Teaching—for what?

As classroom teachers and, concomitantly, students of the teaching of elementary school mathematics, we readily accept the idea that learning is accomplished on a variety of levels. Further, our teaching methods generally reflect this. When we want quick, automatic response to number facts, we use flash cards, timed tests, and the like; but when we want to develop understanding and the abiUty to solve problems, we use quite different procedures. In the latter instance we are likely to pace instruction more evenly, provide many concrete materials, encourage pupil inquiry through group discussion and give appropriate attention to the mathematics of our environment. Experience has taught us, however, that when we attempt to evaluate such instructional efforts, we seem much less effective than we assumed ourselves to be during the original teaching process. Our instruments are often haphazard and ill-conceived, and our interpretations of test results can be equally vague and sterile. The material that follows presents a hi erarchical framework for evaluation—a framework that can help us systematically examine different levels of learning and subsequently determine appropriate evaluation activities more closely associated with tbe original purposes of the instruction.