Many mathematics educators are aware of the challenge of integrating computational components into a classical first year mathematical curriculum. The diversity of student backgrounds, large class sizes, resource limitations and the fact that the curriculum is already full provides for difficulties. Even worse, teaching efforts can be rejected by students who perceive that the computational components are extraneous add-ons that are irrelevant to the core material. On top of this, the concept of ‘computational components’ is sufficiently broad in first year mathematics courses to induce debate amongst educators. (Tonkes et al, 2005)
Regrettably, an analysis of recent efforts to introduce computer algebra systems (CAS) into engineering mathematics reveals that we are still faced with the same issues that Tonkes et al. identified by in 2005. In this blog I present a ten-step proposal to smoothening the incorporation of MATLAB teaching within a typical first year undergraduate engineering mathematics module. Although my focus is on MATLAB, I believe that these ten steps are also relevant to other CASs, including Maple and Mathematica.
The Ten Steps
- Have some criteria for deciding which topics to include and to leave out of an Engineering Mathematics module when you introduce MATLAB. The integration of MATLAB into first year mathematics increases the dilemma of selecting which topics to leave out of mathematics modules to make room for MATLAB teaching. When it comes to topic selection, Wolfram (2008) suggests that a topic should only be included in a module if:
- It is useful for a technical job.
- It is useful for everyday living.
- It is culturally highly significant.
- It is leading to any of the above.
- Be aware of the prevailing student dispositions towards Maths and MATLAB (Cretchley, 2000; Tonkes et al., 2005):
- Students have a strong preference for mastery of basic concepts and techniques on paper before using MATLAB.
- Students will resist the introduction of MATLAB if they cannot surmount the initial hurdle of mastering the MATLAB syntax.
- Students will resist the introduction of MATLAB if they do not see its relevance in their future studies and work
- Highlight the relevance of MATLAB throughout the mathematics module ((Colgan, 2000; Majid et al, 2012; Tonkes et al, 2005):
- Explicitly mention the acquisition of MATLAB computational and visualisation skills in the aims and objectives of the module.
- Identify and include MATLAB skills in the module’s intended learning outcomes, including specifying how the attainment of such skills will be assessed.
- Use MATLAB to illustrate concepts within lectures so as to reinforce the message that it is an integral component of the module.
- Make continual references to MATLAB in all aspects of the modules, including using MATLAB to generate any plots and graphs used in lecture notes, and making students aware of this.
- Ensure that students perceive MATLAB as integral to the mathematics module. Students need to view MATLAB as a necessary accompaniment to the mathematical content, and not just as an add-on.
- Carefully consider your approach to blending MATLAB tuition into your lectures (Cretchley, 2000; Majid et al., 2012; Mathematics in Education and Industry, 2008;
- Carefully design each lecture so as to explicitly identify the specific points in the lecture at which specific MATLAB concepts and illustrations should be introduced.
- Blend mathematics concepts and MATLAB techniques to ensure that students are able to master the mathematical concepts that are being taught, and to effectively apply them to mathematical modelling.
- Use MATLAB to extend and support the learning of mathematical concepts, and seek to reduce, rather than to replace, the manual manipulation of mathematical concepts.
- Use a graded two-step approach to introducing mathematical concepts in lectures (Hosein, 2009)
- First emphasise the mastery of the necessary procedural steps to enable students to master the mathematical concepts and to enable them to think mathematically, i.e. in terms of mathematical symbols.
- Then, only after the students have become accustomed to the terminologies and procedural steps, introduce MATLAB to enable the students to focus on applying the new mathematical concept to solving problems without being bogged down by computational details.
- Use a scaffolding approach to teach MATLAB (Tonkes et al., 2005)
- Early in the course, place more emphasis on introducing single line commands at the MATLAB command interface.
- Follow this up by introducing visualisation and plotting using the MATLAB graphical user interface.
- Gradually introduce basic MATLAB programming through the use of examples. Do this by providing pre-written MATLAB programs that emphasise the core material. Encourage the students to adapt and extend the program code to other related examples.
- Finally encourage the students to develop their own programs without relying on pre-written programs.
- Build in MATLAB support into lectures, tutorials and assignments (Cretchley, 2000; Majid et al., 2012; Tonkes et al., 2005):
- Provide adequate technological support for MATLAB throughout the module.
- At the onset of teaching, provide students with hands-on training sessions to familiarise them with MATLAB and its syntax.
- For each topic area, provide a summary sheet containing lesson objectives, the necessary MATLAB commands for the topic area, and examples of their use.
- Within lectures emphasise mathematical problem-solving in your teaching, and enhance this through the use of MATLAB.
- Within workshops build students’ confidence and overcome the majority of initial syntax problems by including introductory MATLAB examples. Use a step by step approach to demonstrate each example, and let the students reproduce each of these steps prior to attempting similar problems.
- Provide continuity between lectures and supporting workshops (Majid et al., 2012):
- Introduce and explain mathematical concepts within lectures.
- Work out examples in lectures, first going through the steps manually, and then with MATLAB, and then have the students to do some in-class problems.
- Within the workshop sessions, provide worksheets with similar examples and problems. Let students solve these problems manually and with MATLAB, as appropriate.
- Provide a structured sequence of workshop exercises that require increasing proficiency with MATLAB. (Majid et al., 2012, Tonkes et al., 2005):
- In each workshop exercise sheet provide a summary of the key mathematical concepts covered in the topic, supporting MATLAB commands, a sequence of examples, and lastly the student exercises.
- Prior to issuing out each workshop sheet, confirm that it fits into the workshop time duration, and have its difficulty and accuracy validated by an external party.
- Design module assessments that reflect the inclusion of MATLAB (Colgan, 2000; Majid et al., 2012; Mathematics in Education and Industry, 2008; Tonkes et al., 2005):
- Assessing MATLAB competence helps to motivate students to invest time in its use.
- Assess students understanding and performance using both MATLAB-based and paper-based assessments.
Colgan, L. (2000). “MATLAB in first-year engineering mathematics.” International Journal of Mathematical Education in Science and Technology, 31(1), 15-25.
Cretchley, P., Harman, N., Ellerton, G., and Fogarty. (2000). “MATLAB in early undergraduate mathematics: An investigation into the effects of scientific software on learning.” Mathematics Education Research Journal, 12(3), 219-233.
Hosein, A. (2009). Students’ approaches to mathematical tasks using software as a black-box, glassbox or open-box, PhD thesis, The Open University.
Majid, M. A., Huneiti, Z. A., Al-Naafa, M. A., and Balachandran, W. “A study of the effects of using MATLAB as a pedagogical tool for engineering mathematics students.” Presented at Interactive Collaborative Learning (ICL), 2012 15th International Conference.
Mathematics in Education and Industry. (2008). Computer Algebra Systems in the Mathematics Curriculum – Report of the invitation MEI seminar supported by Texas Instruments.
Tonkes, E. J., Loch, B. I., and Stace, A. W. (2005). “An innovative learning model for computation in first year mathematics.” International Journal of Mathematical Education in Science and Technology, 36(7), 751-759.
Wolfram, C. (2008). “Building a curriculum ground-up with computer maths – a new vision, new challenges “, in Mathematics in Education and Industry, (ed.), Computer Algebra Systems in the Mathematics Curriculum – Report of the invitation MEI seminar supported by Texas Instruments. Mathematics in Education and Industry.)