Students' preferences when solving quadratic inequalities.

Abstract

We address the question whether to present students with a single method or with a number of methods for solving quadratic inequalities. Twenty 10th graders were presented with three differently sequenced methods, i.e., the graphic, the sign-chart, and the logical connectives method. We examined participants' preferences when solving related tasks, and their success in doing so. Almost all students correctly solved the different inequalities, and most liked being presented with several methods. The method most frequently used was the graphic method, and many students preferred the method they had studied first. Some conclusions and educational implications are drawn.

Students' Preferences When Solving Quadratic Inequalities

When planning instruction, teachers may face the dilemma of whether to present their students with a single method for solving a specific type of mathematical problem or with a number of methods. When a single method is chosen, a question arises about the criteria for this choice. If, on the other hand, several methods are going to be presented, one may wonder whether there is any significance to the order of presentation. A related question is, what should such didactic decisions refer to? For example, should we only consider students' success (as defined by the ability to answer correctly) or should we also attend to the methods students choose either when solving related mathematical tasks or when responding to explicit questions regarding their preference? We address these issues with regard to the process of solving quadratic inequalities.

The literature commonly presents three major methods for solving quadratic inequalities: the graphic method, the sign-chart method, and the logical-connectives method. The graphic method involves the interpretation of graphic representations, e.g., using parabolas to solve quadratic inequalities. The sign-chart method involves finding the zeros of an equivalent equation and using a sign chart to determine the solution of the inequality. The logical-connectives method involves the translation of the inequality into a system of linear inequalities, which are connected to each other by "or" and "and" connectives (see example in Figure 1).

Two approaches for teaching quadratic inequalities can be identified in the literature: the single-method approach, presenting the students with only one method, and the multiple-method approach, presenting the students with any combinations of the two or all three above-mentioned methods for solving this type of inequality. For example, Dreyfus and Eisenberg (1985) preferred the graphic method and provided mathematical and didactical argumentation for their claim:

[FIGURE 1 OMITTED]

  this [graphic] approach appears to make the solution of many
  inequalities easier for average and weaker than average students who
  have had some experience with graphing functions. It also provides
  quite a bit of insight into what it means to solve an inequality and
  in what sense inequalities are related to functions. (p. 653)

McLaurin (1985) and Dobbs and Peterson (1991) identified the sign-chart method as the best method for teaching quadratic inequalities. They explained that "one of the most appealing aspects of sign charts is that they serve as a uniform and relatively easy method for solving what many consider to be more complicated inequalities" (p. 664). In fact, the researchers' claims were actually more far-reaching, since both McLaurin and Dobbs and Peterson suggested that the method they offered provided students with a powerful tool for solving not only quadratic, but any type of inequality. However, their papers report no research that supports their conclusions, which seems to suggest that these conclusions are based on the authors' impressions from their own teaching.

Piez and Voxman (1997), on the other hand, supported the multiple-method approach for solving inequalities. Their claim was that "students need to be strongly encouraged, possibly required, to work with multiple representations ...", because, "In the long run, students who have more flexibility will be more successful in solving a wide range of problems" (p. 166). Their recommendations were based on a structured teaching-experiment but they provided no details regarding either the intervention or the inequalities the students were asked to solve.

Our study examined students' performances and their preferred method of solving quadratic inequalities after being presented with the three above-mentioned methods. We asked: Do students who have been presented with three methods for solving quadratic inequalities use these different methods in their solutions? Of the three methods, is there one method that students prefer, and if so which one, and why? Do students appreciate learning different methods? And what are students' common errors when using each of the methods?

Study Design

Participants

Twenty 10th grade students, who were studying in a comprehensive school, participated in this study. These students were attending the highest of three mathematics levels, that is, they intended to major in mathematics in the 11th and 12th grade and to take final mathematics examinations. Success in these examinations is a condition for acceptance to science, mathematics or engineering faculties in academic institutions. The participants had previously studied the topic of algebraic equations, including linear equations, system of linear equations, quadratic, and absolute value equations. They had also studied how to investigate and graph quadratic functions, and how to solve linear inequalities and linear systems. However, they had no experience with graphic calculators or with computer algebra.

Process

The teacher (MR, the second author of this paper), allotted four 45-minute lessons to teaching the topic of quadratic inequalities in her class, and she encouraged students to approach her during, as well as between, class sessions to clarify difficulties and errors. It should be noted that the time-frame, as well as the number and type of tasks given to the students, were similar to what was commonly done in her classes when dealing with this topic. In order to present the participants with the three above-mentioned methods in a manner that would avoid foregrounding any one of them, they were presented as follows:

(1) Lesson 1: Learning a Method -- The class was divided into three small groups. Each group was presented with three examples fully solved by means of single method (graphic, sign-chart, or logical-connectives). Inequalities were of the a[x.sup.2] + bx + c > 0 and a[x.sup.2] + bx + c < 0 type. In two cases a[x.sup.2] + bx + c could be presented in the factored form, while in the third inequality this was impossible. During the lesson, the teacher assisted students in each group when asked.

(2) Practice: Home Assignment -- After the lesson, the students were given 12 quadratic inequalities as a home assignment. The teacher corrected submitted solutions in detail, and invited students to offer individual, face-to-face explanations of the method they had studied. They also received mimeographed sheets containing several related examples. In order to make sure that the students understood their first-studied "solving method", for every error found they were given an additional task of the same type, to be solved and presented to the teacher before the following lesson.

(3) Lesson 2: Learning in Mixed Groups -- The class was re-divided into four new groups, so that each group included at least one representative of each of the three methods. Each student was asked to teach his/her peers in the small group to solve quadratic inequalities by the method (s)he had been taught in the previous lesson. They initially worked on the same inequalities as in Lesson 1 and on their home assignment. Subsequently, 10 additional similar quadratic inequalities had to be solved (starting in class and concluding at home) using all three methods.

(4) Lesson 3: The Teacher's Summary -- The next lesson was devoted to a summary, presented by the teacher, who solved several quadratic inequalities. Each inequality was solved three times, using each of the three methods. Each time the three methods were sequenced differently. The teacher refrained from evaluating the methods and emphasized that every student could use their method of choice. Nor did she make any comment regarding advantages or disadvantages of studying or using a single method or multiple methods for solving quadratic inequalities.

(5) Lesson 4: The Final Assignment -- One day after the summary lesson, during a mathematics lesson, the students were administered a questionnaire that included 12 quadratic inequalities and some questions regarding their preferences regarding the method(s) for solving such inequalities. We examined students' preferences by looking at their performance when solving the 12 quadratic inequalities and by examining their statements when evaluating the multiple-method presentation approach.

Tools

The students were administered a questionnaire that included 12 quadratic inequalities, which were chosen because they cover a diversity of types of such inequalities (see Table 1).

The literature reports various difficulties that students have experienced when solving quadratic inequalities. For example, students were found to incorrectly multiply both sides of a given inequality by a negative number without changing the direction of the inequality sign (e.g., Tsamir & Almog, 2001; Tsamir & Bazzini, 2002). Students encountered difficulties when having to deal with logical connectives, exchanging 'and' and 'or' or leaving the solution with no connectives all together (e.g., Tsamir & Almog, 2001; Tsamir, Almog, & Tirosh, 1998). There is also data regarding students' difficulties when solving inequalities that result in a single value, inequalities that result in [empty set], and inequalities that have 'R' solutions (e.g., Linchevski & Sfard, 1991; Tsamir & Almog, 1999; Tsamir & Bazzini, 2001). These findings were considered when designing the questionnaire.

An analysis of the chosen tasks may be done with reference to the following criteria:

1. The structure of the given expressions (is it presented in a factored form or not, and does [x.sup.2] have a negative coefficient or a positive coefficient): The tasks included two inequalities in a factored form (see tasks [1] and [7] in Table 1), and in Inequality 1 the coefficient of [x.sup.2] is positive, while in Inequality 7 it is negative. The other inequalities were presented in a non-factored form way, so that five inequalities (Table 1, [2]-[6]) had positive coefficients to [x.sup.2], and the other five ([8]-[12]) had negative coefficients.

2. The number of solutions of the related equations, that is to say, the sign of the discriminant - discriminant > 0 two solutions, discriminant = 0 one solution, and discriminant < 0 no solutions to the related equation: Six tasks (1, 2, 3, 7, 8, 9) had a discriminant > 0, two tasks had a discriminant = 0 (6 and 10), and four tasks had a discriminant < 0 (4, 5, 11, 12).

3. The types of solutions of the inequalities ("and" or "or" intervals, [empty set]-empty set of solutions, or R--any real number, and {x | x = a}--single value solutions). The tasks included six inequalities with "interval type" solutions--three inequalities that had solutions of the type "and" {x | m<x<n} and three of the type "or" {x | m > x OR x > n}. Two inequalities were of the type {x | x = t}, two resulted in R and the other two in [empty set].

The students were also asked to indicate (a) whether they preferred learning one method or a number of methods for solving quadratic inequalities, and (b) to specify which method they preferred.

Students were asked to explain their solutions and their responses.

Results

The results are discussed in the order in which the questions were posed earlier in the introduction.

1. Do students who are presented with three methods for solving quadratic inequalities use any or all of these different methods in their solutions?

Each inequality was solved by each student in no more than one way, which is to say, students who solved a certain task did so by means of a single method and no one tried to solve a task in two ways, for instance, to validate its solution. Table 2 shows that 13 out of 20 participants used a single method for solving all the inequalities and seven used at least two methods. We could answer the posed question by relating to the latter seven students for whom familiarity with multiple methods of solving quadratic inequalities afforded a means of solving different inequalities in different ways. An alternative way to examine the use of different methods is by relating to those who consistently used a single-method in all their solutions. While our findings point to the fact that most students used only one method for solving inequalities, further analysis revealed that among these students no single method was used exclusively throughout problem solving. Different students chose different methods as the single problem solving method. That is to say, students consistently chose one method to solve all problems, but methods chosen could be any of the three. Consequently, all three available methods were applied for solving the inequalities.

2. Among the three methods, is there one method students prefer, and if so which one, and why?

The examination of students' preferences regarding a method to solve quadratic inequalities was done by looking at their choices when solving the given inequalities, as well as by looking at their declarations when explicitly asked about these choices. Table 2 shows that the use of the graphic method was the most prevalent. Among students who had only used one method, the most frequent one was the graphic method (10 students); two students used the sign-chart and one student used the logical connectives method. The graphic method was also most prevalent among the students who used more than one method: it was used by all of these students at some point. Among the students who used two methods, four used the logical connectives and the graphic methods and two used sign-chart and the graphic methods. One student used all three methods.

In response to the question whether they had a preferred method and if so, which, 16 out of 20 students pointed to the graphic method, and even those who did not choose it as their preferred method highlighted its advantages. They presented pragmatic reasons--"It is the easiest", "It is the shortest", clarity reasons--"I can actually see the solution", and reasons of priority--"It is the first of the three methods I studied, so I feel I know it better than the others." Indeed, further examination of solutions with reference to first-studied method shows that all students who were first introduced to the graphic method used that method exclusively. The others used either the method they had studied first or the graphic one. One question this raises is: in the case of students who used the two methods, which inequality did they solve by means of their first-studied method and which by the graphic method?

Table 3 shows that students who first studied the logical connectives tended to use this method for inequalities resulting in a single value, R or [empty set]. That is to say, the students who first studied the logical connectives method used their first studied method when solving inequalities that were identified in the literature as being the most problematic (e.g., Linchevski & Sfard, 1991; Tsamir & Almog, 2001; Tsamir & Bazzini, 2001). They shifted to the graphic method in the "more friendly" cases where the corresponding parabola had two points of intersection with the x-axis.

There is no such clear-cut pattern regarding the two students who first studied the sign-chart method. It is possible that familiarity with the graphing of parabolas made use of the sign-chart and graphic methods equally available to the students. The one student who used all three methods in his solutions had first studied the logical connectives method. He explained that he chose a method randomly for each task, with the general aim to exercise and to exhibit his understanding of the three methods.

Most students (12) expressed the merits of the method they had studied first and they explained that, since it was presented first, they understood that method better than the others. Of these twelve students, five first studied the sign-chart, five first studied the graphic method and two first studied the logical connectives method. They referred either to the way it was studied: "I practiced the first method more than the other ones", "I studied the first method in greater depth", "I paid more attention when studying the first method", or to its being easier than the other methods: "My first-learned method is the simplest of them all." An examination of the findings reveals that eight of these twelve students actually used their first-studied method only, that is to say, five graphic students used only the graphic method, two sign-chart students used only the sign-chart method, and one logical connectives student used only the logical connectives method.

3. Do students appreciate being taught different methods for solving quadratic inequalities?

In response to the question whether they preferred learning one method or a number of methods for solving quadratic inequalities, the participants expressed appreciation of their familiarity with the different methods. Sixteen, including those who consistently used only one method in all their solutions, stated that they were in favor of studying several methods. Their reasons were usually student-related and sometimes mathematics-related. In relation to students, they explained that it is useful to be familiar with multiple methods because it allows students to use methods suited to their personal inclination, e.g., "For every inequality the student may now choose the method he finds to be the most suitable or which is, in his opinion, the easiest." They further explained that familiarity with several methods increased their confidence in their ability to solve given tasks. Typical explanations were, "If I don't understand a certain method, I can always choose an alternative one that will enable me to solve the task"; or, "It reassures me to know that if I get stuck using one way, I'll have an alternative way of solving the inequality."

From the mathematical perspective, students explained their appreciation of the multiple-method presentation approach as follows: "It is good to know different methods, because some inequalities are easily solved by means of one method and others by means of another method." They also appreciated the use of different methods as a possible tool for validation: "If I can solve an inequality by two methods, I can verify my solution." Still, none of the students actually solved any of the inequalities by means of more than one method, and none tried to validate his/her solution in this way.

Four students stated that it would be better to study just one method and suggested that "it is the responsibility of the teacher to choose the simplest one." Three of them had first studied the logical-connectives method and one had first studied the sign-chart method. It should be noted that all four expressed a fear that "different methods might cause confusion". Still, two students who initially studied the logical-connectives method used different methods in their solutions to the inequalities. One of them used two methods and the other used all three methods, contrary to his declarations.

4. What errors are most common when using each of these methods?

No significant differences were found among the rates of correct solutions given to the different inequalities, neither among those given to inequalities solved in a different way (graphic, sign-chart, logical-connectives), nor among inequalities solved by different, "first-studied" methods. Most (93%) of the solutions were correct. In this section we focus on identifying students' errors when solving quadratic inequalities, with reference to the methods they used. Some errors were method-related. For example, typical graphic method errors were the graphing of a parabola with a minimum point instead of a maximum point (but rarely vice versa). Most errors occurred in response to the inequality -[x.sup.2]-6x-9 [greater than or equal to] 0. Typical errors were of the following type,

[FIGURE 2 OMITTED]

Typical logical connectives method errors included incorrect application of logical connectives and incorrect/no solution when the quadratic expression was non-factorable. For example, when solving the non-factorable inequalities, a student wrote that the (corresponding) equation had no solution and then framed the (ax+b)[.sup.2]+c for c>0 expressions she reached in the same way as she framed all her other final solutions:

[X.sup.2]-3x+4[less than or equal to]0

(x-[3/2])[.sup.2]+1[3/4]

[x.sup.2]-3x+10>0

(x-[3/2])[.sup.2]+6[3/4]

[x.sup.2]-2x+5<0

(x-1)[.sup.2]+4

This frame expressed that in her opinion this was the final stage in the solution of the process, but she was actually leaving the inequalities unsolved.

In all three methods, errors occurred when students factored the given expression, or when they solved the corresponding equation. Most of these errors were found for the expression a[x.sup.2]+bx+c where a<0, and they may be rooted in students' reported difficulties when multiplying an inequality by a negative number (see also Tsamir & Almog, 2001).

In conclusion, students usually solved the inequalities correctly. It is possible that when they encountered difficulties while using one method, their familiarity with multiple methods enabled them to shift to another method, thus enabling them to correctly solve the inequality. However, since the number of participants in this study was rather limited and since participants relatively rarely chose the logical connectives and sign-chart methods it is impossible to deduce which method was more problematic in terms of correct application.

Discussion

We began this paper by addressing some general didactic dilemmas, posing the question whether in the teaching of certain mathematical topics teachers should present a single method or various methods for solving related tasks. We also asked: If we decided to present a single method, how should the choice of this method be made? And upon deciding to present several methods, how should teachers sequence presentation? The discussion addresses these questions with reference to quadratic inequalities.

1. Should we teach quadratic inequalities by presenting a single method or by presenting various methods for solving related tasks?

Our findings indicate that familiarity with multiple methods for solving quadratic inequalities was useful for most students. First and foremost was the high rate (93%) of correct solutions. True, our study included no control group that studied a single method to determine whether the high level of achievement could be attributed to teaching multiple methods. Still, in previous studies which investigated the performance of students who had studied only one method, i.e., the graphic method, we found that only about 60% correctly solved quadratic inequalities (e.g., Tsamir & Almog, 2001; Tsamir & Bazzini, 2001). It seems that when encountering difficulties when working in a certain way, our students could opt for an alternative method, thus enlarging their chance of success.

An examination of students' methods for solving the given inequalities shows that while most (13) participants consistently used a single method for all their solutions, not all students applied the same method. This was another indication that students benefited from their familiarity with multiple methods for solving inequalities. In addition, familiarity with multiple methods of solving quadratic inequalities provided more than one third of the participants with alternative routes for problem solving.

Many participants were aware of these advantages and appreciated the multiple-method approach they had experienced. This was not just the case for those who implemented the various methods but also for students who used only one method. The latter explained that familiarity with different methods provided them with optional routes for solution and gave them more confidence in their ability to come up with the required solution. Therefore, it seems reasonable to recommend, as did Piez and Voxman (1997), to present students with multiple methods when teaching quadratic inequalities.

2. How should the presentation of the multiple methods be sequenced?

Our findings point to the role played by the graphic method, and by the first-studied method in students' performance and in their declared preferences. In line with Dreyfus and Eisenberg's (1985) recommendations and contrary to Piez and Voxman's (1997) findings, our findings suggest that the graphic method was the preferred method. Piez and Voxman reported that when students were presented with two analytical methods and one graphic method making use of graphic calculators, they preferred the analytical methods. Still, it should be noted that, while in their article, Piez and Voxman emphasized the students' (mis)use of graphic calculators, their general tendency to use the graphic method was ignored. Consequently, the students' attitude towards this method was confused with problems they might have experienced in using the calculators.

In our study, the graphic method was clearly the method most frequently used and the one most frequently designated "the easiest." All students used the graphic method at least three times in their solutions, and most of them (all those who studied it first and four others) used the graphic method exclusively. It is therefore possible that the difficulties encountered by Piez and Voxman's students when using the graphic method were rooted in technical problems they experienced with the graphic calculators. However, research findings indicate that the use of graphic calculators may contribute much to students' mathematical performance. We therefore agree with Piez and Voxman's recommendation to promote students' ability to use calculators and to master the graphic method.

When more than one method was used, solutions tended to include the graphic method and the method first studied. Students regarded the first-studied method, whether it was graphic, logical-connectives, or sign-chart, as the best understood. A possible conclusion might be to open a multiple-presentation sequence with the graphic method. By studying the graphic method first students may benefit from both its visual aspect as well as from the impact of its being presented first. These two aspects may enhance students' ability to correctly solve quadratic inequalities.

On the other hand, and quite paradoxically, it seems reasonable to suggest the opposite. That is to say, to suggest a multiple-presentation teaching sequence that ends with the graphic method. In this way, students may have a grasp of two methods, the first studied one as well as the graphic method, by way of an extra, for solving quadratic inequalities. Clearly, the order in which the methods should be taught is a matter for further research.

3. If we decided to present a single method, which one should it be? Posing this question after the previous conclusions may seem rather strange.

However, in day-to-day reality it may happen that while intending to present a multiple-method approach, time limitations may restrict a teacher to present only one method.

When choosing a single method the responsibility of the teacher is much greater. In addition to the criteria of being "most easily understood", "most frequently chosen" or explicitly mentioned as the "best choice", a teacher has to evaluate the likely contribution of each method to the class as a whole and to the global mathematical knowledge of individual students, for example, a method's likely contribution to students' ability to solve related mathematical tasks and to other mathematical topics.

We would say that of the three methods, it seems reasonable to recommend the graphic method, which allows rich connections to functions and graphs and (in addition) is helpful for solving various types of inequalities (see also Dreyfus & Eisenberg, 1985). However, this recommendation comes with a reservation: While the advantages of this method are quite apparent, it cannot readily be taught to students who lack a background in graphing functions. We would therefore suggest that research be conducted in this area.

4. Some points for further thought

These suggestions are made here with some caution, since our study related to a small population and looked at the narrow topic of quadratic inequalities. One may wonder whether the graphic method, which was the preferred method here, would also be the preferred one with regard to other types of inequalities (e.g., rational inequalities)? Is the multiple-method approach preferable also for any other mathematical topic? Does the first-studied method tend to be more appreciated and used in other mathematical topics, too? What sequence of instruction is best for what type of student? How can we identify students who prefer being taught the single method? And could students' preference of a single method be changed by giving them feedback on this study's results? Such questions and the need to examine the present findings with larger populations call for further research.

References

Dobbs, D., & Peterson, J. (1991). The sign-chart method for solving inequalities. Mathematics Teacher, 84, 657-664.

Dreyfus, T., & Eisenberg, T. (1985). A graphical approach to solving inequalities. School Science and Mathematics, 85, 651-662.

Linchevski, L., & Sfard, A. (1991). Rules without reasons as processes without objects-The case of equations and inequalities. In F. Furinghetti (Ed.), Proceedings of the 15th Annual Meeting for the Psychology of Mathematics Education, Assisi: Italy. (Vol. II, 317-324).

McLaurin, S.C. (1985). A unified way to teach the solution of inequalities. Mathematics Teacher 78, 91-95.

Piez, C.M., & Voxman, M.H. (1997). Multiple representations-Using different perspectives to form a clearer picture. The Mathematics Teacher, 90, 164-166.

Tsamir, P., & Almog, N. (1999). "No answer" as a problematic response: The case of inequalities. In O. Zaslavsky (Ed.), Proceedings of the 23rd Annual Meeting for the Psychology of Mathematics Education, Haifa: Israel. (Vol. I, 328).

Tsamir, P., & Almog, N. (2001). Students' strategies and difficulties: The case of algebraic inequalities. International Journal of Mathematics Education in Science and Technology, 32, 513-524.

Tsamir, P., & Bazzini, L. (2001). Can x=3 be the solution of an inequality? A study of Italian an Israeli students. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Annual Meeting for the Psychology of Mathematics Education, Utrecht: Holland. (Vol IV, pp. 303-310).

Tsamir, P., & Bazzini, L. (2002). Algorithmic models: Italian and Israeli students' solutions to algebraic inequalities. In A.D. Cockburn, & E. Nardi (Eds.), Proceedings of the 26th Annual Meeting for the Psychology of Mathematics Education, Norwitch: UK. (Vol. IV, pp. 289-296).

Tsamir, P., Almog, N., & Tirosh, D. (1998). Students' solutions of inequalities. In A. Olivier and K. Newstead (Eds.), Proceedings of the 22nd Annual Meeting for the Psychology of Mathematics Education, Stellenbosch: South Africa. (Vol. IV, pp. 137-144).

Pessia Tsamir

Maya Reshef

Tel-Aviv University

Table 1: The twelve inequalities with reference to the representation of
the quadratic expression in factored form, the sign of the discriminant
and their solutions

                                FACTORED      SIGN
THE TASK                        FORM          OF D*  THE SOLUTION

 1] (x-3) (x+7) < 0             Factored      D > 0  {x | -7 < x < 3}
 2] [x.sup.2] + 2x - 15 < 0     Non-factored  D > 0  {x | -5 < x < 3}
 3] [x.sup.2] - 10x + 21 > 0    Non-factored  D > 0  {x | x < 3 or
                                                       x > 7}
 4] [x.sup.2] - 3x + 10 > 0     Non-factored  D < 0  R
 5] [x.sup.2] - 2x + 5 < 0      Non-factored  D < 0  [empty set]
 6] [x.sup.2] - 4x + 4          Non-factored  D = 0  {x | x=2}
  [less than or equal to] 0
 7] (x+2) (5-x) < 0             Factored      D > 0  {x | x < -2 or
                                                       x > 5}
 8] -[x.sup.2] + x + 20 > 0     Not Factored  D > 0  {x | -5 < x < 4}
 9] -[x.sup.2] - x + 20 < 0     Non-factored  D > 0  {x | x < -5 or
                                                       x > 4}
10] -[x.sup.2] - 6x - 9         Not Factored  D = 0  {x | x = -3}
  [greater than or equal to] 0
11] -[x.sup.2] + 3x - 4         Non-factored  D < 0  [empty set]
  [greater than or equal to] 0
12] -[x.sup.2] + 2x - 10 < 0    Non-factored  D < 0  R

*D stands for the discriminant

Table 2: Frequencies of using 1, 2, or 3 methods with reference to the
first method studied (1)

                         No. of Methods Used
                    1 method  2 methods  3 methods

1st method studied  13        6          1

G*                   6 (G)    --         --
L**                  1 (G)    4 (L+G)    1 (L+SC+G)
                     1 (L)
SC***                3 (G)    2 (SC+G)   --
                     2 (SC)

*G = Graphic method
**L = Logical-Connectives method
***SC = Sign-Chart method
(1) in parentheses are the methods (graphics, logical-connectives,
sign-chart) used by the students

Table 3: Distribution of methods used when using more than one method
with reference to the first method studied.

                                TASK
No.                 1      2      3     4   5            6
Type of solution    "and"  "and"  "or"  R   [empty set]  x=a

1st method studied

L**

                    L      G*     G     --  --           L
                    G      G      G     L   L            L
                    G      G      G     L   L            L
                    G      L      G     L   L            G
                    SC     SC     L     G   G            L

SC***
                    G      G      G     G   SC           G
                    SC     SC     SC    SC  SC           SC

                                TASK
No.                 7     8      9     10   11           12
Type of solution    "or"  "and"  "or"  x=a  [empty set]  R

1st method studied

L**

                    G     G      G     L    --           --
                    G     G      G     L    L            L
                    G     G      G     L    L            L
                    --    G      G     G    L            L
                    SC    L      SC    L    G            G

SC***
                    G     SC     SC    G    G            SC
                    SC    G      G     G    SC           SC

*G = Graphic method
**L = Logical-Connectives method
***SC = Sign-Chart method