School of Computer Science THE UNIVERSITY OF BIRMINGHAM CoSy project CogX project

Meta-Morphogenesis and Toddler Theorems: Case Studies
(DRAFT: Liable to change)

Aaron Sloman
School of Computer Science, University of Birmingham.


Installed: 7 Oct 2011
Last updated: 11 Sep 2013
28 Sep 2012; 10 Apr 2013 (including re-formatting); 8 May 2013; 7 Aug 2013;
9 Oct 2011; 21 Oct 2011; 29 Oct 2011; ....; 7 Jul 2012; ... 23 Aug 2012;

This web page is
Also accessible as:
A messy automatically generated PDF version of this file is:
It is one of a set of documents on meta-morphogenesis, listed in
Also accessible as:

A partial index of discussion notes is in



  1. Introduction: What are Toddler Theorems?

    For a long time I have been arguing that __________________________________________________________________________________________



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  3. Background




    Core ideas (no claims are made here about novelty):


    Many psychologists are brought up to think that all scientific evidence
    must include numbers, correlations, and graphs.

    That is a result of very bad philosophy of science.
    I'll outline some alternatives.

    Much research on children (and other animals) is restricted to looking at patterns of
    responses to some experimenter-devised situation. This is like trying to do zoology or
    botany only by looking in your own garden, or doing chemistry only by looking in your own
    kitchen. It is based on a failure to appreciate that many of the most important advances
    in science come from discovering what is possible, i.e. what can
    occur, as opposed to discovering laws and correlations. This is explained in more
    detail in Chapter 2 of The Computer Revolution in Philosophy (1978)

    How to discover relevant possibilities:
    First try to find situations where you can watch infants, toddlers, or older children
    play, interact with toys, machines, furniture, clothing, doors, door-handles, tools,
    eating utensils, sand, water, mud, plasticine or anything else.

    Similar observations of other animals can be useful, though for non-domesticated animals
    it can be very difficult to find examples of varied and natural forms of behaviour. TV
    documentaries available on Cable Television and the like are a rich source, but it is not
    always possible to tell when scenarios are faked.

    Some videos that I use to present examples are here:
    More examples are presented or referenced below. Some are still in need of development:
    more empirical detail and more theoretical analysis of possible mechanisms.

    Compare Robert Lawler's video archive described below.

    [To be continued.]

    Doing science requires formulating deep questions,
    and, if possible, good answers.
    Without good questions it's unlikely that the answers will turn up.

    Many of the research questions commonly investigated are very shallow:
    Which animals can do X?
    At what age can a human child first do X?
    What proportions of children at ages N1, N2, N3, ... can do X?
    Under what conditions will doing X happen earlier?
    What features of the situation make it more likely that a child, or animal,
       will do X?
    Which aspects of ability, or behaviour, or temperament are innate?
    To avoid shallow questions, learn to think like a designer:

    Sometimes that requires thinking like a mathematician, as illustrated below in several
    examples -- a designer needs to be able to reason about the consequences of various design
    options, in a way that covers non-trivial classes of cases (as opposed to having to
    consider every instance separately).

    That normally involves thinking like a mathematician -- especially discovery of, and
    reasoning about, invariants of a class of cases. For example, an invariant can be a
    feature of a diagram that represents reasoning about all possible circles or all possible
    triangles, in Euclidean geometry. Usually that does not require the diagram to be
    accurate. When schoolkids are taught to measure angles of a collection of triangles to
    check the sums of the angles, they are NOT being taught to think like a mathematician.

    Sometimes people who are not able think like a designer or a mathematician resort to doing
    experiments (often on very small and unrepresentative groups of subjects). I have compared
    that with doing Alchemy, here: (Is education research a form of alchemy?)

    Unfortunately, the educational experience of many researchers includes neither learning to
    think like a mathematician nor learning to think like a designer.

    E.g. many people who can state Pythagoras' theorem, or the triangle sum theorem have no
    idea how to prove either, and in some cases don't even know that proofs exist, as opposed
    to empirical evidence obtained by measuring angles, areas, etc.

    [To be continued.]

  7. Some Common Types of Erroneous Thinking
    (In researchers, not their subjects!)

    Domains for toddler theorems (and some post-toddler theorems)
    These examples provide fragmentary evidence for the diversity of domains of
    expertise and the kinds of knowledge transformations they make possible.
    Some of the examples illustrate portions of the process of information
    re-organisation (perhaps instances of what Karmiloff-Smith means by
    "Representational Redescription"?).

    This list of examples is a tiny sample. I shall go on extending it.
    (Contributions welcome.)

    NOTE: The order of the examples presented here is provisional.
    Later I'll try to impose a more helpful structure.
    Some of the examples were inspired by this wonderful little book:
      J. Sauvy and S. Sauvy,
        The Child's Discovery of Space: From hopscotch to mazes --
           an introduction to intuitive topology,
        Penguin Education, 1974,
        Translated from the French by Pam Wells,

    A provisional collection of examples follows.
    (Needs to be re-ordered):

    (To be extended and re-organised.)

    Brief Examples of Use of Knowledge About Physical Objects

  9. Added 7 Aug 2013: Robert Lawler's video archive
    Bob Lawler has generously made available a large collection of video recordings of
    three children over many years here:

    I have not yet had time to explore the videos in any detail, but I expect there are
    many examples relevant to the processes and mechanisms involved in discovery of
    toddler theorems.

    The first video I selected at random "Under Arrest"
    illustrated many different things simultaneously, including how two part-built
    information processing architectures at very different stages of construction, with
    an adult out of sight, could interact in very rich ways with each other, some
    physical some social, and to a lesser extent with the adult through verbal
    communication. The older child clearly has both a much richer repertoire of spatial
    actions and a much richer understanding of the consequences of those actions. He also
    has some understanding of the information processing of the other child, including
    being able to work out where to go in order to move out of sight of the younger
    child. However the younger child does not forget about him when he is out of sight
    but is easily able (thanks to the help of a wheeled 'walker') to alter her orientation
    to get him back in view.

    How a child moves from the earlier set of competences to the later set, is a question
    that can only be answered when we have a good theory of what sorts of information
    processing architectures are possible, and how they can modify themselves by
    building new layers of competence, in the process of interacting with a rich
    environment -- partly, though not entirely, under the control of the genome, as
    outlined in Chappell & Sloman 2007).

    The ability to be able to model such transitions in robots is still far beyond our
    horizon, despite all the shallow demonstrations of 'progress' in robot training

  10. Kinds of dynamical system:
    Moved to a separate file (10 Aug 2012)
    Replaced by a more up to date version:
    A Multi-picture Challenge for Theories of Vision
    Including a section on types of dynamical system relevant to cognition.
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  11. Some relevant presentations and papers

    Example presentations and papers on this this topic written over the last 50 years,
    especially since the early 1990s.



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Maintained by Aaron Sloman
School of Computer Science
The University of Birmingham