Since several decades, researchers in science and mathematics problem-solving consider the distinction between intuitive and analytic forms of reasoning as crucial. Pioneering work on intuition was conducted by Fischbein (1987), resulting in the intuitive rules theory (Stavy & Tirosh, 2000). Recently, intuitively-rooted mathematical knowledge is studied from conceptual-change perspectives (Vosniadou & Verschaffel, 2004), and intuitive/analytic distinctions are made from situated-cognition viewpoints (Vinner, 1997).

Parallelled by this dualism, dual-processing accounts originated in (cognitive) psychological research, including decision making (Kahneman, 2002), social cognition (Chaiken & Trope, 1999), and reasoning (Evans, 2003). Roughly, such accounts contrast heuristic processes (associative, rapid, automatic, effortless) with analytic processes (rule-based, slow, sequential, controlled). Both may compete for the control of behaviour.

The symposium aims at discussing how recent dualist cognitive-psychological theories and methodologies can deepen our understanding of problem-solving processes, and considers possible instructional outcomes of such findings.

Inglis et al. found that mathematicians respond better to a classical logic task than the general well-educated population. Their inspection-time eye-movement data showed that mathematical education correlates with the use of analytical processes that can override preconscious attention biases.

Three other contributions use recent methodologies – so far applied in cognitive psychology and neuroscience – to address the intuitive and analytical reasoning processes related to problem solving.

Gillard et al. characterised proportional reasoning as intuitively-based, by experimentally restricting students’ reaction times while solving word problems.

Stavy et al. used reaction time and fMRI measurements to show how participants reason when overcoming intuitive interference and addressed the role of salience and working memory.

Babai el al. showed that conflict training activates control mechanisms to overcome intuitive problem-solving interference, but elongated reaction times indicated that these reasoning processes were effortful.

The symposium shows that such methodologies could deepen our understanding of students' reasoning processes, enabling to develop and to evaluate instruction.

Two experiments are reported that examine successful mathematicians' responses to the Wason Selection Task, a classic logic task designed to interrogate reasoning behaviour (Wason, 1968).

Experiment 1 found that the range of answers given by mathematics undergraduates and researchers was different to the typical range of answers made by the general well-educated population, both in terms of the proportion selecting the logically correct answer, and in terms of the non-logical responses that they made.

In Experiment 2 these differences were investigated further using an inspection time eye-movement methodology. It was found that mathematicians spend longer inspecting the cards mentioned in the rule that they reject (the rejected matching cards) than the general population.

These findings are analysed in terms of the heuristic-analytic dual process theory (Evans, 2006). It is argued that studying advanced mathematics is correlated with the use of analytical processes on the Wason Selection Task.

Background and Aims

In the standard version of the Wason Selection Task (Wason, 1968) participants are presented with four cards:

D          K          3          7

They are told that each card has a letter on one side and a number on the other, and are given a conjectured rule about the cards:

If a card has a D on one side, then it has a 3 on the other.

They are then instructed to select all the cards, but only the cards, that need to be turned over to discover whether the rule is true or false.

The logically correct answer is to pick the D and 7 cards (P and not-Q for the rule ‘if P then Q’) as, according formal logic, the only case that can falsify a conditional is where both P and not-Q are true. Consequently the P and not-Q cards need to be selected to ensure there is not a not-Q and a P on their respective other sides. Few participants make this selection, less than 10% according to most published research. Instead the most common responses are to pick D and 3 (46%), or D (33%) alone. The primary goals of this research were to (i) explore how successful mathematicians respond to the task, and (ii) to compare the processes by which mathematicians and non-mathematicians reach their answers.

Methodology and Findings

Three groups were asked to solve the Wason Selection Task, via an experimental website: (i) mathematics undergraduates (N=289), (ii) history undergraduates (N=168) and (iii) research active mathematicians (N=68). Groups (i) and (ii) were studying at a highly rated UK university and were recruited by means of an email from their respective departments. Group (iii) were subscribers to a mathematics research newsgroup.

The distribution of the three main selections is shown in Table 1. The mathematics undergraduates selected the normatively correct response (D7) more often than the history undergraduates, c²(1)=12.6, p<0.001, and the overall range of answers also differed between the groups, c²(6)=43.3, p<0.001. Fewer mathematics undergraduates selected the matching D and 3 cards than the history group. The pattern of responses from the mathematics researchers was broadly similar to that of the undergraduates. When collapsed across card selections, it was found that the mathematics students selected the 3 card less frequently than the history students, 29% vs. 52%, c²(1)=24.2, p<0.001, but selected the 7 card more frequently, 35% vs. 24%, c²(1)=5.82, p<0.05.

Table 1: The distributions of the three main selections made by the three groups

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In Experiment 2 an eye-tracking methodology was applied to determine each participant’s inspection times for the various cards. The sample consisted of mathematics and statistics students (N=30) and arts students (N=28). Participants were asked to solve the task whilst wearing an eye-tracker.

The distribution of responses in Experiment 2 was similar to that of Experiment 1 (with the two groups having a different range of responses, Fisher’s Exact Test, p<0.05). The total times a participant spent fixating on an area around each card were calculated. A ANOVA was conducted to compare dwell times on cards which participants selected and those which they did not. Dwell times were longer on selected cards than non-selected cards, F(1,32)=9.66, p<0.01, however there was no significant card-type (selected/non-selected) ´ group interaction.

Of particular theoretical interest are the processes by which participants reject cards. An ANOVA was conducted to compare how participants rejected matching (the cards mentioned in the rule: 3) and mismatching cards (K and 7). (The D card was excluded from this analysis because of the large proportion of participants who selected it.)There was a significant card-type (non-selected K/3/7) ´ group interaction, F(2,116)=3.46, p<0.05, with the mathematical group spending longer rejecting the matching 3 card than the history group, and both groups having similar dwell times for the mismatching K and 7 cards. These data are shown in Figure 1.

Background and Aims

Proportional reasoning is useful in many everyday life situations. It takes a central place in school mathematics, where it is often trained with missing-value problems (where three numbers are given and a fourth is unknown). Research (e.g., Van Dooren, De Bock, Hessels, Janssens & Verschaffel, 2005), however, demonstrated that students often over-use proportional methods. E.g., most 6th graders answer “90” to the non-proportional problem

Ellen and Kim are running around a track. They run equally fast but Ellen started later. When Ellen has run 5 rounds, Kim has run 15 rounds. When Ellen has run 30 rounds, how many has Kim run?

In the present study we apply a dual process framework (e.g., Evans, 2003; Stanovich & West, 2000) to the over-use of proportionality in mathematical word problem solving.

According to dual-process theorists, there are two distinct cognitive reasoning systems:

-                      S1 (heuristic system): Automatic, associative, and undemanding of computational working memory capacity.

-                      S2 (analytic system): Controlled, deliberate and effortful, thus heavily demanding of computational working memory capacity.

S1 responds rapidly, based on similarity to stored prototypes, whereas S2, which operates on ‘decontextualized’ representations, is serial and time-consuming (Sloman, 1996). The fast S1-heuristics often provide correct responses, but sometimes the two systems cue different responses. In these cases, S2 needs to override S1-responses to obtain correct responses. Hence, a failure to provide normatively correct answers may be attributed to S1’s pervasiveness and S2’s failure to intervene. An important processing claim from the dual process framework is that S1 operates faster than S2. Otherwise stated, S1-responses require less processing time than S2-responses.

We suggest that, due to the omnipresence of proportionality, and the extensive training of proportional methods at school, a ‘proportional heuristic’ is created as part of S1, which is triggered by contextual features (e.g. the missing-value format of a word problem). Students’ over-use of proportionality can then be attributed to S1’s pervasiveness and S2’s failure to intervene. Therefore, we expect that manipulations like limiting students’ response time or burdening their working memory capacity will result in an increase of S1-based proportional responses, – as was already successfully shown for some paradigmatic cases in dual process literature, like the Wason selection task (see De Neys, 2006).

As proportional strategies lead to correct answers to proportional word problems, we expect no effect of time restriction on these problems. For non-proportional problems, however, we anticipate an increase in inadequate proportional answers, and, thus, a decrease of correct answers. The current experiment focuses on verifying the first factor, i.e., the effect of limiting the available time to solve a word problem.

Methodology

111 sixth graders solved six proportional problems (i.e., a proportional strategy leads to the correct answer) and six non-proportional problems (i.e., an additive strategy leads to the correct answer) with a missing-value format. They were presented in random order on a computer. Students were offered two response alternatives – a proportional and an additive solution – and they were asked to choose the correct one. The reason for giving response alternatives in stead of asking students to solve the problems themselves is that arguably the choice for a proportional strategy is S1-based, while the actual calculations still require S2-intervention.

For examples of word problems and solutions, and the way both were presented, see Figure 1.

The available time for solving the word problems was manipulated as follows:

-                      L(ong)-condition (n=55): sufficient time to respond thoughtfully (90 seconds).

-                      S(hort)-condition (n=56): sufficient time to respond but severe time pressure (25 seconds).

A bar at the bottom of the screen indicated the remaining time (see Figure 1).

Findings

As a first step in the analysis, we removed all trials in the L-condition for which the response was given before the time limit of the S-condition (because with such short reaction times, these responses may also be purely S1-based). With this criterion, 35.6% of the L-condition trials were kept for further analysis.

As expected, accuracy for proportional problems was significantly higher than for non-proportional ones, F(1, 891)=34.50, p=.00, confirming again students’ strong tendency to over-use proportional methods. Under time pressure, there was a decrease of accuracy, F(1, 891)=4.37, p=.04, but, contrary to our expectation, this was observed for non-proportional and proportional problems (see Figure 2A); so, no interaction effect was observed, F(1, 891)=.74, p=.39.

During the data collection, however, we noticed considerable differences between classes; this became clear after separating the data of the six classes (see Figure 2B). In half of the classes (i.e., 1, 2, and 5), proportional reasoning was indeed clearly S1-based: Under time pressure, accuracy for non-proportional problems decreased, whereas accuracy for proportional problems remained intact. In the other classes (i.e., 3, 4, and 6), we did not find the expected effect, probably because of a floor effect in the L-condition: Accuracy to non-proportional problems was already so low that a further decrease under time pressure became almost impossible. Moreover, in class 6, accuracy in the S-condition was about 50% for both problem types, suggesting pure guessing behaviour under time pressure.

Discussion and Significance

Our study provided partial evidence for the conclusion that time pressure causes a decrease in accuracy on non-proportional word problems (and an increase in the choices for proportional answers), while accuracy on proportional problems remains unaffected. When students already performed at floor level on non-proportional problems without time pressure, this effect was not observed.

Future research needs to yield further evidence that proportional reasoning in word problem solving is S1-based. In this research, floor effects in accuracy to non-proportional word problems need to be avoided, for example, by stimulating students’ reflectivity or by working with older subjects. As previous studies have shown, 11-12-year olds are at the peak of the over-use of proportionality, so, S1-stimulating manipulations may have little effect on them. Older students still have a tendency toward inadequate proportional reasoning, but may be more capable of suppressing it. So, S1-stimulating manipulations may be applied more effectively in these students, and thus show the heuristic character of proportional reasoning more clearly.

Stavy and Tirosh (2000) proposed that specific task features activate a particular type of rule which has the characteristic of being intuitive (e.g., more A [salient feature] -- more B [the property in question]). This framework was employed here to study the nature of the reasoning processes associated with overcoming the interference caused by this intuitive rule in the context of comparison of perimeters of geometrical shapes. Accuracy of responses, reaction times and brain activity (using fMRI) were measured while participants were asked to compare the perimeters of two geometrical shapes, in two conditions: in-line with the intuitive rule (congruent: larger area – larger perimeter) and counter-intuitive (incongruent: larger area – same perimeter).

It was found that in the incongruent condition accuracy dropped and reaction time was longer than in the congruent condition. Moreover, increasing the salience level of the irrelevant feature 'area' or increasing the complexity of shapes resulted in increased interference in the incongruent condition. Evidence for the engagement of bilateral parietal lobes during congruent trials was found by fMRI. Activation of bilateral orbital frontal cortex was evident when subjects inhibited the interference associated with processing the irrelevant salient feature 'area' and correctly completed the comparison of perimeters in the incongruent condition.

These results indicate that the reasoning processes underlying correct responding to this problem involve overcoming the interference by inhibiting the irrelevant feature, area. The results indicate that in addition to control mechanisms, other factors such as: congruity, salience, and working memory load have effect on problem solving and point to the need to take them into consideration when teaching science and mathematics.

This study was carried out in collaboration with Vinod Goel, York University, Toronto, Canada; Hugo Critchley and Ray Dolan, Wellcome Department of Cognitive Neurology, London, UK.

Background and Aims

The intuitive rules theory has recently been suggested to explain students’ incorrect responses (Stavy & Tirosh, 2000). Stavy and Tirosh proposed that salient task features induce students to rely on particular type of intuitive rules. We focus here on the intuitive rule: More A – more B in the context of comparison of perimeters of geometrical shapes.

Previous studies (e.g. Babai, Levyadun, Stavy & Tirosh, 2006) indicated that participants tend to incorrectly argue, in line with this intuitive rule that the perimeter of a given rectangle is larger than that of a polygon derived from it by removal of a small square from one of its corners (Figure 1B, simple).

The aim of the current study was to explore the nature of the reasoning processes associated with overcoming the interference of the intuitive rule and to explore the impact of different factors on solvers' responses (congruity, salience and complexity of shapes). We measured accuracy of responses, reaction times and brain activity while solvers responded to trials in line with the intuitive rule and to trials that are counter-intuitive. The trials differed also in the level of salience and complexity of the presented shapes.

Methodology

Eighteen participants, average age of 27 years, took part in the study. All had university level education.

We used event-related fMRI, while participants compared the perimeters of two geometrical shapes presented sequentially. Participants had to decide whether the perimeter of the second shape was larger/equal/smaller than that of the previous shape. They responded by pressing the appropriate button in a response box. Their brain activity, accuracy of responses and reaction times were recorded.

There were two types of trials:

1.    Congruent - 48 trials - in which the correct response is in line with the intuitive rule (larger area – larger perimeter; see for example Figure 1A).

2.    Incongruent - 80 trials - in which the correct response is not in line with the intuitive rule (larger area – same perimeter; see for example Figure 1B).

The shapes in half of the trials were filled, in the other half unfilled; half of the trials were simple (one shape is a rectangle the other a derived polygon), the other half were complex (both shapes are polygons, one derived from the other). Participants were trained with 30 similar trials before the experimental conditions.

The current study focuses on comparison of areas and comparison of perimeters of geometrical shapes in the framework of the intuitive rule more A – more B. Our recent findings suggested that conflict training could improve students' performance. Here we used such training and compared accuracy of responses and reaction times before and after training.

Two test conditions were examined, congruent: in which correct response is in line with the rule and known to elicit correct responses, and incongruent: in which correct response runs counter to the rule leading to low rate of success (this condition included two types of tasks: incongruent-inverse and incongruent-equal). Two eighth's grade classes took part in the study. One class, the experimental group, received cognitive conflict training while the other, the control group, did not.

The results of the study show that in terms of accuracy of responses in the incongruent condition, the experimental group significantly benefited from the intervention as compared with the control group. In addition, it seems that the cognitive conflict training activated control mechanisms that are effortful as was evident from the reaction times results. We believe that researchers in science and mathematics education would benefit from applying cognitive psychology techniques, as was done in the current study. Using such methodologies could lead to a deeper understanding of students' difficulties and reasoning processes, enabling to develop and to evaluate improved instructional strategies.

Background and Aims

The focus of the current study is the intuitive rule: more A -- more B. Recently the reasoning process involved in intuitive responses of the type more A -- more B was studied by employing the reaction time technique. Focusing on the comparison of areas and comparison of perimeters of geometrical shapes Babai et al., (2006) presented high-school students with two task conditions:

1) Congruent - correct response is in line with the rule (Figure 1A).

2) Incongruent - correct response runs counter to rule (Figure 1B1 - incongruent-inverse; Figure 1B2 - incongruent-equal).

Almost all students provided correct responses to the comparison-of-area in all conditions and to the comparison-of-perimeter, congruent trials. About 99% and 54% correct responses were obtained to the comparison-of-perimeter incongruent-inverse and incongruent-equal, respectively, both with significantly longer reaction times for correct responses, compared with the reaction time for correct responses in the congruent condition (Babai et al., 2006).

These findings suggested that conflict training could improve students' performance. In the current study we used such training and compared accuracy of responses and reaction times before and after training.

Methodology

Forty seven eighth's grade students from two classes of the same school participated in the study. One class serves as experimental group (N=23) and the other as control group (N=24).

We used the comparison-of-area and comparison-of-perimeter test (Babai et al., 2006).  The test consists of 16 congruent, 16 incongruent-inverse and 16 incongruent-equal trials. In each trial two polygons, similar to the ones shown in Figure 1, were presented on a computer screen, until the participant gave a response (right is larger; left is larger; both are equal) by pressing an appropriate key.

Each participant was twice presented with the entire test, once as a pre-test and the other, after a month, as a post-test. The participants were individually tested in two sessions (comparison-of-area and comparison-of-perimeter) on two different days, week apart.

The experimental group received cognitive conflict training (for 45 min) that included three steps: 1) Students individually compared perimeters of three types of tasks: congruent; incongruent-inverse; incongruent-equal; 2) Evoke of cognitive conflict during class discussion; 3) Students individually exercised the comparison-of-perimeters (10 tasks).   

Findings

Pre-test

Pre-test results (Table 1) were similar to those reported by Babai et al, (2006).