1. Competence Orientation in Practice Part II: Development of tasks for competence acquisition
Burkhard Alpers, Hochschule Aalen – Aalen University
Burkhard Alpers

2. Overview Presentation: Group activity: Break
Ways to find tasks for competence acquisition
Concept and examples for:
Application oriented assignments (weekly assignment sheet)
Application oriented mini-projects (accompanying regular lectures)
Application oriented medium projects (project as central activity)
Some further considerations and caveats
Group activity:
Finding relevant tasks:
Laser beam device
Exchange of practical tasks
Break
Plenary: Collecting results
Burkhard Alpers

3. Ways to find concepts and usages in relevant contexts and situations
Examination of textbooks, manuscripts of colleagues
E.g. engineering mechanics, control theory
Looking for mathematical concepts, representations, argumentations
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)
Look around in the “wild” (video: soft-drink supply)
Industry guidelines
Reports on internships, Bachelor thesis written in industry
Burkhard Alpers

4. Application oriented assignments (1): Criteria
Criteria for application-related tasks for weekly assignment sheets (Wolf)
Math-topic-oriented: Should offer opportunities to train current topic of math lecture
Authentic: background should not be artificial but meaningful
Modelling-oriented: one or more back-and-forth steps between math and application subject
Clearly presented (pictures, sketches, short descriptions) and suitable difficulty level
Not requiring additional knowledge (beyond that which has been learned already in application subjects)
Transferable to similar problems
Reference:
P. Wolf & G. Oevel: A conception for developing special application-oriented tasks for mathematics for engineering, published on ResearchGate 2015 (longer version of a conference paper)
Burkhard Alpers

5. Application oriented assignments (2): 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 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). t s 3 5 ?
Burkhard Alpers

6. Application oriented assignments (3): Examples
Stressing a structural element (P. Wolf, extract): A structural element whose bearings are made for an equivalent force of R=981kN with a direction angle of aR=70°, will be stressed in a new situation by the forces F1, F2, F3.
How large can the forces be without demanding an increase of the bearings?
For what direction angles are the forces in this setting unique and determinable …
Burkhard Alpers

7. Mini-projects (1): Concept
Mini-project accompanying the traditional course Mathematics I, II
Topic: Related to application subject running in parallel (here: engineering mechanics)
Group work: Up to 3 students
Timeframe: 6-10h
Prerequisite for final exam, no contribution to exam grading
Burkhard Alpers

8. Mini-projects (2): Example 1
Design, build, and compute forces in a truss
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9. Mini-projects (2): Example 2
Motion design: The milling head of a milling machine should move along the curve depicted below. When going through the parabolic part, the milling process takes place. Therefore, the tangential velocity should be constant (0.6 m/s). The maximum velocity is 1 m/s, the maximum positive acceleration is 0.2 m/s2, the maximum negative one is m/s2.
Construct a motion function v(t) where the milling head starts from rest and end up in a rest (v=a=0). The overall time should not exceed 10s.
1,5 m
0.3 m
0,75 m
Start
Ziel
Burkhard Alpers

10. Medium projects (1): Concept
Mathematics III: Numerical procedures
Lecture is moved to first weeks, then project for gaining CPs
Criteria for project specifications:
Should require application of important mathematical concepts
Should be concerned with meaningful application topic
Should include a data acquisition phase (e.g. by taking measurements)
Should require the application of mathematical software
Should be open
Should be time-adequate
Should be modular (for delegation of subtasks)
Group work: Up to 4 students
Timeframe: 30h
Documentation and presentation
Reference:
B. Alpers: Mathematical Application Projects for Mechanical Engineers – Concept, Guidelines, and Examples. Proc. ICTMT 5, Klagenfurt 2001.
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11. Medium projects (2): Example 1
Example: Measurement, redesign and reproduction of a part
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12. Medium projects (3): Example 2
Motion analysis: In the vending machine positioned in the cafeteria (see picture to the right), a grabber performs „rest-to-rest“ motions. Videotape the motion and analyse it using video analysis software. Investigate whether the motion is performed according to one of the motion laws specified in the guideline VDI 2143 (issued by the German Association of Engineers).
Burkhard Alpers

13. Medium projects (4): Example 3
Drop tower: What is the final velocity of the weight when it hits the test specimen?
Energy model (without friction)
Model with friction: videotape the motion, compute approximation parabola, compute velocity, iterate.
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14. Large Projects Large projects as with some mathematical content
Setting: 5 CP, documentation and presentation required
Example: Design of a laser cutting device
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15. Further thoughts about the relationship between mathematics and application subject teaching
What is the split of responsibility? Shouldn’t there be a clear cut: mathematics education is responsible for math concepts, application subject education is responsible for how to apply mathematical concepts?
Not an “either or” situation: Mathematics is for motivation and initial understanding, application subjects for deepening the usage and meaning
Mathematical competence is growing over time; it is not there after mathematics education and holds forever
When students see more applications they might be able to see clearer the abstract kernel (how are they assumed to abstract when there is nothing to abstract from?)
Burkhard Alpers

16. 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

17. Task for group work Form 6 groups Group discussion:
Identification of tasks regarding the laser beam device
Exchange of tasks: Do you already include tasks in relevant engineering contexts in your teaching?
Goal: Train to find tasks for understanding doing and using mathematics in relevant contexts and situations.
Burkhard Alpers