Redish, E. & Smith, K.A. (2008). Looking beyond content: Skill development for engineers. 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.
Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings or subjective views.