(Page 1 of 3) - Investigating Teaching and Learning Subtraction that Involves Renaming using Base Complement Additions. authored by McCarthy, Peter.

INVESTIGATING TEACHING AND LEARNING OF SUBTRACTION THAT

INVOLVES RENAMING USING BASE COMPLEMENT ADDITIONS

Peter McCarthy

University of Toronto

## email not listed ##

This article reports on how “base complement additions” was used to help research participants,
in grades 4 and 5, do compound subtraction (subtraction that involves renaming). The
participants’ understanding the use of “numeration cards” helped them to do addition,
subtraction and to convert compound subtraction to simple subtraction. Findings revealed that
participants prefer using base complement additions to the decomposition (borrowing) strategies.

In the history of mathematics education, teaching children to subtract has long been

considered a problem area. Winch (1920), noted that “no methods give more trouble and are less
successful than those of teaching subtraction” (p. 20). Thorndike (1921) believed that the
controversy regarding how children should be taught to subtract centered on the argument of
whether to use the “subtractive” or “additive” method. Today, learning subtraction that involves
renaming continues to be a source of difficulty for many young students and the question of how
best to teach it to the youngsters is gaining renewed interest. Currently, both teachers and
students in the United States, Canada, and other parts of the world predominantly use the
decomposition (D) strategy, though there might be other student-developed strategies. There is
one very important advantage for using D strategy: the ability to demonstrate the regrouping
procedure using bundles of sticks, straws and other manipulatives. This has made the strategy
very popular, especially following the “meaningful teaching” at the turn of the century (Brownell
& Moser, 1949). However, the method appears to have many inherent drawbacks, which
adversely affect pupils in their performance on arithmetical tasks involving compound
subtraction. The most obvious drawback of the strategy is that it takes too much time for students
to learn the prerequisite concepts like place value, expansion, and renaming. Many students end
up using “buggy” strategies due to limited preparation in time (Gyening, 1993).

This paper reports on grades 4 and 5 students’ learning of the base complement additions

(BCA) strategy for solving compound subtraction of whole numbers. This study is an attempt to
shed some light on the teaching and learning of compound subtraction by BCA; and to verify if it
will promote meaningful understanding of compound subtraction using BCA.

The Base Complement Additions Strategy

The base complement additions (BCA) is a strategy for solving compound subtraction

problems by changing both the minuend and the subtrahend. The strategy transforms compound
subtraction to simple subtraction by way of base complement(s). By base complement of a whole
number n relative to the base b, we mean the positive number that should be added to n to obtain
b. In other words c is the base complement of n if n + c = b, where b is the base and b, c, and n
are whole numbers. The rationale for BCA approach is based on compensation. That is, if the
same number is added to both the minuend and the subtrahend the difference is not changed
though there is change in arithmetic expression.

Methodology

The study started on January 2000 and ended on August 2001. To know the participants’

entry behavior they were first given a set of test items on subtraction to do. This was then