Posted by: Alexandre Borovik | October 12, 2011

“Mathematical needs”

The ACME policy document “Mathematical Needs” is a case of missed opportunities. Interesting data collected in workplace interviews are compromised by careless and methodologically flawed analysis.

For an example, one can look at the following case study.

“6.1.4 Case study: Modelling the cost of a sandwich
The food operations controller of a catering company that supplies
sandwiches and lunches both through mobile vans and as special
orders for external customers has developed a spreadsheet that
enables the cost of sandwiches and similar items to be calculated.
It was necessary as part of this work to estimate the cost of onions
in hamburgers, which was done by finding out how many burgers
can be filled from one onion. The most difficult parameter to
estimate for the model is the cost of labour.”

This example illustrates one of the fundamental flaws of ACME’s approach: factually interesting case studies are interpreted via the the skewed prism of “modelling” agenda. Meanwhile, anyone who ever did a spreadsheet of complexity of a sandwich should know that the key mathematical skill required is a basic ability of manipulating brackets in arithmetic and algebraic expressions, something that Tony Gardiner calls “structural arithmetic” and Michael Gove calls “pre-algebra”. At a slightly more advanced level working with spreadsheets requires understanding of the concept of functional dependency in its *algebraic* aspects (frequently ignored in pre-calculus): if the content of cell B10 is SUM(B1:B9) and you copy it in cell C10, the content of this cell becomes SUM(C1:C9) and thus involves *change of variables*.

Intuitive understanding that SUM(B1:B9) is in a sense the same as SUM(C1:C9) is best achieved by exposing a student to a variety of algebraic problems which would convince him/her that a polynomial of kind x^2 + 2x + 1 is, from an algebraic point of view, the same as z^2 + 2z + 1.

ACME never tried to look at the actual mathematical content of workplace activities, and therefore their recommendations for education are based on entirely false premises. That the mathematical content is missing from their analysis is further confirmed by an important observation found on page 2 of the document:

“Employers emphasized the importance of people having studied
mathematics at a higher level than they will actually use. That
provides them with the confidence and versatility to use
mathematics in the many unfamiliar situations that occur at
work.”

ACME missed a chance to ask a correct question: why was indeed this happening? Instead, they appear to accept the employers’ vague hint that this is something about emotional maturity of their employees. But this is not about emotions; indeed, it is fairly obvious that a person’s “confidence” is directly linked to person’s *understanding* of what he or she is doing; meanwhile, the word “versatility” directly points to some mathematical skills involved in solving practical problems; the last point was lost (or even never even looked at) in ACME’s analysis.

Next, we cannot avoid commenting that the intellectual vacancy of the concept of “modelling” as it is used by ACME is obvious from

“6.1.2 Case study: Mathematical modelling developed by a
graduate trainee in a bank [...]
1. Modelling costs of sending out bank statements versus going
online.”

In 19th century there was of course no option of going online, but in a similar situation they would simply say “comparing costs of sending out bank statements by post versus hiring an in-house courier”.

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