NEGATIVE SOLUTIONS IN THE CONTEXT OF ALGEBRAIC WORD
PROBLEMS
Aurora Gallardo and Teresa Rojano
Centro de Investigación y Estudios Avanzados, IPN, México.
ABSTRACT
Resistance to accepting negative solutions to equations and problems is found
both in the history of the development of mathematics and in students who are
beginning the study of symbolic algebra, in the latter case, avoidance of the
negative solution is usually accompanied in practice by a separation of the
operational manipulation of positive or negative numbers and the solution of
algebraic equations. This article reports the results obtained by a clinical study
carried out with secondary school students. The results reveal some of the
conditions which propitiate the acceptance of negative solutions in algebraic
word problems. Analysis in this study is guided by an earlier analysis concerning
the acceptance and rejection of this type of solution in the history of mathematics.
INTRODUCTION
Studies such as those carried out by Bell (1982), Carraher (1990),
GaIlardo/Rojano (1990) and Resnik (1989), among others, indicate the difficulties
students confront when they are faced with the conceptualization and operational
manipulation of positive or negative numbers. The results of some of these studies show a
separation of the development of operativity of whole numbers and their use in certain
contexts. Thus, for example, in the case of the solution of algebraic equations, there are
subjects that show resistance to the acceptance of a negative solution despite possessing
fluid operativity with the numbers (see Gallardo, 1987). Teaching is one of the multiple
causes to which this separation can be attributed, since it is quite common to find that
school mathematics curricula treat the theme of whole numbers fundamentally as an
extension of natural numbers, emphasizing the operative and paying little attention to the
role these numbers play in the extension of the numerical domains of coefficients and
solutions to algebraic equations.
In the search for other fundamental causes of the difficulties encountered in the
conceptualization and operational development of positive and negative numbers,
historical-epistemological, philosophical and psychological theoretical analyses have all
been brought into play. Such is the case of the research carried out by Glaesser (1981),
Schubring ( 1988) and Dreyfus/Thompson (1988). In the project entitled “The status of
negative numbers in the solution of equations”1, the research starts from a historical
analysis of the acceptance and rejection of negative solutions in the context of the
solution of algebraic equations. One of the most important conclusions of this analysis is
that the acceptance of the first negative solutions presupposes a certain level of
development of the language of symbolic algebra (advanced syncopation) as well as full
operativity, advanced levels of interpretation of positive and negative numbers and the
development of ad hoc methods in the case of negative solution (Gallardo, Rojano,
Carrión, 1993). This type of conclusion has served as a basis for the formulation of
hypotheses at an ontological level concerning the conditions in which it is feasible to go
from primitive stages of conceptualization to stages of consolidation and formalization of
the notion of relative number in the above mentioned context. This article reports on the
results of the second part of the project in which some of these conjectures are put to the
test by means of a clinical study with students 12 and 13 years of age, who were being
introduced to the syntactic manipulation of algebra and the solution of algebraic word
problems at the moment the study was carried out.
THE CLINICAL STUDY. Solution of word problems and negative solutions.
Data was gathered in two ways: a group of 25 students at 2nd grade of secondary
school in México City were asked to answer questionnaires; individual clinical interviews
were recorded by video and analyzed. The results of the questionnaire were used to select
a group of students for clinical observation. The blocks of items presented in the
interview dealt with the following themes:
1) Operativity in the domain of whole numbers.
2) Translation of situations expressed in words to symbolic language.
3) Use of pre-symbolic and symbolic language in the context of equations.
4) Solution of word problems.
This article gives the results of the clinical interview referring to the block
“solution of problems". The dimensions of analysis in this part of the interview were:
method or strategy/problem solution/interpretation of the solution.
The methods used by the students to solve word problems were as follows.
PROBLEMS OF AGES. Luis is 22 years old and his father 40 years old, how many years
must pass for his father to be twice the age of his son?
Method of Two. (Used by four students). The student finds the problem impossible
because "the 2 is always there". This refers to the difference of 2 in the units of the data
given in the problem as the ages of father and son advance. Example: One student
established two lists of numbers, increasing the ages starting with the ages given in the
problem: 22 years, 40 years. He writes: 23, 41; 24, 42; 25, 43; and so on. He notes that
the difference in the figures of the units of each pair of numbers is always 2. He
concludes that the problem does not have a solution.
Method of Duplication. (Used by three students). The student arrives at the correct
solution, 18 and 36, the ages of son and father respectively, but he duplicates the ages and
thinks that 36 and 72 is the true solution. There were also cases where the student thought
that 36 x 2=72 and 72 x 2 = 144, is also a solution to the problem.
Method of the Difference. (Used by four students). The student finds the difference in
ages, that is 40-22=18. He deduces from this that the son is 18 years old and consequently
the father is 36.
Method of Altering the Difference. (Used by two students). The difference between the
ages (18) is divided in half, 9, and this value is then added to the son's age, 22. The
answer to the problem thus given is 31.
Ascending/Descending Method. (Used by four students). The student increases the ages
of the father and the son and finds that the problem cannot be solved. He then decides to
decrease the ages and arrives to the correct solution.
Algebraic Method. (Used by two students). Spontaneous formulation of the equation that
solves the problem.
PROBLEM OF PURCHASING GOODS. A salesman has bought 15 pieces of cloth of
two types and pays 160 coins. If one of the types costs 11 coins the piece and the other 13
coins the piece, how many pieces did he buy of each price?
Method of one equation. (Used by 15 students). The student looks for multiples of 11 and
13 that add up to 160. (This is equivalent to solving the equation 11x+13y=160. The
existence of x+y=15 is ignored). When the student does not find the multiples needed to
solve the problem, that is 11x11+13x3=160, he uses an additional interpretation to
explain his results, for example,
Student 1. He writes 66+91=157, and says, "he bought 6 pieces costing 11 coins and he
had 3 coins left over'.
Student 2. Me writes 154+0=154 and explains, "He bought 14 pieces costing 11 coins
each and none costing 13 coins".
Student 3. He writes 154+13=167 and says, "He owed 7 coins".
Additive Method. (Used by one student). The problem of the purchase of goods is
modified such that the figures are smaller in order to facilitate the solution. The equations
which model the problem in this case are: x+y=3; 2x+3y=40. The student assumes that
each one of the pieces of cloth has a price different from that established in the statement
of the problem in order to adjust the total price. He writes 1x2+1x3=5, thus, 40-5=35. He
then says "the salesman bought 3 pieces: 1 costing 2 coins, another costing 3 coins and a
third costing 35 coins".
Sharing out Method. This is also found in the modified version (x+y=3; 2x+3y=40). A
student divides the total price, 40, by two. The result of the division, 20, is used with the
other data of the problem 2, 3, and he formulates the sums: 18+2=20; 17+3=20. His
answer is “He bought 18 pieces worth 2 coins each and 17 pieces worth 3 coins each".
It is important to point out that, contrary to what might be expected, the modified
version of the statement (with small numbers) renders the problem impossible for many
students. The conflict is accentuated since the solution is sought in the positive domain
and the lack of adjustment between the data of the problem is more notorious than in the
previous version (x+y=15; 11x+13y=160) where the magnitude of the numbers tends to
hide the conflict. This obstacle disappears when it is suggested to the student that he use
algebra to solve the problem.
Algebraic Method. (Used by two students). Spontaneous formulation of a system of
equations to solve the problem.
Let us now look at the case of a student who, by using the process of substitution
of the solution in a system of equations, managed to solve the problem which at first he
had thought impossible. The student formulates the equations 11x+13y=160; x+y=15. He
obtains the solution x=17.5. The following dialogue then ensued:
Student: Totally impossible.
Interviewer: And now how are you going to find y?
Student: It can't be done, totally impossible.
Interviewer: Let's try anyway. (Student finds y = 2.5.
spontaneously he substitutes the values in the
equations).
Student: It worked!
Interviewer: What happened then?
Student: Instead of buying, he gave 2 and a half pieces
of both to the person he was going to sell
them to, and bought 17.5 of the other. That is,
the buyer gave the seller 2 and a half pieces
of cloth and the seller gave him 17.5 pieces.
It's like barter.
Interviewer: Why did you say it was impossible before?
Student: Because it's impossible with positive numbers.
Below there are two tables that summarize the dimensions of the analysis:
method or strategy/problem solution/interpretation of the solution, for the two problems
presented in this article.
PROBLEMS OF AGES
Method or Strategy
Method of Two
Solution
Does not exist
Method of Duplication
Positive
(one of various)
Positive
Method of Difference
Negative
Method Altering the
Difference
Method of Half
Positive
Positive
Positive
Ascending/Descending
Method
Negative
Negative
Algebraic Method
Interpretation of the solution
Impossible
Occurs various times in the
lives of father and son: 36, 72
: 72, 146
Impossible
18, 36
4 years ago.
-4 years
4 years
+ 4 years
Luis is 31 years old
36, 18
4 years before.
-4 years
x=-4
PROBLEMS OF PURCHASING GOODS
Method or Strategy
Method of one equation
Solution
Positive
Debt
Surplus
Additive and Sharing Out
Method
Positive
Algebraic Method
Negative
Interpretation of the solution
Impossible
Change of problem data so as
to adjust quantities involved
Interpretation of barter of
goods
CONCLUSIONS OF THE STUDY
From the analysis of the different problems exhibited in the study2 the following
was concluded:
1. It is possible to solve the problem without expressing the solution in negative terms.
The problem of ages (method of duplication, method of difference, ascending/descending
method).
2. The creation of specific methods from problem to probtem occurs. Problem of ages (6
methods). Problem of purchasing goods (3 methods).
3. The choice of the appropriate method requires the acceptance of a negative solution
which is interpreted in the context of the problem. Problem of ages (method of
differences, ascending/descending method, algebraic method). Problem of purchasing
goods (algebraic method).
4. When faced with problems with negative solutions the student turns to changes or
adjustments in the data of the problem statement as well as the construction of sources of
meaning which allow him to give plausible interpretations to the solution obtained.
Problem of purchasing goods (method of one equation, additive method, sharing out
method).
5. A problem that can appear impossible to solve with arithmetical methods, is thought of
as possible using algebra, once the negative solution is validated by being substituted in
the corresponding equation or equations. Problem of ages (algebraic method). Problem of
purchasing goods (algebraic method).
FINAL DISCUSSION
As we pointed out in the introduction to this analysis, historical-epistemological
analysis carried out with respect to negative numbers in the solution of algebraic
equations has guided this study at an ontological level and has allowed us to establish
some of the conditions which propitiate the acceptance of the negative solution of word
problems by secondary school students. We can thus affirm, at an ontological level, that
for a student to accept a negative solution, the following are necessary: the use of ad hoc
method, the consolidation of meaning appropriate to the context of the problem, and,
fluid operativity of negative numbers. In the case of some problems of a commercial type,
the algebraic language becomes indispensable for the possible arrival at the negative
solution
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Acknowledgements
We wish to thank Hermanos Revueltas school in México City for providing the
research study. We also wish to acknowledge the support of the Departamento
Educativa of CINVESTAV (México).