Landscapes, Maths, Line of Sight c/o Penteado & Skovsmose Eds. (2022)

The care / knowledge domains of

Hodges' model as landscapes:
as 'initial sketches'?

'Line of sight/thought?'

Individual
|

      INTERPERSONAL    :     SCIENCES               

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

 SOCIOLOGY  :    POLITICAL 

|

Group

reflection critical reflection cognitive/conceptual landscapes

maths and sciences = hard subjects ?

"We understand that mathematics is especially relevant in knowledge processing, and operates in the process of globalisation,4 i.e., it interferes in several aspects that integrate with society. We admit that globalisation refers to all aspects of life and that, depending on how it is questioned and operationalised, it may or may not be beneficial. Therefore, globalisation “has to do with the construction, codification, and distribution of knowledge that turns into goods for sale” (Skovsmose, 2014, p. 130). This way, mathematical knowledge is involved as part of the foundations of society, making it necessary to question its position in this laborious civilising equation (Civiero and Bazzo, 2020). In this context we delegate some power to critical mathematics education (CME) by considering that it can contribute to the formation of critical individuals by promoting reflection on this process. Bazzo (2019) considers us to experience a civilising equation whose variables need to be discussed in schools." pp.296-297.

"The word citizenship, a contested concept, emerges in different discourses that can hide contrasting interests. Differences in its meanings are enhanced with the changes in society through history. This chapter presents a discussion about mathematics education for citizenship and its outcomes. School presents itself as a stage of dispute and a space for expressing different ideologies that can configure ways of thinking and structuring an education on the topic of citizenship. Mathematics education is shown to be supportive of different citizenship discourses. In a globalised world with deep inequalities, a type of mathematics education for citizenship that matches with inclusion and diversity is required; one which considers global issues. The proposed landscape of investigation Global Visibility Matters is a possible support aid for mathematics classes, based on enabling students to develop global citizenship. By reading and interpreting a social situation as being open to change, it becomes possible for the students to be recognised as—and to act as—global citizens." p.133.

Chapter 8 Global Citizenship, Manuella Carrijo, pp. 133-147.

Chapter 18 Critical Mathematics Education in Action:  To Be or Not to Be. Paula Andrea Grawieski Civiero and  Fátima Peres Zago de Oliveira. pp. 295-321.

Penteado, M. & Skovsmose, O., editor. (2022) Landscapes of Investigation: Contributions to Critical Mathematics Education. Cambridge, UK: Open Book Publishers. [Pdf] Retrieved from the Library of Congress, https://www.loc.gov/item/2021385926/

landscape

https://hodges-model.blogspot.com/search?q=landscape

sketch

https://hodges-model.blogspot.com/search?q=sketch

'line of sight'

https://hodges-model.blogspot.com/search?q=line+of+sight

Book: Landscapes of Investigation [open access]

"Creating landscapes of investigation is a primary concern of critical mathematics education. It enables us to organise educational processes so that students and teachers are able to get involved in explorations guided by dialogical interactions. It attempts to address explicit or implicit forms of social injustice by means of mathematics, and also to promote a critical conception of mathematics, challenging the assumption that the subject represents objectivity and neutrality. Landscapes of Investigation provides many illustrations of how this can be done in primary, secondary, and university education. It also illustrates how exploring landscapes of investigation can contribute to mathematics teacher education programmes. 

This edited volume is the result of a collaboration established through the Colloquium in Research in Critical Mathematics Education, which took place in 2016, 2018, and 2019 in Brazil. Its twenty-eight contributors are young researchers from Brazil, Chile, Colombia, India, Mexico and the USA, who are dedicated to the further development of critical mathematics education. 

Organised in eighteen chapters, the volume presents examples of engaging students from a diversity of social and economic backgrounds, age ranges, and abilities across different countries. The chapters present original findings on the social aspects of all levels of mathematics education. Landscapes of Investigation is of particular relevance to those with an interest in the potential of mathematics education to challenge social injustices."

Penteado, M. & Skovsmose, O., editor. (2022) Landscapes of Investigation: Contributions to Critical Mathematics Education. Cambridge, UK: Open Book Publishers. [Pdf] Retrieved from the Library of Congress, https://www.loc.gov/item/2021385926/

Review:

David M. Bowers
"Book Review: Landscapes of Investigation: Contributions to Critical Mathematics Education (2022) (M. G. Penteado & O. Skovsmose, Eds.)". Journal for Theoretical & Marginal Mathematics Education, vol. 2, no. 1, 2023. doi:10.5281/zenodo.10440243

My source:

https://x.com/dylanwiliam/status/1602776823225540610?s=20

Previously

https://hodges-model.blogspot.com/search?q=mathemacy

James's 'terrible algebra'

“You are right in your consciousness that we are all echoes and reverberations of the same, and you are noble when your interest and pity as to everything that surrounds you appears to have a sustaining and harmonizing power. Only don’t, I beseech you, generalize too much in these sympathies and tendernesses — remember that every life is a special problem which is not yours but another’s, and content yourself with the terrible algebra of your own. Don’t melt too much into the universe, but be as solid and dense and fixed as you can." 

Henry James.

Source - and in full: 

https://www.goodreads.com/quotes/1242340-you-are-right-in-your-consciousness-that-we-are-all

My source: Berwick, I. Over the limit, Life&Arts, FTWeekend, 20-21 January, 2024, p.10. (Review of "How We Break").

Why not a 'care algebra' - across the five domains?

 Individual
   |

      INTERPERSONAL    :     SCIENCES               

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

 SOCIOLOGY  :   POLITICAL 

|

Group

reasoning thought James's ... 'terrible algebra'	THEORY <gap> PRACTICE mathematics & logic abstraction algebra
PRACTICE <gap> THEORY Applied Mathematics in the Social Sciences	Equity, Equality, Efficiency, Effectiveness, Efficacy Difference, Comparison Inclusion, Exclusion

My emphasis.

Humanistic motivations for study - c/o Harris et al.

 "Asking '[I]n what sense . . . can mathematics be considered a democratic regime . . .' open to all, Pimm and Sinclair quote (Supergiant) Henri Poincaré to the effect that 'only mathematicians are privy to the aesthetic sensibilities that enable' the decision of 'what is worth studying.' The article, published in a journal for educators, is motivated by the 'view that mathematics can do something for me in a humanistic sense that repays the the careful attention and deep engagement it may require; one that may expose students to a fundamental sense and experience of equality . . . and provide them with another sense of human commonality.'" p.33.  

Harris, M. (2015) Mathematics without Apologies: Portrait of a Problematic Vocation. Princeton University Press.

Hodges' model - Axes - Structure :: Domains - Content

ii SNAP! Abstract [working] Hodges’ model as a mathematical object, a lens for social care and inclusion: category theory or category mistake?

Snap! 

As suspected the elasticity in the holistic bandwidth of the working abstract just failed.

The abstract rejected, I'm not surprised.

[ Prev. https://hodges-model.blogspot.com/2023/03/category-theory-or-category-mistake.html ]

I hoped 'no news' ... might fuel progress for a little longer.

Clearly (again), holistic bandwidth is never enough - focus, focus.

I'm grateful to learn from the call however, that there is a need to address:

How can prevention in social care be better conceptualised?

So I've a start at least ...

Grateful too, in the call, for Wright Mills sociological imagination:

“Know that the human meaning of public issues must be revealed by relating them to personal troubles—and to the problems of the individual life. Know that the problems of social science, when adequately formulated, must include both troubles and issues, both biography and history, and the range of their intricate relations. Within that range the life of the individual and the making of societies occur; and within that range the sociological imagination has its chance to make a difference in the quality of human life in our time.” Wright Mills (1959:226). [My emphasis.]

Clearly the current and previous governments have not got a clue!

I have rarely felt the fury that I feel at todays @DHSCgovuk “announcement” on social care. A massive retreat from what was already bare minimum first steps in long term reform. So on top of previously announced delays to the cap and a better means test, here’s what we now have..

— Sally (@so_says_sally) April 4, 2023

Abstract [working] Hodges’ model as a mathematical object, a lens for social care and inclusion: category theory or category mistake?

There's a clearer idea now - an abstract - something to aim for in July:

Call for Papers - Community Development and Preventative Care With Older People:
New Values and Approaches

Not yet fully fledged, but then Spring [N] still beckons ...

Hodges’ model as a mathematical object, a lens for social care and inclusion:
category theory or category mistake?

Abstract

The health and social care sectors include disciplines that inevitably fall under the combined aegis of the sciences and humanities. This paper examines social inclusion across Snow’s ‘Two Cultures’. The method is interdisciplinary and descriptive utilising a generic conceptual framework known as Hodges’ model. Rather than Hodges’ model acting as a model of, and for care, here the model is used to investigate how social care can be better conceptualised. While mathematics is a mandated competency in healthcare, the subject, as for the general population, is one preferably left at school. The question posed is: What happens when Hodges’ model is treated as a mathematical object? A challenge for reader and author alike, the purpose is to seek new insights into social inclusion, value, values and development through a relational and dialectic strategy with diagrammatic support, also signposting future avenues of study.

Given ongoing demographic pressures for many nations, is there an additional danger of two intergenerational cultures? Is prevention enough; and can we improve understanding of what can be termed ‘legacy issues’ in health and social care? Can we simultaneously reduce the focus to Simmel’s ‘dyad’, and yet open new avenues for discourse and description? Can we expose the epistemological and ontological dimensions of social inclusion and the life (and death) experience of older adults? Inclusion and exclusion are implicit parameters within the model, its original purposes being person-centredness and recognition of health as political. References and resources, in the form of a template and bibliography are provided.

<>

Still much more to do ... heaven knows what the result will be - and do? Even as I keep the call 'in mind', it may not 'fit' the intended journal at all. Non-predatory, and no article processing charges is essential. Brevity, conciseness ... the order of the day. I can't even hide behind "A little knowledge is a ...": I know nothing - but I'm sure there is some-thing here and in #h2cm.

If anyone is interested in writing something quite different and challenging, then please get in touch. There may (realistically imho) be two-three papers here?

From the call:

"As Rapoport highlighted in the 1960s, however, translating the unified view of prevention associated with public health into social welfare is inherently problematic. This remains the case. Preventative social care and support necessarily operate in complex and dynamic systems, generally where knowledge of causation and the consequences are unclear, and an imaginative application of care needs and contexts is required.

Though current policy direction across many countries suggests opportunities for re-imagining how prevention may be best conceptualised, numerous studies have highlighted that there remains considerable confusion and disparity in how this plays out in practice."

I have held on to the following article from Computing, March 19th 1987 and still learning from Durham and colleagues' journalism:

 "Sometimes a highly abstract, unifying mathematical theory underlies a whole set of scientific or engineering subjects. ...

Category theory is one such subject. Sometimes described as an algebra of algebras, it plays a unifying role in discrete mathematics. Discrete mathematics is a general heading for all the mathematical subjects which deal with jumpy, lumpy entities. Most of the mathematics used in the theory of digital computing is of this kind.

Another unifying subject, homology theory or differential topology, is less well known to computer scientists ... Homology theory is a unifying theory in continuous mathematics, which is the name given to everything that deals with smooth and stretchy entities.

Durham, T. (1987) Over the Horizon.
Working out the algebra of algebras, Computing, 19 March. pp.28-29.

[According to Bowden,] 'category theory and homology theory are fairly close things. The big difference is that homology theory includes information about the topology of the space that it describes.

Category theory is just about information. Homology theory is about information and structure, the structure of something.'

Space itself is not something we usually think of as possessing structure. But space does impose limitations on the kinds of behaviour that can be exhibited by things like electric fields and currents, or electromagnetic waves." p.28.

Durham, T. (1987) Over the Horizon. Working out the algebra of algebras, Computing, 19 March. pp.28-29. (Sorry no issue / volume nos.).


See also:

Call for Papers (i) Community Development and Preventative Care With Older People: New Values and Approaches

Call for Papers (ii) Community Development and Preventative Care With Older People: New Values and Approaches

Call for Papers (ii) Community Development and Preventative Care With Older People: New Values and Approaches

 I drafted this with (i) but didn't want to distract from the call and journal itself.

 With three working titles in a 250 word abstract in response to:

"How can prevention in social care be better conceptualised?" - 

I am open here (as ever) to expressions of interest towards collaboration. Support (help) would be invaluable, whether advice/co-author - to develop ideas on Hodges' model as a mathematical object(?). Even if this includes telling me to stop the barking and leave the tree alone. That point aside, the reading and videos are a challenge, but really enjoyable too.

In reading the call, Quaternary prevention will be added to:

"While applied in the context of complex social problems, discourses of prevention are most coherent and established in the realm of public health (e.g., tertiary, secondary, and primary prevention). As Rapoport highlighted in the 1960s, however, translating the unified view of prevention associated with public health into social welfare is inherently problematic. This remains the case. Preventative social care and support necessarily operate in complex and dynamic systems, generally where knowledge of causation and the consequences are unclear, and an imaginative application of care needs and contexts is required."

The result will be abstract. With “'re-imagine' social care" taken literally. 

Since there are quinary industries ( 'ours' no less), is there, should there be - quinary prevention? I think so, and this - and several legacy issues encountered in my career and ongoing today (also accentuated in the future?) will form the basis of discussion. Early days, but also need to decide upon the manuscript type:

• Article: a paper containing original research results that has not been published elsewhere. Articles shall have a maximum length of 6,000 words (the word count limit includes title, abstract, tables, figures, and references list). During a potential revisions stage, after peer-review, authors can extend the article length to a maximum of 8,000 words to better address the reviewers and editors’ comments.
• Commentary: an opinion piece providing a critical evaluation of a published article or topic of interest to the readership of the journal. Letters to the Editor and replies should be submitted as Commentaries. Commentaries shall have a maximum length of 2,000 words (the word count limit includes title, abstract, tables, figures, and references list).
• Review: a paper which comprehensively sums up the current state of research on a particular topic. Reviews shall have a maximum length of 6,000 words (the word count limit includes title, abstract, tables, figures, and references list). During a potential revisions stage, after peer-review, authors can extend the article length to a maximum of 8,000 words to better address the reviewers and editors’ comments.

The completed draft may not be 'finished' by July, but the exercise will help develop ideas to further understanding and awareness of Hodges' model. I'm almost too close to Hodges' model and I can't discuss ideas on a daily - work basis. The other impediment, is standing on my foot, pressing me against the walls (whichever domain) is the fact that models of/for care, and nursing theory have little currency these days. Finally, having explained I have no funding, encouragement to write is appreciated. This effort is also informed by threads on (and off) the Spirit of 1848 ListServe.

Mathemacy and Mathematical archaeology c/o Ole Skovsmose

(Ack. quoted at length and hopefully justified?)

"Mathemacy

The notion of “literacy” has been developed by Paulo
Freire to mean much more than just being able to read
and write. Literacy also includes competence in interpreting
social life. The notion of mathemacy can be developed
in a similar way to mean more that an ability to
calculate.

In order to do this, it is important to pay attention to
the notion of reflection. When a calculation, based on a
mathematical task, is carried out, it is possible to reflect
on the actual result. Such a reflection has mathematical
concepts and algorithms as its objects. A question guiding
such a mathematics-oriented reflection can be: Are the
calculations made correctly?

When setting up a mathematical model, in order to
solve a non-mathematical problem, we face many difficulties.
Thus, any modelling process presupposes that
certain simplifications are established. This means paying
attention to certain aspects of “reality” and neglecting
others. This is, for instance, what takes place when a
population model is used for the purpose of forecasting.
Reflections referring to a modelling process are of a different
kind than mathematics-oriented reflections. While
this kind of reflection has mathematics calculations as its
object, a model-oriented reflection has the relationship between
mathematics and an extra-mathematical reality as
its object. A question guiding a model-oriented reflection
can be: Is the output of the modelling process reliable?
Model-oriented reflections are carried out with an interest
in improving the model, i.e. with a technological interest.
Such reflections concern the validity of the model." p.199.

 INDIVIDUAL

|

 INTERPERSONAL    :     SCIENCES               

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

SOCIOLOGY  :   POLITICAL 

|

GROUP

reflection "Be the difference!" ... Said, one number to another - hopefully!

reflection

"Mathematical archaeology Social structuration, at macro or at micro level, often takes place with reference to mathematics. Such an interplay between knowledge and power is referred to by the thesis of the formatting power of mathematics. As mathematics is basically an “invisible” part of social structuration, we need strong analytical tools to capture the role of mathematics. This leads to the notion of mathematical archaeology." p.199.

"A reflection guided by such questions can be called a context-oriented reflection. “Context” is here understood as a political, social or cultural context." p.199.

Hodges' Health Career - Care Domains - Model

'health career' - 'life chances'

"lifeworld-oriented reflection"p.200.

Skovsmose, O. Linking mathematics education and democracy: Citizenship, mathematical archaeology, mathemacy and deliberative interaction. Zentralblatt füur Didaktik der Mathematik 30, 195–203 (1998). https://doi.org/10.1007/s11858-998-0010-6

 #HealthTheory