구체적 교수법과 추상적 교수법을 결합한 두 자리 수 곱셈 문제 해결 전략 지도
Combining of Concrete and Abstract Instruction for Teaching Strategies for Solving Two-digit Multiplication Problems
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초록
사람들은 학교에서 익숙히 배운 계산법 및 문제별 특징에 알맞게 고안된 유연하고 적합한 풀이 전 략 등을 활용하여 여러 자리 수 곱셈 문제를 해결한다. 이 중 분배 법칙을 활용하여 곱하는 수를 재구 성하여 여러 자리 수 곱셈 문제를 해결하는 것은 매우 효과적이고 편리한 문제 해결 전략이다(예: 12 × 17 = 12 × (10 + 7) = 120 + 84 = 204). 본 연구에서는 구체적 교수법과 추상적 교수법을 결합 한 교수 방법이 분배 법칙을 활용한 곱셈 문제 해결 전략을 지도하는 데 얼마나 효과적인지 비교 분석 하였다. 이를 위하여 서울 남부의 2개의 초등학교에서 참가자를 모집하였으며, 참가에 동의한 3학년 학생들은 4개의 실험 집단으로 임의 배정되었다. 각 실험 집단에 속한 참가 학생들은 구체적 교수법 (연산 감각 기반 또는 넓이 기반) 과 추상적 교수법 (방정식 기반) 의 두 개의 교수 모듈이 결합된 워크 북을 구체적 교수법 선행(연산 감각 기반-방정식 기반 또는 넓이 기반–방정식 기반 순서) 또는 추상적 교수법 선행 (방정식 기반–연산 감각 기반 또는 방정식 기반–넓이 기반 순서) 의 순으로 완성하였다. 연구 결과 교수법 순서에 따른 곱셈 문제 해결 능력 향상에는 차이가 없는 것으로 나타났다. 하지만 연산 감각에 기반한 구체적 교수법의 경우 넓이에 기반한 구체적 교수법보다 훨씬 더 두 자리 수 곱셈 문제 해결 능력을 향상시키는 것으로 나타났다. 다른 유형의 복잡한 곱셈 문제 해결 능력(예: 방정식 형태의 곱셈 문제)과 관련된 학습 효과의 전이 측면에서는 모든 교수 방법에서 비슷한 효과를 보였다. 마지막으로 개인별 한 자리 수 덧셈 뺄셈 연산 유창성의 경우 분배 법칙을 활용한 곱셈 문제 해결 전 략 학습 정도를 예측하는 데 유의한 변인인 것으로 나타났다. 교육적 함의 및 제언으로서는 효과적인 교수 학습 지도 이론 및 연산 문제 풀이 해결 전략 방법에 대해 논의하였다.
Abstract
People solve multi-digit arithmetic problems using a combination of routine calculation procedures and adaptive strategies. An important adaptive strategy for multi-digit multiplication is decomposition via the distributive law, e.g., 12 × 17 = 12 × (10 + 7) = 120 + 84 = 204. The current study investigated the combining of concrete and abstract instruction for teaching the distributive law. To achieve the goal of the study, we recruited third graders from two elementary schools in Seoul. Participated third graders were randomly assigned to one of the four instructional conditions. They completed concrete instruction (either arithmetic or area) and abstract instruction (algebra) in one of two orders: concreteness fading (either arithmetic-algebra or area-algebra) or abstraction instantiation (either algebra-arithmetic or algebra-area). Results found that there was no effect of instructional sequence on performance on the target 2D × 2D multiplication problems at posttest. However, there was greater performance on the target 2D × 2D multiplication problems at posttest for concrete instruction that was arithmetic-based versus area-based. Transfer to multiplication problems of broader ranges and different (i.e., algebraic) formats was comparable for all instructional conditions. Finally, individual differences in addition and subtraction fact fluency predicted performance on the target 2D × 2D multiplication problems at posttest. The implications of these findings for theories of learning instruction and for future research on arithmetic strategies are discussed.
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