Dirk De Bock

Not Everything Is Proportional: Effects of Age and Problem Type on Propensities for Overgeneralization

Previous research (e.g., De Bock, 2002) has shown that-due to the extensive attention paid to proportional reasoning in elementary and secondary mathematics education-many students tend to overrely on proportional methods in diverse domains of mathematics (e.g., geometry, probability). We investigated the development of misapplication of proportional reasoning with the age and the educational experience of students. A paper-and-pencil test consisting of several types of proportional and nonproportional arithmetic problems with a missing-value structure was given to 1,062 students from Grades 2 to 8. As expected, students tended to apply proportional methods in cases in which they were clearly not applicable. Although some errors of overapplication were made in the 2nd grade, their number increased considerably up to Grade 5 in parallel with the growing proportional reasoning capacity of the students. From Grade 6 on, students started to distinguish more often between situations when proportionality was applicable and when it was not, but even in 8th grade, a considerable number of proportional errors were made. The likelihood of error varied with the type of nonproportional mathematical model underlying the word problems.

The Illusion of Linearity: Expanding the Evidence towards Probabilistic Reasoning.

Previous research has shown that – due to the extensive attention spent to proportional reasoning in mathematics education – many students have a strong tendency to apply linear or proportional models anywhere, even in situations where they are not applicable. This phenomenon is sometimes referred to as the ‘illusion of linearity’. For example, in geometry it is known that many students believe that if the sides of a figure are doubled, the area is doubled too. In this article, the empirical evidence for this phenomenon is expanded to the domain of probabilistic reasoning. First, we elaborate on the notion of chance and provide some reasons for expecting the over generalization of linear models in the domain of probability too. Afterwards, a number of well-known and less-known probabilistic misconceptions are described and analysed, showing that they have one remarkable characteristic in common: they can be interpreted in terms oft he improper application of linear relations. Finally, we report on an empirical investigation aimed at identifying the ability of 10th and12th grade students to compare the probabilities of two binomial chance situations. It appears that before instruction in probability, students have a good capability of comparing two events qualitatively, but at the same time they incorrectly quantify this qualitative insight as if the variables in the problem were linked by a linear relationship. Remarkably, these errors persist after instruction in probability. The potential of this study for improving the teaching and learning of probability, as well as suggestions for further research, are discussed.

From Addition to Multiplication ... and Back: The Development of Students' Additive and Multiplicative Reasoning Skills.

This study builds on two lines of research that have so far developed largely separately: the use of additive methods to solve proportional word problems and the use of proportional methods to solve additive word problems. We investigated the development with age of both kinds of erroneous solution methods. We gave a test containing missing-value problems to 325 third, fourth, fifth, and sixth graders. Half of the problems had an additive structure and half had a proportional structure. Moreover, in half of the problems the internal and external ratios between the given numbers were integer, while in the other cases numbers were chosen so that these ratios were noninteger. The results indicate a development from applying additive methods “anywhere” in the early years of primary school to applying proportional methods “anywhere” in the later years. Between these two stages many students went through an intermediate stage where they simultaneously applied additive methods to proportional problems and proportional methods to additive problems, switching between them based on the numbers given in the problem.

Just Answering … or Thinking? Contrasting Pupils' Solutions and Classifications of Missing-Value Word Problems

Upper primary school children often routinely apply proportional methods to missing-value problems, even when it is inappropriate. We tested whether this tendency could be weakened if children were not required to produce computational answers to such problems. A total of 75 sixth graders were asked to classify 9 word problems of three types (3 for which proportionality is an appropriate model, 3 implying an invariant additive relationship, and 3 for which the result is constant) and to solve a parallel version of this set of problems. Half of the children first found the solution and then did the classification task (SC-condition), while for the others the order was the opposite (CS-condition). On the word problem test, children often overused proportional methods, but those in the CS-condition performed better than those in the SC-condition, suggesting a positive impact of the classification task. On the classification task, most pupils took into account the underlying mathematical models, but they did not always distinguish proportional from non-proportional problems. Students in the SC-condition performed worse than those in the CS-condition, suggesting that solving the word problems first negatively affects later classifications.

Studying and Remedying Students’ Modelling Competencies: Routine Behaviour or Adaptive Expertise

First, we summarise some studies on students’ overuse of the linear model when solving problems in various domains of mathematics, showing to what extent they are led by routine behaviour in mathematical modelling. Second, we discuss a teaching experiment that aimed at enabling 8th graders to adaptively choose between a linear, a quadratic or a cubic model while solving geometry problems. The results show that, after the experiment, the students applied the linear model less automatically, but tended to switch back and forth between applying it either “everywhere” or “nowhere”, indiscriminately.

How Students Connect Descriptions of Real-World Situations to Mathematical Models in Different Representational Modes

The translation of a problem situation into a mathematical model constitutes a key – but not at all obvious – step in the modelling process. We focus on two elements that can hinder that translation process by relating it to the phenomenon of students’ overreliance on the linear model and their (lack of) representational fluency. We investigated: (1) How accurate are students in connecting descriptions of realistic situations to “almost” linear models, and (2) Does accuracy and model confusion depend on the representational mode in which a model is given? Results highlight that students confuse linear and non-linear models, and that the representational mode has a strong impact on this confusion: Correct reasoning about a situation with one mathematical model can be facilitated in a particular representation, while the same representation is misleading for situations with another model.

Early Experiments with Modern Mathematics in Belgium. Advanced Mathematics Taught from Childhood

During the early phases of the New Math reform, several series of classroom experiments were set up in Belgium by leading figures of the reform movement. Although there are no primary data documenting the experiments, some information did appear in scholarly publications. Moreover, the experiments were often referred to as powerful arguments in favour of the reform. We discuss the experiments set up by the structuralist school of Georges Papy, and their reception among Belgian and international reviewers. These experiments are compared to the very different approach advocated by Paul Libois. We conclude with some observations on the role of these experiments in the New Math reform in Belgium.

Students’ overreliance on linearity in economic applications: A state of the art

Students’ overreliance on linear models is well-known and has been investigated empirically in a variety of mathematical subdomains, at distinct educational levels and in different countries. We present a state of the art of students’ overreliance on linearity in economic applications. We illustrate the widespread but sometimes debatable use of linearity in economics, discussing the treatment of demand and supply functions and of the Phillips curve in major economic textbooks. Next, we provide an overview of instances of, and comments on, this phenomenon in the economic education research literature. Typically, the phenomenon is described in the margin of economic studies whose primary focus is elsewhere. Finally, a study having students’ overreliance on linearity as its main research focus is discussed in some detail.

How Students Connect Mathematical Models to Descriptions of Real-World Situations

Research has shown that problem posing, in a sense the “inverse activity” of problem-solving, can positively affect students’ problem-solving skills. We report the design and results of an empirical study in which the potential positive effect of a specific problem-posing variant, “inverse modelling”, (i.e. the selection of a real-world situation given a mathematical model), on modelling was investigated. Eighty 11th grade students were randomly divided into two equal-sized subgroups, one first receiving a modelling task and then an inverse-modelling task. The other subgroup received both tasks in reverse order. Results indicated that inverse modelling did not have an overall positive effect on modelling: Only for affine functions with negative slope, accuracy scores for modelling significantly improved after inverse modelling.

Word Problem Classification: A Promising Modelling Task at the Elementary Level

Upper primary school children often routinely apply proportional methods to missing-value problems, also when this is inappropriate. We tested whether this tendency can be broken if children would pay more attention to the initial phases of the modelling process. Seventy-five 6th graders were asked to classify nine word problems with different underlying mathematical models and to solve a parallel version of these problems. Half of the children first did the solution and then the classification task, for the others the order was opposite. The results suggest a small positive impact of a preceding classification task on students’ later solutions, while solving the word problems first proved to negatively affect later classifications.