Abstract:

The teaching of arithmetic mean has been included for many years in the mathematics curriculum for secondary school and, more recently, in the last grade of primary school in Spain. Understanding this concept is a requisite for getting competence in exploratory data analysis and understanding other concepts, such as median, percentiles and quartiles or standard deviation. Moreover, generalizations of this concept develop in the idea of population mean, which is a parameter in the definition of models of distributions such as the normal, exponential or binomial distribution.

Although research on the understanding of average is wide, it has mostly focused on university or high school students or even in prospective primary school teachers. Moreover, most of this research has only considered isolated elements of the meaning of the concept.

In this paper, we describe an exploratory study oriented to evaluate understanding of the arithmetic mean in a sample of 84 first grade secondary school students, who were given a questionnaire with seven open tasks. Our research is based on theoretical framework termed onto-semiotic approach to mathematics education in which the following types of mathematical objects are considered:
1. The field of problems from which the object (in this case the mean) emerges.
2. The different representations of the object.
3. The procedures and algorithms to manage the object or solve related problems.
4. The different definitions and properties of the object.
5. The arguments associated to the same.

Our questionnaire takes into account a variety of these objects, such as verbal, numerical and symbolic representation, colloquial and mathematical language, simple arguments, algebraic and numerical properties and intuitive definitions of the mean. In the paper, the particular objects linked to each item are made explicit.

Once the questionnaires were collected, we performed a content analysis of the students' responses, following a qualitative methodology. In the paper we report the percentages of correct, incorrect and partly correct responses in the different items and of the objects related to arithmetic mean that are better or worse understood by these students. We also describe and classify the semiotic conflict that appear in the partly correct or incorrect responses, that is, the interpretation of representations, procedures, language or properties that do not coincide with mathematical interpretation. In addition, we compare our results with previous research, which has mostly been carried out with upper level students. In this sense, we provide original information, since research on students’ understanding of average at this age is scarce.

Keywords:

Secondary education, students' understanding, arithmetic mean.