To help with planning the sequence of interactives, I am moved to set down some of the design considerations, opportunities for minimizing the amount of work involved, and underlying cognitive principles which seem important to me and which will guide my work.

At the elementary level, the material to be conveyed is well represented by the Singapore curriculum, though there will also be 'targets of opportunity' not in the Singapore curriculum which can easily be taught given the interactive format available, and so should be developed also. This material can be seen as a series of concepts and associated procedures, each generally dependent on its predecessor, as for example counting to 10, counting to 100, counting to 1000, counting to the millions and billions, counting by 10s, counting by 100s, counting by 1000s, counting by millions and billions, comparing numbers, adding single digits, adding numbers with multiple digits, subtracting single digits, subtracting numbers with multiple digits, multiplication, division, fractions and their arithmetic, comparing fractions, decimals and their arithmetic, comparing decimals, converting fractions to decimals, percentages. Concepts along the way might include the commutative, associative, and distributive rules for addition and multiplication, subtraction as the inverse of addition, etc. There is certainly more material which is worth trying to squeeze in, e/g/ bar charting and pie charting, aspects of estimation (a favorite problem of mine is "roughly how many baby carriages are sold in New York City daily?"), graphing, some elementary set theory, etc.

Although some key aspects of this material certainly call for drill and memorization, the sequence of interactives to be developed will also aim to let the student understand the reason for each of the facts encountered, e.g. why is 2 + 2 = 4?

Opportunities to minimize work

Applications

Estimation

Budgeting

Carpentry

Elementary mathematics has two aspects, experiential and linguistic. To teach experientially is to make the student familiar with facts of mathematical significance taken from the real or from some artificial world. For example items can be put into collections and counted; collections can be joined and separated; objects have shape and can be measured, cut into pieces, and reassambled; when lines meet a corner is formed; objects can retain their basic shape while they grow and shrink in size; the positions of objects can change and objects can be rotated; tossed coins obey certain statistical laws. Mathematics' linguistic side envelops basic facts of this kind in various specialized notations, i.e. languages, which describe them concisely and allow new facts to be derived by formal manipulation of notations. The first of these specialized languages which the child encounters is the decimal notation for numbers, which comes with a set of formal procedures for addition, subtraction, etc. Experience with the stuff of mathematics supplies intuitive underpinning for subsequent formalization. Such experience invests the otherwise bare notations with enough meaning to make work with them comfortable and their rules of manipulation reasonable.

Another basic aspect of mathematics is its concern for statements universaly true rather than for isolated facts. That 3 + 2 = 2 + 3 is far less interesting to the mathematician than the fact that m + n = n + m for all integers. Though basic, this concern emerges only gradually in the course of mathematical education, which must initially concentrate the translation of English-language sentences into sequences of mathematical operations, and on the techniques for manipulating integers, fractions, and decimals. The child who cannot add is ill-prepared to appreciate the fact that m + n = n + m is always true.

In some areas the ability of children to learn is immense, much greater than that of their teachers. Set an American six-year old and their teacher down in Peking or Rabat and come back six months or three years later. After six months the child will be speaking Chinese or Arabic, learned informally from playmates, with a flawless accent, while the teacher will still be struggling to master the rudiments of the language with the help of daily tutoring. After three years the child will be indistinguishable from a native speaker, while the teacher may have given up on the language and come, like many immigrant parents, to rely on the child to handle as many ordinary Chinese-language interactions as possible. The problem of education is to harness the child's immense but specialized ability to learn to the goal of conveying the hard-won accomplishments of our common culture.

The limited world-knowledge and short attention-span of children impedes this. For example, the teacher knows that some mastery of numbers and their properties, say an ability to make change, scrutinize bills, and determine whether furniture will fit into an available space, is essential to comfortable adult living. The child has no experience of any of this, so for the child arithmetic, when it first appears in the classroom, is just a puzzling game of unfamiliar words that, for some unknown reason, the teacher wants to play. And a boring game at that: if the child's psychological clock runs at 5 times that of an adult, then a 30 minute arithmetic class is for the child what a 2 1/2 hour lecture on an obscure, and perhaps opaque, topic in advanced mathematics would be for the teacher. Much nicer for both to romp with friends!

How can this fundamental difficulty be overcome? Keying the arithmetic or other skills to be taught to the real-world problems to which these skills will eventually be applied cannot help, since these problems, e.g. making change, reviewing bills, and ordering furniture, are just as foreign to the child as multiplying and extracting square roots, and if anything less accessible. I still remember the puzzlement which surrounded a unit on 'How to write checks' when it appeared for a week or two in one of my 9-th grade arithmetic classes. I grew up in a working class family and neighborhood. Neither my family, nor any of the families of my friends, had checking accounts. I had no idea of what checking accounts were for, or how they worked. My mother had a bank savings passbook that I sometimes saw, but as a child I had no understanding of the role it played in the workings of our household. So, faced with an opportunity to write nominal checks, I would have much preferred to multiply two numbers, where at least I understood what I was doing, if not why.

Over its multi-year course, mathematics instruction ought to aim at conveying some of the mathematician's habits of mind, which are of value even in non-mathematical settings. Of these, the most important is the use of sequential reasoning to uncover facts that at first seem to be quite out of reach. Connected to this is the systematic use of iterative procedures to generate results progressively or by successive approximation. Understanding of the role of languages and notations specially tailored to handle situations for which ordinary language is insufficient is another mathematical fundamental. Yet another is use of the 'isomorphism principle', i.e. the fact that problems can often be seen in several,initially quite different, ways, some of which may instantly reveal what others leave obscure. Related to this is the potential intellectual importance of trenchant analogies. Other mathematical techniques include the study of extreme cases of a problem, drastic simplification where appropriate, and systematic attempts to extend and generalize methods and approaches once they start to seem successful.

Active involvement in a subject is necessary for acquiring and mastering it. Months of riding as passenger in a car generally teach one far less about the roads traversed than a single experience as driver. Nevertheless pure 'discovery learning' approaches, pushed to extremes, become either salted mines or mockeries of the hard-won accomplishments of human culture. For thousands of years the ancient Egyptians, even while undertaking enormous national construction projects, remained committed to a 'unit fractions' system for the arithmetic of fractions now seen as bizarrely misconceived. Neither did centuries of Greek astronomical calculation, some quite sophisticated, and the genius of Euclid, Aristarchus, and Archimedes lead to discovery of a system of elementary arithmetic as good as that learned by today's first graders. Can we expect Jimmy and Mary, left without guidance, to accomplish what these great figures and centuries did not?

A clear proof of the bogus nature of extreme discovery leaning approaches lies in that fact that no one proposes to apply them to anything considered important. Basketball coaches do not insist that team candidates discover their own offensive and defensive formations or dribbling techniques. Medical students are not invited to rediscover anatomy or biochemistry, nor are police cadets invited to re-invent ways of subduing suspects or identifying fingerprints.

Geometry

What then can bring children to focus on what we want to teach?

Systematic use of hot-wording allows much more extensive cross reference between text and figures than is possible in a book, reducing the unskilled reader's need to decode cross-references.

Things to do: adapt "arranging numbers" to alphabetization