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

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A Commentary on Freudenthal's Didactic Phenomenology of the Mathematical Structures Associated with the Notion of Measurement The Mathematics Enthusiast Volume 14 Number 1 Numbers 1, 2, & 3 Article 3 1-2017 A commentary on Freudenthal’s didactic phenomenology of the mathematical structures associated with the notion of measurement Omar Hernandez Rodriguez Jorge Lopez Fernandez Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation Rodriguez, Omar Hernandez and Fernandez, Jorge Lopez (2017) "A commentary on Freudenthal’s didactic phenomenology of the mathematical structures associated with the notion of measurement," The Mathematics Enthusiast: Vol. 14 : No. 1 , Article 3. Available at: https://scholarworks.umt.edu/tme/vol14/iss1/3 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. TME vol. 14 nos. 1, 2 and 3 p. 3 A COMMENTARY ON FREUDENTHAL'S DIDACTIC PHENOMENOLOGY OF THE MATHEMATICAL STRUCTURES ASSOCIATED WITH THE NOTION OF MEASUREMENT OMAR HERNANDEZ´ RODR´IGUEZ GRADUATE SCHOOL, FACULTY OF EDUCATION, UNIVERSITY OF PUERTO RICO AT R´IO PIEDRAS JORGE M. LOPEZ´ FERNANDEZ´ MATHEMATICS DEPARTMENT, UNIVERSITY OF PUERTO RICO AT R´IO PIEDRAS Abstract. This paper discusses Freudenthal's didactical phenomenology for the mathematical structures related to measurement. Freudenthal starts with the set G of all objects having the attribute of weight, an attribute that is to be measured. He proposes two operations, the first one among the measurements them- selves, called \addition". Addition is interpreted in terms of the mental actions associated with measurement and, in the case of weights, it consists on the process of placing weights on one of the two dishes of a balance in order to balance them with a predetermined gauge, its fractions or multiples thereof, which are placed on the other. Their total weight is obtained by comparison with a predetermined gauge. The second operation, between weights and positive integers, allow weights to be amplified (by adding as many times as desired their weight measure). A related operation to amplification is that of contraction (by dividing the weight into as many equal parts as desired.) Freudenthal then extends his definitions in order to assign a meaning to the multiplication of a measure by, first, any positive rational number, and, second, by completeness, to any real number. Freudenthal's didactical phenomenology is based on the historical phenomenology of measurement, fractions, decimals and percentages. It is a central tenet of Realistic Mathematics Educa- tion (RME) that didactical phenomenologies of mathematical structures constitute proposals from which to make cognitively advantageous teaching sequences for the relevant mathematical structures. We discuss how Freudenthal's phenomenology of the mathematical structures for measurement gets translated into the mathematical models associated with the rational numbers. Finally, we make reference to a paradigmatic teaching unit which incorporates coherently Freudenthal's didactical phenomenology. Keywords: Realistic Mathematics Education, fractions, percentages, decimals, models, didactical phenomenol- ogy. 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 1The Mathematics Enthusiast, ISBN 1551-3440 vol. 14, nos. 1, 2 & 3, pp. 3-14, 2017 c The Author(s), Department of Mathematical Sciences & The University of Montana. Hern´andez& L´opez, p. 4, 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 × G ! G (called \addition") and an order relation < ⊆ G × G such i. < satisfies trichotomy: for all a; b 2 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 2 G, if a < b and b < c then a < c. iii. + is associative: for all a; b; c 2 G,(a + b) + c = a + (b + c). iv. + is commutative: for all a; b 2 G, a + b = b + a. v. + satisfies the cancellation law: for all a; b; c 2 G, if a + c = b + c, then a = b. Also, vi. For all a; b 2 G, a < b if and only if for some c 2 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 2 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 2 G and each integer n ≥ 1 there is b 2 G (uniquely determined) so that n · b = a. We write 1 b = · a: n 1 1 1 A straightforward argument using the definition proves that n · ( m · a) = nm · a, for all n; m positive integers and all a 2 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. Clearly ≈ and < are relations on G. The relation ≈ is an equivalence relation and < satisfies the conditions of Definition 2.1. An important observation is the following: Proposition 2.2. If n and m are positive integers, then 1 1 (n · a) = n · a ; m m for all a 2 G. Proof. First of all, it is true and straightforward to check that p · (q · a) = (p · q) · a for all integers p; q with p ≥ 1, q ≥ 1 and for all a 2 G (note that the various occurrences of the symbol \ · " can have different meanings in this relation. So suppose that a 2 G and that n; m are integers with n ≥ and m ≥ 1. Let 1 b = m · a be the unique element of G satisfying a = mb. It is also clear that for all integers p ≥ 1 and a 2 G, TME vol. 14 nos. 1, 2 and 3 p. 5 1 a = p · (p · a). Then 1 1 · (n · a) = · (n · (m · b)) m m 1 = · ((n · m) · b) m 1 = · (m · n) · b m 1 = · m · n · b) m = n · b 1 = n · · (m · b) m 1 = n · · a : m n All this goes to show that for a 2 G and integers n and such that n ≥ 1 and m ≥ 1, the expression m · a 1 1 is well defined, either as n · ( m · a) or m · (n · a) Remarks 2.3. a. Using the axioms and mathematical induction, it is easy to ascertain that the following properties hold for positive integers n, m and a; b 2 G: i. (n + m) · a = n · a + m · a; ii. n · (a + b) = n · a + n · b; iii. n · (m · a) = (nm) · a b. It can also be shown that for each positive integer m, 1 1 1 · (a + b) = · a + · b: m m m Using this fact it is easily verified that ai-aiii also hold for when n and m stand for positive rational numbers. c. A consequence of the above considerations is the relation for the addition of fractions. Of course, \ad- dition" does not refer to some previously defined algebraic rule, but rather to the addition of weights in G as postulated by Freudenthal. Students have a hard time understanding why not define addition of fractions by n n0 n + n0 + = rather than the, perhaps, unexpected m m0 m + m0 n n0 nm0 + n0m + = : m m0 mm0 1 It is straightforward to prove from the above properties that n · a = m (n · m) · a for every pair of positive integers n; m and every a 2 G.
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