ERIC EJ961653: How Problem Solving Can Develop an Algebraic Perspective of Mathematics

2011·0.51 MB·English

by  ERIC

How problem solving
can develop an
algebraic perspective of
mathematics
Introduction
wIll wINDSOR
Problem solving has a long and successful
provides a brief history in mathematics education and
is valued by many teachers as a way
overview of problem to engage and facilitate learning within
their classrooms. The potential benefit for
solving and links using problem solving in the development
of algebraic thinking is that “it may
it to algebraic broaden and develop students’ mathematical
thinking beyond the routine acquisition
thinking. Readers are of isolated techniques and procedures
often associated with secondary school
encouraged to try the
algebra” (Booker, 2007; Kaput, 2008;
Carraher & Schliemann, 2007; Lins, Rojano,
tasks with their own
Bell & Sutherland, 2001). Furthermore,
the thinking required to solve problems
students and then
can be extended from methods tied
to concrete situations—the backbone of
re-read the article.
primary school mathematics—to experiences
that develop an ability to solve problems
using abstractions based on the relationships
within the problems. By establishing
an algebraic perspective of problem
solving it acknowledges that “students
can adapt their ways of thinking, they
can express mathematical generalisations
and it can provide an entry to algebraic
symbolism that is meaningful” (Carraher &
Schliemann, 2007).
8 APMC 16 (4) 2011
How problem solving can develop an algebraic perspective of mathematics
Using mathematical problems to to be, it is not until the solver actually
develop algebraic reasoning engages in its solution that the nature of the
thinking comes to life.”
This article describes a lesson that was The two problems are sometimes
undertaken in a Year 7 class. Six groups, each classified as algebra problems because they can
with four students, were given structurally be solved using a system of simultaneous
similar problems. The approaches taken equations (Lenchner, 2008). However,
by two of the groups are presented, to this interpretation limits and possibly
demonstrate how algebraic reasoning can predetermines a procedural approach that
build from their experiences, discussions and simply mimics the processes of the teacher or
interpretation of mathematical problems. text book. It should be noted, that sometimes
Their voices explain the strategies and the classifications of problem solving
thinking they used to solve two structurally strategies are used by experts when giving
similar problems. “after-the-fact explanations of their own or
others problem solving behaviours” (Lesh
Brush Up & Zawojewski, 2007). The two examples
At an art store, brushes have one price and described in this paper demonstrate how
pencils have another. Eight brushes and three students can solve problems from an algebraic
pens cost $7.10, but six brushes and three pens perspective when mathematical objects are
cost $5.70. How much does one pen cost? considered relationally and not simply as
specific numbers. This change in student
A good Sport thinking is necessary as students develop the
At the local sports store, all tennis balls ability to think algebraically (van Amerom,
are sold at one price and netballs are sold 2002)
at another price. If three netballs and two
tennis balls are sold for $47.00, while two Isolating one unknown to find a solution
netballs and three tennis balls are sold In solving the problem Brush Up, Dougal,
for $38.00, what is the cost of a single a Year 7 student, identified the relationship
tennis ball? between the brushes and pencils using blue
and yellow counters and two calculators.
Solving problems using relational thinking As shown in Figure 1, the yellow counters
Many primary school teachers might regard represented the paintbrushes and the
solving this type of problem as a difficult blue counters represented the pens, and,
task, only accessible by those students with interestingly, he recorded the cost of each
an aptitude in mathematics. Conversely, statement on the two calculators. He then
students are simply shown an algebraic removed six yellow counters and three
procedure with little understanding of the blue counters from the top statement and
meaning behind the symbols. By analysing the subtracted $5.70 from $7.10. Once the
processes required for using two relationships counters were removed, Dougal explained
simultaneously, there is a greater likelihood (pointing at the 1.4 on the display), “Divide
of understanding the algebraic sequences by two is 70, so we know that one brush equals
encountered in secondary school algebra. 70 cents.” Once he had found the cost of one
The two examples describe the thinking the brushes, he logically found the cost of the
used by students to isolate the objects and pens. His use of the two calculators helped
show what the students did in order for the him to reduce his cognitive load and freed his
problem to be less complex. As Lins, Rojano, thinking to concentrate on the mathematical
Bell and Sutherland (2001) note, “no matter objects rather than computational processes.
how suggestively algebraic a problem seems Furthermore, the materials helped his group
APMC 16 (4) 2011 9
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students and her pedagogical understanding
was crucial in guiding Holly and Amelia’s
mathematical thinking as well as developing
an algebraic perspective that acknowledges
the inter-connectedness of mathematics.
After Holly explained how she solved
common fraction addition problems, Amelia
pointed at the statement she had written
about the netballs and tennis balls. She
suggested that the relationship between the
Figure 1. Dougal using counters and calculators to solve balls could be maintained if each statement
the problem Brush Up. was renamed by a factor of two and three.
Figure 2 demonstrates Holly and Amelia’s
to see the problem clearly and they developed thinking with the coloured text highlighting
their mathematical argument in a strategic what the two girls actually recorded. The
and organised manner. table has been extended to show the girls
thinking in more detail and how as each
Using factors to isolate one unknown statement is renamed by a factor of 2, 3, 4 or
At the beginning of the lesson, Holly and 5, the relationship between the netballs and
Amelia had solved a structurally similar tennis balls can be maintained.
problem to Dougal’s group and throughout
the prior lessons both girls, especially Amelia, Relationship x2 x3 x4 x5
had demonstrated the capacity to think about
6 Netballs 9 Netballs 12 Netballs 15 Netballs
a variety of problems from a generalised and 3 Netballs
4 Tennis 6 Tennis 8 Tennis 10 Tennis
2 Tennis Balls
relational perspective. Amelia wrote down Balls Balls Balls Balls
$47
$94 $141 $188 $235
the statement:
2 Netballs 4 Netballs 6 Netballs 8 Netballs 10 Netballs
3 netballs and 2 tennis balls = $47
3 Tennis 6 Tennis 9 Tennis 12 Tennis 15 tennis
2 netballs and 3 tennis balls = $76
Balls Balls Balls Balls balls
Unlike the problem Brush Up, both girls found $38.00 $76 $114 $152 $190
the problem A Good Sport demanding and Figure 2. A summary of Holly and Amelia’s thinking.
difficult to understand because it required two
mathematical statements to be manipulated Once they established a “like part” or
simultaneously. Their teacher, having seen common coefficient, in this case the six
Amelia’s statement, asked them to reflect on netballs, the two girls could logically solve the
their prior understanding and to consider how problem. The two girls reasoned that if they
they solved addition or subtraction problems took six netballs and four tennis balls ($94)
involving “unlike common fractions.” Holly, away from the statement “six netballs and
who was proficient at arithmetic, proceeded nine tennis balls” ($114) then only five tennis
to explain to Amelia that when both balls would remain ($114 – $94 = $20), thus
common fractions were “unlike fractions” isolating one of the variables. Knowing that
3 1
(e.g., and ), she renamed each fraction five tennis balls cost $20, the two girls simply
4 3
as a “like fraction” or an equivalent fraction ascertained that the cost of one tennis ball
using a factorisation method. Interestingly, was $4. The girls presented their thinking to
during this short episode, the teacher did not their classmates, as shown in Figure 3; their
set about explaining a solution but asked them explanation was enthusiastically applauded
to reflect on the mathematics they already by everyone in their class, but, importantly,
knew and allowed the girls to develop their other students used the girls’ ideas that
own solution. The teacher’s knowledge of the they had to build their own understanding,
10 APMC 16 (4) 2011
How problem solving can develop an algebraic perspective of mathematics
isolating the mathematical objects. Using
coloured counters may help to illustrate
this process and allow students to ‘see’ the
relationships by reducing their cognitive
load and allowing them the opportunity
to compare, identify and manipulate the
amounts of each statement more readily and
with greater understanding.
In the problem A Good Sport, what is
known can be presented using two different
Figure 3. The solution Holly and Amelia presented to the class. coloured counters and the price recorded
beside each statement. Similarly to Dougal’s
and this was reflected in the work of other example (see Figure 1), a calculator can
students following the girls’ explanation. On be used to record the price beside each
reflection, the applause and enthusiasm for statement. As shown in Figure 4, black and
the both Dougal’s and Amelia and Holly’s white counters can show how the original
explanations highlights the value that students statement, “Three netballs and two tennis
place on “mathematically significant” events balls cost $47 and two netballs and three
when they are actively involved and engaged tennis balls cost $38” can be renamed. Using
in teaching and learning. the Amelia and Holly’s thinking the “three
netballs and two tennis balls” can be renamed
Using materials by a factor of two, thus: six netballs and four
As Dougal, Holly and Amelia demonstrated, tennis balls will cost $94. The “two netballs
once an object or value has been isolated, and three tennis balls” can be renamed
students can logically solve for the other by a factor of three, thus: six netballs and
unknown. However, difficulties arise in nine tennis balls will cost $114. Now both
developing the thinking required to isolate one statements have a “like amount” or common
of the objects, especially when the coefficients coefficient, in this case the six netballs.
are not equal. Building on Dougal’s example, Figure 4 illustrates how the like terms from
counters can be helpful for showing the both statements are removed, thus leaving
thinking that Holly and Amelia used for only five tennis balls (five white counters) at
renaming their mathematical statements and a cost of $4 per tennis ball.
the thinking required for eliminating and
$47.00 $38.00
$94.00 $76.00
$141.00 $114.00
Figure 4. Holly and Amelia’s explanation using counters.
APMC 16 (4) 2011 11
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Conclusion Pencils and Pens
In a stationary store, pencils have one price and
Developing algebraic thinking initially relies pens have another. Two pencils and three pens
on students' ability to consider the structures cost 78 cents, but three pencils and two pens cost
of problems from a generalised and relational 72 cents. How much does one pen cost?
perspective. From this position, students
can develop increasingly sophisticated
ways of thinking about the problems and References
their solutions, as well as organising and
Booker, G. (2007). Problem solving, sense making and
manipulating their thinking in order to solve thinking mathematically. In J. Vincent, J. Dowsey
the problems. As demonstrated by Dougal, & R. Pierce (Eds), Mathematics — Making sense of
our world (pp. 28–43). Melbourne: Mathematical
Holly and Amelia, students are capable of
Association of Victoria.
exploring sophisticated problem solving Carraher, D. W. & Schliemann, A. D. (2007). Early
algebra and algebraic reasoning. In F. Lester (Ed.),
approaches in a coherent and organised way.
Second handbook of research on mathematics teaching
The two student examples present their own
(Vol. 2, pp. 669–705). Charlotte, NC: Information
unique approaches yet their thinking has Age Publishing.
Kaput, J. J. (2008). What is algebra? What is algebraic
elements often associated with the algebra of
reasoning? In J. J. Kaput, D. W. Carraher & M.
secondary school. One example shows how L. Blanton (Eds), Algebra in the early grades (pp.
materials and calculators can complement 235–272). New York: Lawrence Erlbaum Associates.
Lechner, G. (2005). Creative problem solving: In
students thinking as they grapple with new
school mathematics. Wahroonga, NSW: Australian
ideas. The other highlights how simple Primary Schools Mathematical Olympiad.
mathematical ideas such as renaming and Lesh, R. & Zawojewski, J. S. (2007). Problem solving
and modelling. In F. Lester (Ed.), Second handbook
equivalence can be used by students to
of research on mathematics teaching (Vol. 2 (pp. 763–
understand concepts often associated with 804). Charlotte, NC: Information Age Publishing.
Lins, R., Rojano, T., Bell, A. & Sutherland, R. (2001).
algebra. The problems provide a context in
Approaches to algebra. In R. Sutherland, T. Rojano,
which to introduce the ideas of algebra but
A. Bell & R. Lins (Eds), Perspectives on school algebra
more importantly facilitate a generalised (pp. 1–11). Dordrecht, NL: Kluwer Academic
Publishers.
way of thinking that can transfer to other
van Amerom, B. A. (2002). Re-invention of early algebra:
situations beyond mathematics. Can you use Developmental research on the transition from arithmetic to
the thinking and strategies that Dougal, algebra. Unpublished doctoral thesis, Netherlands:
University of Utrecht.
Holly and Amelia used to solve these two
problems?
Flourish & Botts
At the Flourish & Botts bookstore, the first Will Windsor
wizard book and the second wizard book together Griffith University
cost $45. Two copies of the first wizard book <[email protected]>
and three copies of the second wizard book costs
APMC
a total of $125. At this book store how much is
the first wizard book?
12 APMC 16 (4) 2011