Dina Tirosh

Intuitive rules in science and mathematics: the case of ‘more of A ‐‐ more of B’

In the last twenty years researchers have studied students’ mathematical and scientific conceptions and reasoning. Most of this research is content‐specific. It has been found that students often hold ideas that are not in line with accepted scientific notions. In our joint work in mathematics and science education it became apparent that many of these alternative conceptions hail from the same intuitive rules. We have so far identified two such rules: ‘The more of A, the more of B’ and, ‘Everything can be divided by two’. The first rule is reflected in students’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.), in all tasks related to intensive quantities (density, temperature, concentration, etc.), and in tasks related to infinite quantities. The second rule is observed in responses related to successive division of material and geometrical objects, and in successive dilution tasks. In this paper we describe and di...

Challenging and Changing Mathematics Teaching Classroom Practices

This chapter summarizes general factors influencing change, and discusses matters which need to be considered by those working to achieve change in mathematics teaching practices. In the first section, we discuss two major sources of impetus for teacher change: (1) values and beliefs, and (2) technological advances. In the next two sections we explore the question: What important considerations in implementing planned changes in mathematics teaching are suggested by the existing body of knowledge? We focus on two seemingly essential dimensions of anyplanned effort to change teacher classroom practices: organizational approach and the nature of the professional development. Examples presented in the text are drawn from a variety of countries. We conclude with comments on the role of the specific goals of the planned change in framing the professional development and in assessing the change effort, and on the impact of culturally based views on the goals, content and availability of professional development. We acknowledge, throughout this chapter, that when considering research on either challenging or changing mathematics teaching classroom practices, readers should remember that care needs to be exercised in generalizing conclusions reached in one culture to another.

Is It Possible to Measure the Intuitive Acceptance of a Mathematical Statement

The main purpose of the present research was to check the possibility of measuring the feeling of “intuitive acceptance”, experienced by a subject when he offers an intuitive solution to a problem. It was postulated that two dimensions have to be considered and combined: The level of confidence and the degree of obviousness. Almost all the questions asked referred to the notion of infinity. The subjects were pupils belonging to grades 8 and 9. Three main categories of problematic situations have been identified:(a)Problems which got high percentages of correct solutions and high levels of intuitive acceptance.(b)Problems which got two types of contradictory solutions, each of them being accepted with moderate intuitiveness.(c)Problems which got low frequencies of correct solutions and high frequencies of typical incorrect solutions, the second category presenting higher levels of intuitive acceptance than the first (counter-intuitive problematic situations).

The role of representations in students’ intuitive thinking about infinity

Initial investigations suggest that students’ intuitive decisions concerning the equivalency of two given infinite sets are largely determined by the way these sets are represented. So far the effects of two types of representations were investigated: the numerical‐horizontal and the numerical‐vertical representations. Our study was mainly aimed at determining which representations (numerical‐horizontal, numerical‐vertical, numerical‐explicit or geometric) yielded higher percentages of one‐one correspondence reactions. For these purposes, 189 middle class 10th to 12th graders were asked to react to 14 problems dealing with comparing infinite sets. The problems presented different representations of the same infinite sets. It was found that one‐one correspondence justifications were mainly elicited by numerical‐explicit and by geometric representations. The discussion suggests ways of adjusting these findings to two different approaches to teaching: analogy and conflict.

Cognitive conflict and intuitive rules

This paper is a part of an extensive project on the role of intuitive rules in science and mathematics education. First, we described the effects of two intuitive rules ‐‐ ‘Everything comes to an end’ and ‘Everything can be divided’ ‐‐ on seventh to twelfth grade students’ responses to successive division tasks related to mathematical and physical objects. Then, we studied the effect of an intervention, which provided students with two contradictory statements, one in line with students’ intuitive response, the other contradicting it, on their responses to various successive division tasks. It was found that this conflict‐based intervention did not improve students’ ability to differentiate between successive division processes related to mathematical objects and those related to material ones. These results reconfirmed that intuitive rules are stable and resistant to change. Finally, this paper raised the need for additional research related to the relationship between intuitive rules and formal knowledge.

Intuitive rules in science and mathematics: a reaction time study

It has been observed that students react in similar ways to mathematics and science tasks that differ with regard either to their content area and/or to the type of reasoning required, but share some common, external features. Based on these observations, the Intuitive Rules Theory was proposed. In this present study the framework of this theory was employed and the reaction times of two types of responses were measured: those that are regarded as intuitive and those that are viewed as counter-intuitive. The motivation behind this study was to empirically address the immediacy characteristics of intuitive responses in the context of science and mathematics. The focus was on the comparison of area and perimeter of geometrical shapes, in the context of the intuitive rule more A – more B. The main findings showed that the reaction times of intuitive responses were, indeed, shorter than reaction times of counter-intuitive ones.

Intuitive rules in science and mathematics: the case of ‘Everything can be divided by two’

In the last twenty years, researchers have studied students’ mathematical and scientific conceptions and reasoning. Most of this research is content‐specific. It has been found that students often hold ideas that are not in line with accepted scientific notions. In our joint work in mathematics and science education, it became apparent that many of these alternative conceptions hail from a small number of intuitive rules. We have so far identified two such rules: ‘The more of A, the more of B’, and, ‘Everything can be divided by two’. The first rule is reflected in students’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.), all tasks related to intensive quantities (density, temperature, concentration, etc.), and tasks related to infinite quantities. The second rule is observed in responses related to successive division of material and geometrical objects, and in seriation tasks. In this paper we describe and discus...

Infinity - the never-ending struggle

Infinity has fascinated mankind since time immemorial. Zeno revealed that, whether we consider space and time to be infinitely divisible or consisting of tiny indivisible atoms, in both cases paradoxes appear. Despite this uncomfortable problem, practical mathematicians continued to use a range of infinitesimal and indivisible methods of calculation through to the 17th century development of the calculus and beyond. At the beginning of the 19 century, infinitesimal methods were still widely used. Dedekind’s construction of the real numbers suggested that the real line consists only of rationals and irrationals with no room for infinitesimals. He began with the set Q of rational numbers and proceeded to construct a set R of ‘cuts’ of the set Q which consist of two subsets A, B where every element of A is less than every element in B. He showed that these cuts were of two types. The first type corresponded to a rational number r with rational numbers less than r in A and rational numbers greater than r in B. (In this case the rational number r could be in either A or in B.) The second type did not have a rational number sitting between A and B. He showed that the set of cuts formed a system with elements of the first type corresponding to rational numbers and elements of the second type corresponding to irrational numbers. This construction ‘completed’ the real line by adding irrational numbers to ‘fill in the gaps’ between the rational numbers. In such a number line, there is ‘no room’ for infinitesimal quantities. The arithmetization of analysis by Riemann confirmed this view that no number a on the real line could be ‘arbitrarily small’, for if 0 < a < r for all positive real numbers r, then 2a is positive and even smaller than a. Infinitesimals therefore did not fit into the real number system. When Cantor constructed the concept of infinite cardinal and ordinal numbers, he developed a remarkable extension of counting finite sets to define the cardinal number of an infinite set with an operation of addition corresponding to the union of two sets and multiplication corresponding to the Cartesian product of two sets. Two infinite sets are said to have ‘the same cardinal number’ when they can be put in one-one correspondence. This was not without its difficulties. For instance, in the infinite case, a set and a proper subset could now have the same cardinal number, which contradicts finite experience and continues to cause confusion in those learning the theory today. The arithmetic of cardinals also has no use for infinitesimals because infinite cardinals do not have multiplicative inverses. By the beginning of the twentieth century, infinitesimal ideas were theoretically under attack, but they still continued to flourish in the practical world of engineering and science, often as a ‘facon de parler’, representing not a fixed infinitesimal quantity, but a variable that could become ‘arbitrarily small’.

Intuitive Beliefs, Formal Definitions and Undefined Operations: Cases of Division by Zero

In this chapter we describe a study in which we explore secondary school students’ adherence to the perform-the-operation belief in the cases of division by zero. Our aims were: (1) to examine whether secondary school students identify expressions involving division by zero as undefined or tend to perform the division operation, (2) to study the justifications given for their approach, and (3) to analyze the effects of age (grade) on their responses. A substantial number of the participants argued, in line with the perform-the-operation belief, that division by zero results in a number. This intuitive belief was also evident in the justifications of students who correctly claimed that division by zero is undefined. Performance on division by zero tasks did not improve with age. Possible causes and educational implications of these findings are described and discussed.

To Define Or Not To Define: The Case Of (-8)1/3

This paper discusses two possible approaches to (-8)1/3. The first is that (-8)1/3 = 3√(-8) = -2. The second is that (-8)1/3 is undefined. The pros and cons of each of these approaches are considered and some implications to teacher education are specified.