followed by interviews. The researcher met with the participants for 8 contact periods during
which they were given instructions on both D and BCA (not at the same time). The duration for
the intervention was approximately 35 minutes for each contact period. Fourteen days after the
eighth contact period post-intervention tests and interviews followed consecutively. All the tests,
instructions and interviews were centered on subtraction, especially that which involves
renaming. All the interviews were audiotaped; and the transcribed audiotapes, the test papers and
the working papers formed the bases for the data required for the study.
The numeration cards were used for development of the instructional intervention. These
were mainly used for developing meaningful understanding of place value, base complements,
compound subtraction expressions, and transformation of compound subtraction to simple
subtraction expressions. The numeration cards were displayed and named as small (ones), long
(tens) and flats (fifty and/or hundred) to help participants understand BCA strategy.
The learning situation was first presented in free play before it became structured (Dienes,
1973). The representations covered concrete materials, followed by image formation of the first
representations. Sketches, diagrams, and pictures were used for the second phase of the
instructional intervention before the final phase, which was symbolic (Bruner, 1966; Dienes,
1973).
Findings and Discussion
The pre-intervention test and interviews indicated that participants used more than one
strategy for simple subtraction but not for compound subtraction. The post-intervention
interviews revealed additional strategy they used, for solving compound subtraction, in which
they were more confident with.
Participants made many errors when working with “borrowing”
strategy. But with the introduction of BCA they were seen to make only few errors with respect
to multi-digit subtraction computations. Before the intervention participants could not easily
determine errors in compound subtraction computations. After learning BCA they could easily
look at subtraction expressions and determine whether they are right or wrong and why. These
findings give a clear indication that BCA is an excellent tool for computing subtraction that
involves renaming; and it makes sense to the participants.
Compound subtraction that involves at least two renaming appears complex when borrowing
strategy is applied. It involves a series of subtracting and adding before the actual subtraction is
done. Thus, when the student forgets to keep track of the procedure, errors result. With BCA,
however, one uses only simple addition and simple subtraction hence calculations become easier
for the user.
From the findings of the study the learning of compound subtraction using BCA, in grades 4
and 5, could be conceptually based using manipulatives (numeration cards) for meaningful
understanding of place value, subtraction concepts, and the idea of equivalence. From the
findings of the study standard algorithms could be meaningfully taught at grades 4 and 5 so that
students understand the concepts and reasoning associated with the procedures (Ashlock, 2002).
This means paper-and-pencil procedures could involve both conceptual and procedural
knowledge at grades 4 and 5.
Conclusion
The learning of BCA offered participants an additional tool for their mathematical toolkit.
From this toolkit they chose the most appropriate strategy based on their comfort level. Thus, in
grades 4 and 5, students’ learning of BCA may expand their knowledge base and improve upon
their subtraction skills.
Convention