RATIO COMPARISON IN TWO DIFFERENT CONTEXTS.
A METHODOLOGY FOR THE STUDY OF INTUITIVE STRATEGIES

 

Silvia Alatorre

National Pedagogical University, Mexico City

alatorre@solar.sar.net  and  alatorre@servidor.unam.mx

 

A methodology, originally constructed for the study of intuitive strategies used in double urn probability tasks, was applied in an experiment involving two ratio comparison tasks: a probability task and a mixture task. This methodology, previously described in Alatorre (1999), consists of two parallel systems: one for classifying the answers given by subjects (strategies), and one for designing the questions to be asked (situations) in order to observe those strategies. The results are analyzed for comparison between both contexts from three viewpoints: strategies, situations and performance. The concentrate task is easier for the adult subjects considered in this study than the probability task. The methodology can be effectively used in both contexts. It can then be asserted that it is useful for the anticipation and study of strategies used in any mathematical problem involving ratio comparison.

 

PURPOSES AND THEORETICAL FRAMEWORK

The main aim of this work was to apply a methodology reported earlier (Alatorre, 1999) for the study of strategies used by adults in probability experiments, in another ratio comparison frame, for which Noelting’s (1980) classical orange juice experiment was chosen, and to compare the results obtained in both.

As a theoretical framework several studies were considered. Worthy of mention are Piaget’s (1951) classical work on the acquisition of the concept of randomness, Fischbein’s (1975) concept of intuition, Kahneman and Tversky’s (1982) studies on adults’ intuitions in probabilistic problem solving, Falk’s (1978) work with probability in young children, Noelting’s (1980) work on strategies used in proportional reasoning, and Maury’s (1986) research with high-school students in a probability task. The contributions of these authors to the framework of the research are more detailed in Alatorre (1999). Also considered for this work is Tourniaire and Pulos’ (1985) comment about the lack of research comparing proportional reasoning in different contexts.

 

METHODOLOGY

In this work the same ratio comparison problems were posed to subjects, in two versions (see figure 1). One of them consisted of the probability problem of two open urns with simple extraction, equivalent to the cards task reported in Alatorre (1999); the other one was adapted to Mexican context from Noelting’s orange juice experiment. The graphical disposition of the figures shown to subjects was similar, and the differences among both versions consisted in the randomness in the first vs. the mixture in the second, and the discreteness of the first vs. the continuousness in the second. In both versions subjects were asked to justify their choices in written form, and in some cases a clinical interview was conducted afterwards.

Figure 1

In the juice concentrate task glasses containing concentrate and water would be poured inside two jars; it was asked which of the jars had a strongest taste (or if it was the same). In the probability task white and black marbles would be thrown inside two bottles; one bottle would be closed and agitated and one marble would come out of it; it was asked which bottle would the subject choose (or if it was the same) if the desired result was a black marble.

 

The methodology constructed for the study of strategies used in probability tasks reported in Alatorre (1999) was applied. The following paragraphs summarize it in the terms of the application to both tasks considered in the research here reported. It involves two parallel lines: one designed to interpret the answers given by subjects and the other designed to pose the adequate questions in order to observe those answers. For the methodological construction, each problem is defined as an array of two ordered pairs of favorable (f) and unfavorable (u) cases. Also defined are the total cases or glasses n = f+u, the differences d = f–u, and the ratios p = f /n.

The first methodological line contemplates the construction of categories for the interpretation of the answers given by the subject. A strategy was identified in each answer, which could be of a simple or composed form. Simple strategies can be centrations or relations: a centration is the observation of only the favorable cases, the unfavorable ones or the total ones. In a relation two of those three elements are observed and compared by means of an order relationship, a difference

Table 1: STRATEGIES

 

Coding

Name and description

CENTRATIONS

{CN–}

negative centration in total cases: choosing the side where there is a smaller amount of marbles (glasses)

{CN=}

equality centration in total cases: saying “it is the same” because in both sides there is the same amount of marbles (glasses)

{CF+}

positive centration in favorable cases: choosing the side where there is a larger amount of black marbles (concentrate glasses)

{CF=}

equality centration in favorable cases: saying “it is the same” because in both sides there is the same amount of black marbles (concentrate glasses)

{CU–}

negative centration in unfavorable cases: choosing the side where there is a smaller amount of white marbles (water glasses)

{CU=}

equality centration in unfavorable cases: saying “it is the same” because in both sides there is the same amount of white marbles (water glasses)

RELATIONS

{ROlw}

lose-win order relation: choosing the side where there are more chances of winning than of losing (more concentrate glasses than water ones), whereas in the other one there are more chances of losing than of winning (more water glasses than concentrate ones)

{ROld}

lose-draw order relation: choosing the side where there are as many chances of winning than of losing (as many concentrate glasses as water ones), whereas in the other one there are more chances of losing than of winning (more water glasses than concentrate ones)

{ROdw}

draw-win order relation: choosing the side where there are more chances of winning than of losing (more concentrate glasses than water ones), whereas in the other one there are as many chances of winning than of losing (as many concentrate glasses as water ones)

{RO=}

lose-lose or win-win order relation: saying “it is the same” because in both sides there are more chances of losing than of winning (more water glasses than concentrate ones), or because in both sides there are more chances of winning than of losing (more concentrate glasses than water ones)

{RD+}

largest difference relation: choosing the side where the difference of black minus white marbles (concentrate minus water glasses) is the largest

{RD=}

equal difference relation: saying “it is the same” because the difference between black and white marbles (concentrate and water glasses) is the same in both sides

{RP+}

largest quotient proportionality relations: choosing the side where the quotient f/n or the quotient f/u is the largest

{RP=}

equal quotient proportionality relations: saying “it is the same” because the quotient f/n or the quotient f/u is the same in both sides

 

or subtraction, or a proportion. The main centrations and relations are displayed in table 1. Composed strategies include two or more simple strategies joined by a logical operation (of which there are four types); the intervening simple strategies may be dominant or dominated.

Strategies are also classified according to their correctness. {RP} strategies are always correct, as are most {RO} strategies, with the exception of {RO=}, which is incorrect. Some centrations may be correct in certain situations, e.g. {CU–} in a possibility-certainty situation (no white marbles or water glasses in one of the two sides). Some composed strategies may also be correct, e.g. {CF+ & CU–} (choosing the side where there is a larger amount of black marbles or concentrate glasses and where there is the smaller amount of white marbles or water glasses).

The second methodological line is the construction of the situations: the specific amount of black and white marbles or concentrate and water glasses in each collection for each problem posed. They were built in order to allow the detection of the simple strategies described above. The different possible arrays were grouped in categories determined by several variables. The combination is a succession of the possible results of the order relationship between these five elements of sides A and B of the array: n, f, u, d and p. The location unites the possible results of both ratios, each in five forms: “surely lose” (p=0), “lose” (0<p<0.5), “draw” (p=0.5), “win” (0.5<p<1) and “surely win” (p=1). Some locations do not exist in some combinations; Piaget’s original 10 categories are broken down in 85 different situations, which allows a finer analysis.

 

RESULTS AND DISCUSSION

The research was done in the National Pedagogical University in Mexico City. 65 first year university students aged 17-28, majoring in Educational Psychology and without any prior instruction in probability participated in the study. Each received two paper and pencil tests with 16 items each, first the concentrate task and then the probability task; in both the arrays were the same and in the same order and graphical disposition (see figure 1). The situations of the 16 arrays were chosen so as to favor the happening of different strategies, as shown in table 2.

Table 2. Expected Simple Strategies in the 16 Items

 

 

S t r a t e g i e s    l e a d i n g    t o    c h o o s i n g

Item

(f1,u1)(f2,u2)

Side A

Side B

It is the same

1

(3,3)(4,6) *

{CN–}, {CU–}, {ROld}, {RD+}, {RP+}

{CF+}

 

2

(6,2)(5,5)

{CN–}, {CF+}, {CU–}, {ROdw}, {RD+}, {RP+}

 

 

3

(4,3)(5,4)

{CN–}, {CU–}, {RP+}

{CF+}

{RO=}, {RD=}

4

(2,5)(2,7)

{CN–}, {CU–}, {RD+}, {RP+}

 

{CF=}, {RO=}

5

(2,2)(3,3)

{CN–}, {CU–}

{CF+}

{RD=}, {RP=}

6

(2,5)(3,7)

{CN–}, {CU–}, {RD+}

{CF+}, {RP+}

{RO=}

7

(2,4)(4,5)

{CN–}, {CU–}

{CF+}, {RD+}, {RP+}

{RO=}

8

(6,4)(7,3)

 

{CF+}, {CU–}, {RD+}, {RP+}

{CN=}, {RO=}

9

(2,0)(4,2)

{CN–}, {CU–}, {RP+}

{CF+}

{RD=}, {RO=}

10

(6,1)(7,3)

{CN–}, {CU–}, {RD+}, {RP+}

{CF+}

{RO=}

11

(2,3)(4,6)

{CN–}, {CU–}, {RD+}

{CF+}

{RO=}, {RP=}

12

(2,4)(4,4)

{CN–}

{CF+}, {ROld}, {RD+}, {RP+}

{CU=}

13

(1,2)(2,3)

{CN–}, {CU–}

{CF+}, {RP+}

{RO=}, {RD=}

14

(2,1)(6,4)

{CN–}, {CU–}, {RP+}

{CF+}, {RD+}

{RO=}

15

(4,1)(8,2)

{CN–}, {CU–}

{CF+}, {RD+}

{RO=}, {RP=}

16

(2,3)(5,4)

{CN–}, {CU–}

{CF+},{ROlw}, {RD+}, {RP+}

 

* Note: Item Nº 1 is the one displayed in figure 1

 

The comparison of the results obtained for the probability and the juice concentrate tasks was carried by means of three analysis procedures, centering in strategies, situations and performance.

Strategies. The expected strategies (table 2) did happen in both contexts, but their occurrence (percentage of times a strategy occurs among the situations where it can possibly happen) was different in both tasks. The results are displayed in table 3.

Table 3. Occurrence of strategies in both tasks

Strategy family

Concentrate task

Probability task

{CN} centrations

3%

12%

{CF} and {CU} centrations

22%

18%

{RO=} (incorrect)

17%

27%

Other order relations (correct)

76%

60%

{RD} relations

15%

12%

{RP} relations

20%

12%

Situations. The percentage of times a given situation was correctly solved in each task (whatever the correct strategy used) was analyzed, and the results compared for both contexts. The main result is that in 10 of the 16 situations considered in the tests, the juice task problems are significantly better solved than the probability ones (items 3,4,7,8,10,11,12,13,14 and 15); only in one situation the reverse is true, and this is the possibility-impossibility situation (item 9).

Subjects’ performance. The consistency showed by subjects between their answers to both tasks was analyzed. The answers given by the same subject to the same array in both contexts were compared: in only 46% of the cases one can speak of a consistency, but these results are similar to the ones obtained in a previous work (Alatorre, 1999) where only a version of the probability task was considered. However, there is significantly less consistency in locations of the “lose-lose” or the “win-win” types.

 

CONCLUDING REMARKS

1) Although the methodology was originally constructed for the study of strategies used in ratio comparison in a discrete probability task, it proved to be useful also in the context of a continuous mixture problem: to select the items’ situations, to predict and examine the strategies used by subjects, and to analyze the results. It could be ventured that the same methodology can be applied to any ratio comparison problem, with the possible exception of primitive strategies which could be used by younger subjects and which were not considered in this exposition.

2) The two ratio comparison problems considered in this work are not equivalent: the juice concentrate is an easier context than the probability task. This was apparent in the three analysis procedures. On the first hand, all correct strategies of the {RP} family and the correct order relations occur more in the context of the juice concentrate than in the probability task, whereas all incorrect strategies, including {RO=}, occur more in the probability task. On the second hand, with one exception the situations in which the probability task proved to be more difficult than the concentrate task are precisely those in which the incorrect relation {RO=} can be applied: locations of the “lose-lose” and the “win-win” types. Finally, it is in those same situations where the subjects show more inconsistency.

3) Although the mixture task was presented as discrete amounts of glasses, there seemed to be no transfer among contexts (few students realized that the questions were equivalent). This could have implications in the design of learning experiences.

4) Some remarks about the {RO} and the {RP} families may be of interest. The incorrect strategy {RO=} is more appealing to subjects in the probability task than in the concentrate task, but this works the other way around for the other order relations, {ROlw}, {ROld} and {ROdw}, which are correct: they are more attractive in the juice concentrate task than in the probability task. As for {RP}, one possible explanation for the larger occurrence of this family in the juice concentrate context could be that liquid is a continuous medium as opposed to the discrete number of marbles in the probability task: it is thus easier to think in terms such as “one and a half glass of water for each glass of concentrate”. Further research should then compare the results obtained in discrete mixture problems with the two open urns problem, and continuous mixture problems, such as the juice concentrate, with a continuous probability problem.

 

REFERENCES

Alatorre, Silvia (1999). Adult’s intuitive answers to probability problems: a methodology. Proceedings of the 21st Annual Meeting of PME-NA. Cuernavaca: Centro de Investigación y de Estudios Avanzados del IPN and Universidad Autónoma del Estado de Morelos.

Falk, R. (1978). Analysis of the concept of probability in young children. Second International Conference for the Psychology of Mathematics Educa­tion, 1978.

Fischbein, E. (1975). The Intuitive Sources of Probabilistic Thinking in Children. Dordrecht: D. Reidel Co.

Kahneman, D., Slovic, P. & Tver­sky, A. (eds) (1982). Judgment Under Uncertainty: Heuristics and Biases. Cambridge Univer­sity Press.

Maury, S. (1986). Contribution à l'étude didactique de quelques notions de probabilité et de combinatoire à travers la résolution de problèmes (Thèse de doctorat d'Etat). IREM de Montpellier et Université des sciences et techniques du Languedoc.

Noelting, G. (1980). The development of proportional reason­ing and the ratio concept. Part I, Differentia­tion of stages. Part II, Problem-structure at successive stages; problem-solving strategies and the mecha­nism of adaptive restructuring. Educational Studies in Mathematics, 11, 217-253 & 331-363.

Piaget, J. & Inhelder, B. (1951). La genèse de l'idée de hasard chez l'enfant. Paris: Presses Universitaires de France.

Tourniaire, F. & Pulos, S. (1985). Proportional reasoning: a review of the literature. Educational Studies in Mathematics, 16, 181-204.