| 年齢 | ピアジェ | ブルーナー | 性質 |
| 0- | 感覚運動期 | 動作的 | 反射的反応、繰返し再現させようとする行動、新規な現象への興味、物の永続性の理解 |
| 1.5- | 前操作期 | 言語的適応の始まり、質量の保存認識は未獲得、長さの保存認識の獲得 | |
| 7,8- | 具体的操作期 | 映像的 | 数の保存認識、事象の変換前後での不変なものを認識、推移律 |
| 11,12- | 形式的操作期 | 記号的 | 力学的釣り合いの理解, 仮説演繹的推論／諸法則の帰納 |
"3 + 5"
"4 + 4"
"16 - 8"
"4 2"
"8"
| The life work of Jean Piaget is both wide and deep enough to defy any cursory summation. Since there exist summaries and critiques (e.g. Furth: Piaget and Knowledge: Theoretical Foundations), a more selective strategy is in order. |
| Two of Piaget's fundamental notions are attractive from a computer scientist's point of view. |
| The first is that knowledge, particularly in the young child, is retained as a series of operation models, each of which is somewhat ad hoc and need not be logically consistent with the others. (They are essentially algorithms and strategies rather than logical axioms, predicates and theorems.) It is much later in development that logical is used and even then through extralogical strategies. |
| The second notion is that development proceeds in a sequence of stages (which seems to be independent of cultural environment), each one building on the past, yet showing dramatic differences in the ability to apprehend, generalize and predict casual relations. Although the age at which a stage is attained may vary from child to child, the apparent dependency of a stage on previous stages seem to be invariant. Another point which will be important later on is that language does not seem the mistress of thought but rather the handmaiden, in that there is considerable evidence by Piaget and others that such thinking is nonverbal and iconic. |
| Both Piaget and Bruner have coined names for the stages of development. Bruner's are a bit more descriptive so they are included here also. |
| Age | Piaget | Bruner | Properties |
| Sensorimotor | Enactive | Reflexive, REVERSIBILITY, Discrepancy/Novelty, Object Consernation | |
| 1 1/2 | Preoperational | Speech starts, Mass: Not, Length: Conserved | |
| 4 | Concrete Operational | Iconic | Length, Conservation, Negation (Inversion), f(x) = y, f'(y) = x, Reciprocity |
| [?] | Formal | Symbolic | Multi v. Conservation, Hypotheses/Deductions |
| 年齢 | ピアジェ | ブルーナー | 性質 |
| 0- | 感覚運動期 | 動作的 | 反射的反応、繰返し再現させようとする行動＜注:REVERSIBILITY って何でしょうね？＞＜よく分かりませんでした。参考にした本だと「可逆性」の理解はまだ身に付いてないはずなんですけど・・・＞、新規な現象への興味、物の永続性の理解 |
| 1.5- | 前操作期 | 言語的適応の始まり、質量の保存認識は未獲得、長さの保存認識の獲得 | |
| 7,8- | 具体的操作期 | 映像的 | 数の保存認識、事象の変換前後での不変なものを認識、推移律 |
| 11,12- | 形式的操作期 | 記号的 | 力学的釣り合いの理解, 仮説演繹的推論／諸法則の帰納 |
| If the stage dependency is real, if may be worse than useless to try cramming context from upstream stages down children's throats until they are ready. For instance, it is now fashionable to teach children (in "new math") point set topology on a 2-dimensional cartesian coordinate system at the earliest possible age. The wisdom of this is contradicted by a series of Piaget's experiments which show that children in the operational stage simply do not grasp the notion of a coordinate system until later on. However, they do have very sophisticated notions of topology, connection, enclosure and grouping - all RELATIVE concepts. These facets were used by Papert and Goldstein to teach geometry and topology without reference to a global coordinate system - a much more satisfying state of affairs. |
| If we believe in the veracity of "operational" (semantic) rather than "predicative" (logical, syntactic) models, a quarrel has to be made with the very syntactic notions currently in favor in "New Math". For example, in natural numbers: |
"3 + 5"
"4 + 4"
"16 - 8"
"4 2"
"8"
| are said to be "numerals" for the number eight. |
| This concept is not only misleading and nonsemantic, it is also wrong. (What number is "8/3" a "numeral" for?) |
| Minsky has noted: "The trouble with new math is that you have to understand it everytime you use it". (20) |
| Piaget's and others' work on the bases and forms of children's thought is a fairly convincing argument for believing that computers are an almost ideal medium for the expression of a children's epistemology. What is an "operational model" if not an algorithm, a procedure for accomplishing a goal? Algorithms are fairly informal and not necessarily logically consistent (as anyone who has ever spent a few hours debugging a program well knows). This fits in well with the child's viewpoint which is global and interested in structure rather than strict implication of "truths". ON the other hand, the computer also aids in the formation of skills concerning "thinking": strategies and tactics, planning, observation of casual chains, debugging and refinement, etc. Rarely does a child have a chance to practice these skills in an environment that is patient, covert and fun! |