When is Arithmetic Meaningful?

Percent: A Privileged Proportion

“…Upon combining these two mixtures, one must find the actual amount of salt in the total volume of the mixture to calculate the solution percent. Instruction in mathematics often emphasizes how to compute rather than when it makes sense to compute (Brownell, 1945), a situation which has been especially true in the domain of percent (Kircher, 1926). As in the previous example, cases in which the addition rules appear to be violated are not actually addition problems in the first place!…”

Methods and Materials in the Teaching of Mathematics

“…He distinguished between meaning of and meaning for in the case of arithmetic and suggested four categories under which the meanings of arithmetic might be grouped (6). Pointing out a general confusion resulting in the teaching of social applications of arithmetic rather than the meanings and rational principles which make arithmetic a phase of mathematics, Brownell listed objections to teaching meanings, followed by reasons for teaching them (10). He explained and illustrated his concept of meaning (7,10), and listed results of experimental situations which favored his meaningful method of teaching borrowing to third-grade pupils, over mechanical methods taught to another like group (10).…”

“…Pointing out a general confusion resulting in the teaching of social applications of arithmetic rather than the meanings and rational principles which make arithmetic a phase of mathematics, Brownell listed objections to teaching meanings, followed by reasons for teaching them (10). He explained and illustrated his concept of meaning (7,10), and listed results of experimental situations which favored his meaningful method of teaching borrowing to third-grade pupils, over mechanical methods taught to another like group (10). Implications for research in arithmetic were outlined (8) In a critical analysis of the backgrounds of arithmetic, Renfrow (55) pointed out the impossibility of substituting algebra for arithmetic, the importance of the concrete approach in arithmetic, the need for mastery rather than omission of number processes, and the necessity for drill.…”

“…The nature and place of meaning in the teaching of arithmetic was dealt with by Brownell (6,7,8,10). He distinguished between meaning of and meaning for in the case of arithmetic and suggested four categories under which the meanings of arithmetic might be grouped (6).…”

A Survey of Mathematical Education

“…To develop better methods is really the task at a more adequate psychology of thinking and learning." It seems likely that the developmental concepts of Piaget (e.g., see 165,166,167) [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. Among the mathematical questions he has looked into are the multiplication combinations (33), borrowing in the sUbtraction of integers (37), and mathematical readiness (38).…”