Academic journal article
*The Mathematics Enthusiast*

# A Commentary on Freudenthal's Didactic Phenomenology of the Mathematical Structures Associated with the Notion of Measurement

## Article excerpt

(ProQuest: ... denotes formulae omitted.)

1.INTRODUCTION

1 In [1, Freudenthal (1973)] a phenomenology for the mathematical structures relevant to measurement is presented. Freudenthal's exposition is paradigmatic of the process of "modelization" at two levels. First, he starts with the description of the mental actions relevant to the phenomena, in this case, measurement of weights, (horizontal mathematization) to then begin a process of a more abstract formalization which subsumes his original description of the "phenomena" (vertical mathematization). Hence, Freudenthal's original model in terms of the mental actions germane to measurement constitutes a descriptive model of his didactical phenomenology, and he transforms it into a prospective model for the mathematical structures of measurement. We would like to discuss this point since it illustrates that the vertical mathematization implicit in evolution of descriptive models into prospective models, both in general and in the study of measurement in particular, not only introduces new formal elements into the descriptive model to transform it into the prospective one, but it also, incorporates the corresponding mental proceses and associated logical interconnections. This can be regarded as part of the verticalization process and, in our view, it is, in part, an ingredient of what is generally called "number sense." Implicit in this verticalization there are elements of automatization of structural model features (formal features) as well as mental processes. Such automatizaton empowers students to add to their knowledge efficiently as they are able to muster the resources automatized and use them as basic or given elements in the subsequent use of models to describe new mathematical situations. We will give examples of this using the Realistic Mathematics Education (RME) approach to fractions, percentages and decimals, as presented, for instance, in [3, Keijzer et all (2006)]. Since in RME, didactical sequences are to be based on an initial proposal suggested by a didactical phenomenology of the appropriate mathematics, we begin by discussing first Freudenthal's didactic phenomenology for measurement, as discussed in [1, Freudenthal (1973), pp. 193-212].

2.MATHEMATICAL FORMULATION OF MAGNITUDES ACCORDING TO FREUDENTHAL

Definition 2.1. A set of magnitudes is a set G with and operation + : G x G ^ G (called "addition") and an order relation < Ç G x G such

i. < satisfies trichotomy: for all a,b ∈ G one and only one of the following conditions hold: a < b or a = b or b < a.

ii. < satisfies transitivity: for all a,b,c ∈ G, if a < b and b < c then a < c.

iii. + is associative: for all a,b,c ∈ G, (a + b) + c = a + (b + c).

iv. + is commutative: for all a,b ∈ G, a + b = b + a.

v. + satisfies the cancellation law: for all a,b,c ∈ G, if a + c = b + c, then a = b.

Also,

vi. For all a,b ∈ G, a < b if and only if for some c ∈ G (necessarily unique) a + c = b.

vii. Measurements can be amplified by whatever integral factors desired:

The symbol "n * a", for n a positive integer and a ∈ G, is defined by induction as 1 * a = a and for each positive integer n, (n +1) * a = n * a + a. We call n * a the amplification of a by the factor n.

viii. Measurements can be divided into as many equal parts as desired:

For all a ∈ G and each integer n > 1 there is b ∈ G (uniquely determined) so that n * b = a. We write

...

A straightforward argument using the definition proves that ..., foralln,mpositive integers and all a ∈ G.

Freudenthal (ibidem) describes an example of a set G consisisting of all objects having the attribute of weight. We can define two relations as follows: two objects a and b are the equivalent if they both can be used to level the balance; we write a « b in such a case; the weight a is smaller than the weight b if a is the lighter of the two; we write a < b. …