Mathematical Needs, a report by ACME: A brief review of the methodology of the report

I start with a disclaimer:

All opinions expressed in the present post are of the author and no-one else. More specifically, all views expressed here may or may not represent the position of the London Mathematical Society which does not bear any responsibility for the content of this post.

So, in my opinion, the ACME policy document Mathematical Needs: Mathematics in the workplace and in Higher Education is a case of missed opportunities. Interesting data collected in workplace interviews appear to be compromised by 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 narrow 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”. (I have heard from a graduate of my University that his co-workers’ mastery of spreadsheets is inhibited by “brackets overload”; this expression is interesting because it has been coined at the actual workplace.) At a slightly more advanced level working with spreadsheets requires interiorisation  of the concept of functional dependency in its algebraic aspects (frequently ignored in pre-calculus).

To illustrate this point, I prepared a very simple spreadsheet. Look at the picture above: if the content of cell C14 is SUM(C8:C13) and you copy cell C14 into cell D14 (see the next picture),

the content of cell D14 becomes SUM(D8:D13) and thus involves change of variables. What is copied is an algebraic expression, not its value: notice that the value 85 became 130 when moved from cell C14 to cell D14!

Please notice that I am using the word “interiorisation” rather then “understanding”: people can manipulate brackets (or their cognitive equivalents) even if they are not able to clearly explain what they are doing; however, more basic principles like “every openning bracket should be matched by a closing one” perhaps can be formulated by anyone able to do a spreadsheet. Please notice and yet another qualifying remark: I am talking about “cognitive equivalents” of brackets: they have many forms, say, branching rules in tree-like structures, parsing, and primitive forms of recursion, to name a few.

Intuitive understanding that SUM(C8:C13) is in a sense the same as SUM(D8:D13) is best achieved by exposing a student to a variety of algebraic problems which convince him/her that a polynomial of kind $latex x^2 + 2x + 1$ is, from an algebraic point of view, the same as $latex z^2 + 2z + 1$, and that in a similar vein, the sum

$latex C_8 + C_9 + C_{10} + C_{11} + C_{12} + C_{13}$

is in some sense the same as

$latex D_8 + D_9 + D_{10} + D_{11} + D_{12} + D_{13}$ .

It looks as though ACME’s researchers never tried to look at the actual mathematical content of workplace activities, and therefore their recommendations for education are based on 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.

In my opinion, here ACME missed a chance to ask the right 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 at all; 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 has been lost (or even never been looked at) in ACME’s analysis.

Next, I cannot avoid commenting on the intellectual vacancy of the concept of “modelling” as it is used by ACME:

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 the 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”. Use of the trendy word “modelling” is superficial here, it only distracts from seeing the actual mathematical content of the activity.