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Although problem based learning (PBL) has been around for almost 50 years, following its introduction and development in medical education at McMaster in the late 1960s and early 1970s, it has remained relatively unknown until fairly recently. Putting this into perspective, in the period 2000 – 2010, very few engineering academics had ever heard of PBL – fast forward to today, and PBL has become almost synonymous with the term “engineering education”.

However, despite the ubiquity of PBL within the engineering academic community, it remains a challenge to come across academics who are comfortable in effectively implementing PBL in their own practice. A key reason for this may be the failure of academic development centres to keep pace with the rapid adoption of 21^st century-focussed curricula within engineering education, and within higher education in general.  Another contributory factor may be the prevailing, and thoroughly misplaced, belief that academics do not really need any pedagogic training at all.

For the engineering academic needing to get up to speed with PBL-oriented pedagogic practices, it may be a challenge just to have an idea of where to turn to. In this piece, I highlight some of the online resources that an individual can access.

1. Problem based Learning: Northern Illinois University, Faculty Development and Instructional Design Center – This is a short guide that explains what PBL is, how it differs from traditional teaching, and how you can get started implementing PBL in your own teaching.
2. Introductory Course On PBL In Higher Education – Free Online Course: Aalborg Centre for Problem Based Learning In Engineering Science And Sustainability Under The Auspices Of UNESCO (UCPBL) – A free online course that gives an overview of the PBL process as well as links to pertinent resources.
3. The Aalborg Experiment: Project Innovation in University Education. This is an evaluation of PBL implementation at Aalborg University carried out in 1994. Although somewhat dated, it gives important insights into the practical issues that need to be addressed if PBL implementation is to be successful.
4.  Problem and Project Based Learning: Goodhew, Peter. “Teaching engineering.” The School of Engineering’s Active Learning Lab at The University of Liverpool(2010). In section  5.4-5.6 of this textbook, which is available for download, Peter Goodhew gives pertinent instructional advice on how one can get started on problem and project based learning. He also provides relevant engineering examples that one can try out.
5. Problem-Based Learning:Stanford University, Speaking of Teaching Newsletter Archive. This is a short guide on PBL that provides guidance on designing problem sets, structuring PBL classes, and offers advice on assessment design.
6. The Tutor in Problem Based Learning: A Novice’s Guide: This is a fairly comprehensive guide from McMaster University outlining how you can design and implement PBL. It also provides common problems that arise in PBL implementation and offers suggestions on resolving them.
7. Revolutions and Re-iterations: An Intellectual History of Problem-based Learning: Virginie Servant PhD Thesis – Who were the “Founding Fathers” of PBL? What were the issues and arguments they had to grapple with? How did PBL come to be the way it is today? If you are grappling with any of these questions, this thesis is your best starting point.

It is no longer enough for higher education to focus just on the transmission of information and the retention of facts. Rather, we now expect students to come out of higher education equipped with a range of high-level skills and abilities such as:

• critical thinking skills to enable them to handle and interpret concepts, evidence and ideas
• the ability to think and act as experts
• innovation and creativity to enable them to produce original insights and valuable knowledge for the benefit of society (Imperial College, 2019).

In addition to adopting learning and teaching techniques that foster the development of these high-level skills, we also need to transform higher education assessment practices as well. Within engineering mathematics, where the end of year exam still predominates, this requires careful selection and design of exam questions to ensure students are assessed for competence in both the basic and high-level mathematical skills. In this piece, I look at two taxonomies that aid the development of such exam questions. These two are the Mathematical Assessment Task Hierarchy (MATH) taxonomy (Smith et al., 1996) and  Galbraith and Haines’ Mechanical-Interpretive-Constructive taxonomy (Galbraith and Haines, 2000).

The MATH taxonomy

The MATH taxonomy groups mathematical skills into three groups, A, B, and C.

Group A skills

Smith et al defined Group A skills as the standard, routine procedures taught to students along with the factual information they have to recall. They further divided this group into three categories:

• Factual knowledge – the ability to recall previously learned information in the way in which it was given
• Comprehension – the ability to use standard techniques to solve a problem. This includes the ability to recognise symbols in a formula and to substitute into the formula using information learnt previously.
• Routine procedures – the ability to use routine procedures to solve a given problem in the same way previously learnt to solve similar problems. This requires students to have worked on similar problems beforehand.

Group B skills

Group B skills focus on the ability to use mathematical information in new ways, for example, applying routine procedures to new situations. These skills fall into two categories:

• Information transfer – includes the ability to transform mathematical information from one form to another, for example from verbal to numerical (or vice versa), or from algebraic to graphical etc. This category also includes the ability to explain the relationships between component parts of a mathematical problem, the ability to recognise whether or not a particular formula or method can be used in a new context. In addition, this category also includes the ability to explain mathematical processes, including summarising in non-technical terms for non-mathematical audiences.
• Application in new situations – the ability to choose appropriate methods and information and apply them to new situations. This includes modelling real life situations, extrapolating new procedures to new situations, and proving previously unseen theorems and results.

Group C skills

Group C skills are the skills that enable students to apply previously learned concepts to the analysis and solution of mathematical problems for which no routine procedures have been provided. These skills fall into three categories:

• Justifying and interpreting – the ability to justify and/or interpret a given result, or a result derived by the student. This includes proving a theorem in order to justify a result, the ability to find errors in reasoning, recognition of unstated assumptions, and recognising the limitations of a model and being able to decide on the appropriateness of a model.
• Implications, conjectures, and comparisons – the ability to make comparisons, with justifications, in different mathematical contexts, as well as the ability to draw implications and make conjectures, with justification, when given or having found a result.
• Evaluation – the ability to judge the value of material for a given purpose based on definite criteria. This includes the ability to make judgements, the ability to select for relevance, the ability to argue coherently on the merits, or otherwise, of an algorithm, organisational skills and the ability to rearrange information and draw previously unseen implications from it.

The Galbraith & Haines’ taxonomy

Galbraith and Haines arrived at their taxonomy independently of the Smith et al MATH taxonomy. However, after reviewing both taxonomies they suggested that their taxonomy could be interpreted as a summary of the MATH taxonomy (Galbraith and Haines, 2000).

Like the MATH taxonomy, the Galbraith & Haines taxonomy also has three levels, although they have a different terminology. The three levels are the mechanical, interpretive and constructive levels.

Mechanical skills

These skills refer to the routine use of mathematical procedures as cued by the wording of the question. They are equivalent to Group A skills in the MATH taxonomy.

Interpretive skills

These skills refer to the ability to retrieve and apply conceptual knowledge. They are equivalent to Group B skills in the MATH taxonomy.

Constructive Skills

These skills refer to the ability to arrive at a solution or conclusion using a range of mechanical and interpretive tasks without the necessary guidance for doing so. This essentially involves the construction of a solution rather than simply selecting between given alternatives. These skills are equivalent to the Group C skills on the MATH taxonomy.

Concluding remarks

Most mathematics exams only assess Group A skills, with only a few assessing Group B skills as well, and virtually none assessing Group C skills (Brown, 2010). However, Pounteny et al (2002)  suggest that we should be teaching undergraduate students up to the level of Group C as this is the skill level at which students are able to demonstrate “understanding to the point of justifying and explaining knowledge, being able to evaluate actions, and the development of new knowledge”. These, according to Pounteny et al, are the skills that we associate with being a mathematician and a problem solver. Moreover, it is clear that these are the skills expected of higher education graduates in the 21^st century. Hence, to ensure that the end-of-year exam remains relevant in the 21^st century,  we have to ensure that a significant proportion of the exam questions is  pitched at the Group C / Constructive level.

References

BROWN, R. G. 2010. Does the introduction of the graphics calculator into system-wide examinations lead to change in the types of mathematical skills tested? Educational Studies in Mathematics, 73, 181-203.

GALBRAITH, P. & HAINES, C. 2000. Conceptual mis (understandings) of beginning undergraduates. International Journal of Mathematical Education in Science and Technology, 31, 651-678.

IMPERIAL COLLEGE. 2019. Project Xeper -The future of engineering teaching [Online]. Imperial College,.  [Accessed 28/07/2019 2019].

POUNTNEY, D., LEINBACH, C. & ETCHELLS, T. 2002. The issue of appropriate assessment in the presence of a CAS. International Journal of Mathematical Education in Science and Technology, 33, 15-36.

SMITH, G., WOOD, L., COUPLAND, M., STEPHENSON, B., CRAWFORD, K. & BALL, G. 1996. Constructing mathematical examinations to assess a range of knowledge and skills. International Journal of Mathematical Education in Science and Technology, 27, 65-77.

 

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

Abstract for paper presented at the 8th Research in Engineering Education Symposium (REES 2019) held in Cape Town, South Africa, from 10–12 July 2019. The extended abstract can be downloaded HERE:

CONTEXT

Traditionally, mathematics and engineering principles course modules constitute the bulk of undergraduate teaching in the first and second year of the engineering curriculum. These introductory course modules play a critical role in the study of engineering by laying the foundational engineering and mathematics principles that underpin the study of more advanced engineering modules in the latter stages of the undergraduate programme. However, students often perceive these foundational modules as being disconnected from the “real” engineering that they came to study, and this often leads to heightened student dissatisfaction and disengagement from further engineering studies. To alleviate this situation, most engineering education researchers and educators have advocated the adoption of computational modelling within these foundational engineering course modules.

PURPOSE

MATLAB has been an integral part of teaching in first and second year engineering mathematics at University College London since 2013. This has improved student engagement with mathematics and raised their awareness of its relevance within their chosen engineering disciplines, and subsequently led to the adoption of MATLAB in other introductory course modules.. However, this adoption has not been uniform.  In the study that I report in this paper, I set out to establish the reasons why MATLAB adoption patterns are so variable amongst first and second year introductory engineering course modules. Specifically, I sought to establish the pedagogic and learning environment structural features that facilitate or impede the successful adoption of MATLAB as a teaching tool within the early-stage foundational engineering course modules.

APPROACH

To address these research questions, I adopted a multi-dimensional case study approach that incorporated interviews with students and lecturers in the first and second year of undergraduate engineering across five engineering departments, peer observation of teaching, evaluation of student performance and feedback, and interviews with programme leaders and directors of education. To facilitate this analysis, I framed the study as an innovation diffusion case study. This enabled me to evaluate the data and to frame the outcomes using a variety of innovation diffusion models that have been applied to the study of curriculum change in various educational establishments.

RESULTS

Whilst the study is still very much a work-in-progress, key findings that are emerging are largely consistent with emerging views of innovative learning diffusion within educational practices. Specifically, my findings point to the need for effective learning support mechanisms that incorporate both top-down and bottom-up approaches to driving innovation. The study also highlights the detrimental effects of institutional focus on disciplinary research on the adoption of innovation in learning and teaching practice. In addition, the study also highlights the need to open up the undergraduate education landscape to emerging educational practitioners such as learning technologists and education administrators.

CONCLUSIONS

The teaching and delivery of first and second year undergraduate engineering course modules can be substantially improved by incorporating computational modelling, visualisation and analysis techniques. However, this adoption process is not straightforward, and requires cultural and structural changes within engineering schools if it is to be effective. For this to happen, institutional leaders have to demonstrate a willingness to go beyond rhetoric by actively advocating and supporting the adoption of innovative practices within the undergraduate engineering curriculum.

KEYWORDS

Computational modelling and analysis, visualisation, undergraduate engineering education

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

Mathematics can pose significant challenges and obstacles to students who are in the early phases of their studies at university (Hernandez‐Martinez and Williams, 2013). This also includes engineering students, even though they, along with the rest of the engineering community, believe that the study of mathematics is indispensable to the study and practice of engineering (Sazhin, 1998). Students find the transition to university mathematics difficult for a number of reasons. One of these reasons is the lack of mathematical resilience, the subject of my discussion today. For this, I shall be relying on the work of Sue Johnston-Wilder and Clare Lee, who have popularized the concept of mathematical resilience within pre-university mathematics education.

Mathematical resilience

An individual is said to be resilient if they achieve “good outcomes in spite of serious threats to adaptation or development”(Masten, 2001). Previously, educators and researchers believed that resilience was an attribute that only a few people possessed. However, Masten’s studies have since debunked this notion. Instead, her studies posit that, far from being exclusive, resilience is an ordinary trait that any individual can acquire. This new understanding renders invalid prevailing deficit models of intellectual ability. Indeed, using this new understanding, Yeager and Dweck (2012) have shown that the resilience of students improves when they are redirected to see intellectual ability as “something that can be developed over time with effort, good strategies, and help from others.”

Johnston-Wilder and Lee use the construct of mathematical resilience to describe “a learner’s stance towards mathematics that enables pupils to continue learning despite finding setbacks and challenges in their mathematical learning journey” (Johnston-Wilder and Lee, 2010). Often, learning mathematics is associated with negative emotions, such as avoidance, anxiety and helplessness (Lee and Johnston-Wilder, 2017). These negative emotions can be exacerbated by poor mathematics teaching methods, enduring cultural beliefs that mathematics is only for the gifted few, and fixed mindset beliefs that if someone is not good at mathematics, then there is no way they can improve their mathematical ability, even if there is good teaching and support (Lee and Johnston-Wilder, 2017). Mathematical resilience challenges these negative notions, and seeks to engender within the minds of learners the knowledge that that they can grow their mathematical abilities (Lee and Johnston-Wilder, 2017).

Characteristics of mathematically resilient students

There are four characteristics for students with mathematical resilience. These are growth mindset, personal value of mathematics, knowing that mathematics requires struggle and knowing how to recruit support in pursuing (Lee and Johnston-Wilder, 2017).

Growth mindset (Yeager and Dweck, 2012): The notion of growth mindset suggests that the brain is malleable. Consequently, students with a growth mindset believe that their mathematical abilities improve with practice, persistence and support from others (this includes peers and academic staff).

Value: Mathematically resilient students believe that mathematics is a valuable subject and is worth studying. Such values can be fostered within engineering by making explicit the connections between mathematics and engineering theory and practice.

An understanding of how to work at mathematics: Mathematically resilient students are aware that one has to persevere if they are to make progress in learning mathematics. They are aware that everyone experiences difficulties and challenges when learning mathematics; that mistakes are part of the learning process, and that one has to persevere and learn not to succumb to the negative emotions that are a part of learning something new.

Knowing how to gain support: Mathematically resilient students are aware that learning mathematics can be challenging, and that sometimes they need to seek support in order to get ahead. They are also aware of the support structures for mathematics within and outside their university. They are also aware, and adept, at constructively working and studying with each other.

Lessons for mathematics educators

Teaching mathematics is not just about delivering subject content. It is also about developing a supportive, inclusive, collaborative learning environment that challenges and encourages students to do the best they can. Indeed, achieving this is the holy grail of learning and teaching in engineering mathematics.

References

HERNANDEZ‐MARTINEZ, P. & WILLIAMS, J. 2013. Against the odds: resilience in mathematics students in transition. British Educational Research Journal, 39, 15.

JOHNSTON-WILDER, S. & LEE, C. 2010. Mathematical Resilience. Mathematics Teaching, 218, 38-41.

LEE, C. & JOHNSTON-WILDER, S. 2017. Chapter 10 – The Construct of Mathematical Resilience. In: XOLOCOTZIN ELIGIO, U. (ed.) Understanding Emotions in Mathematical Thinking and Learning. San Diego: Academic Press.

MASTEN, A. S. 2001. Ordinary magic: Resilience processes in development. American psychologist, 56, 227.

SAZHIN, S. 1998. Teaching mathematics to engineering students. International Journal of Engineering Education, 14, 145-152.

YEAGER, D. S. & DWECK, C. S. 2012. Mindsets That Promote Resilience: When Students Believe That Personal Characteristics Can Be Developed. Educational Psychologist, 47, 302-314.

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In a book published in the year 2000, entitled Learner-Centered Assessment on College Campuses: Shifting the Focus from Teaching to Learning, Mary Huba and Jann Freed asked the question:

 “Tomorrow’s citizens, tomorrow’s leaders, tomorrow’s experts are sitting in today’s college classrooms. Are they learning what they need to know? Are faculty using teaching methods that prepare them for future roles?”

Then, the main mode of teaching in engineering schools, and pretty much everywhere else within the university system, was the time-honoured, traditional, teacher-centred, lecture model. The main characteristic of this method was the reciting of material from textbooks, and passive, rote-learning from students. Then, the main assessment method was the end of course exam. And this was characterised by a tendency to prioritise rote learning as opposed to informed engagement with the study material.

Fast-forward to the year 2019, Mary Huba and Jann Freed would have to rephrase their question to ask:

 “Today’s citizens, today’s leaders, today’s experts are sitting in today’s college classrooms. Are they learning what they need to know? Are faculty using teaching methods that prepare them for present and future roles?”

Why? Because tomorrow has arrived. And it doesn’t look pretty.

Unlike in 2000, this time around changes are sweeping through higher education, and new forms of student-centred, active learning approaches are now in vogue. Yet, the traditional approach, characterised by lectures, passive learning and exams still dominates in large swathes of the university system.

As engineering academics, we can easily forget that the world that our students are graduating into is no longer the same world that we graduated into thirty or forty years ago. Occasionally, we do get shocked when the evening news announces, yet again, the collapse of a high street brand name. However, come the next day, we go back to our academic duties in the world-acclaimed universities that we belong to, and continue teaching our students the same way that we have taught them in the past ten years. We are used to this, and we like it. We know our lecture notes by heart, to the extent of even knowing when to make that dramatic pause, and when to recite that yesteryear joke.

Our lectures might be boring, but they are predictable, and our exams are also boringly predictable. Our students love it. Our exams have not changed in years. Yes, exams, because we do not believe in any other form of assessing our students. Continuous assessment is anathema to us. And our students pretty much know the questions that will appear in our end of year exams – last year it was this question from this paragraph in the lecture notes, the year before it was this other question, and the year before that it was this other question, so this year it is definitely going to be this question. If you are not in the know, and you look at the exam, it may look as if only the Einsteins and Brunels of this world can crack it.  But guess what, by the time the students come to the exam, they will have practised similar examples over and over again – for them it is a matter of recall, and nothing more. And they go on to get first class and upper second class degrees in Engineering, but they are definitely short of the preparation they need to be effective in industry.

Employers have no choice but to hire our not-fit-for-purpose graduate engineers. They are the only offering on the plate, and there is nowhere else to turn to.  They are the product of a well-honed, boringly predictable, first class engineering education, but they are barely of any use to the engineering world of today. We, the engineering academics know it; the students know it, the government knows it, parents know it, and the employers know it. That’s how things  are, but for how long?

Today’s complex world is crying out for nimble-minded graduate engineers whose education has given them the ability to critically analyse, assess and offer viable technological solutions to the myriads of engineering problems that routinely crop up every day. Instead, our graduates, bred on traditional engineering education, are hopelessly incompetent. In all their university lives, they have never had to face the challenging, unpredictable, academic problem sets meant to prepare them for the bold world of 21^st century engineering. Suddenly, faced with the reality of modern day engineering, their first class and upper second-class degrees lose their shine. Even the illustrious names of the universities that offered them these degrees suddenly look very unimpressive.

The world has to move on, and even now, employers are beginning to look elsewhere for the talent that they need. That talent might not come from an acclaimed engineering school, or from a world-class university, but it will be the talent needed for the engineering world of today. Engineering education has to change; otherwise, our current engineering schools are doomed.

The retail world has been here before us. Century-old retail giants have been collapsing left, right and centre, whilst newer, more technologically attuned retail enterprises take over. If the engineering school of today does not adapt to the changing winds, then a new breed of engineering school will take over, and we and our students will soon go the way of the dinosaurs, only at a much faster pace.

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In everyday conversations, the UK weather is by far the safest bet for non-controversial small-talk. This is because the variability and relative unpredictability of UK weather easily ensures that you can find loads of points of agreement with your fellow conversationalist. Within engineering education we also have another favourite topic for small talk – this is the A Level mathematics qualifications. This is because, until recently, it was relatively easy to find something within the A Level mathematics qualifications that any two members of the engineering academic community could agree to mourn about. This is all about to change, however. A Level mathematics has recently undergone a complete transformation, and the refurbished qualifications are already being delivered in schools, with the first batch of A level graduates expected to start their university studies in September 2019.

Why the A Level mathematics qualifications had to be redesigned

Over the past thirty years or so, there have been wide ranging complaints about the quality and perceived shortcomings of A level mathematics. Michael Gove, the then Secretary for Education, adequately summarised these shortcomings in a letter that he wrote to the Chief Executive Office of Qualifications and Examinations Regulation [DfE, 2013]:

• A level mathematics fails to provide the solid foundation that students need to prepare them for degree-level study and for vocational education
• A level mathematics students lack the deep understanding and/or the necessary skills to make connections between mathematics topics, and between mathematics and other subjects
• Assessment practice in A level mathematics is overly structured and encourages a formulaic approach to mathematical thinking instead of using more open-ended questions that require advanced problem-solving.
• Universities have had to adapt their teaching approaches to include remedial mathematics tuition for underprepared first year undergraduates.

A survey of mathematics departments within universities also revealed the following shortcomings [ALCAB, 2014]:

• The mathematical thinking of the most able students is not developed.
• The distinction between A and A* grades, i.e. the top most grades, seems to be based on the avoidance of careless slips rather than genuine mathematical ability.
• It is not clear what applied mathematics students have learnt.
• Current statistics provision tends to focus on routine calculations at the expense of interpretation and understanding.

Purpose of the new A Level Mathematics

According to the Department for Education, the primary purpose of the new A level mathematics qualifications is to [DfE, 2016]:

• Build from GCSE level mathematics and introduce calculus and its applications
• Emphasise how mathematical ideas are interconnected and how mathematics can be applied to model situations mathematically using algebra and other representations
• Helping students to make sense of data
• Helping students to understand the physical world and to solve problems in a variety of contexts, including social sciences and business
• Preparing students for further study and employment in a wide range of disciplines involving the use of mathematics.

Structure of the new A Level Mathematics qualifications

With the exception of further mathematics, all students studying A level mathematics now have to cover statistics and mechanics in addition to pure mathematics [Ofqual, 2016a]. This now provides universities with certainty as to the exact nature and level of mathematics covered by A level mathematics graduates.

In addition, students are now also expected to demonstrate the following overarching knowledge and skills across the entire content detailed in the A Level mathematics specification [DfE, 2016]:

• Mathematical argument, language and proof
• Mathematical problem solving
• Mathematical modelling.

Ofqual has also provided guidance on mathematical problem solving, modelling and the use of large data sets in statistics. This guidance also includes advice on assessing these key aspects, together with example questions [Ofqual, 2016c].

The new A Level Mathematics objectives and assessment format

The new A level mathematics qualifications have been specifically designed to deliver the following objectives [Ofqual, 2015]:

• Using and applying standard techniques: The new qualifications seek to ensure that learners are able to select and correctly carry out routine procedures, and to accurately recall facts, terminology and definitions.
• Reasoning, interpreting and communicating mathematically: Learners should be able to construct rigorous mathematical arguments (including proofs), make deductions and inferences, assess the validity of mathematical arguments, explain their reasoning and use mathematical language correctly.
• Solving problems within mathematics and in other contexts: Learners should be able to translate problems in mathematical and non-mathematical contexts into mathematical processes, interpret solutions in the context of a problem, and, where appropriate, evaluate their accuracy and limitations. Learners should also be able to translate situations in context into mathematical models. They should also be able to use mathematical models, evaluate the outcomes of modelling in context, recognise the limitations of models and, where appropriate, explain how to refine them.

Ofqual recommends that these three objectives should be assessed in accordance with the weightings outlined in Table 1 below [Ofqual, 2016b]:

Table 1: Assessment weighting for A and AS level mathematics

Objective	Weighting (A Levels)	Weighting (AS Levels)
Using and applying standard techniques	50%	60%
Reasoning, interpreting and communicating mathematically	25%	20%
Solving problems within mathematics and in other contexts	25%	20%

The new A Level Mathematics qualifications compared to First Year Engineering Mathematics

The new A Level mathematics qualifications are closely aligned to engineering mathematics as taught in the first and second year of university. For example, the objectives of the new qualifications closely agree with the intended learning outcomes of our first year mathematical modelling and analysis course module for engineering at University College London, whereby we state that by the end of the course module, students will be able to:

• Recognise the connections between mathematics and engineering, and how mathematical ideas are embedded in engineering contexts;
• Represent real-world systems from engineering in a mathematical framework;
• Identify and draw upon a range of mathematical concepts, including Calculus, Linear Algebra and Differential Equations to analyse specific problems and identify the appropriate mathematics to realise a solution;
• Employ appropriate computer programming and modelling techniques and statistical analysis to efficiently solve and evaluate the performance of engineering systems;
• Use estimation, approximation and dimensional analysis to reduce complexity;
• Relate the behaviour of the output of mathematical models to the underlying physical or conceptual models of interest;
• Carry our engineering problem solving both collaboratively in a team and independently;
• Present and interpret mathematical results in effective and appropriate ways to varied audiences, including non-mathematical engineering audiences.

Given such close alignment, we expect that from September 2019 onwards it will be much easier for first year students to adapt to university-level teaching. This is likely to reduce the time and effort spent by academic staff in remedial mathematics teaching. Instead, we hope that academics responsible for first and second year engineering mathematics teaching will use that time to focus on bringing engineering mathematics into closer alignment with other course modules in engineering.

References:

DfE (2013). Reform of GCE A Levels – Letter from Secretary of State to Glenys Stacey. Available at: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/278146/SoS_January_2013_ofqual_letter_alevels_v2.pdf [accessed 25 August, 2018]

ALCAB (2014). Report of the ALCAB panel on mathematics and further mathematics. The A Level Content Advisory Board. https://alevelcontent.files.wordpress.com/2014/07/alcab-report-on-mathematics-and-further-mathematics-july-2014.pdf [accessed 25 August, 2018]

DfE (2016). Content for mathematics AS and A level for teaching from 2017. Department for Education. Available at: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/516949/GCE_AS_and_A_level_subject_content_for_mathematics_with_appendices.pdf [accessed 25 August, 2018]

Ofqual [2015]. AS and A Level Mathematics and Further Mathematics: Consultation on Conditions and Guidance. Ofqual. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/481981/as-and-a-level-mathematics-and-further-mathematics-consultation-on-conditions-and-guidance.pdf [accessed 25 August, 2018]

Ofqual [2016a]. GCE Subject Level Conditions and Requirements for Mathematics. Ofqual. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/517726/gce-subject-level-conditions-and-requirements-for-mathematics.pdf [accessed 25 August, 2018]

Ofqual [2016b]. GCE Subject Level Guidance for Mathematics. Ofqual. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/514465/gce-subject-level-guidance-for-mathematics.pdf [accessed 25 August, 2018]

Ofqual [2016c]. Report on Mathematical Problem Solving, Modelling and the Use of Large Data Sets in Statistics in AS/A Level Mathematics and Further Mathematics. A Level Mathematics Working Group, Ofqual. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/481857/a-level-mathematics-working-group-report.pdf [accessed 25 August, 2018]

1. Professor Peter J Goodhew CBE FREng, Emeritus Professor of Engineering, formerly Dean of Engineering and Pro-Vice-Chancellor at the University of Liverpool

It might be helpful to clarify what engineering education is not. It is not about the acquisition of specific practical skills, however useful or interesting they might be to any individual. It is not about training people to run CFD codes or send CAD designs to a CNC machine or to grow crystals or to sign off structural steelwork. It is about the conceptual, planning and design skills which should precede all these activities. It is about imagining and understanding and predicting, as quantitatively as possible, why and how an engineering objective can be realised and delivered. (Goodhew 2014)

2. Professor Emanuela Tilley, Director of the Integrated Engineering Programme, University College London

Engineering education is no longer solely about specific content anymore or, indeed, traditional knowledge. It’s much more about processes and the students’ application of the knowledge. (AECOM 2018)

3. Professor Jeremy Watson CBE FREng, past President of the Institution of Engineering Technology

We need to train a new generation of engineers in skills that are genuinely relevant to the new industrial values of flexibility, technical advancement and on-going innovation. Single discipline specialism and theory will no longer cut it in the modern world.  (The Institution of Engineering and Technology and The Engineering Professors’ Council 2017)

4. Professor Janusz Kozinski: Founding President & Vice-Chancellor of the Hereford University of Technology and Engineering, formerly Founding Dean of York University’s Lassonde School of Engineering

The stars are aligned for an Engineering Renaissance here in the UK and throughout the world. We as educators need to seize this moment to work collaboratively with students and employers to co-create a whole new set of models to reflect their needs. In doing so, we can turn a fear of change and flux created by technology and disruption, into a new era of enlightenment for engineering education. (Koziński, J. A., and Evans, E. F., 2017).

5. Massachusetts Institute of Technology

5. Successful engineering education change is characterised by (1) a leadership style that articulates a clear educational vision and demonstrates a personal commitment to establishing a new paradigm for engineering education at the institution; (2) a distinctive ‘spirit’ or culture of collegiality and common purpose that pervades the faculty, (3) student engagement in and understanding of new educational approaches, (4) in-house development of new tools and resources to support and advance the educational approach. (Graham 2018)

References

AECOM. (2018). The future of infrastructure: Expert opinions from around the world on the challenges and opportunities ahead. AECOM.

Goodhew, P. J. (2014). Teaching Engineering: All you need to know about engineering education but were afraid to ask, London: Royal Academy of Engineering.

Graham, R. (2018). The global state of the art in engineering education. Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

Koziński, J. A., and Evans, E. F. (2017). “An Engineering Renaissance”, New Approaches to Engineering in Higher Education. London: The Institution of Engineering and Technology, The Engineering Professors’ Council.

The Institution of Engineering and Technology, and The Engineering Professors’ Council. “New Approaches to Engineering in Higher Education.” Presented at New Approaches to Engineering in Higher Education, London.

We are now approaching the end of the year, and attention has shifted to the next year. Barring unexpected departures, most departments have probably now concluded their hires for the next year. And as for you, congratulations if you are one of those who have secured your first academic position. You should give yourself a pat on the back. After all, you have made it ahead of at least seventy other competitors who applied for the same role.

By now your new department will have been in touch with you to discuss your teaching for the next year. You would have demonstrated in the interviews both your passion and your growing expertise in engineering education practice. Naturally, you will be eager to demonstrate and cement your position as an innovative teacher who aspires to excellence. A word of advice, however: teaching quality can be highly subjective, and excellence in education is context-dependent. So before you rush ahead to implementing all the whistles and bells that you can think of in your education practice, you need to first consider the following three questions:

• What exactly is it that you will be teaching?
• Who exactly will you be teaching?
• Who else will be teaching with you?

What exactly is  it that you will be teaching?

Your new Head of Department may have contacted you to tell you of the good news. They will also have discussed with you the course modules that you will be teaching. However, it’s not just that you will be teaching some specific course module, like Maths101, Fundamentals of Engineering 103 or some other course unit. Even if you may already have some experience delivering these course modules, and you are confident that you will do a good job, you really need to delve deeper into the exact course content that you need to deliver.

Engineering departments are not all the same. They have different missions, and different teaching philosophies, beliefs and values. You may have heard that there is a visible curriculum, and there is a hidden curriculum.  The two are not necessarily the same, and what matters most is the hidden curriculum. It is this hidden curriculum that is in tune with the goals and aspirations of the department, and  the one that determines the specific topics that you should cover, the depth that you are expected to go in each topic, as well as how exactly you are to assess the students.

Degree programmes normally comprise course modules that feed into each other, or that provide knowledge and skill sets that are designed to complement each other. In this way,  module units help to achieve a coherent set of learning outcomes at each level of the degree programme. In practice, the written down syllabus does not provide the whole story. A lot of assumptions, implied and non-implied go into the development of a degree programme. It is your duty to find out all this, and then work out what you need to deliver in your assigned course module. Start talking to people within the department, and go through the historical records of your assigned course modules.

Who exactly will you be teaching?

Again, students are not all the same. Different universities tend to attract different types of students. As a result, students attending a particular university may have expectations that are quite different from those attending some other university. For instance, students at a highly selective, traditional university may have been primed for highly mathematical and theoretical content. They may even be expecting “the sage on the stage” approach to teaching, together with the traditional exam-oriented assessment format. They may even be more focussed on the mark they are going to get, and not necessarily on what they actually learn in your class. In such a situation, rushing to implement fancy stuff like team-based, collaborative learning may serve only to alienate them from your teaching.

On the other hand, more innovative, forward-looking universities usually attract students who are keen to experience “real” engineering practice. Such students usually expect a hands-on, design-based approach that integrates theory and practice. In such an environment, adopting a hands-off “sage on the stage” approach is likely to hasten your departure from the university.

Who else will be teaching with you?

You need to find out quickly the nature of your soon-to-be colleagues. Are they conservative and suspicious of any innovations in education practice? Or are they open to new ideas in education, and enthusiastic enough to experiment with the new? Or are they somewhere in between? You need to style your teaching accordingly. If you must innovate, only do so when you have gained your colleagues’ respect. You may be eager to get good student feedback, but if this ends up exposing your colleagues’ not-so-good teaching, you will have no-one to thank you. Instead, your excellent teaching may be cruelly re-cast as some form of dumbing down, or worse. On the other hand, there may be colleagues in your new department who are passionate about their education practice. If this is the case, get to know them, learn from them and partner with them in the noble goal of ensuring good departmental education practice.

Concluding remarks

How then should you conduct yourself in the first few months and weeks of your new academic career? Simple, if the department is conservative, then tread carefully, gain the respect of your colleagues and students, engage them in discussions on education practice. Think carefully through any changes that you wish to make, and establish a consensus amongst both students and academics alike. If you must make changes to the curriculum, or to your own education practice, then consider going first for those changes guaranteed to generate positive outcomes with high visibility. Then, as you win support, you can go on to implementing bolder changes. If, on the other hand, your department has a culture of innovation and excellence in education, then tuck in, and learn as much as you can, and studiously incorporate this learning into your own practice.

Either way, let the pursuit of excellence in education practice be your primary goal. Expand your networks with other like-minded engineering educators, both within your institution and elsewhere. Take every opportunity to learn about engineering education, and don’t shy away from teaching others as well. Be an engaged member of your department when it comes to matters relating to education and the student experience. Do this consistently, and over time, you will  become an integral member of both your  department and the world-wide community of engineering educators.  Then you will become established in higher education.

 

Introduction

In engineering education, as in all other aspects of higher education, we are constantly bombarded by a stream of new acronyms and concepts. CDIO is one such concept that has been around for a number of years, but one which most people are only just becoming aware of. My intention in this blog is to present a quick overview of the CDIO approach to engineering education reform. This should be adequate for anyone who needs a quick introduction, and is particularly ideal for busy senior academics and administrators.

What is CDIO?

The acronym CDIO stands for Conceive, Design, Implement, Operate. It is an approach to designing, running and coordinating undergraduate engineering education programmes with the objective of producing work-ready graduates equipped with the necessary professional and technical skills they need to hit the ground running when they move into employment.

The CDIO Approach

The CDIO approach is based on the premise that the conceive – design – implement – operate product/systems lifecycle approach is the basis for engineering practice, and as a result, every engineering graduate should be able to

conceive-design-implement-operate complex value-added engineering systems in a modern team-based environment (Crawley 2002).

The CDIO approach to engineering education is delivered in the context of the product/system conceive-design-implement-operate lifecycle, and it is designed to ensure that students are adequately grounded in the fundamentals of their engineering discipline. This approach is characterised by the following features (Crawley et al. 2014):

• It involves stakeholders in developing learning outcomes.
• It constructs a sequence of integrated learning experiences that offer authentic learning opportunities for students to encounter and experience situations that engineers encounter in their profession.
• It ensures that learning activities simultaneously facilitate student learning of critical personal and interpersonal skills, and product, process, and system building skills, as well as enhancing the learning of engineering fundamentals.

CDIO driving factors

The main driver for the CDIO approach is the recognition that engineering education is characterised by an ever-increasing amount of technical knowledge. In addition, engineering graduates now need to be equipped with the necessary personal, interpersonal and professional skills they need to become effective from the very first working day (Crawley 2015). To address these two contradictory issues within the existing timescales for completing an undergraduate engineering programme, CDIO has opted for an approach to education that is based on the engineering problem solving paradigm.

CDIO historical Context

The historical context behind the CDIO approach is that following the end of the second world war, engineering education has evolved to a point where it has become too focussed on engineering science at the expense of engineering practice. This has led to a situation whereby engineering graduates lacked the skills that industry was seeking (Crawley et al. 2014). This state of affairs is a consequence of the long-term changes in the composition of engineering academics. Before the 1950’s, most engineering academics came from a practitioner background, and as a result, engineering education was primarily practitioner-oriented.

From the 50’s onward, incoming academics increasingly came through the graduate school route, and they were more well-versed in engineering science than engineering practice. Consequently, their teaching tended to draw mainly from engineering science. The resulting balance between practice and science in the 1960’s led to engineering graduates who had the appropriate mix of science and practice skills and insights that industry required.

However, from the 70’s onwards, as engineering scientists became the majority, engineering education became more and more focussed on engineering science, and increasingly dissociated from engineering practice. This has led to a situation where graduating students lack the skills required by industry.

Implementing a CDIO curriculum

The CDIO approach envisions a curriculum that is organised around mutually supporting disciplines, with CDIO activities highly interwoven between them (Crawley et al. 2014). Learning should be rich with student design-build experiences, and delivered in classrooms and student workplaces that are sufficiently equipped and specifically designed to support active and experiential learning. This should be accompanied by assessment and evaluation processes designed to ensure constant improvement.

The first step in developing a CDIO curriculum is to identify the skills and attributes that students need to attain by the time they graduate. This should be done in consultation with stakeholders, who include current and former students, employers, academics and the society at large. These attributes and skills will constitute the programme syllabus.

Once the syllabus has been designed and agreed, the necessary learning activities needed to achieve the identified learning outcomes are then developed, alongside with appropriate assessment methods (Crawley et al. 2014). In practice, this may require modification of the existing curriculum and course modules, redesign of learning environments, and adopting student-centred, active and experiential learning approaches to teaching. Assessment methods also need to be evaluated and redesigned to ensure that they are fit for purpose.

The design and development of all these learning activities should be developed with reference to the 12 guiding principles that the CDIO Initiative has developed to describe CDIO programs. These principles are termed the CDIO Standards, and together they define the key features of a CDIO programme. These standards serve three primary objectives, namely:

• providing guidelines for educational programme reform and evaluation
• specifying programme benchmarks and goals
• providing a framework for continuous programme improvement.

Table 1: Guide to the 12 CDIO Standards

CDIO Aspect	Addressed by
Program philosophy	Standard 1
Curriculum development	Standards 2, 3 and 4
Design-build experiences and workspaces	Standards 5 and 6
New methods of teaching and learning	Standards 7 and 8
Academic staff development	Standards 9 and 10
Assessment and evaluation	Standards 11 and 12

Where to get additional Information

The best way to start off on your journey towards a deeper understanding of CDIO is by visiting the CDIO website: http://www.cdio.org/. There you will find relevant publications, notices for CDIO-related meetings, and you can also view a list of institutions that have adopted the CDIO paradigm.

References

Crawley, E. (2015). “The CDIO Syllabus: A Statement of Goals for Undergraduate Engineering Education, 2001”. City: Worldwide CDIO Initiative. http://www.cdio.org: CDIO Knowledge Library. Cambridge, MA; .

Crawley, E. F. “Creating the CDIO syllabus, a universal template for engineering education ” Presented at 32nd ASEE/IEEE Frontiers in Education Conference; 6–9 November, Boston, MA.

Crawley, E. F., Malmqvist, J., Östlund, S., Brodeur, D. R., and Edström, K. (2014). “The CDIO Approach”, Rethinking engineering education. Springer, Cham, pp. 11-45.

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

It is no longer in dispute that engineering education has to change if it is to produce graduates who can face up to the challenges of the 21^st century. Moreover, it’s no longer a case of “Are there any curriculum transformation strategies that I can look at?” Instead, it’s now a case of “Which transformational strategy should I adopt for my engineering school.”

Ten year ago, conferences and journals focussing on engineering education were scarce and infrequent. This is no longer the case. In fact, we are now spoilt for choice. Multiple engineering education conference proceedings and journals are now crammed full with ideas and examples of curriculum change by a multitude of authors from engineering schools all over the world. This now leaves the Director of Education wishing to transform their engineering curriculum with the following questions:

1. Whose voice should I listen to if I am considering curriculum change in my own school?
2. Which successful institutions, worldwide, should I turn to for guidance?
3. Of these successful institutions, which ones closely match my own, in terms of size, operational environment and institutional education mission?
4. Of the plethora of engineering education models out there, which ones are likely to stand the test of time, and which one are just passing fads?

At the best of times these are very challenging questions. However, the publication of an MIT-sponsored report by Ruth Graham entitled “The global state-of-the-art in engineering education: Outcomes of Phase 1 benchmarking study” will make it much easier to address these questions. The report came out of MIT’s desire to have a clear insight of the current state of cutting edge engineering education globally and how this was likely to pan out in the future.

Individuals currently at the forefront of engineering education reform

In order to come up with the report, Ruth first needed to identify and interview some of the leading figures in engineering education. To do so she selected 50 individuals from 18 countries from across the world. The selected individuals were either pioneers in engineering education research, policymakers in the field and/or university leaders with direct experience of delivering the world’s most highly regarded engineering education programmes. This list has been placed in one of the report appendices, and it is a good starting point if you need to talk to someone with current experience in engineering education transformation.

Ten institutions at the forefront of engineering education reform

One of the key objectives of the report was to identify institutions that are leading in engineering education innovation. The report identifies ten institutions that are currently acknowledged as world leaders. These institutions include MIT, Stanford, Olin College of Engineering, University College London, and, with the exception of the National University of Singapore, all of them are based either in the USA or Europe. With the exception of Olin, these institutions are typically well-established public universities that are renowned for research excellence, and that cater for relatively large cohorts of undergraduate engineering students. Also, without exception, all the ten institutions  actively engage in disseminating their ideas and practices across the international higher education community.

The report also identifies key pedagogical features that are common to these leading institutions. Typically, these institutions offer student-centred, hands-on, experiential learning, with opportunities for engaging with the university’s research activities. Design-based learning is also a feature of their curricula, and all the institutions have well established partnerships with industry that inform the engineering curriculum as well as the engineering research agenda. In addition, these institutions also offer their students opportunities to engage in student-led, extra-curricular activities and experiences.

Ten institutions emerging as leaders in engineering education reform

The report also identifies ten institutions that are viewed as emerging leaders in engineering education, with the most cited being Singapore University of Technology and Design (SUTD) and Olin College of Engineering. Other institutions that make it into the list include the Pontifical Catholic University of Chile, and Tsinghua University in China. Compared to the earlier list, this one is more globally distributed, with only four institutions coming from the USA and Europe, which are the traditional strongholds.

The report also identified a “watch list” of some of those institutions that did not make it into the emerging world leaders list. This includes New Model in Technology & Engineering (NMiTE), UK, Lassonde School of Engineering at York University, Canada, B.V. Bhoomaraddi College of Engineering and Technology, India, and Insper, in Brazil.  Apart from B.V. Bhoomaraddi, all these other institutions have been established within the last five or so years.

The engineering curricula in the emerging world leaders share a number of common traits. First, all the ten institutions have opened up entry to students with non-conventional entry requirements, and they have all put in place selection processes that take into consideration the aptitudes of prospective students towards engineering. In addition, programmes at these emerging institutions place significant emphasis on integrating work-based learning into their curricula, as well as blending off-campus online learning with on-campus intensive experiential learning. Another common characteristic is that programmes offered at these institutions place dual emphasis on both engineering design and student self-reflection, and both these are integrated across the entire curriculum. Finaly, in addition to the formal curriculum, student-led, extra-curricular activities are a key feature of these institutions.

Additional remarks

The report also highlights contextual features that have driven curriculum reform in these institutions that have been identified in this report. This includes government initiatives, and local labour market requirements.  The report also highlights the likely future trajectory of engineering education, as well as identifying the likely key ingredients necessary for effective, sustainable engineering education reform within individual institutions. It is definitely a report worth reading, even if you are not contemplating making changes to your own curriculum in the near future.