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
Students can fail to make the connection between symbols and their physical meaning and their relationship to empirical measurement.

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.
 Sfard, A., Prusak, A. (2005) Telling identities: In search of an analytic tool for investigating learning as a culturally shaped activity, Educational Researcher, 34, 4, 14 – 22. 
 
Identity is a term thrown around a lot in educational and other literature. Sfard and Prusak felt that a definition of identity that is operational is needed if it is to be useful. First and foremost, they define identity as stories about people. The notion of identity differs from other terms such as character, nature and personality in that it is not as “tainted” with undertones of biological determinism and as such is better suited to research located in a sociocultural context. 
 
“[I]n particular, identity features prominently whenever one addresses the question of how collective discourses shape personal worlds and how individual voices combine into the voice of a community” (p. 15) [so group to individual and individual to group] and “the question of the mechanisms through which the collective and the common enter individual activities also lies at the center of educational research on learning” (p. 15). It is known that culture shapes learning. There is a “complex dialectic” between learning and its sociocultural context and identity seems tailor-made to bridge the gap. However, identity can only be really useful if it has an operational definition. The authors says “it is the activity of identifying rather than its end product that is of interest to the researcher”, whose focus is not on identity so much as the “complex dialectic between identity-building and other human activities” (p. 17). 
 
But why not beliefs or attitudes as the bridge between learning and sociocultural context? Beliefs are tricksy things as they imply an existence which is discourse-independent yet how is an analyst to perceive and objectively identify them? Attitudes are similarly tricksy. 
 
Sfard and Prusak refer to Blumer’s test of admissibility, which is something I have not read the original of and probably should seek out. For a concept to be applicable in research it has to meet 3 conditions: we have to know what to look at in order to pinpoint the concept, we need to know how to identify things which are not the concept, and it needs to “enable accumulation of knowledge” (p. 15). Beliefs and attitudes don’t meet these criteria and, in fact, identity is at similar risk unless an operational definition can be found. The authors criticise others’ usage of the word identity without an explanation of what it is or how to use it, however they like Gee and other’s use of the term or idea of narrativisation, of people telling stories about who they are. The flaw they find in this is the postulation of there being some sort of internal thing about which you are telling stories. They call this an “essentialist vision” of identity and consider the notion of identity being something that is extra-discursive and independent of action to be both untenable (it gives us no idea of what to look for) and potentially harmful (descriptors can become self-fulfilling). 
 
Instead of beliefs or attitudes, Sfard and Prusak support the use of identity as a descriptive tool. They provide a collection of definitions which operationalise identity. 
Identity: “we suggest that identities may be defined as collections of stories about persons or, more specifically, as those narratives about individuals that are reifying, endorsable and significant” (p. 16) 
Reifying: the use of verbs such as be, have or can (rather than do) and with the adverbs always, never, usually, etc. that stress repetitiveness of the action. 
Endorsable: the “identity-builder” would agree that the story “faithfully reflects the state of affairs in the world” 
Significant: any change in the story is likely to affect the storyteller’s feelings towards the identified person. Often the most significant stories are those about inclusion in or exclusion from a community. 
 
Actual identity: stories about the actual state of affairs; present tense; formulated as fact 
Designated identity: stories about a state of affairs expected to be the case in the future, having the potential to become part of one’s identity; future tense, or expressed in words evoking obligation, commitment, necessity, wish; not necessarily desired, but always binding; sometimes the person is not aware that there are alternatives; the most important source is narratives authored by others; 
Critical stories: “those core elements that, if changed, would make one feel as if one’s whole identity had changed” (p. 18); without these elements the person would find it hard to tell which stories about him/her were endorsable or not. 
Significant narrators: influential people, “carriers of those cultural messages that will have the greatest impact on one’s actions” (p. 18). 
 
They then define learning as closing the gap between actual and designated identities. 
 
Any narrative can be seen as a triple BAC where A is the identified person, B the storyteller and C the recipient. Without going on any further it is clear that there are therefore multiple identities for any one person, which really has to be part of any definition of identity. Whether any of these three are the same person or not allows us to define three “levels” of identity narrative: 
AAC: the storyteller is the identified person: first-person identity 
BAA: the story is told to the identified person: second-person identity 
BAC: all three are different people: third-person identity 
AAA is a particularly special identity, a first-person self-told identity which is possibly what many people mean when they speak of identity – it has potentially the most impact on one’s actions. Second- and third-person identities have the potential to be incorporated into our first-person designated identities (p. 18). 
 
A merit of this narrative definition of identity is that “human agency and the dynamic nature of identity are brought to the fore” (p. 17); the focus of attention is on things being said, not on something existing behind what is said. As stories, identities are human-made, have authors and recipients, and can change according to need, they are accessible and investigable. A criticism of this narrative definition is that “text” cannot replace experience. Wenger in particular stresses that the words used to express or represent identity are not the identity itself. Sfard and Prusak argue that, while they agree with Wenger that identities originate in the “experience of engagement”, it is the stories one tells about experiences, or the visions of these experiences, that constitute identities, not the experiences themselves. 
 
Of all the theoretical texts on identity which I have covered so far, it is this narrative one I like the best. One reason is that I am wanting to use theory to look at some data I have and that data is largely in the form of interviews. Perhaps on a different data set I might prefer a different theoretical tool. Still, this definition of identity as story, with a split into actual and designated identities with learning drawing those two together really works for me. I shall try to apply this to my data and see what comes out! 
 
Potential future reading
Blumer, H. (1969) Symbolic Interactionism: Perspectives and Methods. Englewood Cliffs: Prentice Hall.
 
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.

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