Puzzling over Papert’s ‘Humanist’ Maths

It is about whether personal computers and the cultures in which they are used will continue to be the creatures of “engineers” alone or whether we can construct intellectual environments in which people who today think of themselves as “humanists” will feel part of, not alienated from, the process of constructing computational cultures (5)

Seymour Papert’s vision of a ‘humanist’ mathematics sat at the heart of his belief that teaching and learning required a fundamental change if it was to realise the potential of living in a ‘computer culture’.Viewing his ‘utopian’ views of a ‘humanist’ mathematics, at the heart of teaching and learning mathematics, showed Papert’s opposition to 1970s school culture and why he felt he needed to move beyond Piaget’s stage theory, which was foundational to his understanding of educational psychology.

Papert believed ‘that the computer may serve as a force to break down the line between ‘two cultures’ (38): of the humanities vs science and mathematics. Ironically, recent revisions of Papert’s philosophy and work reveal (Barba, 2016; Albion, 2016) that he may have needed to reconcile ‘two cultures’ in his work as an educational philosopher and a commercially-focused inventor of LEGO robotics.  This article raises two issues related to Papert’s view of ‘humanist’ mathematics, as emblematic of the ‘kind of internal intelligibility’ through which ‘computer worlds’ offers children ways of dealing with ‘greater complexity’ than is usually possible in the physical world’(117).   

  • The first pertains to Papert’s construction of ‘humanist’ mathematics as a form of ‘embodied cognition’ and ‘unconsciousness’ that validates interacting with technological complexity itself.
  • The second issue relates to Papert’s relevance as a model for using new technologies in designing and delivering curriculum today.

The 1980 publication of Mindstorms occurred in the same year as George Lakoff  & Mark Johnson’s Metaphors We Live By and Humberto Maturana and Francisco Varela’s Autopoiesis and cognition. Assuming the events are coincidental, their shared philosophical aim of overturning the “Cartesian dualistic person, with a mind separate from and independent of the body” (Lakoff & Johnson, 1999, p.5) is remarkable. It is not too far-fetched to suggest, therefore, that it is useful to view Papert work alongside cognitive scientists and linguists who seek to validate human cognition as both ‘embodied cognition’ and predominantly the ‘unconscious’. 

The Power Of Papert’s Ideas

Papert’s opposition to what he viewed as ‘artificial and inefficient’ classrooms of his day (8) allows us to understand the nature of the revisions he sought to make to Piaget. In short, Papert’s forged a new theoretical framework for a ‘computer culture’: one which aimed to overturn how mathematical knowledge, dissociated from human knowledge, existed within a patchwork of territories that were not separated by ‘impassable iron curtains’ (39).

If then we ask what value does Papert hold for us through his work in the LOGO laboratory and Turtle Geometry, we might look to Lorena Barba (2016) arguments that his philosophy should guide our efforts in advocating a ‘natural mode of learning’. Barba highlights, for instance, how Papert’s approach was not about proving that ‘computational thinking’ was the domain of the computer scientist but everyone using computers “as an extension of our minds, to experience the world and create things that matter to us”.  

This resonates with how Papert sees the ‘power principle’ as making his invention of Turtle Geometry learnable. His demonstration of drawing simple squares and stars with the Turtle, in fact, carries the important ideas of ‘angle, controlled repetition, state-change operator’. Accordingly, this gives us ‘a more systematic overview of what children learn’ by distinguishing between two kinds of knowledge. One kind is mathematical, and the other kind of knowledge is mathetic: knowledge about learning (63). Interestingly, meta cognition is not a term ever used in Mindstorms: its absence, with what we have come to understand today by this term, is also worth pursuing.

In any case, for Papert, the ‘power principle’ that calls up syntonic learning in the child seems more important that the intellectual splitting of the self from itself. Instead, he offers a sense of ‘body syntonic’ about a child’s sense and knowledge about his or her own body and ‘ego syntonic’ in relationship to intentions, goals, desires, likes, and dislikes. Turtle Geometry is learnable because it is syntonic (63).

Syntonicity & A Humane Mathematics

From the vantage point of syntonicity, it is possible to view a ‘humanist’ and humane mathematics as conversing with technologies through the metaphor of the child teaching the computer ‘a word’. Arguably such an analogy is at odds with the way we imagine relating in an interdisciplinary way ‘across the curriculum’. However, Papert asked very different questions about our cognitive interactions with the computer: in particular, whether the brain and the computer are discrete from one another and how knowledge is modularizable (171).

The modular nature of knowledge to which Papert alludes in Chapter 7 LOGO’s Roots: Piaget and AI thus gain vital importance today through the ‘thinking skills’ movement. For instance, Barbara Oakley’s A mind for numbers: How to excel at math and science: even if you flunked algebra (2014), provides the structure for one of Coursera’s most successful MOOCs. The course, Learning How to Learn: Powerful mental tools to help you master tough subjects, is facilitated by Oakley, who is Professor of Engineering, Industrial & Systems Engineering, Oakland University and Dr Terrence Sejnowski, Francis Crick Professor at the Salk Institute for Biological Studies and Director of the Computational Neurobiology Laboratory. The term used by Oakley and Sejnowski for the mind’s modular building block is ‘the chunk’ which she illustrated below.

Oakley and Sejnowski’s pedagogical style emphasises the use of metaphor as a way of bringing concrete and formal thinking operations far closer together that is customarily practised in traditional mathematical educational practices.  The illustration, for instance, shows the primary metaphor used of an old-fashion pinball machine that does not form a ‘chunk of knowledge’, depicted as neurons that fire together, unless the pattern is ‘meaningful’. The MOOC itself is a study of ‘mind-size bites’ which are not ‘logically arranged’ but through which the learner, as a bricoleur, systematically enter the ‘conservation-combinatorial gap’ (175-6) to create meaning. 

The notion of Papert’s modular nature of knowledge also resonates with John Hattie’s advocacy of ‘solo taxonomy’ over Bloom’s hierarchical levels of intellectual complexity. This is evident through his meta analyses of Piagetian learning in Visible Learning (2009), as well as his influence as head of the Australian Institute for Teaching and School Leadership (AITSL) through which he discusses his preference for Piaget throughVisible Learning website FAQs. His curriculum leadership has also given rise to associated trends through the publication of Solo Taxonomy classroom resources like those by Pam Hooks (2011).   

Papert key insight on the way memory works in retrieving in ‘bricolage’ fashion and putting in place a ‘patchwork theory’, drives the argument paradoxically into areas that seemingly resist a systematic form of ‘curriculum design’. However, as Papert discusses the case of Bill and his apparent ‘poor memory’ of numbers in Chapter 3 of Mindstorms, we find the evidence that Barba (2016) points to about Papert and his ‘revolutionary’ belief that the difference on what Bill “could” and “could not” learn did not depend on the content of the knowledge but his relationship to it. It allowed Bill, in fact, to relate to content more like he did to songs than to multiplication tables. As a result, Papert observed that Bill’s progress was spectacular (65). Bill mathematical knowledge and his knowledge of songs come as a form of ‘deep learning’ where rhythm, movement and navigational knowledge meaningfully create new mathetic knowledge.  

Papert is very clear throughout Mindstorms that ‘the computer by itself cannot change the existing institutional assumptions that separate scientist from educator, technologist from humanist. Most ironically, he concludes that the kind of ‘deliberate action’ which would put ‘humanist’ mathematics at the heart of educational change ‘could, in principle, have happened in the past, before computers existed (189).  Speaking from my own experiences as a performing artist, I appreciate how Papert also believes that technology-rich environment give participants ongoing opportunities to learn, work, and if called on, teach, in and between cognitive domains. This is certainly how I find the technology-rich theatrical space.

The problem is paying for the time it takes for ideas to be realised and cognitive connections to be made, explained and their effectiveness evaluated.  It is a path in education, however, which is crucial now to investigate in detail. In particular, what do we dare say about building educational practices through ‘specialist’ and ‘generalist’ knowledge that also powerfully fit with how children really learn? Papert’s utopian relationship with transformational change in education, begs many questions about realising the potential of technology and the form and content of delivering an educational treatise for a ‘digital age’.