Thomas Lingefjärd PME_NA Athens, GA, USA October 2002 1 Assessing the teaching & learning of mathematics in the mechanical engineering program at Chalmers

Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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Assessing the teaching & learning of Assessing the teaching & learning of mathematics in the mechanical engineering mathematics in the mechanical engineering program at Chalmers Technical Universityprogram at Chalmers Technical University

Thomas LingefjärdChalmers Technical University & Göteborg University

Sweden


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THE CDIO projectConceive-Design-Implement-Output

In October 2000, with support from the Wallenberg Foundation, four universities launched an international collaboration designed to improve undergraduate engineering education in Sweden, the United States, and worldwide. This is a closely coordinated program with parallel efforts at the Royal Institute of Technology (KTH) in Stockholm, Linköping University (LiU) in Linköping, Chalmers University of Technology (Chalmers) in Göteborg, and the Massachusetts Institute of Technology (MIT). The vision of the project is to provide students with an education that stresses engineering fundamentals set in the context of Conceiving-Designing-Implementing-Operating (CDIO) real-world systems and products.


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An earlier published paper from this study can be downloaded from www.cdio.org, where the CDIO project is carefully described in detail.


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The project strategy to implement CDIO has four themes:

1. curriculum reform to ensure that students have opportunities to develop the knowledge, skills, and attitudes to conceive and design complex systems and products

2. improved teaching and learning necessary for deep understanding of technical information and skills

3. experiential learning environments provided by laboratories and workshops

4. effective assessment methods to determine quality and improve the learning process.


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The CDIO Framework

Curriculum

Teaching & Learning

Workshops andLaboratories

Assessment

Engineering education


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Intended Outcomes of the CDIO Project

CurriculumTeaching

and Learning

Laboratories &

Workshops

Assessment

Program Models for curriculum

structure and design

Understanding and addressing

barriers to student learning

Models for the design and

utilization of labs/workshops

Tools and processes for

program evaluation

Student Experience

Curricular materials for

CDIO education

Active, experiential

learning with enhanced feedback

Workshop-based

educational experiences


assessing student

achievement


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Intended Outcomes of the CDIO Project

CurriculumTeaching

and Learning

Laboratories &

Workshops

Assessment

Program Models for curriculum

structure and design

Understanding and addressing

barriers to student learning

Models for the design and

utilization of labs/workshops


program evaluation

Student Experience

Curricular materials for

CDIO education

Active, experiential

learning with enhanced feedback

Workshop-based

educational experiences


assessing student

achievement


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Methods: faculty

The first year of the CDIO project was focused at teachers, administrators, and participating students at the four different universities:

Workshops with discussions about epistemological issues, learning theories, examples of taxonomies, writing measurable objectives, different assessment techniques, and so forth.


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Methods: students Two surveys about the algebra course, two surveys with the Force Concept Inventory (FCI).

Interviews consisting of concept questions, conceptual maps, discussions about course content, teaching & learning, assessment, and so forth.

Interviews Yr 1: First 15 students Algebra

(Yr 2 in CDIO) Second 10 students Analysis

Third 7 students Mechanics

Fourth 5 students Analysis

Visits in the lecture halls and in the classrooms.


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Students from each of the four institutions participate in the four theme areas, as well as contribute as a separate student group...

…in any change of any program, it is of extreme importance to involve the students in the whole process of change…

…to use the students as a way to quickly distribute and implement new ideas from the four themes…

…to improve students' conceptual understanding of technical subjects...

…to make the students more interested and aware of the objectives of their program…


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To understand a concept

We all just reach stages of understanding of a concept and there is no final understanding of any concept. (Vollrath 1994). Read also Vygotskij, Ausubel 1963, Hiebert and Lefevre 1986, Sfard 1991, Tall 1994, Confrey and Costa 1996, Novak 1998, among others.


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To think mathematically

Thinking mathematically is about developing habits of mind that are always there when you need them - not in a book you can look up later.


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What is a concept question?

Either ask a student about what she or he thinks about a concept, how they define a concept, etcetera.

Or:

Ask a student to solve a problem that involves conceptual thinking, i.e. not a routine question.

Note: What is a routine question for one student may very well be a concept question for another student.


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Is this an conceptual question?

For what values for a and b will the equations system AX = B have a parameter solution when

and ?

Determine the rank of the matrix A and for the (3 4) matrix [A B] for all values on a and b.

74

42

121

a

aA

b

bB

2

9


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Solve the magic square of order 3 by placing the numbers 1 - 9 in the given way. Try to do it in 5 minutes.

Try to find all magic squares of order 3 where all the partial sums are 0. Use real numbers.


0

0

0

0 0 0 0 0


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Solve the magic square of order 3 by placing the numbers 1 - 9 in the given way. Try to do it in 5 minutes.

Try to find all magic squares of order 3 where all the partial sums are 0. Use real numbers.

8 1 6 15

3 5 7 15

4 9 2 15

15 15 15 15 15



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Concept Maps(Mintzes, Wandersee, & Novak, 1998; Zeilik, 2000)

• Two-dimensional, hierarchical diagrams that show the structure of knowledge within a discipline

• Composed of concept labels, each enclosed in a box or oval, a series of labeled linking lines and general-to-specific organization.


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(A concept map of concept maps by Joseph Novak).CONCEPT

MAPS

KNOWLEDGE

REPRESENT

CONCEPTS

IS

PERCEIVEDREGULARITIES

ARE

OBJECTS

PROPOSITIONS

IS

COMBINETO FORM

HIERARCHIALLYSTRUCTURED

LABELED

WORDS

SYMBOLS

CONTEXT DEPENDENT

TEACHING

LEARNING

CROSSLINKS

CREATIVITY

INTERRELATIONSHIPS

DIFFERENTMAP SEGMENTS

IS

TO AID

IN

ARE

AIDSIS A BASIS FOR

MAY BE

TO SHOW

IN

NEEDED TO SEE

WITH

ARE

EVENTS


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Student response: Algebra (level 1)


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Algebra

HON base

Area The Room volume

The plane -- Rn

Coordinate systemorigin of coordinates

Change of base

Diff Equations ---- homogenousinhomogeneous

angles

RulesAntiderivativeDifferentiationDiff Equations

Integral

Calculus

Mean value theorem

Derivative

Limit

RulesGaussScalar productCramer

Vectors ---- scalars

Functions, graphs, asymptotes, maximum/minimum



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The third and forth map are more detailed and the third map has a more clear structure. What is evident from the fourth map, is that concept maps easily become too complicated and do not serve as a progression mirror any longer.

It seems necessary to complement with concept questions.


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1. How would you like to explain, describe, and relate the concepts function, continuity and differentiable?

2. Describe the appearance and behavior of the function below. Try to explain how you think about the problem.

3. Evaluate the calculation

Try to explain what criteria you use to do the evaluation.

4. What strategy would you use to calculate

1000)(

x

exf

x

3

3

2 )sin(3)cos()sin( dxxxxxx

211

1

1

1

1 2

x

dxx

Are these conceptual questions?


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Results so farGeneral tendencies:Changes in concepts is an individual process, the pace of students’ development is quite different.

One student can follow a course without changes in concepts while another seems to change substantially.

Students are vague and not enough specific when they try to explain how they understand a concept.

Knowledge that students express through a concept map seems to be lasting.

The complexity of the map and the impossibility in developing it further could be a criteria of “conceptual maturity”.


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Surprising resultsGeneral view or opinion: The mathematicians understand mathematics, engineering people use mathematics.

Results: In mathematics courses students learn techniques for calculation, in engineering courses they learn to understand the mathematics…

The major conceptual growth regarding mathematical concepts seem to occur during engineering courses…


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For more information, please feel free to contact me at

Thomas Lingefjärd <[email protected]>

Or go to www.cdio.org

Documents

Thomas Lingefjärd PME_NA Athens, GA, USA October 2002 1 Assessing the teaching & learning of mathematics in the mechanical engineering program at Chalmers