Taking away and determining the difference—a longitudinal perspective on two models of subtraction and the inverse relation to addition

Subtraction can be understood by two basic models—taking away (ta) and determining the difference (dd)—and by its inverse relation to addition. Epistemological analyses and empirical examples show that the two models are not relevant only in single-digit arithmetic. As curricula should be developed in a longitudinal perspective on mathematics learning processes, the article highlights some exemplary steps in which the inverse relation is discussed in light of the two models, namely mental subtraction, the standard algorithms for subtraction, negative numbers and manipulations for solving algebraic equations. For each step, the article presents educational considerations for fostering a flexible use of the two models as well as of the inverse relation between subtraction and addition. In each section, a mathe-didactical analysis is conducted, empirical results from literature as well as from our own case studies are presented and consequences for teaching are sketched.     

Addition and Subtraction of Three-digit Numbers: German Elementary Children's Success, Methods and Strategies

The present paper describes the success, the methods (mental, informal written, standard algorithm) and the strategies of informal written arithmetic to be observed when 300 elementary students worked on six addition and six subtraction problems with three-digit numbers. These twelve problems were administered repeatedly by means of a class test: in February (grade 3; nine-year-olds) before the standard algorithms were introduced, in June after they had been dealt with and in October at the beginning of grade 4. This revised version was published online in July 2006 with corrections to the Cover Date.     

Building on Children's Mathematics - a Teaching Experiment in Grade Three

There rarely seems to be any disagreement that teaching should build on children's mathematics, but there are many possible ways to put this demand in concrete terms. In this sense, the present paper describes a teaching experiment in grade three based on Treffers' idea of vertical mathematisation. Central components of the developed course are discussed, with focus on the use of the empty number line and of maths diaries as well as on the idea of conducting maths conferences. The teaching experiment is illustrated by means of children's own productions and finally put into a broader context as contributing to the design science approach of mathematics education.