1. Designing a mathematics curriculum for a user programme
Burkhard Alpers, Hochschule Aalen – Aalen University
SEFI Mathematics Working Group
MatRIC IAB
Burkhard Alpers

2. 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
Burkhard Alpers

3. 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):
1992: “Goals” as lists of topics ordered according to field:
Linear algebra
Linear systems of equations
Vector spaces …
Calculus
Functions
Continuity
2002: Topics with desired ability attached:
“Students should be able to … “ (e.g. solve a linear system of equations)
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)
Burkhard Alpers

4. 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 ( Curriculum)
Burkhard Alpers

5. 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.
Burkhard Alpers

6. 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.
Burkhard Alpers

7. 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, 3D-printing device; milling machine; curve-driven clutch plus gears)
Industry guidelines
Reports on internships, Bachelor thesis written in industry
Burkhard Alpers

8. 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
Burkhard Alpers

9. 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:
The set of curves depicted below (copied from a book on machine elements) gives information on the value the so-called notch number ak of a notched round bar. ak is a function of x=r/d and y=D/d. For a few values of y ak is represented as a function of x. Determine ak (0,1; 1,1).
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? t s 3 5 ?
Burkhard Alpers

10. 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
Burkhard Alpers

11. 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.
Burkhard Alpers