The NCTM Curriculum and Evaluation Standards give a broad definition of procedural knowledge which includes knowing when and why an algorithm is appropriate. While I agree these are important, for the nature of this comparative discussion, I am limiting my definition of procedural knowledge to the ability to perform a particular task. What makes this different from conceptual task performance is that the behavior is often learned by associating a stimulus (e.g. “3 x 5”) with the correct response (e.g. “15”). At its worst, this type of learning is merely memorization. At its best, however, it is a necessary part of learning. Progress in mathematics depends on the ability to abstract, but abstraction depends on the ability to capsulize concepts (Skemp). If a student has a concept tied to a symbolic procedure, it is not always necessary for the depth of that concept to be recalled every time he wants to use the algorithm. In this context, associative, habitual, or procedural knowledge is an important factor in the efficiency and abstraction in mathematical learning.
Conceptual knowledge is the product of incorporating a new idea into an established schema, or the re-organization of an existing schema to fit a new idea. It is also goal-directed, as opposed to merely responsive (Skemp). The distinction here is in the process of “getting an answer.” The student applying procedural knowledge is recalling rules and applying them. It is an efficient and accurate process if the student’s memory is good. The student applying conceptual knowledge, on the other hand, approaches every question as a problem to be solved, not a fact to be remembered. The process is one of interpreting the situation, perceiving a goal, and accessing various resources to help reach that goal. One of those resources may very likely be a piece of procedural knowledge, but if that is not available, the student will be able to develop alternative routes to the given end. This process requires less memorization and is more adaptable, but can often require more cognitive effort.
Consider the case of assessing a student’s knowledge about adding
fractions with unlike denominators. Procedurally, you could give
a standard question, “Find the sum of 2/3 + 3/4.” You could ask students
to show each step, or even explain each step, but essentially you are looking
to see if they have remembered and executed the algorithm correctly.
This is important. In addition, however, you might also want to know
what kinds of numerical values the students are thinking about (if any)
when they add these two numerals. Questions that get at students’
conceptual ideas might include: “Estimate the sum of 2/3 + 3/4. Is
it more or less than one? Why?”; or “A student claims that 2/3 +
3/4 = 5/7. What would you say to him?” Through this last question,
in particular, we are trying to see if the student is thinking of 2/3
and 3/4 as parts relative to the same whole. Such a question could
be an open-ended test question, but would perhaps be less threatening as
a journal entry. If students showed difficulty with the idea, a performance
task using fraction bars could be mixed with interviews to help students
make the connection between the objects and the language. A
multi-modal writing assessment as described in Stix’s article is an excellent
opportunity for each student to express their ideas in the format with
which they are most comfortable.
While procedural knowledge can be easily assessed through standard tests, a conceptual assessment of this idea would require a verbal or demonstrative performance beyond symbolic manipulation.
Stix, Andi. “Pic-Jour Math: Pictorial Journal Writing in Mathematics.” Arithmetic Teacher. National Council of Teachers of Mathematics. January 1994. pp264-269.
NCTM. Curriculum and Evaluation Standards for School Mathematics. Reston, VA. 1989. p228.
Skemp, Richard. Mathematics in the Primary School. Routledge, London.
1989. pp1-20, 32-71.