Calculating the derivative of piecewise functions

Exercises involving the calculation of the derivative of piecewise defined functions are common in calculus, with the aim of consolidating beginners’ knowledge of applying the definition of the derivative. In such exercises, the piecewise function is commonly made up of two smooth pieces joined together at one point. A strategy which avoids using the definition of the derivative is to find the derivative function of each smooth piece and check whether these functions agree at the chosen point. Showing that this strategy works together with investigating discontinuities of the derivative is usually beyond a calculus course. However, we shall show that elementary arguments can be used to clarify the calculation and behaviour of the derivative for piecewise functions.

An investigation into the tension arising between natural language and mathematical language experienced by mechanical engineering students

This investigation is grounded within the concept of embodied cognition where the mind is considered to be part of a biological system. A first year undergraduate Mechanical Engineering cohort of students was tasked with explaining the behaviour of three balls of different masses being rolled down a ramp. The explanations given by the students highlighted the cognitive conflict between the everyday interpretation of the word energy and its mathematical use. The results showed that even after many years of schooling, students found it challenging to interpret the mathematics they had learned and relied upon pseudo-scientific notions to account for the behaviour of the balls.

Inspiring young people to engage in engineering education:The Aston University Engineering Academy Birmingham

This paper initially reports concerns about the falling interest in engineering and mathematical disciplines and looks at some of the reasons for this. It then discusses the aims of the Engineering Diploma - a qualification for 14-19 year olds in the UK - and the pedagogical research that that has informed the design and development. The paper highlights the key learning theories that support the delivery of this qualification and provides an example of how this pedagogy has been applied effectively through the curriculum partnership that has been developed between a consortium of schools in the Birmingham local authority, Aston University and employers. It establishes the importance of aligning the curriculum and articulating clear engineering progression routes from the age of fourteen to enable young people to be inspired and motivated towards careers in engineering. The paper presents the view of parents, teachers and pupils involved with the Diploma, during the first year, and the way in which the partnership is informing future developments in the delivery of engineering curriculum within the region. The success of this regional partnership model has resulted in the Department of Children, Schools and Families agreeing to fund the development of the Aston University Engineering Academy Birmingham. This is a school for 14-19 year olds that will open in 2012 on the Aston Science Park adjacent to the University. The final part of the paper looks at the benefits to the young local engineers of this initiative. © 2009 Authors.