*PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies*,

*23*(2), 133-140.

The author argues for an ordering of topics in a multivariable calculus course which brings the three big theorems as early as possible. The textbook he uses is a standard maths text, with the three big theorems coming last. He lists the topics to be covered before Divergence Theorem can be covered, locating it (by my estimate) a bit less than halfway through the course. Thereafter he covers a few more topics and get to Stokes’ Theorem (probably about 2/3 of the way through the course). Green’s Theorem is presented as a special case of Stokes’ Theorem. The benefits of this approach are argued for convincingly and a few drawbacks are also covered (such as parametrised surfaces before parametrised curves). This is the second paper I have read which recommends Schey’s (2005)

*Div, Grad, Curl and All That: An Informal Text on Vector Calculus*, so I really must track that text down. The practical interpretations of div and curl are emphasised as in so many papers I’m reading. I found this paper intriguing and I also greatly appreciated that the author broke the course down into sufficient detail that I, or someone else, could easily structure a course as he has done.

*Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understandings and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.*

*PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies*,

*9*(1), 21-28.

Here we have another paper lamenting (justifiably) the difference in the way vector calculus is taught in maths and physics. The authors emphasise how practical applications and situational geometry are far more important in physics (or engineering) than in maths. For example, they discuss how vectors are defined as ordered triples in maths while as arrows in space in physics. Also, div and curl are defined as differential operations on vector fields in maths but in physics are defined first in terms of their physical meaning as represented by Stokes’ Theorem and Divergence Theorem. The coordinates used in maths are almost invariably rectangular coordinates, the authors argue, while physics situations frequently have circular or spherical symmetry and hence use spherical coordinates to simplify the maths. (Some of the paper’s criticisms could apply to my local course, but not all, I think.) The value of the mathematical methods lies in their general applicability, however in physics the types of cases are few and there is an argument for loss of generality in favour of simplification of the common cases. The authors close with an insistence that the relevant departments collaborate closely.

*Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understandings and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.*

*Engineering Education*,

*8*(1), 122-134.

The authors argue for the role of puzzles in the teaching of mathematics. Puzzles are defined as “a problem that is perplexing and either has a solution requiring considerable ingenuity – perhaps a lateral thinking solution – or possibly results in an unexpected, even a counter-intuitive or apparently paradoxical solution.” (p. 122). They show how parallels can be drawn (in certain circumstances) to the more well known problem-based learning. I saw some old favourites here, such as students:professors and the peach problem. They also cover the importance of estimation and of ill-founded problems. I found it interesting that the authors unproblematically accept that word problems are preparation for real-world problems – a point of view with which I disagree. This was a fun paper and had some interesting references to books of puzzles, specifically one by Badger, one of the authors of this paper.

*Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understandings and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.*

*International Journal of Electrical Engineering Education*,

*50*(4), 351-357.

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.

*Physical Review Special Topics – Physics Education Research*,

*8*(010111), 1-15.

The authors focus in this paper on the mathematical difficulties experienced by physics and engineering students in an “upper division” electricity and magnetism course. They categorise the difficulties as (p. 2):

o “Students have difficulty combining mathematical calculations and physics ideas. This can be seen in student difficulty setting up an appropriate calculation and also in interpreting the results of the calculation in the context of a physics problem. (However, students can generally perform the required calculation.)

o Students do not account for the underlying spatial situation when doing a mathematical calculation.

o Students do not access an appropriate mathematical tool. Students may instead choose a mathematical tool that will not solve the relevant problem, or may choose a tool that makes the problem too complex for the student to solve.”

Using the troublesome concepts of Gauss’s Law, various vector calculus techniques, and electrical potential, the authors demonstrate these three categories of difficulty with many examples of specific problems and student responses. Of specific interest to me were the vector calculus issues. They find that students struggle with the “vector nature” of a vector field, struggling to think of magnitude and direction simultaneously. Additionally, students can calculate gradient, divergence and curl easily, but struggle with the physical interpretation of these quantities. The authors hypothesise that the way vector calculus is taught in mathematics class, the students fail to see integrals as “sums of little bits of stuff”, which I would like to think is not true in my maths classes, where the sum nature of integrals is repeatedly emphasised. Another vector integration difficulty is found in students struggling to express dA and dV in suitable coordinate systems.

They discuss methods they have used in classroom pedagogy, out of classroom assistance and transformed resources to address these difficulties. Even with all their changes, they find that certain problems remain challenging for the students. They argue that these concepts are hard to understand and that the instructors are not keeping this well enough in mind. They discuss ways of moving forward. I thoroughly enjoyed this paper, found it pertinent to current and future work of mine, and benefited from the thorough literature review.

*Journal of Engineering Education*,

*101*(1), 138-162.

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)

*Journal of Engineering Education*,

*100*(3), 424-443.

The author investigated the relationship between mathematical anxiety and performance in an electromagnetics course. There is a literature review of studies on mathematics anxiety showing, in general, that there is a correlation between high anxiety and poor performance. The causal relationship tends to be less clear, however, although there is some evidence to show that poor prior performance leads to higher anxiety which in turns impacts negatively on performance in procedural tasks. Two maths anxiety scales are discusses, the Fennema-Sherman scale and the MARS scale. Those scales and others were adapted to make the Electromagnetics Mathematics Anxiety Rating Scale (EMARS) which was used in this study. The scale had several subscales which measured perceived usefulness of the course, confidence, interpretation anxiety, fear of asking for help, and persistency. The data and results are discussed in some detail. Conclusions include that high anxiety students perform less well in procedural work than low anxiety students, but that conceptual performance is less clearly aligned with anxiety. In addition, high anxiety students felt less confident about their maths ability and also self-describe as being less persistent in solving mathematical problems. The authors close with the suggestion that assessment should be more aligned with conceptual understanding rather than procedural processes.

*Journal of Engineering Education*,

*97*(3), 295-307.

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):

- Equations represent relationships among physical variables, which are often empirical measurements.
- Symbols in equations carry information about the nature of the measurement beyond simply its value. This information may affect the way the equation is interpreted and used.
- Functions in science and engineering tend to stand for relations among physical variables, independent of the way those variables are represented

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.

*Engineering Education*,

*2*(1), 47-58.

Loughborough University has a Mathematics Learning Support Centre where students can go to seek help with their maths problems. The authors report on a survey run across staff and students to look at perceptions of difficulties and reasons for attending the centre. The paper reports on a few questions, with staff and student responses. The one of most interest to me was the one on maths difficulties, specifically basic manipulation. Staff see this as a big issue in student difficulties while students see it as a minor issue.

“Regarding basic manipulation, there appears to be a huge chasm between staff and student perceptions. Staff perceive a fundamental weakness, whereas students see a problem with the question being posed, which again indicates that staff need a greater awareness of current school mathematics syllabi and the level at which topics are delivered.” (p. 56)

“An interesting point to emerge is that students do not seem to appreciate that it is often lack of understanding of basic and fundamental mathematics that is at the root of their problems. This has implications for any attempts by staff to encourage students to undertake remedial work since students do not believe that they need to.” (p. 56)

This echoes what I see in my classes as well. Students’ algebraic manipulative skills are very weak, yet this is not recognised as the big issue it is by the students. Encouraging students to attend to this weakness falls on deaf ears. In my case, I have run compulsory assessments on factorising and manipulating logs (for example) with accompanying worksheets. I insist on an 80% pass and students can rewrite as many times as necessary. We have sessions in class working on these topics. Even with all of this, the students do not value basic manipulation and spend as little time as possible developing these skills. These problems appear to be global.

*European Journal of Engineering Education*,

*20*(3), 341-345.

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.