**ERIC Number:**ED342648

**Record Type:**Non-Journal

**Publication Date:**1991-Mar

**Pages:**96

**Abstractor:**N/A

**Reference Count:**N/A

**ISBN:**N/A

**ISSN:**N/A

From Protoquantities to Operators: Building Mathematical Competence on a Foundation of Everyday Knowledge.

Resnick, Lauren B.

This paper proposes a theory that can account for differences between everyday and formal mathematics knowledge and a set of processes by which informal knowledge is transformed into formal mathematics. After an introduction, the paper is developed in five sections. The first section lays out the nature of informal, everyday mathematics knowledge. Examples of children's thinking for several mathematical principles are presented. The second section discusses evidence of systematic difficulty in learning school mathematics, considering two hypotheses to explain why strong and reliable intuitions documented for children and unschooled people are not sustained in school mathematics. The first, called the "syntax-semantics" hypothesis, is that the focus in school on formal symbol manipulation discourages children from bringing their intuitions to bear on school mathematics learning. The second, called the "abstract entities" hypothesis, is that there is an epistemological discontinuity between informal mathematics rooted in everyday behavior and the kind of mathematical reasoning that is sought in school. The third section discusses a theory of layers of mathematical knowledge. The layers are characterized as the mathematics of protoquantities, quantities, numbers, and operators. The fourth section discusses teaching mathematics on the basis of intuitive knowledge, arranging situations of practice and discussion that will help children elaborate their schemas at successively higher levels of mathematical knowledge. The final section presents six principles that guided the development of a progam for elementary classroom instruction which not only produced substantial gains in children's mathematical performance but challenged some long-held assumptions in psychology and education as well. (MDH)

**Publication Type:**Information Analyses; Guides - Classroom - Teacher

**Education Level:**N/A

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**Pittsburgh Univ., PA. Learning Research and Development Center.