Torquetum

 

The authors teach an electromagnetic field course and recognise that physical interpretation of divergence and curl are a difficulty for students, even though the actual calculations are not. They suggest a teaching method which begins with capturing the students’ interest through fictional and theoretical invisibility cloaks. The maths behind the theory involves Maxwell’s equations and hence divergence and curl. The authors suggest ways of teaching divergence and curl through flux and circulation, beginning with the macro and moving to the micro in logical ways.

 

 

The authors develop an instrument to measure performance, confidence and familiarity with both procedural and conceptual problems in mathematics. The students were second-year engineering students in two institutions in two countries – South Africa and Sweden. The authors provide definitions of the relevant terms and take issue with some education literature using terms like “conceptual” and “knowledge” too loosely and conflating them with other terms. The paper presents detailed data and analysis, finding differences and similarities across different groups (read the paper for details), concluding that “the use of mathematics in other subjects within engineering education can be experienced differently by students from different institutions indicating that the same type of education can handle the application of mathematics in different ways at different institutions.” (p. 158/9)

 

 

Redish and Smith summarise key findings in cognitive and neurological research on how learning occurs and is manifested in the brain. They cite Pellegrino’s three main threads of educational research: constructivism, knowledge organisation and metacognition, and his three components of educational practice: curriculum, instruction and assessment. They proceed to link Pellegrino’s summary of educational research with cognitive research into learning, giving what I found was a really useful summary of various key cognitive findings. Their theoretical framework for learning is based on the concepts of activation, association, compilation and control.

 

Activation refers to the activation of neurons, becoming entrained and working together in clusters. Activation is related to the differences between long term memory and working memory and how working memory can only handle about seven “chunks” of knowledge at a time. This essentially constructivist relationship of neurological activation to learning supports the idea that student misconceptions are not rigid but can be changed.

Association refers to entrained neurological pathways becoming associated with one another through repeated practice to create schemas, “skeletal representation[s] of knowledge abstracted from experience that organizes and guides the creation of particular representations in specific contexts” (p. 298). Associational paths link to Pellegrino’s knowledge organisation. Association is highly context dependent: change the context and different associations are made. This can be disadvantageous in educational contexts as correcting a student’s misconceptions in one arena can fail to transfer to another due to different associational paths: “students build up alternative associational paths; one set of knowledge is activated specifically for a physics class but the other intuitive knowledge is not erased but remains for activation in all other situations” (p. 298). The authors report that this idea links to work in cognitive science on “conditioning, attempts to eliminate conditioning (extinction), and its reemergence (relapse)” (p. 299).

Compilation is related to the idea that “items that are originally weakly associated may become very strongly tied together” (p. 299). This is the process by which chunks are created. Experts have many actions strongly tied together, reducing the cognitive load of certain types of problem solving, while novices experience greater cognitive load. Experts have to “reverse engineer” how certain types of problems are solved in order to teach solution processes to students. It can help to observe students solving problems, to see where their difficulties lie.

Control refers to executive control and metacognition. We are constantly (and usually unconsciously) deciding what input from the external world to pay attention to. These decisions are made by control schema which have developed over time and experience. “These control schemas have three important consequences: they create context dependence, they give us a variety of resources for building new knowledge and solving problems, and they control which of these resources we bring to bear in given circumstances.” (p. 299). Context dependence: We all have developed “expectation schemas” which consider available input and decide what is important and how to react.

Epistemological resources: we understand that knowledge can be transmitted and also created. Some knowledge is outside our control and other is within our control.

Epistemological games, or e-games, is the term given to “a reasonably coherent schema for creating new knowledge by using particular tools” (p. 300). Such e-games include an understanding of beginning and end, what information to use, what structure to impose, etc. such as the typical process of solving a physics problem. Choice of e-game is crucial to successful problems solving.

 

Having provided a summary of cognitive research into learning, the authors take a look at the role of mathematics in engineering and how the differences between how maths is typically taught and how maths is used by engineers can create some serious obstacles to learning. In engineering, students are expected to learn the “syntax” of mathematics, but also what it means and how to use it effectively in engineering contexts. How a mathematician or engineer interprets symbols can differ widely. To a mathematician the letters chosen to represent the variables might be arbitrary, but an engineer they carry meaning. “we [engineers] tend to look at mathematics in a different way from the way mathematicians do. The mental resources that are associated (and even compiled) by the two groups are dramatically different. The epistemic games we want our students to choose in using math in science require the blending of distinct local coherences: our understanding of the rules of mathematics and our sense and intuitions of the physical world.” (p. 302). Differences between how maths is used in maths and in engineering include (p. 302):

The authors provide a schematic describing modelling: define your physical system, represent the system mathematically, process that mathematical system as appropriate, interpret your results. Later they report that one view of modelling is that it is inseparable from problem solving. Redish and Smith, however, feel that modelling is easier to teach than problem solving and that, in practising modelling, much problem solving is learned along the way. They support this view by framing problem solving in similar terms to their schematic of the modelling process. I am a proponent of the Polya framework for problem solving, which differs somewhat from the Redish and Smith framework. There are similarities, however, and a fruitful comparison of the two could no doubt be made.

 

While the role of proof in engineering mathematics is not a focus of this paper, the authors do have a few things to say which support a relatively high presence of proof.

“Often what our students learn in our classes about the practice of science and engineering is implicit and may not be what we want them to learn. For example, a student in an introductory engineering physics class may learn that memorizing equations is important but that it is not important to learn the derivation of those equations or the conditions under which those equations are valid. This metamessage may be sent unintentionally.” (p. 297)

“Often, in both engineering and physics classes, we tend to focus our instruction on process and results. When we teach algorithms without derivation, we send our students the message that “only the rule matters” and that the connection between the equation we use in practice and the assumptions and scientific principles that are responsible for the rule are irrelevant. Such practices may help students produce results quickly and efficiently, but at the cost of developing general and productive associations and epistemic games that help them know how their new knowledge relates to other things they know and when to use it. As narrow games get locked in and tied to particular contexts, students lose the opportunity to develop the flexibility and the general skills needing to develop adaptive expertise.” (p. 303)

 

There are some “bad habits” which can be learned in maths class. Giving algorithms without derivation gives the impression that the assumptions and scientific principles behind the rules are not important (see quote above). Another bad “maths” habit is substituting numbers early in the problem-solving process. This tends to hide associations between variables and inhibits reflecting back on the process. Another habit learned in maths class is to elevate the “processing” part of the modelling process above the others, thereby hindering transfer of the skills to other courses more based on the whole modelling package.

 

The authors close their paper with a discussion of using cooperative learning in a course teaching modelling. They give some interesting examples of relating physical reality to mathematical models and vice versa.

 

I thoroughly enjoyed reading this paper. Some papers I can summarise in a paragraph. This one took me two pages simply to summarise and I also have four pages of notes! I already knew a lot of the cognitive findings, but some was new to me, such as epistemic games and conditioning. This is the sort of paper one reads and rereads many times.

 

 

Varsavsky and colleagues ran a survey amongst the convenors of 130 engineering courses at their institution (Monash, Caulfield) of what maths topics were needed, by whom and when. The paper was interesting for its broad overlaps with my local situation: the maths taught is similar and the challenges of diversity are similar, as well as the lack of communication between involved parties. The survey found, as I have found in a similar survey, that the overall maths needs are large, but most courses use very little, or only basics. The paper closes with some sensible suggestions for setting up an engineering maths curriculum. I am interested to see that, once again, proofs don’t get a mention.

 

 

This is quite a lengthy paper which I shall summarise very briefly. The authors disagree (as many do) with the notion that students can transfer problem solving techniques from typical classroom problems (word problems, usually) to workplace problems, which are ill-structured. They conducted interviews with approximately 90 professional engineers about problems they had encountered and how they had solved them. The authors develop 12 themes which emerged from the interviews, such as “most constraints are non-engineering” and “Engineers primarily reply on experiential knowledge”. The authors close with suggestions for education, such as using PBL (problem based learning). This requires huge commitment from staff and wide ranging reform, however, and can be hard to achieve. Other than that they give suggestions on how to make classroom activities more like workplace situations.

 

 

This paper is a very interesting application of vector calculus in a real-life context. Computer Numerical Controlled milling machines often use a spherical bit for the milling of high precision 3d surfaces, such as the gripping surfaces of surgical tissue clamps. If we start with a two dimensional curve approximating the undulating surface, we can find three methods of plotting the path of the centre of the spherical bit: basic calculus, Lagrange multipliers and vector calculus. Extending to the 3d surface, the vector calculus method transfers best. In the process of the calculation the concepts of radius of curvature and cycloids come up.

 

 

The authors describe the first years of the engineering degree at Purdue University. Students enter a “Freshman College” at first, where they cover calculus, physics, chemistry, English and computers. Once they have achieved a C aggregate in these courses, they proceed to register with one of the engineering disciplines. Entering students write an algebra placement exam. If they fail this exam, they do a pre-calculus semester before beginning calculus. Students also have the opportunity to be fast tracked and proceed straight to second semester calculus. In a quantitative longitudinal study, the authors study the effectiveness of the various courses and the retention and graduation rates of different cohorts of students. At a point in the study, a counsellor-tutorial (CT) program was started, providing “additional services” to make the first-year experience “less lethal” (p. 409). This program was found to be successful, and became even more successful when the program was expanded to take on more students (from 80 to over 400). One concern was that providing support in first-year would merely delay drop out, however that pattern was not apparent in the data.

 

“There is a relationship between ultimately graduating in engineering and first obtaining a thorough understanding of basic mathematics and science principles. As the academic skills of the average student are honed, this relationship between these skills and the “world of engineering” becomes clearer to the engineering recruit. At Purdue University, we believe that the courses in calculus, chemistry, and physics supply the collegian with these necessary skills. We also believe that if you treat these courses as high-risk and supply additional assistance to those students with a higher probability of failure, then those students will acquire the critical background skills that will make it possible to persist in engineering.” (p. 410)

 

 

Mustoe asks “How far should an undergraduate education in engineering aim to meet the needs of industry as perceived by industry?” (p. 313). He proceeds to come at this question from multiple directions: the requirements of industry, of engineering academics, of mathematicians, of professional bodies. He finds that the views of what maths is necessary and desirable varies widely across these groups and that it is impossible to meet everyone’s demands. For example, does the engineering mathematics lecturer teach mathematical concepts in context or not? In-context problems at the first-year level are hard to find as real-world problems are messy and ill formed. In addition, the students might not yet be familiar with the engineering context and confusion rather than a sense of relevance will result. Mustoe cites a view that concepts are easier to study in the abstract without confusing contexts, however, many clamour for relevance in the maths curriculum and show how students lose interest due to no clear links between maths class and other courses.

 

Throughout Mustoe’s literature review of a wide variety of viewpoints, there is an emphasis on basic manipulative skills as well as modelling. “Five general areas of competence worth aiming for are: a ‘feel’ for the magnitudes of quantities, an ability to handle fractions without fear, an appreciation of the use of approximations, the ability to manipulate formulae confidently and the skill to cope with calculus.” (p. 320) Mustoe has little patience for long integration problems: “Other than training in patience, persistence and concentration, there is little to be gained from an exercise in integration, for example, which requires three successive substitutions and the use of a trigonometric identity for good measure.” (p. 321). There is not much mention of the role of proof in engineering mathematics courses within this paper, except for a citation of Craggs (1978): “Craggs was concerned about the level of rigour in the mathematics taught to engineers. His conclusion was that the aims for teaching were to encourage accuracy in manipulation, to state theorems with proof where it was easy and with heuristic justification where it was not, to illustrate the possible gains and losses if work was undertaken outside the domain of these theorems and to respect the additional expertise of the trained mathematician.” (p. 318).

 

Mustoe concludes that curricula should be dismantled and rebuilt from the ground up with a clear view of what the role of mathematics is in the degree programme. (Note that this is a 1995 paper, so hardly current.) I loved this paper for its variety of views and broad literature review.

 

 

Badat, S. (2010). Global rankings of universities: a perverse and present burden, in E. Unterhalter, V. Carpentier (eds.) Universities into the 21st Century: Global Inequalities and Higher Education. Whose Interests Are We Serving? Basingstoke, Hampshire UK: Palgrave Macmillan.

 

Several different ranking of “world class” universities exist. Badat mentions the Times Higher Education-Quaxquarelli Symonds (THE-QS) and Shanghai Jiao Tong Institute of Higher Education (SJTIHE) as being the best known rankings. The purposes of a university are many, chief amongst which are the broad goals of production of knowledge, the dissemination of knowledge and cognitive development of students, and community engagement. Neither of the rankings mentioned above is a good measure of all of these purposes and no linear ranking really could be. The purposes of a university are too many and too varied for any one institution to pursue them all and to continually improve across all area. Universities need to choose and build on areas of strength which are aligned with their missions and their goals. Badat is witheringly eloquent about the acceptance of the worth of these ranking systems in the face of overwhelming evidence of their uselessness. Universities of the global South are encouraged to “catch up” to their Western and Northern counterparts. Rankings play a role in this way of looking at university worth and they should not. Badat calls for universities of the global South to join with one another and with other social actors to develop alternative instruments which measure qualities more in line with university purposes and gaols and with the educational endeavour.

 

 

Naidoo, R. (2010). Global learning in a neoliberal age: Implications for development, in E. Unterhalter, V. Carpentier (eds.) Universities into the 21st Century: Global Inequalities and Higher Education. Whose Interests Are We Serving? Basingstoke, Hampshire UK: Palgrave Macmillan.

 

Classical liberalism has as its central philosophy the freedom of the individual from the control of the state. The neoliberal form of government has deregulation, privatisation and competitiveness in common with classical liberalism, but considers the state to have a positive role to play in facilitating the workings of the market. Neoliberalism, as a framework for higher education, encourages a commodification of the output of higher education and encourages increased research output and high student pass rates. Under-represented groups will not be recruited as they will not contribute to the market-related interests of the institution. Similarly disciplines which are more expensive to teach are less attractive than cheaper ones to the institution. Effective teaching and learning are not encouraged by a neoliberal paradigm as good teaching is expensive in time and resources. Naidoo cites Bourdieu as referring to neoliberalism as an unquestionable orthodoxy, something treated as an objective truth. She argues that unquestioning acceptance of a neoliberal influence on higher education impoverishes all higher education by undermining the value of Bourdieu’s “academic capital”, but in particular developing countries suffer from the imposition of such a paradigm. Naidoo calls for “country ownership” of higher education with the foregrounding of the development agenda. She calls for the development of theoretical and empirical research to challenge the orthodoxy of the neoliberal agenda in education.