Recently I have been teaching my Year 9 Science class about equations in Physics. A colleague, Michelle Weightman, has developed a strategy that we use in the department to help develop metacognition alongside the use of equations for solving problems. She calls the strategy ERSCA. It is an acrostic mnemonic to support and model the thinking that should occur when approaching physics problems. E = equation, R = rearrange, S = substitute, C = calculate and A = answer. Let me attempt to model how this works in the classroom.
Here is the sort of physics problem that a student might face:
Using the metacognitive strategy of ERSCA, a student might then think/say the following...
“Ok, E stands for equation and so I know I need to write down the equation that I know and the question is asking for energy transferred and it mentions specific heat capacity. The only equation I know that includes both energy transferred and specific heat capacity is:
change in thermal energy = mass × specific heat capacity × temperature change
Now the R stands for rearrange the equation if necessary. In this case I do not need to, as the question is asking for the energy transferred and the equation is in terms of energy so I am good. The next letter is S which stands for substitute the values from the question into my equation. Now I have done that I need to calculate because this is what the C tells me to do. Having got an answer from the calculator, I now need to check it. A stands for check the answer and the units. I must ask myself does the value calculated seem right, and if so, think about the units and add them in. In this case the answer seems ok and the units are Joules (J) but the exam paper already has that so I am good.”
Their work might look like this...
So far so good. However, I have found that several of my students have struggled with the concept of rearranging the equations. This might have to be done if the question asks for the calculation of a value other than the subject of the equation learnt by the student. Using the above equation, this might be “find the specific heat capacity...” rather than “calculate the energy transferred...”
Following the Metacognition and Self Regulated Learning Guidance report from the EEF I have modelled how to rearrange equations. I have shown them different strategies that they might employ to rearrange equations (algebraically, the line method, the funnel method, formula triangles etc).
I have explicitly taught them about the metacognition cycle of knowledge (knowledge of task, strategy & self) and yet some students are still frustrated in their attempts to grasp the concept of rearranging equations.
Some of them seem to be able to rearrange on one occasion, then struggle on another. When I model (for the umpteenth time) they seem able to ‘mimic’ how to do it, yet independent assessment informs me that they still have yet to fully comprehend the ideas.
This reminds me of a paper I read some years ago, “Cordelia’s Dilemma”. The paper introduced me to the idea of threshold concepts from Meyer and Land’s work. They consider a threshold concept to be “... a portal, opening up a new and previously inaccessible way of thinking about something. It represents a transformed way of understanding, or interpreting, or viewing something without which the learner cannot progress.”
“Threshold concepts are likely to be:
Transformative: they result in a change in perception of a subject and may involve a shift in values or attitudes.
Irreversible: the resulting change is unlikely to be forgotten.
Integrative: it ‘exposes a previously hidden interrelatedness’ of other concepts within the discipline. Bounded: it serves to define disciplinary areas or to ‘define academic territories’.
Potentially troublesome: students may have difficulty coping with the new perspective that is offered.”
Ian Kinchin also describes the punctuated model of conceptual change proposed by Mintzes and Quinn. Learning does not progress linearly. There are moments of stasis in understanding and learning.
“Only years later ... did I conceptualise the possibility that plateaus of stagnation and bursts of achievement might express a standard pattern for human learning” Gould
In terms of metacognition, students need to know that they are grappling with threshold concepts as part of the knowledge of task (the task being the full and proper comprehension of the idea). But, I would like to argue that we also need to explicitly teach them that learning as conceptual change is a punctuated process. There will be periods of learning plateaus and stagnation where the student (and teacher) feels as though they are making no progress at all, that they are wading slowly across an interminable quagmire. My contention is that students must be taught that this is ok, and a natural part of the journey of learning. This is knowledge of self. Students must recognise (or have it pointed out) that they are in the liminal phase.
“Difficulty in understanding threshold concepts may leave the learner in a state of ‘liminality’, a suspended state of partial understanding, or ‘stuck place’, in which understanding approximates to a kind of ‘mimicry’ or lack of authenticity. Insights gained by learners as they cross thresholds can be exhilarating but might also be unsettling, requiring an uncomfortable shift in identity, or, paradoxically, a sense of loss. A further complication might be the operation of an ‘underlying game’ which requires the learner to comprehend the often tacit games of enquiry or ways of thinking and practising inherent within specific disciplinary knowledge practices.” Land, Meye & Baille
“This (liminal) space is likened to that which adolescents inhabit: - not yet adults; not quite children. It is an unstable space in which the learner may oscillate between old and emergent understandings just as adolescents often move between adult-like and child-like responses to their transitional status. But once a learner enters this liminal space, she is engaged with the project of mastery unlike the learner who remains in a state of pre-liminality in which understandings are at best vague. The idea that learners enter into a liminal state in their attempts to grasp certain concepts in their subjects presents a powerful way of remembering that learning is both affective and cognitive and that it involves identity shifts which can entail troublesome, unsafe journeys. Often students construct their own conditions of safety through the practice of mimicry. In our research, we came across teachers who lamented this tendency among students to substitute mimicry for mastery” Cousins
“We would not, however, wish to imply that this relational view has an overly rigid sequential nature. It has been emphasised elsewhere that the acquisition of threshold concepts often involves a degree of recursiveness, and of oscillation, which would need to be layered across this simple diagram.”
In practical terms this means the central green box does not follow a “rigid sequential” pattern but would be characterised by a range of student activity and work: some fully realising and accurate in their equations and rearrangements, some mimicking teacher modelling, some appearing to grasp the concept in a lesson and the next not, some feeling confident in one instant but helpless and lost the next and so on.
Returning to my year 9 science class I can report that I observed all of these variations in progress. Some had clearly passed through the liminal stage and were clear through on the other side of the threshold concept. Others would report that they felt confident, their work looked good and appeared to show that they were capable of the task but on further assessment found their confidence crumble and their conceptual grasp slip away. Some were still stumbling in the darkness of the liminal space, feeling their way towards the speck of light at the end of the tunnel. Of course I have explained to the class about liminality and the non-sequential nature of conceptual change. It is ok for them to be somewhere in the space between not knowing and knowing, between not understanding and fully grasping the threshold concept. Having this knowledge of task and self allows students to be fully metacognitive about their learning journey.
https://www.ee.ucl.ac.uk/~mflanaga/popupLiminality.html#
