'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'

Ann Hutchinson Guest (Obit.), 'Authority on dance notation ... '

'Although there are many forms of dance notation going back to Pierre Beauchamp's system for Baroque dance in the 1680s, Nijinsky had developed his own cryptic version. Ann Hutchinson Guest first tried to reconstruct his work in 1956 after the choreogapher's widow, Romola, gave her his notes. "I spent years trying to work it out," she said, adding with a laugh: And I'm supposed to be the leading authority on dance notation in the world. She told how Nijinsky had revised his notes many times, often contradicting himself. 'We couldn't get anywhere. Romolo was furious. She said it was taking much too long."

    Hutchinson Guest studied dozens of different dance notation systems, translating many of them into Labanotation, a system developed in the 1920s by Rudolf von Laban who was a central figure in European modern dance. According to a  press report in 1954: "The printed Labanotation page looks like a combination of hieroglyphs, pictographs, Morse dots-and-dashes, đoodles, and a musie score turned on edge" 

    It was not until 1980 that she and Claudia Jeschke, a dance historian, succeeded in cracking Nijinsky's code after finding a copy of his annotated score in the British Library.'

Register, The Times.

My source: Ann Hutchinson Guest (Obit.), 'Authority on dance notation known for reconstructing Vaslav Nijinsky's scandalous ballet, L'Après-midi d'un faune'. Register, The Times. 20 April, 2022, p.48.
https://www.thetimes.com/uk/obituaries/article/ann-hutchinson-guest-obituary-hkw8z88ns

INDIVIDUAL

|

     INTERPERSONAL    :     SCIENCES               

HUMANISTIC --------------------------------------  MECHANISTIC      

SOCIOLOGY  :   POLITICAL 

|

GROUP

abstraction representation interpretation observation - translation code - formalism	body (bodies) symbols - forms dance choreography movement - 4D - time
group ensemble 'anthropological composition?' 'diagnostic grouping'? social - cultural category . . .	'collective power & attention  evinced as time' 'The stage: the space for health'^

Hutchinson Guest in 1956
 Source: The Times.

'Dance has been called moving architecture, a truth which the complexities of contemporary choreography may mask. Not only does the dancer's body form shapes and groups of dancers form moving or static designs, but also group arrangements carve up the stage space, establishing areas of open as well as enclosed space which are significant in their changes and contrasting effects.

While the rccording of movement on paper-dance notation-does not look like movement (disregarding primitive stick figure drawings), there is an "architecture" in the sequence of graphic symbols used to record dance patterns. Over the centuries many different devices have been used to capture dance steps on paper. What are the elements that have to be represented?

The process of dance notation requires reducing four-dimensional movement (time being the fourth dimension) to a two-dimensional surface. The parts of the body in action have to be defined, as does the form of movement involved (flexion, extension, rotation, directional placement) and the duration of each in relation to the overall time structure. In group dances the relationship of dancers to one another must be determined and recorded, as well as their location on stage and their paths of travel. In a dance score each performer is like a small orchestra-arms, legs, head, torso, etc. in motion-this is then multiplied by the number of dancers who are performing individual sequences.'

Guest, A. H. (1990). Dance Notation. Perspecta, 26, 203–214. https://doi.org/10.2307/1567163

^Health is always performed in the political domain, make no mistake (excuses, or exceptions). It is not:

1. Acknowledged, masked - covered by the 'set design' (see #2 below);
2. Implied in the medical, biomedical, bio-psycho-social model;
3. The intermission is neither the time, or place, to reflect upon and critique the political in health :: health in politics.

See also: https://archives.nypl.org/dan/19629

https://en.wikipedia.org/wiki/Ann_Hutchinson_Guest

Previously: 'dance' : 'architecture' : 'movement'  : 'notation'

Paper - (finally!) "A generic model and conceptual framework to prime curiosity across health and social care ..."

 It has been over a decade but at last the second* paper has been published in Journal of Evaluation in Clinical Practice:

Jones, P. (2025), A Generic Model and Conceptual Framework to Prime Curiosity Across Health and Social Care Disciplines to Facilitate Lifelong Learning. Journal of Evaluation in Clinical Practice, 31: e70252. https://doi.org/10.1111/jep.70252

ABSTRACT 

Rationale: The Corona virus pandemic highlighted the importance in continuity of the physical and emotional labour across all care sectors. Synergy between healthcare in hospitals, and community services must be allied with social care; and be central to integrated and efficient policy and service delivery. Services were found wanting in the pandemic and in recurring winter crises. The politics of funding for care delivered in rest and nursing homes remains contested, awaiting long‐promised governmental action. The workforce is recognised as dedicated, skilled, compassionate and yet under‐valued and under‐paid; managers face ongoing recruitment challenges, with a national shortage of staff, and high turnover of personnel. A National Care Service is awaited in England and Wales. 

Aims and Objectives: This study explores the ongoing care crisis, using the educational lens of threshold concepts, projected, analysed and synthesised of a generic conceptual framework, known as Hodges' model. Readers will understand Hodges' model and threshold concepts being equipped to explain both. 

Methods: The study is descriptive and uses a conceptual mapping approach supported by discussion and literature. Additional resources and avenues for ongoing study are also provided. 

Conclusion: In conclusion, Hodges' model is a pragmatic, practice‐based tool, that can support and sustain curiosity for workforce learning and development. 

Acknowledgements I would like to thank Messrs Mark Bird and Matthew Graham and the journal's reviewers for comments on drafts of this study; and acknowledge the effort of Dr Michael T. Flanagan in creating the informative threshold concepts resource (referred to above). 

https://onlinelibrary.wiley.com/share/author/IRPNBR2KHJNYJK8XUAW6?target=10.1111/jep.70252

*This paper was originally the first of a two-part work. Feedback from reviewers suggested the papers be standalone. United now, in one journal.

Many thanks to the reviewers, editor, editorial and publications team at JECP.

This paper has been added to bibliography in the blog's sidebar.

Does the model still work - this way?

Does the model still 'work' -

Group
 |

 POLITICAL       :     SOCIOLOGY  

MECHANISTIC --------------------------------------  HUMANISTIC

          SCIENCES    :    INTERPERSONAL

| 

Individual

POLITICAL	SOCIOLOGY
SCIENCES	INTERPERSONAL

- this way ...?

'Anthropologically strange'

INDIVIDUAL

|

INTERPERSONAL    :     SCIENCES               

HUMANISTIC --------------------------------------  MECHANISTIC      

SOCIOLOGY  :   POLITICAL 

|

GROUP

"Phenomenology focuses on the inter-subjective constitution of the social world and everyday social life. Schütz (1964), in a seminal essay on 'The Stranger', shows how a social group has its own cultural pattern of life - folkways, mores, laws, habits, customs, etiquette, fashions and so on - that, as far as its members are concerned, are taken for granted, are habitual and almost automatic. Schütz's stranger provides a model for the ethnographer using participant observation. The ethnographer tries to treat the familiar world of 'members' as anthropologically strange, to expose its social and cultural construction. This is particularly demanding when a researcher is studying a group with which he or she is familiar, but represents an ideal attitude of mind for the researcher to pursue nevertheless." p.246.

Researching Society and Culture

"Anthropological strangeness: The art or mental trick of making a social setting and behaviour within it appear as if the observer is encountered as a stranger. If applied to mundane 'taken-for-granted' events, this can lead to unusual and original insights." p.556, (Glossary).

This ethnographic idea 'strangeness' provides an extension to the role of Hodges' model as a template, a clean slate across disciplines, settings and contexts. A means to 'reset' conceptually, to clear the mind, mindset, attitudes and start afresh.

Walsh, D. (2012). Doing Ethnography. (Chapter 14. pp.245-262) In. Researching Society and Culture, Seale, C. (Ed.). 3rd edition. London: SAGE.

See also:

Schuetz, A. (1944). The Stranger: An Essay in Social Psychology. American Journal of Sociology, 49(6), 499–507. http://www.jstor.org/stable/2771547

Another book student bound.

Is there an implicit theory in Hodges' model?

 INDIVIDUAL

|

 INTERPERSONAL    :     SCIENCES               

HUMANISTIC -------------------------------------  MECHANISTIC    

SOCIOLOGY  :   POLITICAL 

|

GROUP

From early on I've thought that there is a theory in Hodges' model. Maybe not in the structure - the axes, or the (potential and eventual) content - the care (knowledge) domains, but in combination is there an implicit theory?

"Models in the philosophy of science are usually distinguished from theories. However, the distinction is not always clear cut and some authors slip from one to the other. Three examples of philosophers who use the terms differently are: Apostel [16] who allows a theory to be a model, Hesse [216] who argues that a theory is meaningless if it does not include a model and van Frassen who says “To present a theory is to specify a family of structures, its models...” [161] p. 64). On the whole the use of the term ‘model’ implies something relatively concrete, possibly visualisable, approximate, specific and tractable whereas a ‘theory’ implies a more general, abstract and reliable construction. In the social sciences it is more common to use the term ‘model’ for things that would be called theories in the physical sciences (according to Braithwaite, 1962, who deplores the practice)."

Edmonds, B. (1999). Syntactic Measures of Complexity. Doctoral Thesis, University of Manchester, Manchester, UK. http://bruce.edmonds.name/thesis/

[16] APOSTEL, L. Towards the Formal Study of Models in the Non-Formal Sciences. Synthese, 1960, 12, 125-161.

[216] HESSE, MB. Models and Analogies in Science. London: Sheed and Ward, 1963.

[161] VAN FRASSEN, BC. The Scientific Image. Oxford: Clarendon, 1980.

[73] BRAITHWAITE, R. B. Models in the Empirical Sciences. In NAGEL, E; SUPPES, P; TARSKI, A (eds.). Logic Methodology and the Philosophy of Science: Proceedings of the 1960 International Congress. Stanford, CA: Stanford University Press, 1962, 224-239.

Developing a Palliative Care Model for People with Dementia in the Community c/o @PallC4Dementia

Challenge

People are living longer, which means that more of us will get age-related illnesses such as dementia. Dementia is caused by different diseases of the brain. These diseases affect the parts of the brain which are normally used for learning, memory and language. Although Dementia usually affects people as they get older, it’s not a normal part of ageing. Many people don’t realise that Dementia is a terminal illness. People may die from dementia, not just die with dementia. Palliative Care aims to improve the quality of life of people and their families who are facing an incurable, life-limiting illness, such as dementia. Palliative care involves actively looking for symptoms of any kind- physical, social, spiritual, or psychological - and providing excellent care to lessen these symptoms. People with dementia need palliative care just as much as people with cancer, but they often don’t get it.

Research Project (currently ongoing)

This project aims to improve the palliative care of people with dementia, and their families, by developing a “model”. This model aims to support people with dementia to live well at home in the community for as long as possible. The model will map out what services, assessments and treatments the person might need and where, so the health system can better organise their care. To develop a model that will work in Ireland, we must first learn what is already known about “good” palliative care for people with dementia. This mixed methods study will be carried out in several phases:

• We will conduct a review of the literature, to find all available research in this area. This will allow us to see what existing models and services are supported by research evidence.
• We will conduct a survey of healthcare workers and researchers to ask them two things. Firstly, what do they think an ideal model of dementia palliative care would look like? And secondly, what existing services providing dementia palliative care are there?

...

• ... Please visit the project's website for full details....

Phase 1: The research team designed a survey to i) identify existing models for dementia palliative care in Ireland and the UK, and ii) to identify key components of effective models. The survey is currently live on this website. The research team has also developed a protocol for a scoping review to guide their search of the literature for relevant research. The protocol is registered with PROSPERO ID: CRD42018091158, and the search is currently underway.

...

Follow us on Twitter

@PallC4Dementia | #PallCare4Dementia

For further information contact

Email: s.fox AT ucc.ie
Email: info AT aiihpc.org 

• ... Please visit the project's website for full details....

The Fourth Quadrant - Johnson (2010)

For anyone who has read Steven Johnson's Where Good Ideas Come From, you might appreciate that for me it was a case of take-two from the previous post 'The adjacent possible'. How could I not include the chapter "The Fourth Quadrant" especially as the book's 'Conclusion'?

If title was one take, the other was "Oh! How does this relate to h2cm?" After reading and reflecting things fell into place. Johnson wants to represent the history of innovation (no less) using roughly two hundred of the most important innovations and scientific breakthroughs from the past six hundred years, starting with Gutenberg's press, Einstein's theory of relativity, the invention of air conditioning through to the birth of world wide web (pp.218-219).

There are three figures in the chapter spanning 1400-1600; 1600-1800; 1800-present. Johnson's diagram for 1800-present (p.229) is in-part 'mapped' to Hodges' model below.

A few insights into my reasoning. In the Interpersonal domain I am emphasising an individual's primary (and original) motivation for their work and what follows. In the sciences - here PROCESSES are central. As with all models this is an idealisation, so there is a lot of overlap.

Individual

|

INTERPERSONAL : SCIENCES

humanistic / non-market ------------------------------- mechanistic / market

SOCIOLOGY : POLITICAL

|

Network

3. Non-Market/Individual Not devised for fiscal gain, individually devised .. amateur scientist, or hobbyist. Spectroscope           Hormones Bunsen Burner             E - mc2 Rechargable Battery      Special Relativity Nitroglycerine        Earth's Core Cell Nucleus          World Wide Web Stethescope         CT Scan ...	1. Market/Individual Devised for fiscal gain, individually devised i.e. Air Conditioner Mason Jar Tesla Coil Gatling Gun Nylon Vulcanized Rubber Programmable Computer Transistor ...
4. Non-Market/Network Not devised for fiscal gain, collectively devised. Open source envirnoment or academia. Braille      Periodic Table     RNA Splicing Chloroform        EKG    Aspirin Cell Division    Global Warming Benzene Structure        Continental Drift Hereditary       Superconductors ...	2. Market/Network Devised for fiscal gain, collectively devised, multiple firms interact Airplane    Lightbulb    Steel   Automobile Induction Motor      Radio  Contact Lenses    Welding Machine Moving Assembly Line   Motion Picture Camera Locomotive     Vacuum Cleaner Electric Motor     Washing machine Refrigerator      Vacuum Tube Telegraph      Helicopter ...

Johnson, Steven (2010) Where Good Ideas Come From, London: Allen Lane.

Q. What is nursing work? A. Hodges' model

Presenting during the afternoon sessions: What is nursing work? My new model from a meta narrative review at @theRCN's #research2018 https://t.co/kLsGGl9YXb see you there! pic.twitter.com/Wtj6S7fldl

— Jennifer Jackson (@JJackson_RN) April 15, 2018

Image: Jennifer Jackson

The above also reflects the domains of Hodges' model which is derived from the following questions:

Who do nurses (healthcare professionals, carers) care for?

This provides us with the INDIVIDUAL - GROUP (Population) vertical axis.

What types of activities, tasks - work do nurses then have to carry out to deliver care?

This question accounts for the horizontal axis of Hodges model.

individual

|

INTERPERSONAL : SCIENCES

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

SOCIOLOGY : POLITICAL

|

group

COGNITIVE	PHYSICAL CARE
EMOTIONAL	ORGANISATIONAL

Nurses as modellers and informaticians: surely not!

Nursing is still trying to escape and evade the sexual 'Carry On' stereotypes that have plagued the profession in the popular imagination. For the majority of people talk of nursing and models more readily conjures up visions of catwalks than an academic pursuit.

You can still see and hear the response of bright-eyed girls and boys (aged 9-10...) to the age-old question: "What do you want to be when you grow up?" The answer: "I want to be a nurse and help people get better!" Even though youngsters are more sophisticated these days (the reference to girls and boys is not just me being politically correct), they are still most likely primarily motivated by humanistic leanings as opposed to wanting to pursue the necessary studies in the sciences.

Despite efforts worldwide practicing nurses do not all see themselves as data modellers enthralled by IcT. IT isn't usually why they came to nursing, although many mature students may have started their career in the IT sector. Chapter 1 of Programming the Semantic Web highlights how a basic table is a model (p.6-7). So of course gifted with natural language we are all data modellers. Hodges model then is the ubiquitous high-level data model - a two-by-two table and a whole lot more.

An invitation to mine data, gather information and deliver nuggets of knowledge.

Ref:
Programming the Semantic Web: Build Flexible Applications with Graph Data
By Toby Segaran, Colin Evans, Jamie Taylor
Publisher: O'Reilly Media
Released: July 2009