Algebraic Thinking A Problem Solving Approach

L. Sparrow, B. Kissane, & C. Hurst (Eds.), Shaping the future of mathematics education: Proceedings of the
33rd annual conference of the Mathematics Education Research Group of Australasia. Fremantle: MERGA.
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Algebraic Thinking: A Problem Solving Approach
Will Windsor
Griffith University
<w.windsor@griffith.edu.au>
Algebraic thinking is a crucial and fundamental element of mathematical thinking and
reasoning. It initially involves recognising patterns and general mathematical relationships
among numbers, objects and geometric shapes. This paper will highlight how the ability to
think algebraically might support a deeper and more useful knowledge, not only of algebra,
but the thinking required to successfully use mathematics. The paper will highlight how a
deeper analysis of mathematical problems can instigate student discourse, providing
meaningful experiences that can developing algebraic thinking.
Recent calls for reform in mathematics education in Australia have focused on the need
to promote and facilitate improved teaching and learning of mathematics (Council of
Australian Governments, 2008; National Curriculum Board, 2008). A key element of this
reform agenda is the introduction of a national mathematics curriculum. The strand
Number and Algebra will be an integral part of the new curriculum with the middle and
upper primary years emphasising an algebraic perspective of number rather than the formal
algebra familiar to most people. In contrast, secondary school students will undertake the
study of formal algebra having been introduced to algebraic concepts and ideas at the
primary school level.
The algebraic perspective I attempt to illustrate is a perspective that values, enriches
and improves the thinking required to understand algebraic concepts. Consequently, in this
discussion we will adopt the term ‘algebraic thinking’ rather than an algebraic perspective
of number, as it takes into account the variety of activities students can engage in across all
mathematics strands, not simply number. Algebraic thinking is founded on numeracy and
computational proficiency, the reasoning of geometry and skills associated with
measurement- concepts introduced and taught in the primary and middle school (Kaput,
2008). Importantly, it extends the thinking required to solve problems beyond methods tied
to concrete situations. In time, this thinking supports students ability to problem solve
using abstractions and to operate on mathematical entities logically and independently
from the material world.
The approach that I propose involves identifying and using mathematical problems that
promote and advance a generalised perspective of mathematical problem solving. An
ability to consider problems from this perspective can allow individuals to acquire
adaptable ways of thinking, to express the generalisations they have arrived at and leads
into a meaningful use of algebraic symbolism (Carraher, Brizuela, & Schliemann, 2003).
The potential value for using problem solving contexts is that it may broaden and develop
students’ mathematical thinking and provide them with an impetus for understanding a
greater collection of problems of increasing complexity and mathematical abstraction
(Kaput, 2008; Kaput, Blanton, & Moreno, 2008; Schliemann, Carraher, & Brizuela, 2007;
Lins, Rojano, Bell, & Sutherland, 2001). As will become apparent, algebraic thinking
promotes a particular way of interpreting mathematics. It extends the mathematical
thinking of students by encouraging them to interact and engage with the generalities and
relationships inherent in mathematics. Lins et al (2001, p. 3) contend that “no matter how
suggestively algebraic a problem seems to be, it is not until the solver actually engages in
its solution that the nature of the thinking comes to life.”

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Problem Solving and Algebraic Thinking
Using mathematical problems has been advocated as a crucial and motivating
component of learning and understanding mathematics. Schoenfeld (1992) notes that,
when solving mathematical problems, students develop a deeper understanding of
mathematics because it helps them to conceptualise the mathematics being learnt. Stanic
and Kilpatrick’s (1989) review of problem solving indicates that historically, mathematical
problem solving has been instrumental in achieving a variety of goals within the
mathematics curriculum. Furthermore, productive problem solving experiences that move
children beyond the routine acquisition of isolated techniques are fundamental in
developing higher order mathematical thinking and reasoning (Booker & Bond, 2009;
Polya, 1973). In my view, many of the fundamental ideas on which mathematics is built
can make sense to children if those concepts are viewed in meaningful and challenging
contexts.
In the book How to Solve It (1973) Polya is explicit in characterising the heuristics of
effective problem solving. Essentially, he attempts to understand how people think and the
strategies they might use when solving problems. Polya (1973) contends that to solve any
problem, the characteristics and properties of the problem should be analysed. Once the
problem is understood then a plan is devised and strategies are implemented and finally,
opportunities to reflect upon the solution are required. Though Polya emphasises the
heuristics of problem solving, he also acknowledges the idea of mathematical
connectedness and generality, key components of algebraic thinking. He suggests that by
actively engaging with problems students can develop the ability to understand the
generalities associated with problem solving:
In solving a problem of one or the other kind, we have to rely on our experience with similar
problems and we often ask the questions: Have we seen this problem in a slightly different form?
Do we know a related problem? (Polya, 1973, p. 151)
This brief overview of Polya’s work serves to emphasise that the generalities,
relationships and interconnectedness that underpins mathematics can be carried through to
developing algebraic thinking. Furthermore, being aware of and having the capacity to
consider, ascertain and communicate the generalities of a particular problem may
invariably enhance an understanding of formal algebra. As Krutetskii (1976, pp. 334-335)
observed “one must be able to see a similar situation (where to apply it), and one must
master the generalised type of a solution, the generalised scheme of a proof or of an
argument (what to apply).” This perspective appreciates that algebraic thinking and
problem solving are inextricably linked by common skills and mathematical
understandings.
In shifting the emphasis of problem solving, from simply finding a specific answer to
also including a focus on algebraic thinking, I conjecture that it may provide a powerful
way to teach and learn algebraic ideas. Many of the problems found in elementary
arithmetic and geometry have the capacity to support a way of thinking that connects a
range of mathematical content and processes (Booker, Bond, Sparrow, & Swan, 2010;
Bednarz & Janvier, 1996). Extending mathematical problems solving to include the
developing algebraic thinking, educators can facilitate more divergent and adaptive ways
of thinking mathematically. Opportunities arise to engage and extend students’
mathematical experiences that go beyond routine arithmetical solutions. As Silver,
Ghousseini, Gosen, Charalambous, and Strawhun (2005, p. 288) observed, when
discussing the advantages of facilitating a variety of different solutions:

667
An aphorism of unknown origins captures the essence of this idea: “You can learn more from
solving one problem in many different ways than you can from solving many different problems,
each in only one way”.
Developing Algebraic Thinking Using Problem Solving
Teaching algebraic thinking using a problem solving approach can be established amid
the learning experiences that already exist in most classrooms. It is apparent that this
approach evolves and builds upon a child’s ability to consider, see and think about the
mathematical concepts within a problem. Lee’s (2001) analysis of algebraic thinking
highlights some of the underlying strategies which characterise this type of mathematical
reasoning. She observes that when children analyse problems from an algebraic thinking
perspective they may consider:
- Reasoning about patterns (in graphs, number patterns, shapes, etc) stressing and ignoring,
detecting sameness and difference, repletion and order.
- Generalising or thinking in terms of the general, seeing the general in the particular;
- Mentally handling the as-yet-unknown, inverting and reversing operations;
- Thinking about mathematical relations rather than mathematical objects.
It is apparent that the development of algebraic thinking arises from generalising
mathematical thought. Researchers such as Bednarz, Kieran, and Lee (1996) extend this
idea and state “the process of generalisation as an approach to algebra appears ultimately
related to that of justification”. Thus, a classroom environment that values and promotes
collaborative learning situations, student discourse and the opportunities to communicate
mathematical ideas and conjectures can better facilitate algebraic thinking. The resounding
importance of teachers to facilitate algebraic thinking through meaningful discourse can be
observed in the research of Carpenter, Franke, and Levi (2003), Carraher, Schliemann, and
Brizuela (2003) and van Amerom (2002). Each of these research teams encouraged student
discourse so as to promote deeper mathematical reasoning and expedite algebraic thinking.
In focusing on developing algebraic thinking, current thinking suggests students will
progress through three stages of development. At first, many students will describe
generalities and relationships in natural language, this can lead into abbreviating those
ideas by using diagrams and mathematical symbols and finally, these ideas can be
summarised using mathematical expressions and equations, tables of values and graphs
(NCTM, 2000; Mason, Graham, & Johnston-Wilder, 2005). The history of mathematics
would also suggest that to understand and solve problems of an algebraic nature,
individuals operate and constantly manoeuvre their thinking between the before-mentioned
three stages often referred to as rhetorical, syncopated or symbolic stages (Harper, 1987;
Katz & Barton, 2007). Clement, Lochhead, and Monk (1981) describe that most
mathematicians think this way; rarely do they consider their thoughts in a purely symbolic
realm. Instead, describing their ideas as being like pictures — with tables, graphs and
symbols used to interpret those ‘pictures’. However, the thinking required to understand
these thoughts is algebraic because there is a focus on the general rather than the specific.
The work of scholars such as Lannin, Barker, and Townsend (2006) and Booker and
Bond (2009) serve to reinforce the importance of students interacting with the problem,
their teachers and other students at a variety of different levels. Lannin, Barker, and
Townsend describe how social factors, cognitive factors and task factors simultaneously
influence the way students address problems and how this influences their reasoning.
Booker and Bond (2009) document the effectiveness of working collaboratively and the

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value of encouraging multiple perspectives. As they both document, developing algebraic
thinking can be achieved when students are encouraged to use a variety of strategies and
are supported to communicate their ideas, reflect upon solutions and have opportunities to
speculate about the concepts and ideas they have constructed.
Algebraic Thinking within a Classroom Context
This section will illustrate and highlight how the transition from the rhetorical to the
syncopated stage of generalised thought can be promoted. The following three problems
were completed by a Year Six class. The tasks involved finding a pattern and possibly
explaining a method for summing an arithmetic series. The students worked in seven
groups of four and were able to use a calculator, pencil and paper or counters to come with
a solution or explanation.
Problem A - Chiming Clock
An old chime clock strikes one chime at 1 o’clock, two chimes at 2 o’clock, three
chimes at 3 o’clock and so on. How many chimes will it strike in a 12-hour cycle?
Problem B - Counting Coins
Your New Year’s resolution is to save enough money to buy a new bike. You decide to
put $1 away on the first day of the year, $2 on the second day, $3 on the third day, $4 on
the fourth day and so on. How much money will you have after 30 days?
Problem C - A King’s Ransom
A King told his knights that if they could slay the dragon they would be richly
rewarded. He informed them that he would place on a chess board one gold coin on the
first square, two gold coins on the second square, three gold coins on the third square, four
gold coins on the fourth square and so on. How many gold coins will the knights have if
they slay the dragon? There are 64 squares on a chess board.
At first many children simply added successive numbers. However, many had
difficulty in explaining or justifying their solutions in generalised terms. Using their
calculators or pencil and paper many children could achieve the correct solution for
Problem A by simply adding the numbers in the correct sequence. As the children
attempted to understand and solve the other two problems it became obvious that this
method was inefficient and cumbersome. This point can be illustrated by the fact that when
the groups proceeded to the second question five of the seven groups each had a different
answer even though their solution strategies were very similar. Through the ensuing
discussion there came the realisation that Problems B and C were similar to Problem A but
there must be a more efficient way to solve the Counting Coins and A Kings Ransom
problems. One child explained this to her group as follows:
Ashley: There has to be a better way (and takes the calculator). It takes us too long to do the
other two problems. There’s ‘gotta’ be a pattern. It’s like what Mr H showed us the
other day with Pascal’s Triangle.
At this point, the children were encouraged to use counters or a diagram to explore the
first problem again. (See Figure 2). One of the groups who used a table commented that it
reminded them of their rainbow facts.

669
Peter: This is like the rainbow facts we did in grade one.
James: (Laughing) They all add to 13.
Teacher: All of them?
James: Yeah see.1 and 12, 2 and 11, 3 and 10. (Runs his fingers over the connecting lines)
See they all add up to 13.
Peter: You add them?
Teacher: Is there a better way than simply adding 13 each time?
Peter: (Starts tapping his pencil beside each 13).
James: (Looking at the teacher). Is it the same as multiplication?
Peter: (Picks up his calculator and enters 13 x 6) Seventy-eight. It’s the same as our answer.
Figure 2. Chiming Clock solution similar to the “Peter and James” case.
As the lesson continued, three groups used counters to illustrate the Chiming Clock
problem. (See Figure 3.) Each group represented a chime as a counter, with one counter
representing 1 o’clock; two counters 2 o’clock and so on. The teacher moved to a group as
a member from another group also watched the following episode.
Teacher: (Moves the single counter in the first row to the last row.
Then moves the next two counters to the second last row).
Annie: (Moves the next three counters into third last row, then the next four counters and so
on). There’s six rows of 13 so 78 chimes. So all you do is add the numbers and
multiply by the number (pauses) of pairs.
Teacher: Does it work for the other problems?
Figure 3. Using counters to interpret the Chiming Clock problem.
1 3 4 5
13
2 6 7 8 9 10 11 12
13
13
13
13
13

670
Annie and her group were then observed experimenting with other sequences.
Importantly, the teacher let Annie and her group continue their exploration. Her group
spent the remaining 25 minutes making different arithmetical sequences and comparing
their solutions with a calculator. (See Figure 4.) As her teacher was walking past to attend
to another group Annie commented, “And all you need to do is the same for all of the
problems?” Her teacher acknowledged this statement with an approving nod.
Figure 4. An example of Annie’s thinking using the counters.
I infer from Annie’s explanation and her group’s use of the counters that the
multiplicative structures of the problem were more apparent to groups who decided to use
the materials. Using the counters suggested to the children an arrays model for
multiplication that was more easily understood by this class. In the next phase of the
lesson, her group discussed the Counting Coins and A Kings Ransom problems.
Consequently they were also able to explain their solution algebraically and in general
terms at a rhetorical level, moving into the syncopated phase. The group inferred that for
the Counting Coins problem and A Kings Ransom problem that the same ideas existed in
each of the problems respectively. (See Figure 5).
Teacher: How did you find a solution so quickly? Come up to the front and show us.
(points to an overhead projector and hands Paul a pen).
Paul: Well, we knew the numbers added to 31 and divided 30 by two. So it was 31 multiplied
by 15 which is $465 (He writes the number sentences on the overhead projector).
Teacher: Can you do it with the other problems?
Paul: They’re all the same. You can solve them the same way.
Figure 5. An excerpt from a student’s work book.

671
The lesson concluded with the teacher asking “What would happen if we wanted to
sum successive even numbers or successive odd numbers? What would happen if we
started at a number other than one or there was not an even number of values?” The ability
for the class to use the generalisations they had developed during the lesson created
opportunities to shift their thinking from a purely answer focused perspective of
mathematics. The sample episodes show an increasing sophistication in the way students
worked through the problems. In the course of the lessons, the students explained and
justified their responses to each other and were more productive in their capacity to
develop generalisations about the problems.
Discussion and Conclusion
Developing algebraic thinking using a problem solving approach may build upon and
extend the teaching practices used within many classrooms. However, it may compel some
teachers to see problem solving from a different perspective. At a minimum it entails
seeing problem solving as an opportunity to enrich and transform students’ thinking rather
than the ‘ferreting out’ of an answer. One of the distinctive characteristics of this approach
is that it requires teachers to adapt and change the problems, yet maintain the mathematical
generalisations present within the problems. Through appropriate discourse teachers can
encourage students to think algebraically rather than influencing them to use a particular
strategy or procedure. It is through discussion during the solving process that ideas relating
to algebraic thinking and an algebraic perspective of mathematics can be developed.
Encouraging students to reflect on their thinking and share their experiences can assist in
students developing different ways of thinking about problems. As Silver et al (2005,
p. 13) observes:
The presentation of multiple solutions and the consideration of connections between and among
different approaches to a problem could be seen as opportunities to advance the mathematical
agenda.
A central challenge in addressing reforms in algebra can be addressed at many levels.
As Thomas quietly commented at the end of his class discussion, “This is like that algebra
stuff they will teach us at high school. It’s not really different to what we do at the
moment.” If the themes I have identified for developing algebraic thinking address only
students’ perceptions of algebra in a positive manner than the merits of advocating a
problem solving approach should warrant further investigation. Moreover, the benefit for
developing students’ algebraic thinking beyond the mechanics and procedures often
associated with algebra can possibly offer students a more complex and meaningful
conceptualisation of algebra. Using a problem solving approach to develop algebraic
thinking and providing an algebraic perspective of mathematics may enhance the long-
term learning trajectory of the majority of students.
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