Formulae As A Disguise

How do we know what a student knows?  More often than not, the use of formulae (such as perimeter or area) serve as a disguise for the lack of knowledge … a disguise which allows a person to achieve a preponderance of ‘correct answers’ in spite of having no relational or procedural knowledge. 

My motivation, sadly, is personal therapy.  Our beginning algebra classes took a test today, dealing with polynomials.  This is a traditional class, though our work together has focused on meaning and understanding.  One problem on this test is a contrived operation question:

Find the polynomial that represents the perimeter of the figure. [Figure shows a triangle with sides 3a+2, 2a+1, and 6]

A minority of students added the sides.  Two responses predominated the incorrect work — P = 2L + 2W, and A = LW.  Students retrieved these formulae in spite of the visual stimulus indicated that this was not a rectangle.  It is likely that most students had achieved ‘success’ by using these formulae in prior math courses, perhaps where the material was ‘blocked’ (all problems of a similar type, not mixed).

This thought led me to question something at the heart of our current work in this course:  ‘rules’ for operations with exponents.  The formulae for this work have been stated verbally, not symbolically; our class time has been focused on the reasonableness of our rules.  Based on the types of mistakes I see on other items, I suspect that students are storing some of their knowledge in those “formula files” just like the geometry ones.

I am suspecting that a formula in the hands of a novice math student is dangerous, just like some power tools in the hands of novice craftsmen (like myself).  Perhaps we would be better served by avoiding rules in most cases, and avoiding formulae as long as possible, so that all work is done based on some understanding.  Perhaps a student stops learning as soon as there is a rule or formula to remember.  This concern with formulae is related to concerns with PEMDAS:  The presence of a rule which provides sufficient correct answers stops the learning process, and may prevent deeper understanding.

If we are talking about finance formulae involving 6 input variables, I do not see a problem with the formula stopping the learning process.  However, when there is a key mathematical concept involved — whether perimeter or exponents — I think the formulae create enough problems to approach them with reservations.  If anybody knows of related scientific research on the impact of formulae on learning, I would love to hear about it.

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1 Comment

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  By Sue J, November 29, 2012 @ 10:21 am

  http://quantumprogress.wordpress.com/2012/11/29/another-point-in-the-argument-against-timed-tests

  This blog talks about the tendency to substitute something simple and quick instead of slowign down and thinking, and states that “My recollection of Kahneman also is that he said that this leaping to the wrong answer can never be eliminated—it can only be reduced by giving ourselves more time to process, often by deliberately introducing mental speed bumps (like drawing a diagram) designed to slow us down and allow our reasoning brain time to process.”
  Seems formulae work in the opposite direction of speed b umps…

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