Smoothening the Integration of MATLAB into Undergraduate Engineering Mathematics

Prof Abel Nyamapfene Practice and Development

Introduction

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

  1. 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:
    1. It is useful for a technical job.
    2. It is useful for everyday living.
    3. It is culturally highly significant.
    4. It is leading to any of the above.
  2. Be aware of the prevailing student dispositions towards Maths and MATLAB (Cretchley, 2000; Tonkes et al., 2005):
    1. Students have a strong preference for mastery of basic concepts and techniques on paper before using MATLAB.
    2. Students will resist the introduction of MATLAB if they cannot surmount the initial hurdle of mastering the MATLAB syntax.
    3. Students will resist the introduction of MATLAB if they do not see its relevance in their future studies and work
  3. Highlight the relevance of MATLAB throughout the mathematics module ((Colgan, 2000; Majid et al, 2012; Tonkes et al, 2005):
    1. Explicitly mention the acquisition of MATLAB computational and visualisation skills in the aims and objectives of the module.
    2. Identify and include MATLAB skills in the module’s intended learning outcomes, including specifying how the attainment of such skills will be assessed.
    3. Use MATLAB to illustrate concepts within lectures so as to reinforce the message that it is an integral component of the module.
    4. 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.
    5. 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.
  4. Carefully consider your approach to blending MATLAB tuition into your lectures (Cretchley, 2000; Majid et al., 2012; Mathematics in Education and Industry, 2008;
    1. 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.
    2. 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.
    3. 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.
  5. Use a graded two-step approach to introducing mathematical concepts in lectures (Hosein, 2009)
    1. 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.
    2. 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.
  6. Use a scaffolding approach to teach MATLAB (Tonkes et al., 2005)
    1. Early in the course, place more emphasis on introducing single line commands at the MATLAB command interface.
    2. Follow this up by introducing visualisation and plotting using the MATLAB graphical user interface.
    3. 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.
    4. Finally encourage the students to develop their own programs without relying on pre-written programs.
  7. Build in MATLAB support into lectures, tutorials and assignments (Cretchley, 2000; Majid et al., 2012; Tonkes et al., 2005):
    1. Provide adequate technological support for MATLAB throughout the module.
    2. At the onset of teaching, provide students with hands-on training sessions to familiarise them with MATLAB and its syntax.
    3. For each topic area, provide a summary sheet containing lesson objectives, the necessary MATLAB commands for the topic area, and examples of their use.
    4. Within lectures emphasise mathematical problem-solving in your teaching, and enhance this through the use of MATLAB.
    5. 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.
  8. Provide continuity between lectures and supporting workshops (Majid et al., 2012):
    1. Introduce and explain mathematical concepts within lectures.
    2. 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.
    3. Within the workshop sessions, provide worksheets with similar examples and problems. Let students solve these problems manually and with MATLAB, as appropriate.
  9. Provide a structured sequence of workshop exercises that require increasing proficiency with MATLAB. (Majid et al., 2012, Tonkes et al., 2005):
    1. 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.
    2. 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.
  10. 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):
    1. Assessing MATLAB competence helps to motivate students to invest time in its use.
    2. Assess students understanding and performance using both MATLAB-based and paper-based assessments.

References

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.)

Reforming Engineering Mathematics Teaching: A Century Old Debate

Prof Abel Nyamapfene Engineering Education Research, Practice and Development , , ,

Enginmathematics-1550844-639x453eering mathematics is one of those subjects that we, engineering academics, hold particularly strong views about. We worry specifically about how much content we ought to teach, and how to carry out that teaching. It is therefore not surprising that some engineering departments end up taking charge of their own teaching of mathematics. This “taking charge” includes specifying exactly, and policing, any teaching carried out on their behalf by mathematics departments, and in general, it may end up with engineering departments undertaking their own mathematics teaching.

Academics within individual engineering departments hold quite consistent views regarding the content and approach to the teaching of mathematics. This commonality of opinion is largely independent of academic variables such as rank, seniority, alma mater, or personal position on the research-teaching nexus.  It would therefore not be amiss to suggest that mathematics teaching contributes to, and is, in tMinolta DSCurn, affected by the prevailing culture within engineering. Indeed an Australian study on engineering education culture suggests that mathematics contributes to an “engineering way of thinking” by shaping the manner in which truth and reality are proven and described. Similarly, a  UK-based study on the role of mathematics in engineering education suggests that in addition to the direct usefulness of mathematics to engineering education and practice, it also plays “a formative role in the development of an engineer.” An important implication of this is that the content and approach to mathematics teaching plays a significant role in the transformation of the student from being a novice to a full-fledged practising engineer.

Schein defines culture as “… a pattern of shared basic assumptions that was learned by a group as it solved its problems of external adaptation and internal integration, that has worked well enough to be considered valid and, therefore, to be taught to new members as the correct way to perceive, think, and feel in relation to those problems.”

Implicit in this definition is that a group of people, in this case engineering practitioners and educators, are likely to persist in ways of conduct, and in hoengineer-1440190lding certain norms and values despite the application of external pressures to change. At UCL, as part of a faculty-wide, multi-disciplinary curriculum redesign of our undergraduate engineering programmes, we are currently introducing a revised engineering mathematics curriculum that places an emphasis on mathematical modelling and analysis. Our primary motivation for this curriculum change is that engineering students need to master the fundamental mathematical concepts that underpin engineering education and practice, as well as to gain the ability to apply these concepts to solving engineering problems.

Our motivation for the redesign of the mathematics curriculum is not unique to ourselves alone, as several other institutions are undertaking the same changes as well. Not only that, it appears that even a hundred years ago, progressive engineering educators and practitioners shared broadly the same concerns. In 1908, the American Association for the Advancement of Science published a paper entitled “The Teaching of Mathematics to Students of Engineering” that raised most of the concerns that we are grappling with today. It may be that the writers of this paper were far ahead of their time, which is unlikely. What is more likely, however, is that the culture of mathematics teaching within engineering is quite resilient, despite clamours for change. Below, I point out some of the issues raised in this paper.

1.      The importance of mathematics to the practice of engineering

Even a century ago, mathematics was seen as being very important to the practice of engineering. This view is neatly summed up by the statement:  “Mathematics is to an engineer what anatomy is to a surgeon, what chemistry is to an apothecary, what the drill is to an army officer. It is indispensable.”

2.      Criticism of existing teaching approaches

Even then, engineering mathematics was mainly taught without reference to engineering practice. Just as today, progressive academics and practitioners felt that such an approach led to students who only knew “… how to juggle with quantities in order to produce certain results.” This meant that engineering professors in other subjects had to teach again the material covered in pure mathematics classes.

3.      The objectives of teaching mathematics to students of engineering

Then, as now, it was widely accepted that mathematics is a tool for the study and practice of engineering. It was felt that teaching should “be such that the student knows the why and wherefore of each operation-in other words, that he learns to think mathematically.”

4.      Recommended approaches to teaching mathematics

The main method of teaching then, as now, was the monologue lecture method. Then, as now, this method was seen as inappropriate. It was felt that active learning methods were far much better at teaching mathematics as opposed to  “having another man’s mind do the reasoning for him”, as implied by the lecture method. In addition, students had to put time to work: “Midnight oil and the damp towel are for most students necessary accessories to the courses in pure mathematics.” This is consistent with today’s insistence on self-directed independent learning on the part of the students.

Secondly, it was felt that teachers needed to supplement analytical solutions with graphical demonstrations. This is consistent with an insistence on modelling and simulation in today’s engineering education practice.

5.      Who should teach mathematics

As a general rule, it was felt that engineering students were of average mathematical abilities. It was therefore felt that rather than relying on inexperienced instructors as was the norm, competent teachers should be used in the teaching of mathematics to engineering students.

Whilst some felt that teaching should be undertaken by engineers, this was not considered to be a necessary condition. Rather, effective teaching could be carried out by any competent teacher, so long as he was familiar with the potential applications to which the students would put mathematics to use in their later studies and careers.

6.      Achieving a balance between content and depth

As is the case today, teachers had to achieve an appropriate balance between the amount of content to teach and the depth to which teaching had to be undertaken. Mastery of fundamental mathematical concepts and principles was more preferable to teaching a wider range of topics at the expense of depth. It was felt that if students concentrated on a few key topics, then they would have sufficient depth to be able to take on any novel topics in the future.

7.      Pre-engineering school knowledge of mathematics

Then, as now, both academics and practitioners felt that the level and quality of mathematics teaching in high schools was inadequate. This had implications on what could be taught in engineering school as any efforts to implement remedial teaching of mathematics would have knock-on effects on other subjects making up the engineering curriculum.

Concluding Remarks

This discussions clearly indicates that concerns with the teaching of mathematics within engineering have remained largely unchanged over the past century. In addition, the solutions proposed for curriculum reform have been remarkably consistent, despite the passage of time. This is not to say that people have not tried before. Rather, it may be that curriculum reform has failed to take root primarily because of failure to take into account the cultural and other intangible aspects surrounding engineering education and practice.

Re-engineering Mathematics Teaching Within Engineering – Preliminary Reflections

Prof Abel Nyamapfene Reflections

https://orcid.org/0000-0001-8976-6202

Background

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We are now into the second year of introducing a revised mathematics curriculum for undergraduate engineering programmes at UCL.  This is part of a faculty-wide, multi-disciplinary curriculum redesign of our undergraduate engineering programmes. The primary purpose of this redesign is to enhance the student experience by introducing project-based activities across the degree programmes. These activities will run from the first year of the study programme, right up to the final year of study. Our aim in so doing is to ensure that from the first day that students enter our degree programmes, they will be able to study and use mathematics and engineering science in the context of engineering problem-solving.

Positioning Mathematics Teaching Within Engineering Education

Engineering is closely tied to economic and technical development. Prior to the 19th century engineering existed largely as a practice-based vocation passed down from generation to generation primarily through on the-job learning. However, starting from the latter half of the 19th century, university education increasingly became an important entry route into engineering as it became more professionalised.   A key consequence of this move to professionalise engineering was that in addition to practice-based education, science and mathematics became central to engineering education.

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However, determining an appropriate balance between practice and science in engineering education is problematic. Since the early twentieth century debates have raged over how much practice and how much theory to include in engineering education. Prior to the First World War, engineering education in the UK leaned more towards a higher practical content. However, over the years, the pendulum has swung towards a higher theoretical content. Lately, however, engineering employers and students have begun to demand for more practical content within engineering education. The pendulum is therefore swinging back towards more practical content in engineering education. Where does this leave engineering mathematics then?

At UCL we subscribe to the notion that mathematics is central to both engineering education and practice. We are of the viewpoint that engineers need to master mathematical concepts and to be able to apply these concepts to solving engineering problems. However, studies indicate that students often find it difficult to apply their knowledge of mathematics to real world problems. To alleviate this, we have integrated mathematical modelling and simulation in our teaching. To reinforce the link between mathematics and engineering practice, we now use the term “mathematical modelling and analysis” to refer to our mathematics modules.

Our Revised Approach to Teaching Mathematics to Engineers

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We use a blended approach in our delivery of the mathematical modelling and analysis modules. In addition to face-to-face lectures and weekly workshops, we have provided an online suite of mathematical support resources. This includes pre- and post- lecture online quizzes, lecture notes, and MATLAB simulations and demonstrations.  Lectures and workshops are delivered by engineering academics, and wherever possible, examples of current academic research and practice are used to illustrate key mathematical concepts. One outcome of this approach is that mathematics seizes to become a “dry” subject as students begin to see its utility in everyday research and practice. In addition, students get to know and interact with ongoing academic research within engineering.

Students enter into engineering with significant differences in prior mathematical knowledge and competence. In addition, students have different learning and mastery rates. We have introduced a pre-course mathematical quiz to assess individual competence levels in key elementary areas such as calculus and algebra. An additional weekly class has been created to provide additional student support.  Throughout the course, students also have access to a walk-in, student-led support team. This helps to encourage peer-to-peer learning, and to establish connections between undergraduate and postgraduate students.

A Preliminary Assessment

Has this been a walk in the park? Certainly not. Combining theory and modelling in mathematics has been a challenge for both academics and students. For the academics, the main issue has been deciding what content to include and exclude given the constraints of time. In addition, breaking down research to a level where the students can understand and engage with it mathematically requires careful thought. For the students, the main challenge has been the demand for more independent work, both prior and after lectures. In addition, developing a working knowledge of MATLAB has been a challenge.

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Despite these challenges, the emerging results  are encouraging. Compared to previous cohorts at the same stage, students from this cohort appear to have a deeper awareness of their engineering disciplines, and a greater appreciation of the research taking place within the faculty. The module is more activity oriented than the previous module. Amongst the students this is helping to foster a higher level of self-direction and independence. In addition, students show a greater willingness to engage with teaching staff, and a markedly higher assertiveness when it comes to feedback and demands for quality learning. With regard to academics, this module is helping to foster collaboration in curriculum development and teaching between different engineering disciplines. Such interaction between departments can only be good for both the research and teaching within the faculty. Most importantly, the introduction of the module has encouraged academics to adopt a more critical approach to their own teaching. This is leading to greater academic interest in learning and teaching approaches and technologies.

Concluding Remarks

So what are my interim conclusions? The journey has been demanding for both students and academics. But it has been fruitful, and it has generated an excitement and a buzz unlike previous years. Most importantly, the module appears to have increased the engagement of both students and academics. And so, to the next year, here we come.

Formerly my Students, Now my Teachers and Fellow Colleagues in Technology

Prof Abel Nyamapfene Reflections

Almost daily on my commute to and fro UCL Bloomsbury Campus in Central London, I get to meet at least one of my former students. SometimesDCF 1.0 we nod at each other, say one or two pleasantries, and rush off in our different directions. But sometimes we get talking, andsometimes this little chit chat leads to a coffee, and yet more talking. And sometimes this talk-over-coffee leads to heated arguments about technology, university education, and the rights and wrongs of our current approaches to Engineering, and sometimes it focuses on my own personal philosophies and approaches to Engineering Education. No holds barred! I love an argument, I thrive on intellectual argument, and my students know this.

Indeed, London has a strange way of re-uniting academics and their former students. That’s the beauty of London. If you are a long-serving academic, then it is highly likely that there is always a former student within shouting distance of you. Five years at the University of Zimbabwe, four years at the Catholic University in Zimbabwe, six years at Exeter, and 15 months at Bath means that I have a network of former students reaching into all parts of today’s technological frontier. Some are in cutting-edge computer programming, some in telecoms, some in power systems and some in technology consultancy, and even some in banking – I mean that form of banking where they daily roll out these wickedly complex mathematics to drive the world financial systems. I get to hear it all from my students.

I lovingly call them “my students”, but the roles have changed – they are now my teachers, and I am now the student. Not only a student, but the acquisitive, eager, and purposeful student learning the new so as to teach the latest batch of students. Being current, being in the know, especially in things technological is a must for an Engineering academic. But each and every day new technologies are coming online. In fact so fast is the technological innovation in some sectors that yesterday’s “latest” so easily becomes today’s “obsolete”.

Books and publications now struggle to keep pace with technological change. Only those living and working on the technological frontier can hope to keep pace. So is there any hope for an academic like me who has to deal with day-to-day teaching and administration, and various other things that constitute academic life? No hope in hell, you can say.  But that’s where networks of former students come in. For me, these networks are a living encyclopaedia.

As a current asocial-connection-1624773cademic, I am embedded into multiple former student networks, and also immersed into my own academic networks, including my current students. I now live in a world of networks.  I have seized to be a source of knowledge. Instead, I am now a channel of knowledge. I now serve to direct “state of the art” technological knowhow into academia, and to keep my former students connected to higher education, and connected to all my other former students. I am now a node of connectivity.

And when teaching gets tricky, as it sometimes does in such areas like computer programming, project management and software engineering, how do I get by? I can spend a week cramming the latest, so that I can download it onto my innocent class. I still do that, but not always. The best teacher is the one who has experienced it all – after all, experience is the best teacher.  So increasingly, I find myself bringing along my former students to co-teach with me.  And some are better teachers than myself. But we all stand to gain – the current students get to learn from the best, I get good student returns, and my former students get to know what it feels like to be a lecturer.  Their organisations get to be known by the students, and links between the university and industry are strengthened.

But it’s not like my new students are just a sink hole for knowledge. There is no better source of “flash of the bulb” inspiration than the eager minds of students.  Many a time, after having delivered what they thought to be the Oomph Lecture, I have witnessed a visiting lecturer seriously questioning their approaches to technology. An insightful thought, innocently thrown at the visiting lecturer, is enough to convince anyone that we are now all a community of learners. Current students learn from us, I mean from both the full-time academics and the industry experts, and, in turn, we all learn from the current students, and together we build the future. That’s what makes Engineering Education so interesting for me. I am always in the game, but only so long as I stay connected to both my academic and former student networks.