Designing a mathematics curriculum for a user programme Burkhard Alpers, Hochschule Aalen Aalen University SEFI Mathematics Working Group MatRIC IAB Burkhard.Alpers@htw-aalen.de 1
Overview How to specify goals of mathematics education in user programmes: The concept of mathematical competence Example: Aspects of the problem solving competency Ways to find usages of mathematical concepts in relevant contexts and situations Practical consequences of a competence-oriented curriculum Some caveats 2
Goals of mathematics education Historical development of goal specification in the Curriculum Document of the Mathematics Working Group of SEFI (European Society for Engineering Education): 1. 1992: Goals as lists of topics ordered according to field: Linear algebra Linear systems of equations Vector spaces Calculus Functions Continuity 2. 2002: Topics with desired ability attached: Students should be able to (e.g. solve a linear system of equations) 3. 2013: Mathematical competence ability to understand, judge, do, and use mathematics in a variety of intra- and extra-mathematical contexts and situations in which mathematics plays or could play a role. (Niss 2003) 3
Mathematical Competence Specification of mathematical competencies Thinking and reasoning mathematically Mathematical problem solving Mathematical modelling Dealing with mathematical representations Handling mathematical symbols and formalism Communicating mathematically Handling aids and tools Dimensions of goal specification: Degree of coverage (aspects) of competencies Contexts and situations Technical level Example: Mathematics Curriculum for a practice oriented study course in mechanical engineering (http://sefi.htw-aalen.de, Curriculum) 4
Example Posing and solving mathematical problems: This competency comprises on the one hand the ability to identify and specify mathematical problems and on the other hand the ability to solve mathematical problems (including knowledge of suitable algorithms). Aspect 1: Students should be able to solve well-specified computational problems for which algorithms already exist. These problems might show up in an intra- or extra-mathematical environment. Examples: Students should be able to solve an integral where the integrand is a fraction. It is well-known that the method using partial fractions works but the solution scheme can be quite complex for someone who is not accustomed to using it. Students should be able to perform an algorithm for computing the stress occurring in a machine element given a certain load. The algorithm can be found in a machine element book. 5
Example (cont d) Aspect 2: Students should be able to solve problems for which no algorithm is available but the basic problem solving strategies that could help are available and the application of these strategies is not too complex and tricky. Example: In a geometric configuration some geometric data is given and other quantities are looked for. The well-known strategy is to relate the unknowns to the known quantities by using geometric theorems like Pythagoras theorem or sine or cosine theorem. This way one can for example compute in a windscreen wiper the position of the wiper given the rotation angle of the driving motor. 6
Ways to find usages in relevant contexts and situations Examination of textbooks, manuscripts of colleagues E.g. engineering mechanics, control theory Examination of tasks and solutions in textbooks/manuscripts Look around in the labs (videos: Drop tower, parts of machines, 3Dprinting device; milling machine; curve-driven clutch plus gears) Industry guidelines Reports on internships, Bachelor thesis written in industry 7
Practical consequences of a competence-oriented curriculum (1) Classical lectures Illustration for using mathematical concepts Motivation for introduction of mathematical concepts Fourier series and vibrations Continuity, differentiability and motion 8
Practical consequences of a competence-oriented curriculum (2) Application-related tasks as part of weekly assignment sheets Tasks related to subjects running in parallel, no additional application knowledge required Examples: A spacer in a packaging machine waits for 3 seconds for the next product to come along. It then moves the part for 3 dm into a bag sealer and waits again. How should it move? The set of curves depicted below (copied from a book on machine elements) gives information on the value the so-called notch number a k of a notched round bar. a k is a function of x=r/d and y=d/d. For a few values of y a k is represented as a function of x. Determine a k (0,1; 1,1). s 3? 3 5 t 9
Practical consequences of a competence-oriented curriculum (3) Medium projects as major part of a course Setting: 30h, technology use, documentation and presentation required Example: Measurement, redesign and reproduction of a part 10
Finally: Some caveats ( bewares ) Training of routine procedures and letting the students get accustomed to dealing with mathematical objects like matrices and complex numbers is important (e.g. matrix multiplication). Contents are still important. Making mathematics relevant does not mean making mathematics easy (for the majority of students)! Mathematical competence is growing over time; it is not there after mathematics education and holds forever, it further develops in application subjects. But: Following the competence concept lets student experience mathematics as integral part of the study course. 11