How CAS and E-learning change the teaching
and learning of introductory engineering mathematics
-
the ongoing innovation process at Mathematics 1Øresundsdagen, Lund 23-10-2013 Karsten Schmidt:
Overview
1)
12 years with Maple
2)
What do we obtain by using math software
Two examples
3)
The e-Math project
4)
Conclusions
“It is not our task to educate the brothers of Numbskull Jack!” Anders Bondo Christensen 2013, chairman of the Danish teachers’ union
Sanjoy Mahajan Professor at MIT
Street fighting is the pragmatic opposite of rigor (mortis)
Rote learning combines the worst of human and computer thinking
Conrad Wolfram
“A prominent proponent of Computer-Based Math” (wiki)
21st Century Mathematics, Stockholm 2013
Stop teaching by-hand-calculations! Spend the time on modelling and use computer for the calculations!
“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.
21st Century Mathematics, Stockholm 2013
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
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
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
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
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
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
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
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)
12 years with Maple, some key points
The transition problems!
High school
Mathematics 1 Other introductory courses Other advanced courses Advanced math courses DTU
What do the university teachers think?
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.”
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.”
The transition problems!
High school
Mathematics 1 Other introductory courses Other advanced courses Advanced math courses DTU
week
subject
Maple
1 2 3 4 5 6 7 8 9 10 11 1213 Project based exercise in systems: x = Ax + b
.
Matrix Algebra, DeterminantsSystems of Linear Equations
Vectors in Plane and Space General vector spaces
Lin. Transform, shift of bases Complex numbers
(and real numbers!)
Eigenvalue, diagonalization 1. and 2. order ODEs
Systems: x = Ax
.
Thematic exercises
Paper & pencil
First semester redesigned (fall 2012)
Thematic exercises + + + + + +
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)
Difficulties in learning LA
S. Stewart & M. O. J. Thomas:
EMBODIED, SYMBOLIC AND FORMAL ASPECTS OF BASIC LINEAR ALGEBRA CONCEPTS (2007)
week
subject
Maple
Sketchpad
1 2 3 4 5 6 7 8 9 10 11 1213 Project based exercise in systems: x = Ax + b
.
Matrix Algebra, DeterminantsSystems of Linear Equations
Vectors in Plane and Space General vector spaces
Lin. Transform, shift of bases Complex numbers
(and real numbers!)
Eigenvalue, diagonalization 1. and 2. order ODEs
Systems: x = Ax
.
Example
Example
Example Example
Thematic exercises
Paper & pencil
First semester redesigned (fall 2012)
Thematic exercises + + + + + + Example
Geometric vectors
Geometric vectors
week
subject
Maple
Sketchpad
1 2 3 4 5 6 7 8 9 10 11 1213 Project based exercise in systems: x = Ax + b
.
Matrix Algebra, DeterminantsSystems of Linear Equations
Vectors in Plane and Space General vector spaces
Lin. Transform, shift of bases Complex numbers
(and real numbers!)
Eigenvalue, diagonalization 1. and 2. order ODEs
Systems: x = Ax
.
Example
Example
Example Example
Thematic exercises
Paper & pencil
First semester redesigned (fall 2012)
Thematic exercises + + + + + + Example
The eigenvalue problem
GSP: Eigenvalue problems
The eigenvalue problem
Mathematical modelling!
Mathematical theory for integration Geometric object Parameterization Parametric object Calculation Feed back
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
eMath. Philosophy of learning
Improving of individual work and active preperation Better possibilities for finding your own learning styles The most important points are presented in different medias
The eNotes should offer different ways of reading It should be easy to find help by links and video
Flexibility regarding Where and When
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.
e math
