Probabilistic Intuitions in Traditional vs. Alternative ContextS

 

Dale Havill, Ph.D. Educational Psychology

Zayed University, United Arab Emirates

DaleHavill@bigfoot.com

 

 

Research and assessment of probabilistic reasoning has often presented text-only multiple choice problems which ask students to reason about common random generating devices, (e.g., coins, dice). This study presented problems in three contexts: traditional (“COIN/DIE”), pinball diagrams ("GALTON") and database frequencies ("PLANET”). Research questions focused on a common source of students' errors and biases: the distribution of probabilities associated with ordered sequences of independent events. Results indicated that general purpose cognitive heuristics described in previous research change or are not as salient in some alternative problem contexts. Students' problem solving frameworks, heuristics, and atmospheric beliefs were interpreted by extending Fischbein's theoretical framework of primary (everyday) and secondary (instruction-related) intuitions.

 

      In recent years educational organizations have given increased attention to probability curricula and have recommended introduction of probability concepts and activities at earlier grade levels (NCEE, 1995; NCTM, 2000). However, students' difficulty in reasoning normatively1 has posed considerable challenges for curriculum design and implementation (Algren & Garfield, 1991; Jones et al., 1999; Shaughnessy, 1992). These pedagogical challenges are due in part to the fact that, unlike many mathematical topics, naive intuitions (i.e., immediate apprehensions or cognitions2) about probability are entrenched in everyday discourse and activity. Cognitive as well as educational researchers have noted that there are large gaps in our understanding of how these intuitions interact with instruction-based knowledge, and how best to deal with intuitions in order to facilitate students' construction of normative concepts and problem solving skills.

Research Questions

      Traditional cognitive research investigating probabilistic reasoning has often focused on intuitive errors and biases which indicate that human fallibility is associated with natural or everyday thinking. However, many traditional research instruments utilized only limited response sets (e.g., multiple-choice responses). Furthermore, problem situations presented in traditional studies typically involved only standard random generating devices (e.g., coins, dice, spinners, etc.). For example, consider traditional versions of the three problems presented in this study (correct choices for the three problems are 1-c, 2-e, and 3-b):

 

1. Referred to below as the "Four Previous Heads" problem:

 

If a coin is flipped four times and comes up heads every time, which is more likely on the next flip?

a. head

b. tail

c. head and tail equally likely

 

2. Referred to below as the "Equiprobable Ordered Sequences" problem:

 

Which sequence of coin flips is most likely­?

a. HHHTT

b. THTTH

c. HHHTH

d. HTHTH

e. All sequences are equally likely.

 

3. Referred to below as the "Skewed Probability Samples" problem:

 

A die is painted black on five sides and white on one side. What is the most likely outcome of six rolls?

a. 6 blacks and no whites

b. 5 blacks and one white

 

FIGURE 1. Basic multiple choice versions of the problems presented in this study.

 

      A primary motivation for the study was to explore how students' responses differ in alternative contexts. In particular, do naïve intuitive responses found in heuristics and biases research vary across the three contexts? Fischbein (1975) suggested that investigating such differences can help us identify secondary intuitions (i.e., acquired in traditional contexts through school curricula) as well as primary intuitions (i.e., general purpose heuristics and biases acquired through everyday experience). In other words, school-based instruction—as well as general purpose everyday cognitive processes—can be a significant source of misappropriated intuitions. Of particular interest are the Equiprobable Ordered Sequences and Skewed Probability Samples problems in which students appear to misappropriate formal probabilistic knowledge. Intuitions about ordered sequences, (in which multiplicative-analytical skills facilitate understanding that probability is related to conditional dependence of events in a sequence), can erroneously substitute for, or be substituted by, intuitions about unordered samples (in which the probability is directly related to the 'partitive' proportion of independent outcomes in the sample).

Theoretical and Historical Framework

      The traditional normative theoretical framework provided a basic measure for evaluating students’ judgments in this study—a response was considered correct if it was consistent with generally accepted mathematical models of probability. However, even when responses are clearly correct or incorrect in a normative framework, theorists are sometimes divided into opposing camps depending on whether human thinking is characterized as more, or less, rational. On one hand, early cognitive investigations provided evidence which indicated that everyday heuristics and biases are irrational obstacles to normative probabilistic reasoning (Kahneman et al., 1982). On the other hand, some researchers contended that a critical analysis of experimental paradigms is crucial to interpreting research that focused on erroneous heuristics and biases:

 

The paradigms [are] so limited and inadequate that generalisations from current research on heuristics and biases cannot be justified. In particular, the view of people as 'intellectual cripples', who exhibit severe and systematic biases in making judgements, [is] a value judgement on the part of the investigators.

(Phillips, 1983, p. 525)

 

      In general, theories of probabilistic reasoning have differentiated 1) frequentist-distributional vs. local-atmospheric thinking, and 2) intuitive vs. analytical thinking. Interpretation of results in this study was influenced by Fischbein's (1975) theoretical framework which viewed 'intuitional' cognition to be as important for human functioning as formal-operational cognition (cf. the Piagetian focus on formal-operational cognition as the end state in the development of higher order thinking).

Research Method

Virtually all 168 participating undergraduate students were drawn from psychology department subject pools in two U.S. universities. Students were given a question sheet packet and worked alone; most finished the problems in about ten to fifteen minutes. Each question sheet packet contained three problems presented in only one context.

Figure 1 shows the basic structure of problems presented in the traditional "COIN/DIE" context and Figure 2 gives a basic idea of how diagrams looked in the alternative contexts problem (note that the diagrams in Figure 2 are small iconic versions of the diagrams presented to students). Problems in the "GALTON" context showed diagrams of a simple pinball machine in which the path of a pinball was traced through a triangular set of pins (a right/left branch of the pinball is analogous to head/tail in a coin flip). Problems in the "PLANET" context described space exploration probes whose data indicated that there was a 50% chance that planets in a solar system were inhabitable (the 50% chance analogous to head/tail in the traditional context).

In addition to reasoning about the diagrams students were asked to estimate frequencies and percentages for every multiple-choice alternative, and to write a short justification of the alternative they selected. These multiple representations and response modes were designed to facilitate correct interpretation and evoke a more informative response set.

Results

Two general outcomes related to problem format and context will be briefly described, followed by results showing how responses in the alternative contexts differed from the heuristics and biases model of reasoning in traditional contexts. A first general observation in considering the overall response sets is that problem formats with multiple representations and modes of response seemed to have little positive effect. For example, many students gave frequency and percentage estimations that were incoherent (e.g., estimated percentages weren't consistent with estimated frequencies, or the sum of percentage estimations for each alternative was over 100%, even when alternatives were a subset of all possible outcomes).

 

Table 1

Percent Correct On Multiple Choice Responses In Each Context and Problem Type                

 

 

 Context

 

 N

Four Previous Heads

Equiprobable

Ordered Sequences

Skewed Probability Samples

COIN/DIE

56

          93 %

27

38

GALTON

57

63

32

41

PLANET

55

45

  2

62

                                                                                                                                                     

 

A second general observation involved differences across contexts in students' willingness to accept assumptions about the independence of a future event. Performance on the traditional version of the Four Previous Heads problem (93% correct) indicated that most students are familiar with and willing to accept the idea that a future coin flip is independent of previous events. In contrast, students weren't as likely to accept assumptions about independence of events in nontraditional contexts (63% correct in the GALTON context, and 45% correct in PLANET context). Students tended to hypothesize physical causes in the GALTON context and nonprobabilistic dependencies in the PLANET context:

 

GALTON student:   “The ball has momentum and will continue going in that direction."

PLANET student:    “Since one planet is inhabitable the others are more likely to be inhabitable.”

The situated nature of students' reasoning in the alternative contexts was interesting because the study was designed in part to explore how computer-based activities in the alternative contexts might help students reason normatively.

      Significant differences from traditional results may be related to how events are represented as well as students' tendency to hypothesize nonrandom causes or associations in the alternative contexts. Figure 2 shows which outcomes students thought were most likely in the Equiprobable Ordered Sequences problem. First note that results in the traditional COIN/DIE context were consistent with the heuristic and biases model. Outcome #2 (“THHTH”) tends to be viewed as most likely because it is more 'representative' of a random process: 68% of students in the COIN/DIE context ranked "THHTH" among the two most probable outcomes. The other alternatives tend to be viewed as less representative of a random process because “HHHTT” has runs, “THTTT” has a lopsided proportion of heads, and “HTHTH” alternates too regularly (see rankings in Figure 2).

      Students in the GALTON and PLANET contexts more often chose Outcome #1 ("HHHTT" in the traditional context) as most likely. Moreover, the second highest ranking in both alternative contexts was Outcome #3 ("THTTT" in the traditional context) which has a run as well as the least balanced proportion. Thus in contrast to results found in traditional studies, runs of the same outcome in real world scenarios (i.e., a pinball repeatedly branching in one direction, or similar sized planets being inhabitable) may not appear as unnatural.

 


 


FIGURE 2. Differences between contexts in students’ ranking of most likely outcomes in the Equiprobable Ordered Sequences problem.

 

Compared to previous studies a surprisingly high number (57%) of students in the COIN/DIE context ranked Outcome #4 ("HTHTH") among the two most likely outcomes. Thus, in contrast to responses in the alternative contexts, COIN/DIE students seemed particularly prone to view runs of a particular outcome (e.g., "HHH") as indications of a nonrandom process and alternations between heads and tails as indications of a random process. In sum, differences between contexts indicate that strings of letters representing sequences of coin flips appear to have symbolic characteristics which evoke specific situated intuitions—different from intuitions about the most likely paths of a pinball or the most likely patterns of inhabitable planets.

Reasoning about symbolic representations of random events in the COIN/DIE version of Equiprobable Ordered Sequence problem may have been associated with secondary school-based intuitions. Other evidence for secondary intuitions was indicated in students' performance on the COIN/DIE version of the Skewed Probability Samples problem: Students with stronger science/math backgrounds scored lower than students with weaker backgrounds (42% vs. 45% correct respectively). Recall that students were asked to judge the most likely outcome in six rolls of a die that was painted black on five sides and white on one side. COIN/DIE students' justifications for judging six blacks most likely often referred to the high odds for black on each independent roll. This erroneous focus on independence appeared to be associated with secondary intuitions acquired in (school-based) contexts.

In summary, comparison of reasoning across contexts raised some questions about traditional research that suggests specific universal/innate heuristics and biases in probabilistic reasoning. Some of these cognitive processes may in fact be tied to intuitions acquired in formal contexts. Identifying secondary as well as primary intuitions would help educators design early learning environments that instill a more rigorous set of normative conceptual schemata.

 

 

1 Normative, in this context, means "in accordance with mathematical principles of probability."

2 Webster's Collegiate Dictionary (10th Edition)

 

 

REFERENCES

 

        Ahlgren, A., & Garfield, J. (1991). Analysis of the probability curriculum. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters: Probability in education, (pp. 107-134). Norwell, MA.: Kluwer.

 

        Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. Dordrect-Holland: Boston.

 

        Kahneman, D., Slovic, P., & Tversky, A. (Eds.), (1982). Judgment under uncertainty: Heuristics and biases. Cambridge: Cambridge University Press.

 

        Jones, G.A., Langrall, C. Thornton, C., & Mogill, T. (1999). Students’ probabilistic thinking in instruction. Journal for Research in Mathematics Education, 30(5), 487-519.

 

        NCEE. (1995). Performance Standards: Vol. 2, Middle School : National Center on Education and the Economy, Pittsburgh, PA: New Standards.

 

        NCTM. (2000). National Council of Teachers of Mathematics: Principles and standards for school mathematics. Reston, VA: NCTM.

 

        Phillips, L. D. (1983). A theoretical perspective on heuristics and biases in probabilistic thinking. In P. Humphreys (Ed.), Analysing and aiding decisions processes, (pp. 525-543). Amsterdam: Elsevier (North Holland).

 

        Shaughnessy, J. M. (1992). Research in probability and statistics: reflections and directions. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning, (pp. 455-494). New York: Macmillan.