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This paper explored students’ understanding about the relative sizes of exponential numbers. Students usually go through two significant transitions as they progress from primary to secondary classes. One is transition from additive reasoning to multiplicative reasoning and other, which is the focus of this paper, is from multiplicative reasoning to exponential reasoning. Across the globe, exponential reasoning, therefore has been provided a rich avenue for studying students’ alternative frameworks as, in this context, students have to create new meanings for some of the mathematical symbols and operations with which they have been familiar since their primary and lower secondary grades. For example, when a minus sign appears in the power of an exponential number, it gives a new meaning to the expression that is different from the instance in which the minus sign appears at the beginning of the expression, or associated with the base of the expression. Moreover, a fraction appearing in the exponent could make the exponential number qualitatively different, an irrational number, to a number where the fraction is attached to the base of the expression. Another layer of complexity appears when the power of the exponential number is represented as a negative rational number.

This paper, at its core, derive theoretical framework from Sfard’s (1991) notion of learning of mathematical concepts as both developing operational (procedural) understanding and structural (conceptual) understanding. For structural understanding, learners need to develop some mental models which may or may not emerge from their concrete experiences. These mental models situate a particular concept into a larger network of mathematical knowledge. However, I would argue that it is not always the case that one mental model leads students develop different mental models. Rather, different experiences lead students develop different mental models. For example, students who use ‘fair sharing’ model, embedded in cultural practice of fair dealing, to represent division usually cannot make sense of division of a number by fraction as this model cannot help students to extent this prototype to division situation where divisor is a fraction. However, the students who learn the concept by ‘repeated subtraction’ as model for division usually find the structural similarity among different division situations irrespective of type of numbers they carry. This implies that there is a hierarchy of mental models based on their effectiveness to connect a large number of concepts and situations. For example for the division of fraction, researchers have identified advantage of using repeated subtraction model over fair sharing model.

Using the above framework I analysed what mental models of exponents the students had brought to represents exponents with negative and rational numbers.

This paper analyses 30 pairs of higher secondary school (Grade 11) male and female students’ construction for comparing relative sizes of numbers written in exponent form with the purpose to understand their any changes that may have made in their mental models, during the solving of the questions in the setting of the clinical interview. The purpose of this analysis is to understand the role of different conceptual models in students’ understanding of rational and negative exponents.

Data reveals that students generally used repeated multiplication model based on process of counting. That model did not support them to represent exponent with negative integers. Confrey (1994) is a strong proponent of the inclusion of splitting as a representation of the multiplicative situation. Some of the students in this reported study did use the process of splitting to represent exponential expressions, and represented positive, negative and zero indices successfully, but they reverted to the process of counting, or using a number pattern developed on repeated multiplication, when they represented fractional indices. This study did not gain evidence to show why the students reverted to the counting model. One could speculate that it happened because they might have had the experience of the multiplication operation in the early grades through counting, and had developed an understanding of splitting later, and that is why, to deal complex numbers, their primitive model of multiplication surfaced. Alternatively, it could imply that counting is intrinsically associated with the concept of exponent. Researchers from international context need to develop an understanding of this phenomenon on ontological and epistemological grounds through teaching experiments with students from different context meaningful instruction could be designed for the students.