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

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

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

THE CDIO projectConceive-Design-Implement-Output An earlier published paper from this study can be downloaded from www.cdio.org, where the CDIO project is carefully described in detail. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

THE CDIO projectConceive-Design-Implement-Output 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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

The CDIO Framework Engineering education Curriculum Workshops and Laboratories Teaching & Learning Assessment Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

Curriculum Teaching 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 Tools and processes for assessing student achievement Intended Outcomes of the CDIO Project Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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… Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

Is this an conceptual question? 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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

(A concept map of concept maps by Joseph Novak). Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

Student response: Algebra (level 1) Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

Student response: Algebra (level 4) Change of base HON base Coordinate systemorigin of coordinates Algebra RulesGaussScalar productCramer Vectors ---- scalars angles The plane -- Rn Area The Room  volume Mean value theorem Integral Limit Calculus RulesAntiderivativeDifferentiationDiff Equations Diff Equations ---- homogenous inhomogeneous Derivative Functions, graphs, asymptotes, maximum/minimum Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

Are these conceptual questions? 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 Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

Results so far General 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”. Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

Surprising results General 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… Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

For more information, please feel free to contact me at Thomas Lingefjärd <Thomas.Lingefajrd@ped.gu.se> Or go to www.cdio.org Thomas Lingefjärd PME_NA Athens, GA, USA October 2002

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