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What seems more feasible here is that to some extent many respondents have more than one way of understanding, and so thinking about, ionic bonding. So many respondents are capable of thinking about ionic bonding in terms of discrete tightly bound NaCl units, bonded through a process of electron transfer and which are held in the lattice by weaker interactions; and also to appreciate something of the curriculum perceptive of the bonding being no more than the electrostatic binding of the lattice because of the array of positive and negative charges. Given these two ways of understanding the concept, the presented statements may well be judged true if they fi t one or other way of understanding, and so apparently contradictory statements come to be judged to be true. In the UK study, the results from the three groups of students suggest that the relative weighting given to these two ways of understanding ionic bonding shifts towards the curriculum model with increasing level of study (Taber, 1997 ). This interpretation is consistent with what was found in interviews, with students such as Tajinder (see Chap. 5 ), where several ways of understanding the ‘same’ concept were identifi ed, and conceptual change was best understood as a shift in which ideas were cued (Taber, 2001b ), rather than a switch in how a topic was understood (cf. Part IV).
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• Understand the common alternative conceptual framework for force and motion reported in the literature
• Understand the specifi c way a particular student thinks about force and motion This would involve having available as ‘mental resources’ several different ways of making sense of (i.e. understanding) the same phenomenon. For argument’s sake this particular teacher may well be committed to the fi rst of these options, so we could say that she only understands force and motion in one way herself but that she also understands other ways of understanding the concept area.
Meta-understanding and Multiple Understanding
There is an important distinction here, between (a) one’s own understanding, that one is committed to as the best way of making sense of some aspect of the world, and (b) an understanding of other understandings. In the latter case, what is
Fig. 6.3 An individual may have the mental resources to potentially understand something in more than one way
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understood is considered by oneself as something other than (in this example) force and motion. This teacher has a way of making sense of an aspect of the world, a way of making sense of how Aristotle made sense of this aspect of the world, a way of making sense of how (according to researchers) many students make sense of this aspect of the world and a way of making sense of how a particular individual learner makes sense of this aspect of the world.
Although the teacher has alternative ways of understanding the same phenomena available as mental ‘resources’, only one of these is committed to as the appropriate way to make sense of that aspect of the world (and the others are resources used to make sense of other people’s thinking, committed to as appropriate ways to make sense of those specifi c aspects of the world that are the public representations of the contents of other people’s minds , cf. Chap. 4 ). This would seem to be somewhat different from the situation of a learner who has two distinct ways of making sense of ionic bonding, or the relationship between force and motion, and is not strongly committed to either as the best way of understanding that aspect of the world. So there is an important distinction relating to the levels of commitment to particular ideas as representing how some aspect of the world actually is.
Understanding Distinguished from Beliefs
This important distinction can be denoted by distinguishing meta-understanding , that is, the understanding of a possible understanding of a phenomenon, from multiple personal understandings, that is, having several competing understandings for the same phenomenon. The difference between meta-understanding and per- sonal understanding is one of commitment or belief. So the historian or physics teacher can come to understand how Aristotle understood force and motion (Toulmin
& Goodfi eld, 1962/1999 ), that is, to develop meta-understanding of Aristotle’s mechanics, without committing to understanding the world the same way: without believing that is the best way to make sense of force and motion.
In a similar way, a student who comes to school with a strongly committed alternative understanding of some scientifi c topic, perhaps an impetus like under- standing of force and motion, would need to develop an understanding of the curriculum presentation of the topic before there was any possibility of committing to it as a better way of understanding that aspect of the world (Thagard, 1992 ).
Indeed, research suggests that many students who do come to understand the curriculum presentation well enough to use it successfully to answer formal assessment questions still do not commit to it as a better way of making sense of forces and motion in their everyday lives. We might say for these students they have acquired a meta- understanding of the curriculum formalism but do not themselves personally understand the world that way.
Whilst this is a key difference, there is an important relationship between these two situations, as it is not possible to change one’s way of understanding until one has available an alternative. That is, shifting commitment towards a new way of
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understanding is only possible once that understanding is available as a mental resource (see Chap. 15 ). This leaves the question of whether genuine cases of multiple understandings of which the learner is consciously aware as viable alterna- tives, and so not meta-understandings, and commits to, can occur. To be fi rmly committed to more than one such possibility would seem to be logically excluded;
it might be expected to lead to an awareness of a diffi culty, perhaps what Piaget termed disequilibrium (Piaget, 1970 /1972) or what has sometimes been called cognitive dissonance (Chapanis & Chapanis, 1964 ). If a human’s cognitive system is a self-regulating system, then such a state should motivate changes that are intended to remove the ambiguity.
Yet, by the same token, when we consider people as inherently learners, as those whose conceptual systems are still developing and are ‘works in progress’ – from a constructivist perspective as actors in an environment receiving constant feedback to allow them to adjust their mental models to better match expectations to experience (Glasersfeld, 1989 ; Kitchener, 1987 ) – we should not expect the ‘current’ state of a person’s conceptual system to be fully coherent and consistent. We are all, to some extent, such ‘works in progress’, with an ongoing programme of making sense of the complexity of our experiences of the world, including from time to time events that seem completely incongruent with our expectations based on how we under- stand past experiences.
Alternative Interpretations of Perceived Manifold Conceptions
Given these considerations, it would seem that researchers need to show caution in making assumptions when interpreting evidence of learners’ thinking (i.e. the public representations of their thinking; see Chap. 4 ) as demonstrating manifold ways of understanding the same phenomena. The discussion in this chapter suggests that there are various related possibilities here, then, which researchers should look to disentangle:
(a) The individual may have a single internally consistent way of understanding a phenomenon, which may or may not seem coherent when public repre- sentations of her thinking are interpreted from the perspective of canonical knowledge (e.g. what seem similar cases to an observer are within this indi- vidual’s scheme perceived as signifi cantly different on some characteristic imbued with salience by that individual).
(b) The individual may have available several ways of understanding the same phenomenon, and be aware of this, and is committed to one way whilst acknowledging (having meta-understanding of) the different ways others such as a science teacher understand the phenomenon.
(c) The individual may have available several ways of understanding the same phenomenon, and be aware of this, and is not sure yet which is the best way to make sense of the phenomenon.
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(d) The individual may have available several ways of understanding the same phenomenon, and be aware of this, and consider that each has something to offer and so is not motivated to be restricted to one approach: which could in some cases refl ect a more sophisticated epistemological stance that our ways of understanding some things are necessarily limited and imperfect, in which case commitment to one likely imperfect perspective may prove inadequate.
(e) The individual may have available several ways of understanding the same phenomenon, only being consciously aware of one of these, yet sometimes applies other implicit ways of thinking that operate at the preconscious level without noticing this.
If a researcher was only interested in whether a learner offered a ‘correct’
response to a question on an occasion when testing was carried out – which is in effect the stance implied by the nature of many public school examinations – then this complexity is of little relevance. Yet anyone claiming to be undertaking research exploring student understanding in a topic needs to be aware that an account that does justice to an individual’s understanding is likely to need to engage with issues of the status of understandings elicited, in terms of (i) whether they are unitary or part of a manifold of available ways of understanding and (ii) of the level of commitment the learner has to the ways of understanding elicited.