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INTRODUCTION OF ALGEBRAIC THINKING:
CONNECTING THE CONCEPTS OF LINEAR
FUNCTION AND LINEAR EQUATION
V. Farmaki, N. Klaoudatos, P. Verikios
University of Athens, Department of Mathematics
ABSTRACT
This paper is α part of a wider research about teaching and learning of school
algebra to 13 year-old students. The various difficulties and cognitive obstacles
that students face when they are introduced to algebra are well documented in the
relevant literature. Ιn order to avoid these difficulties, we have adopted a functional approach widening the meaning of algebraic thinking. Using this approach
we implemented an introduction to algebra in a class grade 8 and after the course
we had interviews with students of the experimental class in order to investigate
the influence of the teaching to the students. In this paper we concentrate on the
problem of the second interview, that is modelled by linear function y=ax+b, connecting it with linear equation ax+b=c. We analyze the answers of the students,
investigating the advantages and disadvantages of the functional approach in the
solutions of these kinds of problems.
INTRODUCTION
Many students have difficulties making the transition from arithmetic to algebra.
Often these difficulties first appear when students attempt to create algebraic equations to represent word problems. Educators in order to help the students to overcome these difficulties must understand students’ cognitive processes when they
solve these types of problems.
According to the bibliography, algebra is an abstract system in which interactions
reflect the structure of arithmetic (Cooper, Williams & Baturo, 1999). Its processes
are abstract schemas (Ohlsson, 1993) or structural conceptions (Sfard, 1991) of the
arithmetic operations, equals, and operational laws, combined with the algebraic
notion of variable (Cooper, Boulton_Lewis, Atweh, Willss & Mutch, 1997).
Arithmetic does not operate at the same level of abstraction as algebra does, although they both involve written symbols and an understanding of operations (e.g.,
order of operations, inverse operations – Herscovics & Linchevski, 1994), arithmetic is limited to numbers and numerical computations (Sfard & Linchevski, 1994).
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Arithmetic and algebra differ fundamentally in that in arithmetic computational
procedures are separated from the object obtained (Linchevski & Herscovics,
1996). A fundamental requirement of algebra is an understanding that the equal
sign indicates equivalence and that information can be processed in either direction
(Kieran 1981; Linchevski, 1995). It has been noted previously that many students’
understanding of equals is action indication (e.g., “makes or gives” – Stacey &
MacGregor, 1997, p. 113) or syntactic (showing the place where the answer should
be written – Filloy & Rojano, 1989). Misconceptions relating to the equal concept
make it very difficult for students to transform and solve equations (Kieran, 1992;
Linchevski & Herscovics, 1996). Many teachers and researchers know that the
presentation of algebra almost exclusively as the study of expressions and equations can pose serious obstacles in the process of effective and meaningful learning
(Kieran, 1992).
In this paper we refer to a condensed theoretical framework supporting a functional approach to algebra. Using this approach we design a course for introduction
to school algebra. We taught this course in a class grade 8. In order to investigate
the results of this course we had interviews with 8 students from the experimental
class, which solved problems that covered all the subjects of the course. In this paper we focus on the problem from the second interview and analyze students’ answers.
A CONDENSED THEORETICAL FRAMEWORK
Yerushalmy & Schwartz (2000) support that the notion of function is the fundamental subject of algebra and that it ought to be present in a variety of representations in the teaching and learning of algebra from the very start. The concept of
function is one of the central ideas of pure and applied mathematics and many researchers suggest to teaching algebra based on functions (Leitzel, 1989, Thorpe
1989, Kieran 1997). Computers and graphing calculators now make it easy to produce tables and graphs for functions, to construct formulas for functions that model
patterns in experimental data, and to perform algebraic operations on functions.
The coherence of algebra curriculum can be achieved through a central concept,
such as function, which has been considered a unifying topic in both algebra and
other secondary school mathematics courses. The notion of function should be expanded from its view of an abstract object to an understanding that functions describe real-world phenomena.
When adopting such an approach, characteristics terms that are used in tasks in
the traditional teaching of algebra, such as symbols, expressions, and equations,
take new meaning (Kieran et Al 1996). Technology offers an easy way to separate
the traditional meaning “to solve an equation” in two parts - in the typical way of
solving equations by using symbolic processes and in the solution by reading and
interpreting graphs - which are no longer completely connected.
Aligned with “a function approach” is teaching in context and teaching for understanding. For most people, understanding mathematics requires investigations
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into the concepts underlying mathematics. Understanding mathematics is totally
different from memorizing mathematical algorithms. Many of the mathematical
algorithms that are useful in the applied fields and within mathematics are available
on calculators and computers; thus, the need for memorizing the algorithms has
diminished. However, the need for understanding mathematics has increased because of the need to utilize this technology and interpret the results.
According to Lins (1992, p.12), to think algebraically is: (1) to think arithmetically, which means modelling in numbers, and (2) to think internally, which means
reference only to the operations and equality relations, in other words solutions in
the boundaries of the semantic field of numbers and arithmetical operations, and
(3) to think analytically, which means what is unknown has to be treated as known.
A central notion in this view is the intention to make the shift from the situational
context to the mathematical context. Although this view is quite serious, it can be
considered as a base of the traditional way of teaching algebra because, in the educational praxis, this approach is prone to well-known manipulation of symbols according to fixed rules.
Recently approaches have been developed in algebra that broadens the meaning
of algebraic thinking. One of these approaches is the functional one; see for example Kieran (1996), Kieran et al (1996), Yerushalmy (2000). A functional approach
assumes the function to be a central concept around which school algebra can be
meaningfully organized. This means that representations of relationships can be
expressed in modes suitable for functions and that the letter-symbolic expressions
are one of these modes. Thus, algebraic thinking can be defined as the use of a variety of representations in order to handle quantitative situations in a relational way
(Kieran (1996, p.275).
In this paper we adopted the functional approach to algebra which widens the
meaning of algebraic thinking. Then, through problems which are expressed by
equations or inequalities of the form ax+b=c or ax+b=cx+d and ax+b<c,
ax+b<cx+d, we examine the students’ solution processes by the two approaches,
functional and letter-symbolic. Our goal is the investigation of the advantages and
disadvantages of the functional approach in the solutions of problems, which demand the solution of equations and inequalities of this kind.
THE RESEARCH DESIGN
In the Greek curriculum for the second class of 13-year-old students in junior
high school, equations and inequalities precede the functions and in the students’
textbook they are found in two different chapters. The solution of equations
ax+b=c, ax+b=cx+d is presented in a typical way, concentrating on symbol manipulations, while in functions the linear one of the form y=ax+b is the main subject. We point out that the equation chapter contains, as a final paragraph, simple
inequalities of the form ax+b<c, ax+b<cx+d. For this reason investigating the solution of inequalities was another goal of our approach. In this paper we will refer-
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ence to a problem witch modelled by an equation of the form ax+b=c (for the equation of the form ax+b=cx+d see Farmaki, Klaoudatos & Verikios, 2004).
In order to investigate the questions above, we developed a course consisting of
26 lessons of 45 minutes each, four lessons per week. This course replaced the
course on equations and the one on functions. The functional orientation enabled us
to connect various problem situations to graphs, tables and letter-symbolic representations as well as to connect these representations to the notion of equation and
inequality. At the beginning of the solution of a problem, attention was paid to the
graphic representation of it, where x was seen as a variable rather than an unknown
quantity. In this way the symbols as letters, lines or tables, acquired a meaning
from the situational context of the problem. So, problems which traditionally could
be answered only by the solution of an equation were now treated in many ways:
by trial and error working on a table, by the graphic representation or by an equation or inequality. Every answer is not only acceptable but also desirable. Students’
constructions are valuable for their learning.
The solutions came from the work on graphs, with an accurate coordinate system
which was distributed by the teacher, after a class discussion. In general, we used
three strategies to solve equations, according to the three y
y=ax+b
representations of function; firstly, giving a specific value of the
dependent variable on a table and asking for the associate value of the
independent variable x. Secondly, from the graph of y=ax+b, asking
x
for the value of the independent variable x when y had a specific value
(see picture beside). Thirdly, equating the letter-symbolic representations of two
lines, in order to find the coordinates of their intersection point algebraically or
replacing the value of y on the type y=ax+b, that results to the equation
ax+b=c. These cases were presented in class through a problem similar
to the break-even task. The final part of the course was devoted to the
formal solutions of linear equations. In order to achieve the conceptual
understanding of the steps to the solution, the teacher persisted in the justification
of every step, for example he used the metaphor of balance (see picture) for moving e.g. a number from one part of the equation to the other, the role of distributive
low etc.
During the course one PC and a video projector were available to the teacher.
The PC was used to provide graphs and to develop a class discussion on the qualitative aspects of the tasks.
At the end of the course, one and half months later, a post-test was given to the
class, considered from then on as the experimental group, as well as to a second
class, considered the control group, in which the teaching of equations followed the
textbook.
Finally, a number of interviews were implemented by the author. Eight students
from the experimental group participated in five interviews each (apart from two
students who participated only on the first two interviews) which covered all the
subjects of the course. Moreover, two students from the control group participated
in the same interviews.
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The subject of one of these interviews was a break-even task which was modelled by the equation ax+b=c.
THE BREAK-EVEN TASK: Taxi problem
When we use taxi we pay ‘standard charge’ 0.80€ and 0.30€ per kilometer.
1. From what the charge of a route is depended?
2. If we pay y€ for a route of x kilometers, express y as a function of x (in
other words write an equation connecting x and y).
3. Complete the table:
x [route to kilometres] 2 5 8
y [charge in €]
4. Describe how you can construct the graph of this function.
5. George is in the center of Athens in Omonia square. He takes taxi for their
house. He pays for this route 3.5€. How many kilometers is his house
from Omonia?
6. Tom give the taxi driver 5€ and take exchanges. How many kilometers is
his house from Omonia?
In the following we will focus our attention on the break-even task, focusing on
specific students’ answers to the questions 1-5 and we will analyse them, as they
recorded in the interviews, presenting extracts from them. Question 5 refers to the
equation and her connection with the representations of the function. Question 6
will cover the subject of another paper.
ANALYSIS OF STUDENTS’ ANSWERS
QUESTION 1: This question (and the 2, 3, 4) has as purpose to facilitate the student to use a functional approach for questions 5 and 6. Five of the six students
answered without difficulty in this question: “…it depends on how many kilometers
we do” (Sotiris).
One student, Panagiotis, had a difficult to see immediately the answer:
2 P: here says that…from what depends the cost of a route that is the function that
means…that we are going to pay?
3 I: you take a taxi…on what does it depend on how much money you’re paying?
4 P: the standard charges that is 0.8 euros plus the kilometers we are going to do
5 I: the standard charges change or remain stable?
6 P: no, it remain stable
7 I: so what is the thing that changes?
8 P: it is the euro per kilometer
9 I: does the euro per kilometer alter?
10 P: the how many kilometers
11 I: so…
12 P: the cost depends from how many kilometers we are going to make
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The interference of the interviewer with his questions [5, 7, 9] makes Panagioti
to think with terms of variable-stable quantities for the situation problem [6, 8, 10].
Finally he results in the two co-variable quantities [12], the kilometers (independent) and the cost (depended). The difficulty that Panagiotis, who was one of the
best students that participated the interviews, presented, may be due to the misunderstanding of the data of the problem.
QUESTION 2: Situation’s model, that is the functional relation has been constructed with ease from five students and moreover the symbols and the expressions, that were used, had meaning for them, which emerged from the situation
context, which is assumed to be important so as to go on in a more abstract expression. The below transcript from Sotiris’s interview demonstrates the above notion:
S: well ah… it is the y= 0,3 ⋅ x , that is the x are the kilometers, it depends on how kilometers we do and plus 0.8€, which is the standard charge [he writes y=0.30x+0.80].
R: what does illustrate the 0,3 ⋅ x ?
S: it is 0,3 ⋅ x , we say that the 0.3€ is the price for kilometer multiply to x that is…we do
not know how many kilometers we do, so it may be 5 kilometers and 6 and 7.
One student, Olivia, had a difficulty and the interference of the interviewer was
necessary, in order to think with greater caution upon the data of the problem and
the symbolism, that she used so as to result to the model of the situation, that is the
type of the function.
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12
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14
15
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O: …as a function of x…the y equals to…no x…y=x+0.3…
I: can I ask you something? What y represents?
O: how many we are going to give
I: and to what does it express?
O: …
I: so y what is it…kilometers, euros…something else?
O: euros
I: and x what is it?
O: kilometers
I: the x+0.3 what is it?
O: euros
I: …if you add kilometers and euros what are you going to get?
O:… yyyes …
I: does the standard charge play any role?
O: yes, because…ah the 0.8…I confused with 0.3…I have put it correctly…I will add
and 0.8 [she writes y = x ⋅ 0.3 + 0.80 ]
20 I: so…
21 O: …we are going to multiply the x with the amount that we pay for every kilometer
and we are going to add the standard charges
The situation context is helpful to the students, so as he can give meaning to the
symbols in the primary phase of introduction to algebra and they are not abstract
and meaningless. Important role seemed to have the teaching of similar problems
in class.
QUESTION 3: The table has been completed easily. The operations were made either with the help of the type y=0.30x+0.80, replacing on x the value of the table,
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for instance Olivia writes: y = 0.3 ⋅ 2 + 0.8 = 1.4 , or “orally” as Sotiris says without
writing: “0.3 multiply with 2, 0.6, plus 0.8, does 1.4, this is the result, isn’t it?”, or
by writing separately the operations, for example Sia writes 0.30 × 2 = 0.60 and
0.6+0.8=1.4. Panagiotis fills in the table replacing the type and through the questions of the interviewer he understands that only positive values can the table have:
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I: can you take other values except those given?
P: ah yes
I: how many?
P: it depends on how many kilometers will do
I: …how many values could you place?
P: many…but no negative
I: ah no negative values, why?
P: we can not do minus kilometers
QUESTION 4: So as to describe the construction of the graph, all the students
pointed out the need of the construction of two vertical axes, defining which variable will correspond in each axis and also putting numbers on them. It is important
that the students emphasized that the points are these which are indispensable for
the graph and also that every pair of the table is represented (correspond) with a
point.
Every student constructed only the positive semi-axes. The interviewer asked
clarifications from Sotiris:
I: why don’t you put x and y to the other direction too?
S: because it is impossible to do minus kilometers
Sotiris’s answer is like Panagiotis, who used the same expression minus kilometers. Through these, they can “see” the necessity of function’s domain, claiming
essentially that the values of x can not be negatives. It appears in this case how important is the context for concepts first approach.
Finally the graph was constructed from the computer and given ready.
From Olivia’s paper
QUESTION 5: (CONNECTING FUNCTION AND EQUATION). This question is the main question of the situation problem and essentially consist an introduction to equation concept and also connecting it with the concept of function.
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Below we will see and discuss the strategies that had been developed by the students in order to answer this question.
1st strategy: Using the table. The majority of the students used the table in order
to find a solution (see figure from Olivia’s work). Sotiris also gives a very good
answer based on his previous table- as the following extract certifies:
S: ….well...because we said that in 8 kilometers pays 3.2 … so it will be a kilometer more
… because for each kilometer we say that pays 0.3€, it is in 8 kilometers 3.2€ so plus 0.3
more it will go 3.5€
I: so how do you find?
S: 9 kilometers
Sotiris essentially uses here the rate of change of the function so as to find the
next value observing the values from the table.
2nd strategy: Using inverse operations. The second strategy that was developed
was this of the invertial operations, which was used by three of the six students.
Olivia did not only use this strategy but also wrote the operations using symbols,
solving the type of function. Initially, she did not use the parenthesis (she writes
x=y-0.8:0.3) while she did correctly the operations in the way she was thinking
them. She should have had a discussion with the researcher in order to understand
the meaning of the parenthesis and to use it afterwards (she writes x=(y-0.8):0.3).
3rd strategy: Using graph. Every student faced this question using graph, others
more easily, others not. The extract from Sotiris’s paper demonstrates the way
which he processed this specific question: (we notice that Sotiris was one of the
weakest students in the class and one of those who participated in the interviews).
S: well lets go to the value 3.5 over here [ he points 3.5 on the y-axis] and we will go on
to the graph
I: in which way? there are infinite ways to go on
S: we can draw a co-cut line
I: in which way will you draw it?
S: ...
I: is this line correct? [he draws an accidental mental line]
S: naturally no
I: this one?[he draws another accidental line]
S: no
I: tell me how
S: a straight line
I: all these are not straight lines? …. You must say something specific
S: a parallel to x-axis
I: go on
S: until it gets to the graph and it can’t move no longer, and when it reaches this point…
I: yes…
S: we see that we have reached in a point now, from this point we draw a vertical to xaxis and a parallel to y-axis of course
I: ok…move on
S: we see that… ok …it didn’t come exactly, it comes close to 9
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From
Sotiris’s
paper
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In the solution using the graph, essentially we ask to find the value of x for which
the functions y=0.80+0.30x and y=3.5 are equal, that is the equation
0.80+0.30x=3.5. So, even though there is no clear mention, we have a first connection among the concepts of function and equation.
4th strategy: Constructing and solving the equation 0.30x+0.80=3.5. The 4th
strategy that the students used in problem solving was the construction of the equation 0.80+0.30x=3.5 and its solution with typical way, using the properties of
equality and operations. This method appeared to be the most difficult. It was difficult especially for Sia and Sotiris to solve the equation understanding throughout
their actions every step of the solution. The rest of the students appeared not to
have an important difficulty in the solving and explaining of every step they made.
Most students faced the solving of equation based on the properties of equality
(e.g. a = b ⇔ a ± c = b ± c or a / c = b / c ), having as a model this of the balance, which we used in the class. The difficulties and the vagueness’ created by the
typical way of solving an equation is revealed from the below transcript of Sia’s
interview:
1. S: [ she thinks…she has difficulties]
2. I: when we solve an equation which is our goal?
3. S: now I am going back to the… if I isolate the unknown in one side
4. I: ok… what is the first thing that you can do here?
5. S: here to do the opposite operation [she points out 0.30x]
6. I: so, what are you going to do?
7. S: namely to write 3.5-0.8+0.3x … no… basically I want the 0.3 out
8. I: 0.3… eh …
9. S: …
10. I: well, firstly we move out the terms and in the end we go to the multiplication
11. S: so here we do the opposite operation
12. I: which opposite operation?
13. S: the opposite of 0.3
14. I: you will go to the multiplication directly, have we isolate x?
15. S: ahh … yes …
16. I: this is the last step…. which one must go from here?
17. S: 0.80
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18.I: can you continue?
19. S: [ she writes 3.5-0.8=0.8+0.3x-0.8]… this one now it is 2.70=0.30x
20. I: fine, afterwards?
21. S: now… yes… I will put out 0.3…
22. I: so what are you going to do?
23. S: ah yes … division…
24. I: with what will you divide?
25. I: 0.3x to 0.3… x=9
We observe that the essential interferences of the researcher in at leαst two points
[10, 14, 22] helped Sia to solve the equation with the typical way. Her difficulties
were in the distinguish of the “terms” of the equation and even more in the actions
she had to do in order to construct equivalent more simple equation. In contrast,
Helen demonstrated a greater flexibility to interpret and understand the concepts
and her actions in some points, but in other points didn’t demonstrate the same
flexibility and understanding. For instance, she interpreted the subtraction of 3.5
from both sides of the equation and as an addition of -3.5 in different parts of the
interview, showing a deeper understanding of the concept of negative number, but
wrote the equation -0.3-x=0.3-x+x+0.8-3.5 as an equivalent equation to the 3.5+3.5=0,3x+0.80-3,5. She appears hadn’t understand the concepts of ‘term’ of an
equation and unknown’s coefficient. Only after the researcher’s interference she
reflects on her actions and corrects her obscures. The episode below shows our
claims:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
H: [she writes - 3.5+ 3.5=0.3x+0.8-3.5]
I: why is this correct?
H: because when we change side we change sign
I: but I think that you didn’t do exactly this?
H: yes … I put … I take off the same number in both sides
I: when you say ‘put’ or ‘take off’ what do you mean?
H: …
I: usually when we act on an equation we do some operation or something similar.
H: … I added
I: what did you add?
H: the -3.5
I: ahh, you added, have you this right?
13. H: yes, it is like a balance.
She uses the practice rule [3], but explains her action with a property of equivalence, having as prototype balance’s model [13]. Initially she uses informal terminology [5], but, in the discourse with the researscer [6, 8], step by step she develops using more formal terms [9, 11]. In the following she writes -0.3-x=0.3x+x+0.8-3.5.
14. I: the -x how did it locate in the left side?
15. H: since we changed side
16. I: previously you said that we add the same number in both sides … now here which
number did you add?
17. H: the x
18. I: is there x nowhere, alone?
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19. H: the 0.30x
20. I: ahh … is the 0.30x the same with the
21. H: no [she corrects writing -0.3x=0.3x-0.3x+0.8-3.5 and -0.3x=-2.7] … we divide
−0,3χ
−2, 7
with the coefficient of unknown and [she writes
]…
=
−0,3χ −0,3χ
22. I: what do you mean saying ‘the coefficient of unknown’?
23. H: ahh … the 0.3 [she corrects and results to solution x=9]
Helen knows how to construct an equivalence equation [15,16] but here appears
clearly that she hasn’t elucidate the ‘ unknown term’ of an equation [the new equation that she wrote, 15, 17] and the equivalence equation isn’t correct. The researcher’s interposition is important for the continuation [18]. Another term that
Helen hasn’t clarified is ‘the coefficient of unknown’ [21] and the researcher’s
question again is important for the continuation [22]. We observe that the discourse
between Helen and the researcher is eminent for Helen’s learning development.
The interviewer asks from Helen some clarifications about the elimination of the
opposite numbers in the initial equation - 3.5 + 3.5 = 0.3x + 0.8 - 3.5.
1.
2.
3.
4.
5.
6.
7.
I: the way you erase them left side, does anything stays?
H: no
I: so what will you put?
H: the x [perhaps she means that she will transfer the unknown term]
I: … now that you erase them what does it stay there?
H: nothing, leave until there
I: wait, when they leave these two, which will be the result of this operation? [shows
the operation to the left side]
8. H: …
9. I: haven’t got equality with two sides?
10. H: yes…
11. I: haven’t you done an operation to the left side?
12. H: ah … zero
13. I: so why don’t you write it?
14. H: shall I write zero?
15. I: don’t you have?
Helen solves the equation erasing the opposite numbers -3.5 and +3.5 but she
doesn’t write the result, she thinks that there is nothing [6] and it isn’t trouble for
her saying leave until there 6]. Only with the interferences – questions of the interviewer [7, 9, 11] she understands that the opposite number’s eliminate means addition, that’s the result is zero [12], but she believe that it is not indispensable to
write it [14]. Perhaps that means that zero’s acceptance like a number is an obstacle
to Helen. In another equation Sotiris had the same difficulty, which was an obstacle
for him. Specifically Sotiris thinks about the solving of equation 2x=-x+6.
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210
I: what have you got to do over here?
S: to transfer the unknowns in the one side and the knows to the other
I: what act you must do so as to do these you said?
S: … transfer the 2x to the 2nd part
I: and then?
S: …
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I: nice, if you transfer the 2x to the 2nd part what are you going to have in the 1st
S: ...
I: what will remain?
S: nothing
I: …what will it be the next step?
S: I don’t know
I: since we say that we separate the knows from the unknowns?
S: yes, but … since nothing would exist, we could not to continue
I: when you say that nothing would exist, you mean that there would be no number,
so wouldn’t be a part?
220 S: …if we transfer the 2x…it is meant to be left …zero
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213
214
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218
219
Sotiris knows what must do [206], but he does not how to do this 210, 212]. He
does not understand the properties of the equality. When he visualizes the transformation of 2x to the other side, he does not understand that the quantity on the
left side (the result of the addition 2x-2x), is zero [214, 220]. The existence of zero
consist an obstacle for Sotiris when he says ‘nothing’ [214] and ‘in case nothing
would exist, we could not to continue’ [218]. Sotiris visualizes that on the left side
there is nothing, there isn’t equation, and therefore he can’t continue the solution.
Only after the interviewer’s interference Sotiris probably understands that on the
left side is done an operation -addition- who’s the result is zero, as his statement
shows [220].
Unlike, Helen, that said ‘nothing, leave until there’, visualizes an equation with
one side including nothing; she continues the solution regularly, as the one side
being zero.
SUMMARY AND CONCLUSIONS
Based on the analysis of student’s interviews we conclude that our functional approach to algebra helps students to develop their algebraic thinking. This teaching
approach encourages the students to connect the different representations of function and associating each representation with the equation as a strategy of problem
solving. We observed that the situation context was fundamental for students’ conceptual understanding. More specifically, we can report the following conclusions:
The functional approach to algebra that we applied has given the chance to students to solve problems with different approaches and strategies using various representations of the function such as the graph, the table or the letter-symbolic form
of the function and not only with the formulation and solution of an equation.
The equation was faced from the students through the function firstly using table,
secondly graph and finally using the formal way (e.g. the 0.3x+0.8=3.5 faced by
replacing y=3.5 on the type of the function y=0.3x+0.8).
The beginners have often difficulty in understanding that the algebraic symbols
that are referred to the variables are not simple numbers whose values occur to be
unknown (Freudenthal, 1982). The context has been defining, confirming the position in which: the students have often difficulty in understanding the idea of a pos-
13
sible relation among two quantitative variables, unless they see this relation to be
existed in a specific, physical context (Piaget, Grize, Szeminska, & Bang, 1968).
Our teaching approach gave the students the opportunity to see easily the lettersymbol not only as an unknown but also as a variable. The students could seek and
recognize co-variable quantities and to construct the functional relation between
them, initially in an informal way and finally in a more formal one. The context of
the activity was the motive so as the students could confront the problem in a functional way.
The symbols have gained meaning and importance for the students, emerged by
the situation context. This is an important step towards the more formal and abstract form.
We observed that the students have shown a greater understanding when they
solved an equation (of the type ax+b=c ) with the use of a graph or of a table than
in the formal way, based on the properties of the equality and the operations of the
rational numbers, where a lot of difficulties and misunderstandings appeared. This
knowledge was a prerequisite according to the curriculum. It is well known that for
the students of this age a great obstacle for their further mathematical evolution is
the non-consolidation of the operations and the properties of the rational numbers.
The phrases of two students are characteristic: For the operation -3+7, Sotiris since
he has reported some possible results such as e.g. -10, he asked: “how do I know if
the + is an operation or a sign” or in the incitation of the interviewer to change the
order of the additionals he wrote “-7+3”. In the interviewer’s question ‘what do
you think, when you do one such operation’ he answers: ‘I think of the table with
the rules that exist in the book and I try to apply it’.
It is well known that the didactic exchange during an interview, when the members have
an access to each other’s thoughts, it is productive from the mathematical point of view
(Kieran, 2001) while the communication is a target of thought if not the thought itself
(Sfard, 2001). So the interaction with the interviewer gave, during the interview, the opportunity to the student to think upon his/her actions and to alter his primary knowledge
when it came to contrast with the new knowledge. Through the didactic exchange even
the interviewer had the chance to learn a lot in relation to the student’s thought.
The functional approach for the introduction to algebra is not a panacea, however
we support that the shifting from the procedures to the algebraic presentations and
the equivalents in operations upon the functions and the possibility of different representations, can offer more opportunities to the students to pose questions and to
the teachers to offer tasks, that can impel the researchal activity in algebra. A
mathematics curriculum which gives directions for the visualization of mathematical objects is more suitable for problem solving and investigative processes.
Through the interviews of the student we try to examine thoroughly the evolution of
student’s thought. We have indications that with the use of the function as the main concept for the introduction in algebra, we supply the students with a strong instrument in
order to construct their algebraic tasks with meaning. The results from the mathematical
negotiation with these students lead us to the conclusion that the function is not far away
from the conceptual control of the beginners and we hope that this teaching approach pos-
14
sibly will have influence on students’ deeper understanding to other concepts on later
grades.
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