School Entry
1) A Brain Like An Onion Some have the view that the structure of the brain in many ways resembles the build of an onion. Children have access at first to the central core; organising the sheaths as these become, one after he other, more massive and capable, needs development within the child: to be able and to learn how to organise so much new contents of the memory. Our memory contents are organised – indexed – along very many aspects and dimensions. One of them is the temporal one. The regions one has used during infancy (age bracket x) contain the rules, are the rule depositors, for all that what has been learned based on experiences in the age of the infancy (age bracket x). The older the rules, the more archaic they are, and the more they have sub-descendants that spell out implications of the archaic rule. To have access to that region which administers basic truths about numbers, one has to re-involve, by means of phantasies and enactments, some of other age-correlated memories, too. No content without context: we work on re-imagining the situations in which we have learnt to count. 2) What is drawn on the blackboard We see many flowers, apples, bananas, teddies on the drawing by Teacher and are asked: how many of these are apples? What we learn is that one has to a) lift off the specific *type of object* and b) ignore its *place.* The first half-step of the process of abstraction de-individuates the objects by establishing their category within the set of objects, and it creates a distinction between *foreground* and *background. *The apples are now the foreground, all else is in the background. The same stands for the second part of the abstraction process, where we push the *position* of each apple in the *background,* both as a specific apple among apples, and as one of a group of apples contrasted with everything else. 3) What could also be drawn on the blackboard Teacher could also connect the alike objects with colourful lines. The resulting polygons would be quite useful to introduce the concept of time and watches, by pointing out the circularity of the loops. A simplified example would be: this circle shows how often you eat something, and this bigger circle shows how day and night change. If you eat something while morning: that is the breakfast. The concept of a *coincidence *as a recurring type of unit is ready to be cultivated in the developmental stage of a healthy 6 or 7 years old. *How frequently *should be taught before* How many. * We have learnt to imagine the number line as a very long tie. Now we want to play something new and cut the tie up. We stitch the resulting pieces so that we get closed belts. These are like eels which each has its own tail in its mouth. Some are bigger, some even more so. They all make their leisurely loops. When each of them comes up once in a turn it executes for a grasp of air, how will the surface of the pond look? 4) What we do not talk about Every family has a little secret, about which it is impolite to talk, which is best left to a subject avoided. In our family of Sumerian-educated able counters, the troubling little secret relates to a vanishing act, where something disappears as if it had never existed. We have to breach the subject of cuts. As we see Teacher explain 3+4=7 we are never told what had become of that cute little second-level cut which had been separating 3 and 4 and is now no more there, as the 7 have no second-level separation among each other. Do the cuts simply disappear? Do not they do some mischief for our not looking after them? Degradations from second to first level separator on a whim? No talking back? 5) Able pupils We should spend some time with the kids and explain them, that some toys that are in 2 boxes cannot easily mix with other toys in 2 boxes, unless one removes at least one of the boxes. How Teacher manages to deal with the now empty box that had contained one of the summands gives a feedback on the culture of which the techniques she transmits to the pupils. Ignoring the problem would perpetuate a cultural tradition, according to which it is admissible, manageable and considered rational, if one pretends that there are no conflicts at all, reality is that what we define it to be. There have been many voices in the field of mental health who state, that research has shown, that pushing away, repressing, ignoring conflicts that exist in the background will inevitably lead to viewpoints and perspectives that bring forth conflicts, more intense than that caused by the seed would have been. Teacher may make small morsels of the box now superfluous and enhance the first-level separators between the units. She could add the morsels to that part of the box which is empty, making the impression that the box expands, while it in fact integrates, somewhere else, on a different level. It is important that places are *not* simply background: everything is always *somewhere*, and *where *something is, is dependent on *how* the thing is in comparison with other things. Children are able to order and rank things well before they are able to count them. It would be more in line with the onion architecture of the brain to learn to formalise order and sequences, quasi as lighthouses of the abstract landscape, *before* learning the techniques of formalising the quantitative approach to the world of abstract objects. She could also bind the empty box of one of the summands, or of both of them, to the box now carrying the 7. That big box, if it keeps its habits, becomes clumsier and clumsier as it eats up toys from different, smaller boxes, because these are now empty and it has to carry them, the empty ones, with itself. The idea of recycling, connected to the proverb: No trees grow into the sky; also introducing the effect of the cumulation, of implicated negations caused by assertions, above a threshold level. The time will come when this big box will become, in its nature, more a collection of boxes than of toys: in that moment, it will not count as toys in boxes, but as boxes with toys. Then the boxes make an uprising, each standing up and unfolding, all crying: it is us, who determine what is reality, we have been neglected far too long, now we wish to regain our actual sizes, we won’t stay forever in the background; they then expand and we have nice big puff. Our nervous system is purpose-built to recognise patterns. We have the ability to distinguish among patterns. We share this capacity with animals and small children. Then it is safe to conclude, that the idea of patterns, as a subjective concept, is realised, somehow, in the non-subjective outside, in actual Nature. Our goal is to teach the children to use good, useful, state of the art tools of abstraction during their go at understanding the world. If we can transmit the thought, that catching, acknowledging and using patterns is archaic to that detail, how many elements are in a specific part of the pattern, frozen immobile, then we have created mental space wherein to plant the roots of the idea of counting. The new sheath of the onion, somewhere between the core and the outermost peel, is then connected to the contexts of periodicity, cyclicity and rhythm, which rule in the core. We can teach our children to conceptualise in terms of patterns. Learning periodicities, similarities, orders, ranks, paths, compromises and conflicts, continuity and discontinuity, *before* learning the least challenging: the how many, would be a didactic success. Let us offer them a didactic tool, in the form of a numbering system, which generates patterns by itself. This is more complex to imagine than a long tie with spots on it – yet, children’s phantasies are thick enough to abstract from, at the age of 6, also about the appearances attached to rhythm. If only we are able to teach them, what to abstract, from which picture of reality. Embedding in archaic images that the world we abstract from is in its nature a continuously changing one, and there are rules to these changes, would allow children to grow ideas of measuring and counting in the sheath *above *the sheath in which we have already dealt with places, movements, order, changes, periodicities and rhythms. This sequence of didactic steps: first the principle, then the cogs and wheels, would allow the children to create numbering systems, which do not need un-learning some images. The didactic method presently used brings *first* the cogs and wheels of counting to the attention of children, and leaves it to their imagination to come up *later* with a purpose and a principle, what he cogs and wheels are to be used for: not good for the didactic process. We can teach able pupils to learn to count coincidences. While children can be fascinated by wheels and winning results, and may well be able to understand the principle of a Las Vegas one-armed bandit, it would be misjudged to ask them to calculate estimated middle time between coincidences, although we do have an instinctive knowledge of “when”. In fixed interval reinforcement experiments, dogs knew when to expect the next pellet; the brain is prepared to handle the mathematics to it, in a hard-wired way. Our neurology is prepared to deal with differing cycles, like the cycles of the pulse and of the week. These are embedded in periods, like sleep and the year. The rhythm is the interference, co-resonance, coincidence pattern of these two. The simple and complex rules of rhythm are truly archaic. Having the numeric tables at our disposal, based on *a+b=c, *we can create didactic tools that help the pupils to find their way in a consistent explanation of where, when, how many and what kind. The task needs able pupils and good teachers. Due to limits of human capacity, human pupils will need lots of assistance from pupils of a mechanical nature. The cuts could well turn out to be the bridge connecting formal names and semantic meanings. The empty boxes, representing the cuts, that are carried along with the resulting sum, can make summands so much loaded with placeholders, that they become cumbersome to handle, like a big box unto which are attached several smaller boxes. Like if Tetris figures merge, there are step by step less spatial variants available for sums, with a history of big appetite, as summands. The implicated negations, which come with every instance of *a+b=c *(like: one separator second level exists not), are at work in the background. They determine, which entity gains which place. Whether we call it knowledge of the Creator, genetic information, physio-molecular disposition, converging fields, loss of degrees of freedom or simply: information, is a naming of the branches and leaves of the family tree of implications of implicated negations. The roots may well be the cuts. Elementary counting being a basic, actually: archaic, rule of thinking, many rules of thinking have to be changed to be transferred to the newly built fundament. This trouble we grown-ups face can be avoided. We should teach the next generation some non-Sumerian techniques, connected conceptually to the basic, archaic observations and terms of cyclicity, periodicity and rhythmicity of the world, in a well-thought-out order. .
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