How CAS and E-learning change the teaching and learning of introductory engineering mathematics - the ongoing innovation process at Mathematics 1

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1 Karsten Schmidt: How CAS and E-learning change the teaching and learning of introductory engineering mathematics - the ongoing innovation process at Mathematics 1 Øresundsdagen, Lund

2 Overview 0) A look into the future 1) 12 years with Maple 2) What do we obtain by using math software Two examples 3) The e-math project 4) Conclusions

3 Our future role model? It is not our task to educate the brothers of Numbskull Jack! Anders Bondo Christensen 2013, chairman of the Danish teachers union

4 21st Century Mathematics, Stockholm 2013 Sanjoy Mahajan Professor at MIT Street fighting is the pragmatic opposite of rigor (mortis) Rote learning combines the worst of human and computer thinking

5 21st Century Mathematics, Stockholm 2013 Conrad Wolfram A prominent proponent of Computer-Based Math (wiki) Stop teaching by-hand-calculations! Spend the time on modelling and use computer for the calculations!

(wiki) Stop teaching by-hand-calculations!

6 21st Century Mathematics, Stockholm 2013 Charles Fadel (Director of Center of Curriculum Redesign): Don t waste your time on learning Latin if you want to learn roman languages. Begin directly with French, Italian etc Implicitly: Don t learn mathematics to be able to do exploring work later on. Begin the exploring work immediately.

7 About Mathematics 1 at DTU Facts about Mathematics 1: 1. A one year course (20 ECTS) 2. Covers the mandatory curriculum for 900 students on 15 study programmes 3. The ordinary continuous treatment of the math subjects: Lectures (1.5 hours twice a week) Group exercises (supported by 28 TA s and 28 student TA s) Homework exercises (8 times during the year) 4. The project exercises (group work, no lectures): Thematic exercises EX The big 4 weeks project at the end of 2. semester EX

The ordinary continuous treatment of the math subjects: Lectures (1.

8 12 years with Maple, some key points The material environment The debate pro and contra CAS (Robinson Crusoe etc.) The mandatory home work exercise Are we undermined by our own success? The recent debate on the transition problems

9 12 years with Maple, some key points The material environment The debate pro and contra CAS (Robinson Crusoe etc.) The mandatory home work exercise Are we undermined by our own success? The recent debate on the transition problems

10 12 years with Maple, some key points The material environment The debate pro and contra CAS (Robinson Crusoe etc.) The mandatory home work exercise Are we undermined by our own success? The recent debate on the transition problems

11 12 years with Maple, some key points The material environment The debate pro and contra CAS (Robinson Crusoe etc.) The mandatory home work exercise Are we undermined by our own success? The recent debate on the transition problems

12 12 years with Maple, some key points The material environment The debate pro and contra CAS (Robinson Crusoe etc.) The mandatory home work exercise Are we undermined by our own success? The recent debate on the transition problems

13 Rules of thumb for a cautious CAS-use Avoid a banning culture Maple is a universe of opportunities When choosing a Maple method, focus on the learning objectives Do always explain and evaluate Maple outputs Explore where Maple gives most insight Ensure that the students try out diverse methods A students homework paper

objectives Do always explain and evaluate Maple outputs Explore where Maple

14 12 years with Maple, some key points The material environment The debate pro and contra CAS (Robinson Crusoe etc.) The mandatory home work exercise Are we undermined by our own success? The recent debate on the transition problems (2012)

etc.) The mandatory home work exercise Are we undermined by

15 The transition problems! Advanced math courses Other advanced courses Mathematics 1 Other introductory courses DTU High school

16 What do the university teachers think? The overall conclusion from a report from the Danish Ministry of Education (December 2011): What the university teachers emphazise in full agreement is the handling of formal expressions. Typical statements from a university teachers: The most serious problem is the lack of basic skills in manipulating simple formulas I believe we are doing a big disservice, if the students don t understand the basic principles for solving simple equations well enough to master the most simple manipulations without using electronic devices. (..) But CAS is adequate for more complicated equations/expressions.

17 The transition problems! Advanced math courses Other advanced courses Mathematics 1 Other introductory courses DTU High school

18 week subject Maple First semester redesigned (fall 2012) Complex numbers (and real numbers!) Paper & pencil Systems of Linear Equations Matrix Algebra, Determinants Vectors in Plane and Space General vector spaces Lin. Transform, shift of bases Eigenvalue, diagonalization + Thematic exercises + Thematic exercises and 2. order. ODEs + 12 Systems: x = Ax Project based exercise in systems: x = Ax + b

19 Difficulties in learning LA My students first learn how to solve systems of linear equations, and how to calculate products of matrices. These are easy for them. But when we get to subspaces, spanning, and linear independence, my students become confused and disoriented. D. Carlson (1993) S. Stewart & M. O. J. Thomas: EMBODIED, SYMBOLIC AND FORMAL ASPECTS OF BASIC LINEAR ALGEBRA CONCEPTS (2007)

But when we get to subspaces, spanning, and linear independence, my students become confused and

20 Difficulties in learning LA S. Stewart & M. O. J. Thomas: EMBODIED, SYMBOLIC AND FORMAL ASPECTS OF BASIC LINEAR ALGEBRA CONCEPTS (2007)

21 week subject Maple Sketchpad First semester redesigned (fall 2012) Complex numbers (and real numbers!) Paper & pencil Systems of Linear Equations Matrix Algebra, Determinants Vectors in Plane and Space General vector spaces Lin. Transform, shift of bases + Thematic exercises + Thematic exercises + Example Example 10 Eigenvalue, diagonalization + Example Example and 2. order. ODEs + 12 Systems: x = Ax +. Example 13 Project based exercise in systems: x = Ax + b

22 Geometric vectors

23 Geometric vectors GSP: Calculating Vectors

24 Geometric vectors Show that

25 Linear combinations

26 Linear independency

27 Choosing bases in the space

28 Linear Transformations

29 week subject Maple Sketchpad First semester redesigned (fall 2012) Complex numbers (and real numbers!) Paper & pencil Systems of Linear Equations Matrix Algebra, Determinants Vectors in Plane and Space General vector spaces Lin. Transform, shift of bases + Thematic exercises + Thematic exercises + Example Example 10 Eigenvalue, diagonalization + Example Example and 2. order. ODEs + 12 Systems: x = Ax +. Example 13 Project based exercise in systems: x = Ax + b

30 The eigenvalue problem GSP: Eigenvalue problems GSP: Interpretations

31 The eigenvalue problem The enote

32 CAS changes the teaching

33 Volume integrals

34 Volume integrals

35 Volume integrals

36 Volume integrals

37 Volume integrals

38 Volume integrals

39 Introduction to 3D integrals in the enote

40 Mathematical modelling! Parameterization Parametric object Geometric object Feed back Calculation Mathematical theory for integration

41 Advantages When we do not have to stress that the calculations should be easy to do by hand, it is possible to build up the integral calculus strictly with a few key ingredients which many of the students should have a fair chance to understand: The Riemann integral over an axis parallel box (in case of 3D integral) Parameterization and deformation Taylors formula and the Jacobi-function By this method we obtain further: A homogenous introduction to line, surface and volume integrals That the visualization is an active player in the modelling and in the learning of the theory

42 emath. Philosophy of learning The most important points are presented in different medias Better possibilities for finding your own learning styles It should be easy to find help by links and video Improving of individual work and active preperation The enotes should offer different ways of reading Flexibility regarding Where and When An easily accessible reference work

43 Conclusions With CAS and e-learning principles it is possible to: To increase the motivation by a true touch of real world applications To bring the concepts and basic mathematical ideas in focus at the expense of rote learning and tricky calculations To enhance the students ability to prepare for the teaching To strengthen the student s desire to read and enjoy the textual representations of the course materials. To enhance the transfer of the obtained mathematical skills

44 Have we realized the ideal? e math

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which