of Higher Order Understandings
and Processes in Elementary
Although the central concerns of this chapter seem clear from its title -- computer-based assessment of higher order understandings -- each term in that tide is somewhat misleading.
First, the term "higher order" suggests the incorrect notion of ideas which are "up in the air" like nebulous conceptual clouds, or on conceptual mountain tops which can only be addressed after "lower order" facts and skills have been mastered. But the types of understandings that will be emphasized in this chapter might better be characterized using terms such as deeper and broader. They are conceptual cornerstones which provide foundations for the most important mathematical ideas that students should learn; they are not just structurally insignificant conceptual capstones which could have been omitted if time or other instructional resources were unavailable.
The real concern of this chapter is with dimensions of understanding which are especially important, but which neglected, regardless of whether these dimensions seem to extend up, down, out, or in from traditionally emphasized knowledge. The types of deeper/ higher order understandings that will be emphasized must develop on the way to learning foundation-level math concepts and principles. Otherwise, the meanings of these latter ideas will tend to be narrow, superficial and lacking in generalizability and applicability; and they will tend to have pitifully short half- lives in memory.
Hypothesized relationships between basic facts and higher order thinking are especially important because, for real progress to be made in curriculum reform, we must avoid the excessive pendulum swings that have characterized past movements. Today's higher order objectives movement is partly a reaction to the basic skills movement of the early 1980s, which, in turn, was in large part a reaction to the "new math" movement of the 1960s and 1970s. Each of these movements had valid concerns and excesses that should be recognized and taken into account in future reform efforts. Therefore, the policy that this chapter will adopt is that a single framework of objectives must be determined which deals in a balanced and integrated way with both basic facts and skills and higher order understandings and processes; otherwise, neither emphasis is likely to succeed.
Second, the term "assessment" is often considered to refer to a unidimensional, passive indicator of the static, "high-low" state of a nonadapting organism (which may be a student, a teacher, a classroom, a school or a