Skemp takes the role of Devil’s Advocate to justify why teachers teach instrumental mathematics. His first justification is the only one I personally believe to be true, and therefore the one I would like to focus on:
“Within its own context, instrumental mathematics is usually easier to understand; sometimes much easier.”
I will conclude this paper by taking another look at this justification and examining its incompleteness. In the meantime, I would like to examine Skemp’s justifications that teachers may be making against teaching relational mathematics.
Skemp mentions that one of the situational factors which contribute to a teacher consciously choosing instrumental teaching over relational teaching is the difficulty of assessing the student.
“From the marks he makes on paper, it is very hard to make valid inference about the mental processes by which a pupil has been led to make them… In a teaching situation, talking with the pupil is almost certainly the best way to find out; but in a class of over 30, it may be difficult to find the time.”
I do not believe it is classroom size that restricts teachers from judging a student’s understanding and assigning an appropriate grade by talking one-on-one. For parents of students with strictly instrumental mathematical knowledge, they will question why their child receives a failing grade despite passing all homework assignments and tests. As a teacher, justifying a subjective mathematical grade to a parent will be very difficult.
Skemp states another factor as being:
“The great psychological difficulty for teachers of accommodating (re-structuring) their existing and long-standing schemas, even for the minority who know they need to, want to do so, and have time for study.”
I’m going to tie this in with an earlier statement Skemp makes:
“Teaching for relational understanding may also involve more actual content. Earlier, an instrumental explanation was quoted leading to the statement ‘Circumference = pd’. For relational understanding, the idea of a proportion would have to be taught first (among others), and this would make it a much longer job than simply teaching the rules as given. But proportionality has such a wide range of other applications that it is worth teaching on these grounds also.”
I believe these statement to be unfair, putting unwarranted blame on individual teachers. Equations used in projectile motion, which in turn is used in both Physics and Math as an application of Algebra, cannot be derived without using Calculus. However, to require Grade 10 students to take Calculus as a prerequisite to Grade 11 Physics, or to postpone the teaching of a core science until second year university, would be a great injustice to science students, and to blame individual teachers, or even entire administrations, for not drastically altering the fundamental structure of Math/Science education at the high school and university level is simply preposterous.
Skemp partially addresses this concern with the following rationalization:
“An individual teacher might make a reasoned choice to teach for instrumental understanding [on the ground] that a skill is needed for use in another subject (e.g. science) before it can be understood relationally with the schemas presently available to the pupil.”
To respond to this quote, I need to in turn quote Charles Vanden Eynden, author of Elementary Number Theory, to take Skemp’s statement to it’s extreme conclusion:
“Of course the integers are familiar to us from our earliest introduction to arithmetic. We have manipulated them thousands of times, and have formed an intuitive sense of many of their laws. This intuition carries some danger with it, because it may be hard to see the necessity for a proof to verify patterns that we have confirmed by instances over and over again since grade school. Although intuition can be of immense help in mathematics, no accumulation of special cases is sufficient to prove a general proposition.”
To truly understand a concept as simple as the laws governing integers, a number theory class is required to construct all the necessary proofs. Imagine a Grade 8 students reaction to finding out that they need to attend (and pass) a required Grade 8 proof class.
Again, Charles Vander Eynden says it best:
“Anxiety with respect to proofs makes success less likely in most cases. Therefore, the number of computational problems in [Elementary Number Theory] exceeds that of most number theory texts… Computational problems also lead to understanding of the theory behind them.”
I think the last part of this statement is particularly insightful. “Computational problems also lead to understanding of the theory behind them.” Skemp’s first justification stated “Within its own context, instrumental mathematics is usually easier to understand; sometimes much easier.” However he overlooked the simple truth that instrumental teaching can lead to relational understanding.
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