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Concern has been expressed that many commencing undergraduate mathematics students have mastered skills without conceptual understanding. A pilot study carried out at a leading Australian university indicates that a significant number of students, with high tertiary entrance ranks, have very limited understanding of the concept of function, despite the emphasis it receives in the secondary mathematics curriculum. Whilst most students were familiar with families of functions, many were unable to give an appropriate definition or recognize whether a given graph or rule represents a function; and could not make correct connections between function graphs and tables of values.

In this article the authors analyze the written solutions of some first year undergraduate mathematics students from Victorian universities as they answered tutorial exercise questions relating to complex numbers and differentiation. These students had studied at least Mathematics Methods or its equivalent at secondary school. Complex numbers was a new topic for some students. The focus in these examples is on the extent to which students display an awareness of structure in complex number and algebraic expressions. The authors assert that structure sense includes an awareness of structure in numeric as well as in algebraic expressions. They emphasize that the lack of structure awareness reported here was not observed with all students--there were obviously many students who had a sound grasp of numeric and algebraic structure. However their observations suggest that there are also many students for whom this is not the case.

In this article, the authors begin by considering symbolic literacy in mathematics. Next, they examine the origins of misuse of the equals sign by primary and junior secondary students, where "=" has taken on an operational meaning. They explain that in algebra, students need both the operational and relational meanings of the equals sign. When substituting numbers for pronumerals in an algebraic expression, students are able to rely on their operational understanding to evaluate the expression. Similarly, it is possible for students to solve algebraic equations such as 2x + 3 = 11 without recourse to the relational meaning, for example, by a guess and check approach. However, if they fail to understand that the expressions on each side of an equation are equal, then they have difficulty, for example, understanding why x can be subtracted from both sides when solving the equation 2x + 3 = x + 11. Researchers have found that students who understand that the equals sign is a relational symbol of equality are more successful in solving algebraic equations. The remainder of the article examines a sample of the written solutions of first semester undergraduate students demonstrating inappropriate use of the equals sign in a major Australian university's Calculus 1 course. The article concludes that the notion of expecting symbols to have meaning and a habit of checking the meaning of the symbols used is an aspect of working mathematically that needs to be cultivated at all levels: primary, secondary and tertiary. In particular, ensuring that students understand the relational role of the equals sign is an important step in developing symbol sense.

Since the 1990s, computer algebra systems (CAS) have been available in Australia as hand-held devices designed for students with the expectation that they will be used in the mathematics classroom. The data discussed in this paper was collected as part of a pilot study that investigated first year university mathematics and statistics students' understanding of functions and variables, as well as the use of technology in their last year of school (Year 12). Did their teachers discourage the use of CAS for algebra? Did the students actually learn how to use CAS to support their work in algebra or to support their learning of algebra? Did they find that, given the level of algebra, it was faster to work with pen-and-paper than to correctly enter algebraic expressions? The results reported in this paper are based on items included in a pilot survey. They raise questions rather than provide answers. The results do however tell us that, at least from these first year university students' recollection of their Year 12 experience, most or their VCE mathematics teachers made little use of CAS as a pedagogical tool in their classes, despite the institutional approval and encouragement indicated by both the State's curriculum and assessment for the past decade. A better understanding of the barriers to teachers using CAS technology to enhance their pedagogy is needed and then perhaps more effective professional learning programs can be provided for teachers.