FULL TEX:
http://memristors.memristics.com/Kindergarten%20and%20Differences/Kindergarten%20and%20Differences.pdf
http://memristors.memristics.com/Kindergarten%20and%20Differences/Kindergarten%20and%20Differences.html
Kindergarten math: numbers and counting
How it startsWhat are the aims of a standard western Kindergarten education in such abstract disciplines like math, geometry and counting?
The answer is easy found. Simply check the offers of one of the many educational organizations and supporting industries.
They all guarantee the parents a steep learning curve for their children to learn to master the basics of the adult mind set of math.
There is not a single offer that is taking the capacities of children seriously and offers strategies to develop genuine infant-adequate education.
One of the many succesfull companies is the company “Home Schooling for Kids” which offers “KS3, A-Levels, GCSE & IGCSE Courses From £350".
www.OxfordHomeSchooling.co.uk
What’s on offer on the ‘Sure Start’ market?
"The goal of kindergarten math curriculum is to prepare children for first grade math. Please see below a list of objectives and goals for kindergarten math:
To count by rote at least to 20, but preferably a little beyond.
The concepts of equality, more, and less.
To count backwards from 10 to 0.
To recognize numbers.
To be able to write numbers.
To recognize basic shapes.
To understand up, down, under, near, on the side, etc. (basic directions).
To have a very basic idea of addition and subtraction.
It also helps to expose the student to two-digit numbers.
"Children may also get started with money, time, and measuring, though it is not absolutely necessary to master any of that. The teacher should keep it playful, supply measuring cups, scales, clocks, and coins to have around, and answer questions."
http://www.homeschoolmath.net/teaching/kindergarten.php
It is also important to know that the definition of a rational human being is implying the skills of those math topics added with the ability to draw some logical conclusions, say with modus ponens.
All that is instructed in the social context of a schooling program that is confusing learning and training with education.
Failing such skills of adult cognition excludes the person to be qualified as a rational human being (homo sapiens).
Without surprise there is some resistance to the schooling movement.
"I suppose it is because nearly all children go to school nowadays, and have things arranged for them, that they seem so forlornly unable to produce their own ideas.” Agatha ChristieIn this paper, I will not deal with the many approaches of the anti-schooling movements but with the very essentials of conceptual thinking that are accepted by both sides, the schooling and the anti-schooling institutions and movements.
http://studentliberation.com/quotes_1.html
Some background theories for the traditional approach
Principles postulated in the tradition of the Piaget school
The abstraction principle
"The realization of what is counted is reflected in this principle. A child should realize that counting could be applied to heterogeneous items like toys of different kinds, color, or shape and demonstrate skills of counting even actions or sounds! There are indications that many 2 or 3 year olds can count mixed sets of objects.The order-irrelevance principle
"The child has to learn that the order of enumeration (from left to write or right to left) is irrelevant. Consistent use of this principle does not seem to emerge until 4 or 5 years of age (German and Galistel 1978).Constructivism approach
"There is strong evidence that the early teaching of standard procedures for arithmetic problem solving “thoroughly distorts in children’s mind the fact that mathematics is primarily reasoning.” (Kamii et al.1993). In order to address the above problem, new mathematics curricula have been introduced, based on the Piaget theory of Constructivism.
"This approach suggests that logico-mathematical knowledge, apart from empirical or social knowledge (Novick 1996), is a kind of knowledge that each child must create from within, in interaction with the environment, rather than acquire it directly (almost “being donated”) from the environment.”
Natalia Marmasse, Aggelos Bletsas, Stefan Marti, Numerical Mechanisms and Children’s Concept of Numbers
http://web.media.mit.edu/~stefanm/society/som_final_natalia_aggelos_stefan.pdfThese principles are subordinated to the binary question: “To what extent is the sense of numbers innate, and to what extent is it learned?” Nature or nuture?
I learned most, not from those who taught me but from those who talked with me. St. Augustine
Again, Gove’s adviser: “Genes make you smart, not teaching. Genetics outweighs teaching, Gove adviser tells his boss.”
http://www.theguardian.com/politics/2013/oct/11/genetics-teaching-gove-adviser
Without doubt, this poor guy has never studied the miserable arithmetics of modern genetics and DNA research.
There are good reasons to see the decision for a dialog with children as neither belonging to the “nature” nor to the “nuture” camp of the ongoing battle. Dialogs are not in-forming children with educational content but are evoking their own yet hidden capacities of thinking and understanding their own world.
It will turn out that exactly the postulated principles, the principle of abstraction and its corresponding principle of order-irrelevance, has no ‘natural’ foundation in the thinking process of a curious not yet educationally manipulated child. Nor are there any genetical conditions that are forcing to a specific kind of thinking numbers and logic.
Even if there would be strong empirical evidence and verification of a close connection between the concept of number and the genetical prepositions of the humn brain it wouldn’t stop the human mind to surpass such a little handicap.
Up to now we haven't detected any human beings that are able to study the moon with a naked eye nor do we know any genetically privileged children flying around the village without a little helicopter. And, certainly, the whole calculations for the scientific thesis wouldn’t have been realized by non-assisted human brains alone.
Order-relevance is constitutive for the understanding of numbers in the sense of the Stirling abstraction.
A principle of concretization, in contrast to the homogenizing principle of abstraction, is essential for an understanding of numbers in a polycontextural sense.
Gunther’s uncountable objects
The situation today wouldn’t be much different as it was for Gotthard Gunther when he asked, around 1908, his elementary school teacher two serious questions: 1. How is it possible that a simple addition of some single mountains (Berge) results into a mountain range (Gebirge)? That is, 5 Berge = 1 Gebirge, and how works that: 5 = 1!? 2. How is it possible to add different kinds of objects, like 1 church + 1 crocodile + 1 tooth pain + 1 thought together? And how would this relate to the example of the mountain rage (Gebirge)?
Would there be such a monster like a ‘mountain-church-crocodile-tooth pain’ range as a single, albeit complex object, like the addition of mountains is producing a mountain range?
The teacher's answer today will be more or less the same as a child got it a century ago.
Abstraction and enumeration (arithmetization). Obviously, an answer that just moves the question to another level of un-answered questions.
In his biographical text, Selbstdarstellung im Spiegel Amerikas, (1974) he writes:
"Die Arithmetik mußte ganz anderes und Wunderbares leisten können, weshalb er an seinen Lehrer die Frage stellte: Wenn das Zusammensein von vielen Bergen ein Gebirge ergab, was ergäbe dann zahlenmäßig das Zusammensein, wenn man eine Kirche zu einem Krokodil addierte und dazu noch seine Mutter und obendrein ein Zahnweh. (Es ergab sich nämlich, daß gerade zu diesem Zeitpunkt seine Mutter an Zahnschmerzen litt.) Das erschien ihm als eine der Arithmetik würdige und hochinteressante Aufgabe.http://www.vordenker.de/ggphilosophy/gg_selbstdarstellung.pdf
"Als man ihm mitteilte, daß man die vier angeführten Daten eben nur als verschiedene Sachen zusammenzählen könne, hielt er das zuerst für ein Mißverständnis und bestand darauf, daß er keine Sachen, sondern eben Kirchen, Krokodile usw. addieren wolle. Und was ändere sich am Addieren, wenn man das Krokodil durch einen Löwen ersetze? Daß sich dann nichts ändere, wollte er nicht glauben.
"Später vergaß er das Problem. Er mußte fast 60 Jahre alt werden, bis es für ihn in der biologischen Computer-Theorie in neuer Gestalt wieder auftauchte.”
The development of Gunther’s answers to his early questions went through several stages. From the kenogrammatic approach, to the polycontextural understanding and to a concept that is closely related to his theory of negative languages.
How does abstraction work and how are the natural numbers justified for such a counting process of different objects?
Again, we have the luck to ask Philip Wadler from the university of Edinburgh. His answer is ultimative and should stop any such naive questions for ever.
In his lovely text, probably written for his children and some professors of computer science, he makes it crystal clear:
"Whether a visitor comes from another place, another planet, or another plane of being we can be sure that he, she, or it will count just as we do: though their symbols vary, the numbers are universal.
"The history of logic and computing suggests a programming language that is equally natural. The language, called lambda calculus, is in exact correspondence with a formulation of the laws of reason, called natural deduction. Lambda calculus and natural deduction were devised, independently of each other, around 1930, just before the development of the first stored program computer. Yet the correspondence between them was not recognized until decades later, and not published until 1980. Today, languages based on lambda calculus have a few thousand users. Tomorrow, reliable use of the Internet may depend on languages with logical foundations. " Philip Wadler, As Natural as 0,1,2 Evans and Sutherland Distinguished Lecture, University of Utah, 20 November 2002.
http://homepages.inf.ed.ac.uk/wadler/papers/natural/natural3.pdf
Gunther was aware that his kind of thinking, and his way of understanding numbers, made him an alien.
Not enough, in his late years he started to develop a system of arithmetics that not only answered his early two crucial questions but it also will be enjoyed by alien intelligence.
He sincerely told his baffeled longtime friend Helmut Schelsky that he isn’t anymore a human being, he just looks like one.
Also the discovery of the zigzag movement of numbers in a transclassical number system is amazing it would be a sign of a serious lack of understanding Gunther’s attempts towards a ‘dialectical’ number theory to celebrate this zigzagging against the ‘Gänsemarsch’ of linearly ordered natural numbers as the sole achievements of Gunther’s polycontextural constructions of the relation of ‘number and logos’.
What could we learn from this story?
Some primitive questions are not necessarily an expression of a lack of rationality but more a sign or symptoms of another, still hidden, pattern of thinking and understanding the world.
Instead of destroying it, a teacher should be able to accept this ‘deviant’ way of thinking and be able to set it into a broader framework of different kinds of rationality.
Talking to the child and developing together new experiences could lead to surprising insights, relevant for the teacher and the curriculum too.
Math for young dancers: Gaps and Jumps
Gaps and Jumps
Where in all those mathematical concepts of successor functions, induction steps, recursion cycles and deduction trees are the gaps and jumps that are natural to dancers?
It surely would be crazy if our numerical counting process would have to stop somewhere at an obstacle, or falling into a counting gap or would have to jump out of such a paradoxical situation.
Why to trust in continuity?
Also I never was a dancer I believe that life without gaps and jumps is grey. Personally, I was never convinced of this principle of homogeneous continuity necessary for induction, deduction and other step-wise developments of reasoning inside a single paradigm.
On the other hand, if we accept this principle of closure, life gets significantly boring and there is no special motivation to go into it.
Didactical jumps
"First Leah made a jump of three along her number line and then a jump of four. Where did she land?
"Next Leah made a secret jump along her number line. Then she made a jump of five and landed on 9. "How long was her second secret jump?
http://nrich.maths.org/5652
But an intriguing pre-mathematical question arises too: How does the child know on which number line the jump has to land?
The classical supposition that there is one and only one arithmetical number line possible is not self-evident at all.
Why do we not have different number systems? Greens and reds and blacks?
As we know well, our teacher would explain us that all those differently colored number lines represent the same numbers because we can map each number from one color to the corresponding number of the other color. As they say, number systems are isomorphic. In color terms, they are all grey. And paradoxically, grey itself is not considered as a color.
Why should we accept that?
This principle of homogeneous continuity necessary for induction, deduction and other step-wise developments of reasoning inside a single paradigm has never got my enthusiasm.
On the other hand, if we accept this principle of closure, life gets significantly boring and there is no special motivation to go into it.
Also I was never a dancer I believe that life without gaps and jumps is grey.
What do we learn from this not so innocent example of counting with number lines?
There are at least two different kinds of jumps possible: One inside a linear number system, and one between linear number systems.
