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

Øresundsdagen, Lund 23-10-2013

Karsten Schmidt:

Overview

1) 12 years with Maple2) What do we obtain by using math software

Two examples3) The e-Math project4) Conclusions

0) A look into the future

“It is not our task to educate the brothers of Numbskull Jack!”Anders Bondo Christensen 2013, chairman of the Danish teachers’ union

Our future role model?

Sanjoy MahajanProfessor at MIT

Street fighting is the pragmatic opposite of rigor (mortis)

Rote learning combines the worst of human and computer thinking

21st Century Mathematics, Stockholm 2013

Conrad Wolfram“A prominent proponent of Computer-Based Math” (wiki)

21st Century Mathematics, Stockholm 2013

Stop teaching by-hand-calculations! Spend the time on modellingand use computer for the calculations!

“Don’t waste your time on learning Latin if you want to learn romanlanguages. Begin directly with French, Italian etc…”

Implicitly: Don’t learn mathematics to be able to do exploring worklater on. Begin the exploring work immediately.

21st Century Mathematics, Stockholm 2013

Charles Fadel (Director of Center of Curriculum Redesign):

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 programmes3. 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 EXThe ”big” 4 weeks project at the end of 2. semester EX

12 years with Maple, some key points

The material environmentThe debate pro and contra CAS (Robinson Crusoe etc.)The mandatory home work exerciseAre we undermined by our own success?The recent debate on the transition problems

The material environmentThe debate pro and contra CAS (Robinson Crusoe etc.)The mandatory home work exerciseAre we undermined by our own success?The recent debate on the transition problems

12 years with Maple, some key points

The material environmentThe debate pro and contra CAS (Robinson Crusoe etc.)The mandatory home work exerciseAre we undermined by our own success?The recent debate on the transition problems

12 years with Maple, some key points

The material environmentThe debate pro and contra CAS (Robinson Crusoe etc.)The mandatory home work exerciseAre we undermined by our own success?The recent debate on the transition problems

12 years with Maple, some key points

The material environmentThe debate pro and contra CAS (Robinson Crusoe etc.)The mandatory home work exerciseAre we undermined by our own success?The recent debate on the transition problems

12 years with Maple, some key points

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

The material environmentThe debate pro and contra CAS (Robinson Crusoe etc.)The mandatory home work exerciseAre 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 manipulatingsimple formulas”

“I believe we are doing a big disservice, if the students don’t understand thebasic principles for solving simple equations well enough to master the mostsimple 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 handlingof formal expressions.”

The transition problems!

High school

Mathematics 1 Other introductory courses

Other advanced courses

Advanced math courses

DTU

week subject Maple12345678910111213

.Project based exercise in systems: x = Ax + b

Matrix Algebra, DeterminantsSystems of Linear Equations

Vectors in Plane and SpaceGeneral vector spaces Lin. Transform, shift of bases

Complex numbers(and real numbers!)

Eigenvalue, diagonalization

1. and 2. order ODEsSystems: 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 Sketchpad12345678910111213

.Project based exercise in systems: x = Ax + b

Matrix Algebra, DeterminantsSystems of Linear Equations

Vectors in Plane and SpaceGeneral vector spaces Lin. Transform, shift of bases

Complex numbers(and real numbers!)

Eigenvalue, diagonalization

1. and 2. order ODEsSystems: x = Ax

.

Example

ExampleExample Example

Thematic exercises

Paper & pencil

First semester redesigned (fall 2012)

Thematic exercises

+

+

+++

+ Example

Geometric vectors

Geometric vectors

GSP: Calculating Vectors

Geometric vectors

Show that

Linear combinations

Linear independency

Choosing bases in the space

Linear Transformations

week subject Maple Sketchpad12345678910111213

.Project based exercise in systems: x = Ax + b

Matrix Algebra, DeterminantsSystems of Linear Equations

Vectors in Plane and SpaceGeneral vector spaces Lin. Transform, shift of bases

Complex numbers(and real numbers!)

Eigenvalue, diagonalization

1. and 2. order ODEsSystems: x = Ax

.

Example

ExampleExample Example

Thematic exercises

Paper & pencil

First semester redesigned (fall 2012)

Thematic exercises

+

+

+++

+ Example

The eigenvalue problem

GSP: Eigenvalue problems

GSP: Interpretations

The eigenvalue problem

The eNote

CAS changes the teaching

Volume integrals

Volume integrals

Volume integrals

Volume integrals

Volume integrals

Volume integrals

Introduction to 3D integrals in the eNote

Mathematical modelling!

Mathematical theory for integration

Geometric object

ParameterizationParametric object

CalculationFeed 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 keyingredients which many of the students should have a fair chanceto 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

eMath. Philosophy of learning

Improving of individual work and active preperation

Better possibilities for finding your own learning stylesThe 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

An easily accessible reference work

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

e math

Have we realized the ideal?