments with
different approaches to give algebra more meaning, with approaches to strengthen students’
transformational ability with symbols, and with capitalising on new technology both for doing
mathematics and for learning mathematics (Stacey, Chick & Kendal, 2004). Teachers need to find
ways to make algebra a gateway for higher mathematics, rather than a wall that blocks students’
paths. In this paper, I first showed the transitions that students need to make to move from an
arithmetic way of approaching problems to an algebraic way through equation solving. There are
vey substantial changes, related to what an unknown is, what is operated upon (numbers or an
equation), the logical links between the steps. These were summarised in Figure 5. Although it will
always be the case that there are major cognitive gaps that need to be crossed when learning
algebra, it is also increasingly recognised that the teaching of algebraic thinking should not start
when students begin to learn algebra in a formal way. Instead, teachers can have an orientation
towards examining the key ideas underlying algebra, such as generality and reasoning about
structure, within the teaching of arithmetic from the earliest stages. In this way, the fundamental
processes of algebraic thinking can enrich all teaching of mathematics.
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