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Functions and Equations In High School Algebra

Learning Problem-Solving Skills in Algebra

This research study investigates the concept of function developed by a sample of
secondary and university students in England and Malaysia studying mathematics as
one of their subjects. It shows that whilst students may be able to do the 'mechanical'
parts of this concept, their grasp of the 'theoretical' nature of the function concept may
be tenuous and inconsistent.
The hypothesis is that students develop 'prototypes' for the function concept in much
the same way as they develop prototypes for concepts in everyday life. The definition
of the function concept, though given in the curriculum, proves to be inoperative, with
their understanding of the concept reliant on properties of familiar prototype examples:
those having regular shaped graphs, such as x2 or sinx, those often encountered
(possibly erroneously), such as a circle, those in which y is defined as an explicit
formula in x, and so on.
The results of the study in England revealed that even when the function concept was
taught through the formal definition, the experiences which followed led to various
prototypical conceptions. Investigations also show significant misconceptions. For
example, three-quarters of a sample of students starting a university mathematics course
considered that a constant function was not a function in either its graphical or algebraic
forms, and three quarters thought that a circle is a function.
The extension of the study in Malaysia was made with the hypothesis that there is a
significant difference between the concept as perceived to be taught and as actually
learned by the students. Although the intended curriculum emphasises conceptual
understanding, in the perceived curriculum (curriculum as understood by the teachers),
only 45% of the teachers follow this approach. The tested curriculum as reflected in the
public examination questions, only emphasises the procedural skills and the results of
the learned curriculum show that learning of functions is more consistent with the
theory of prototypical learning. Students in Malaysia develop their own idiosyncratic
mental prototypes for the function concept in much the same way as those students in
the UK.

Functions and Equations In High School Algebra

Citations

Learning Problem-Solving Skills in Algebra

What do students learn about functions? : a cross cultural study in England and Malaysia

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