Journal of Statistics Education, v17n1: Ilana Lavy and Michal Mashiach-Eizenberg

Various terms are used to describe mathematical concepts, in general, and statistical concepts, in particular. Regarding statistical concepts in the Hebrew language, some of these terms have the same meaning both in their everyday use and in mathematics, such as Mode; some of them have a different meaning, such as Expected value and Life expectancy; and some have the opposite meaning, such as Significance level. Spoken language plays an important role in shaping how the informal statistical definitions taught in schools are remembered. In the present study we examine the impact of the everyday use of terms on the students’ informal definitions of various statistical concepts. Though all the study participants were familiar with the concepts they were asked to define, a high percentage of them failed to provide correct definitions of the given statistical concepts. Analysis of the incorrect definitions revealed that the everyday use of the terms used to label the concepts, influenced the informal definitions provided by the students. 1. Contextual Framework The formal definitions of mathematical concepts constitute a symbolic language which is independent of any spoken language. Since these definitions in their symbolic format are difficult to teach and understand, spoken language is used to informally define mathematical concepts. Within these informal definitions, mathematical symbols are replaced by proper words from the spoken language. Following this model, statistics has become language dependent and its informal definitions are also an integral part of its language. In addition, a basic understanding of statistics requires one to distinguish between samples and populations. On the one hand, while some of the concepts are clearly population quantities, such as Expected value, some of them, on the other hand, such as Standard deviation, require the addition of an adjective (population or sample) to distinguish the statistical definition from its everyday use. In the case of Hebrew as well, various words from the spoken language are used to describe mathematical concepts. Regarding statistical concepts, some of these terms have the same meaning in their everyday use and in mathematics, such as Mode, some have a different meaning, such as Expected value and ‘Life expectancy, and some have the opposite meaning such as Significance level. Researchers have explored various aspects regarding the understanding of statistical concepts (i.e. Falk, 1986, Haas et al., 2003), in order to see how technology can help students understand, integrate, and apply fundamental statistical concepts (Chance et al., 2000). Pimm (1987) related understanding to the use of everyday words with particular mathematical sense. This connection is also relevant in the case of statistics. He also considered that analogies and metaphors, where everyday words can be given particular meanings, are very important for the construction of mathematical meaning. Difficulties in the acquisition and understanding of mathematical concepts were explored by various researchers (e.g. Vinner, 1983; Vinner & Hershkovitz, 1980; Sfard, 1991). According to Vinner (1983), the representation and organization of many mathematical books are based on the following two assumptions: (a) the acquisition of concepts is done mainly via their definitions; (b) students will use definitions to solve problems and prove theorems. Hence, definitions should be minimal and elegant. Referring to the acquisition of mathematical concepts, Vinner & Hershkovitz (1980) Journal of Statistics Education, v17n1: Ilana Lavy and Michal Mashiach-Eizenberg http://www.amstat.org/publications/jse/v17n1/lavy.html (1 of 9)2/26/2009 9:46:50 AM Journal of Statistics Education, v17n1: Ilana Lavy and Michal Mashiach-Eizenberg distinguished between 'concept definition' and 'concept image.' 'Concept definition' is defined as a form of words used to specify an idea (Tall & Vinner, 1981). The term 'concept image' describes the total cognitive structure associated with the idea, including all mental pictures and associated properties and processes. Difficulties in attaining proper definitions can point to a gap between the concept definition and the concept image of a certain idea. According to Sfard (1991), mathematical conceptions have a dual nature. Namely, mathematical entities and concepts can be perceived as process (structurally), or as object. Moreover, the understanding of a concept structurally, precedes the understanding of a concept on the object level. In the present study we examine the interplay between Hebrew as a spoken language and the informal definitions of specific statistical concepts. A questionnaire, consisting of several statistical concepts and everyday expressions bearing the same meaning as the statistical concepts, was given to second-year college students who had already completed courses in probability and statistics. The students were asked to define several statistical concepts informally, in their own words, and to include an example of the concept's applicability. 2. Theoretical framework Though all the study participants were familiar with the concepts they were asked to define, a considerably high percentage of them failed to provide even a correct informal definition of the given statistical concepts. Some of the difficulties stem from the fact that informal definitions of statistical concepts are language dependent. Since some of the terms used in statistical concepts have different or opposite meanings in spoken Hebrew, we assumed that the common use of these terms had a strong impact on their perceived statistical meaning. Therefore we referred to past research on language and understanding. The inability to provide even informal definitions is related to difficulties the students have in the acquisition and understanding of these concepts. These difficulties stem from the complex nature of mathematical concepts. We decided to interpret the research data using two theoretical frameworks which deal with the essence of the acquisition and understanding of mathematical concepts. According to the first theoretical framework, the understanding of a mathematical concept necessitates the construction of two mental ‘entities’ in the learner's mind: ‘concept image’ and ‘concept definition’ (Vinner & Hershkowitz, 1980; Tall & Vinner, 1981). According to the second theoretical framework, the complexity of mathematical concepts stems from their dual nature: they are both process and object. This means that the acquisition and understanding of mathematical concepts occurs gradually, whereby they change from being perceived as process and become mental objects (Sfard, 1991). Elaboration of the above frameworks will be presented in the next section. 2.1 Language and understanding Mathematical understanding is both a linguistic and a conceptual matter (Vergnaud, 1998). To understand mathematics, one has to be able to not only identify relationships between mathematical symbols, but one has to be able to identify these symbols' relationships to natural everyday language. Learning and understanding of mathematical concepts are required to build associations between symbols and realities which exist independently of our minds (Stadler, 2004). Language and the understanding of mathematical concepts and entities come to fruition in mathematical discourse. Mathematical discourse and its objects are mutually constituted (Sfard, 1998). An activity of mathematical discourse initiates the need for creating mathematical objects, and it is the mathematical objects that influence the mathematical discourse and point it in new directions. When more concepts are introduced, students try to fit them into familiar templates to use in the new discourse. Hence, the introduction of mathematical symbols can be considered an important part of the reification process. Reification is the ability to see the concept as a whole entity. One of the central goals of mathematics teaching in school, is the "understanding" of mathematical concepts. Researchers have distinguished between different kinds of understanding: instrumental and relational (Skemp, 1987; Pesek & Kirshner, 2000). Instrumental understanding refers to developing skills to use algorithms and arriving at the correct answer, while relational understanding refers to the deeper understanding of the content associated with a certain concept. According to Sfard (1991), mathematical understanding can be viewed as a process whereby a mathematical object transforms from process to mental object. A profound understanding of a mathematical object entails not only the manipulation of difficult expressions, but additionally, involves the ability to create a mental picture of an abstract concept. This, apparently, is a major part of mathematical ability. When students are engaged in a mathematical discourse, they use words and terms from the spoken language, while remaining aware that they have to relate to their mathematical meaning. However, since some of the terms used in statistical concepts have a different or an opposite meaning in Hebrew, when students are asked to define these concepts informally, the first denotation that comes to mind is the everyday use of these terms. 2.2. Concept image and concept definition Many researchers have explored the difficulties regarding the acquisition and understanding of mathematical concepts (e.g. Vinner & Hershkovitz, 1980; Tall & Vinner, 1981; Vinner, 1983; Sfard, http://www.amstat.org/publications/jse/v17n1/lavy.html (2 of 9)2/26/2009 9:46:50 AM Journal of Statistics Education, v17n1: Ilana Lavy and Michal Mashiach-Eizenberg 1991; Stadler, 2004). Some of these difficulties might be a consequence of intuitive human thinking. Other difficulties might be related to visual representations or verbal descriptions of the concept that precede its definition. Intuitive thinking, visual intuitions, and the verbal descriptions of a concept are essential for its understanding. However, there might be a gap between the mathematical definition of a concept and the way one perceives the concept. In this case we may say that there is a gap between the concept definition and the concept image. Concept image describes the total cognitive structure that is associated with the idea, including all mental pictures, and associated properties and processes (Vinner & Hershkowitz, 1980). Concept definition is the words used to specify the idea (Tall & Vinner, 1981). In the present study, we examined the students’ difficulties in providing intuitive definitions of basic statistical concepts. However, since they were expected to define these concepts, they provided definitions that described the everyday meaning of the words. Hence, we assume that the concept images of the statistics terms being tested are not independently or wholly structured in the students’ minds. 2.3 The dual nature of mathematical conceptions The difficulty in the acquisition and the understanding of mathematical concepts stems from their complex structure. According to Sfard (1991), mathematical entities and concepts have a dual nature. Referring to the differences between structural and object perception, Sfard said: "Whereas the structural conception is static, instantaneous and integrative, the operational is dynamic, sequential and detailed ... The former [object] is more abstract, more integrated and less detailed than the latter." (Sfard, 1991, p.4) >There are people who conceptualize mathematical concepts as real objects which exist outside the human brain. Indeed, most mathematicians discuss properties of functions and groups the same way scientists discuss the structure of physical materials. It can be said that the mathematicians’ perception of mathematical concepts is structural perception. However, we can find in mathematical textbooks definitions which uncover a different kind of perception: process based. The latter is based on a dynamic imagery of the mathematical concepts. Sfard (1991) also said that the structural perception precedes the object perception. Sfard described the development from an operational to a structural conception as a process of reification. In order to internalize the concept meaning, the student becomes acquainted with it by computing in single steps. These steps connect with each other in the next, condensation phase. Reification is the ability to see the concept as a whole. It is a static state where "the concept becomes semantically unified by this abstract and purely imaginary construct" (Sfard, 1991, p.20). Some of the study participants, in spite of providing an intuitive verbal definition, also described the computational steps needed to calculate the defined concepts, which may possibly imply that the students’ perception of these concepts exists at a structural level.