'Anthropological Theory of the Didactic' ATD - in learning mathematics

INTRODUCTION

Several studies have discussed the specific knowledge taught and learned in precalculus, calculus, and analysis courses, from different perspectives: for example, concept image and concept definition (e.g., O’Shea, 2016), APOS theory (e.g., Martínez-Planell, Trigueros Gaisman, & Mcgee, 2016), and the Anthropological Theory of the Didactic (ATD; e.g., Bergé, 2016). Our starting point is the general and relatively vague question of when in an undergraduate degree in mathematics does a student need (need in the sense of to succeed in the course) to engage in mathematical activities that may substantially, or meaningfully, lead to developing mathematical practices. We consider and frame this question within the ATD (Chevallard, 1999), which provides theoretical tools for modelling any human activity or practice. The semantic distinction between these two words is essential to us. Our hypothesis is that the kinds of didactic constructs to which professors and students are exposed are decisive in fostering the emergence of practices out of collections of local, particular, and relatively short-lived activities. From the theoretical stance we take, this means the development of mathematical knowledge out of local, particular, and relatively short-lived mathematical activities.' p.487.

THEORETICAL FRAMEWORK 

“Activity and practice”

'As mentioned above, we have come to see the semantic difference between activity and practice as pertinent to our work. The ATD’s notion of praxeology provides a fundamental model for defining mathematical practice, which, in the context of the theory, is equated to mathematical knowledge. According to the model, any practice (or piece of knowledge) can be represented by a quadruplet [T, τ, θ, Θ] involving four interconnected components: a type of tasks T, which generates the practice, the corresponding collection of techniques τ developed to accomplish T, the discourse used to describe, justify, explain, and produce the techniques (i.e., their technologies θ), and the underlying theories Θ that serve as a foundation of the technological discourse. As students progress in their studies of mathematics, they engage in numerous activities, which progressively determine the practices they develop'. p.489.

Laura Broley and Nadia Hardy. (2018). A study of transitions in an undergraduate mathematics program. In PROCEEDINGS of INDRUM 2018 Second conference of the International Network for Didactic Research in University Mathematics. pp.487-496. ERME topic conference.
https://indrum2018.sciencesconf.org/data/Indrum2018Proceedings.pdf

THEORETICAL CONSTRUCTS 

'ATD “postulates that any activity related to the production, diffusion or acquisition of knowledge should be interpreted as an ordinary human activity, and thus proposes a general model of human activity built on the key notion of praxeology” (Bosch & Gascon, 2014). The praxeology 𝛱 is represented by a quadruple [𝑇/𝜏 /𝜃/𝛩]: its praxis part (or know-how) consists of a type of tasks 𝑇 together with a corresponding technique 𝜏 (useful to carry out the tasks 𝑡 ∈ 𝑇 in the scope of 𝜏). The logos part (or know-why) includes two levels of description and justification: the technology 𝜃, i.e. a discourse on the technique, and the theory 𝛩, which often unifies several technologies. 

The elaboration of a reference epistemological model (Florensa, Bosch, & Gascon, 2015) as sequences of praxeologies, for a given body of knowledge, is an important step in any research carried out in the ATD framework. It is the tool that will be used by the researcher to describe, analyse, put in question or design the specific contents that are at the core of a teaching and learning process. In order to build such a model, “mathematical praxeologies are described using data from the different institutions participating in the didactic transposition process, thus including historical, semiotic and sociological research, assuming the institutionalized and socially articulated nature of praxeologies” (loc. cit. p. 2637).' pp.498-499.

Charlotte Derouet, Gaetan Planchon, Thomas Hausberger, and Reinhard Hochmuth. (2018). Bridging probability and calculus: the case of continuous distributions and integrals at the secondary-tertiary transition. In PROCEEDINGS of INDRUM 2018 Second conference of the International Network for Didactic Research in University Mathematics. pp.497-506. ERME topic conference.
https://indrum2018.sciencesconf.org/data/Indrum2018Proceedings.pdf

I came across this source as I'd read about the way chemistry as a discipline co-opted some mathematical symbols, but adapted them for their own use.

Previously: 'math' : 'threshold concept' : 'liminal' : 'learning'

iii Picking up sticks: when axes matter

It is one thing for us to break the stick, but quite another from a concept's perspective. They are accustomed to the dynamics at work, when as search terms they are exploded, or not. And, not just any concept but one that can be unique. Consider the concept, and how or where it finds itself [ in Hodges' or another model ]? Let us use the intra-INTERPERSONAL domain, as an example, and ask:

1. Is 'mood' fixed by the I-axis alone?
2. Does the humanistic axis have a role in where in the doomain the user of Hodges' model places 'mood'?
3. Acknowledging person-centredness, if the patient/client had a view, to what extent would this agree (with annonymity assured) with the professional's judgement, or a relatives / guardian?
4. What is as a result, 'mood' has become 'very low mood'? (To continue in this vein , invites complication, but we will continue.)
5. Introducing the 'Other' whatever their role and relationship, pushes us to the humanistic, but whether this is achieved in a person-centred manner (triage excluded) is a matter of opinion and judgement (evaluation).

If mood is fixed by the I-axis then if we determine the issue is 'low mood' are we introducing a combined quantitative and qualitative dimension, a continuum, at the selected point in the interpersonal domain? Writing previously about threshold concepts, I wondered, and proposed compound threshold concepts. There is a progression from the patient's subjective self-assessment, to in all likelihood of an objective measure being introduced, so spanning two domains. Definition of 'compound' aside; there is a compound structure at work.

What points 1-5 above reveal is triangulation. When it is argued that Hodges' model facilitates navigation as per my presentation on 21st April, this is a strategy from mixed-methods research in practice. Many years ago, Hodges' model was described as a 'cognitive periplus', reflecting the way ancient mariners initially charted the coastlines creating the first map, as per their culture and dispora. In the west more recently the value of social approaches and activities, especially in mental health. Care navigation is a role. Literally sign-posting to individually suited (which must be stressed) activities, resources, and agencies. So, in Hodges' model what are our three triangulation points? They are:

1. Axial;
2. Intra- InterDomain; 
3. Person-centred. 

Perhaps, we can describe 1-2 as conceptual anchors at least when dealing with a concept, while #3 is more about approach, values and philosophy. So, there is something of a 'cheat' going on, because each one is compound:

1. Axial: The axial is comprised of four points. The axes create a Cartesian plane (figure). And, as far as a patient is concerned, they mark off a point on the respective Individual and Humanistic-Sciences parts of the axes. This form of triangulation point can claim a general conceptual precision (language) in the applied terminal labels; but what happens in-between can be patently fuzzy. A mix of what may be objective and subjective.

2. Intra- Interdomain: This may be the most imprecise triangulation point. It appears the axes of Hodges' model are composite (figure). They have two-sides as a boundary. To be pedantic, in our example, one plays to the interpersonal domain, the other the sciences.

[The relationship between the axis and the domains in Hodges' model, is far from unique; but I wonder if there is something else here?]

2. Person-Centred: There is a paradoxical nature to person-centred(ness). It is also nebulous in nature, and yet in care it is definitive in practice (safety, purposes, values, professionalism, competency, ...).  It forms locus which can also be classed as a traingulation point, at the centre of the model. This is the point around which Hodges' model is built. This captures the individual's (life, care) context and situation, all influenced by their life chances as consequent health - and other - careers. Salience and our attention^ is key. Is it possible to frame our purpose(s)?

^A focus for [N] autumn.

Marrakech, Le Jardin Secret. 24th April. 1000-1830.

Variety really is the spice of Hodges' model

In the previous post - RCN Congress iii 2025 Accountability for patients in ambulances I wrote of how corridor care has become variegated, as it is bound to, in a system(?).

Variegation means difference, and yet in organisational accountability, there should be reliance upon a procedure, if not a standard operating procedure. The experience of individual patients should be nuanced, standardised and yet catering for individual differences.^ The resolution at congress was questioning the need for guidance amid legal ambiguities.

This is also helpful in two senses. 

1. In exploring ways that Hodges' model might help encompass and explain the dual character category*, [or other, or not] as may apply to person-centredness and service-centredness?
2. Plus, (and potentially confusing for me) in seeking a basic understanding of dual categories in a mathematical sense.

* Book review iii: 'Categories we live by'

from the above post:
Knobe J, Prasada S, Newman GE. Dual character concepts and the normative dimension of conceptual representation. Cognition. 2013 May;127(2):242-57. doi: 10.1016/j.cognition.2013.01.005. Epub 2013 Mar 1. PMID: 23454798.

^This highlights a quality of what it is to care/and caring. It relies upon relationship building, the nurse getting to know the patient, person and vice-versa (to a degree) as befits the emergence of empathy, rapport, generation of trust; and the nurse's ability to anticipate not only care needs (that arise from direct observation), but those that are unique to the patient as an individual. In the previous post which mentioned corridor care, the physical, emotional, social and political environment is not established. Noise, on several levels stymies care, disrupting communication and interpersonal skills - even presence and touch - amid the chaos of the corridor. 

Collective Intelligence in Hodges' model: c/o Röling (2009)

individual

|

INTERPERSONAL : SCIENCES

humanistic ----------------------------------------------- mechanistic

SOCIOLOGY : POLITICAL

|

group - population

Intelligence Quotient Emotional Intelligence Spiritual Intelligence ... Individual beliefs Individual motivation "Can individuals really be rational and modify behaviours accordingly?"

Ozone layer Troposphere Biosphere  HUMANS are a FORCE of NATURE Deepest Ocean bed "Anthropogenics would also be a praxiology - a science that informs decisions and action." p.40.

The 'Enlightment' tradition: Science, Knowledge, Human ideals distilled in history and future history.

"Can we adapt tools forged to create wealth and use them for sustainable development? Is ever-increasing wealth the only viable measure of progress?" "A close understanding of how institutions determine individual behaviour might even curb the enthusiasm for "methodological individualism", the tendency to explain collective things such as the marketplace as a necessary outcome of individual choices." p.41.

[Context: How can agriculture feed the world ..? ]

"One of the things this exercise brought home to me was how poorly we understand humans as agents of planetary change, how little of what we do know is widely shared, and that this knowledge is scattered across disciplines that appear distant to governance. I began to ask if we needed a new discipline, one that could beyond the work of thousands of ecologists, climatologists, economists, anthropologists, psychologists, political scientists, neuroscientists and the like. I decided to appropriate the name "anthropogenics" for my dream discipline. p.40.

My source:
Röling, N. (2009) Profile. A proper study of mankind, New Scientist, 17 January, 201, 2691. pp.40-41.