Algebra Structure Sense Development amongst Diverse Learners

Theoretical and Empirical Insights to Support In-Person and Remote Learning

Teresa Rojano, Teresa Rojano

• Publisher: Routledge
• Year: 2022
• ISBN:
  9781000591538
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

This volume emphasizes the role of effective curriculum design, teaching materials, and pedagogy to foster algebra structure sense at different educational levels.

Positing algebra structure sense as fundamental to developing students' broader mathematical maturity and advanced thinking, this text reviews conceptual, historical, cognitive, and semiotic factors, which influence the acquisition of algebra structure sense. It provides empirical evidence to demonstrate the feasibility of linking algebra structure sense to technological tools and promoting it amongst diverse learners. Didactic approaches include the use of adaptive digital environments, gamification, diagnostic and monitoring tools, as well as exercises and algebraic sequences of varied complexity.

Advocating for a focus on both intuitive and formal knowledge, this volume will be of interest to students, scholars, and researchers with an interest in educational research, as well as mathematics education and numeracy.

1Algebra Structure SenseConceptual Approaches and Elements for Its Development

Teresa Rojano

DOI: 10.4324/9781003197867-1

In order to advance the conceptualization of algebra structure sense and to conceive didactic approaches that promote its development, we posit that there is a need to deepen the knowledge of the internal structure and order of algebraic objects, as well as the elements that come into play in the processes related to awareness of such structure.

Introduction

Research carried out in the 1960s by Vadim Krutetskii reports several cases of gifted children, one of which is of Sonya L., who was observed between the ages of 8 and 10 and demonstrated (among other skills) a great ability to understand and use the generality of a simple rule. In one of the experiments, Sonya was presented with the rule $5a+5b=5(a+b)$ and was then asked to express in a simpler form the expression ${4m}^2\left(2p-q\right)-2m\left(q-2p\right)-2m\left(q-2p\right).$ Her immediate verbal and written response was: ‘In two of the parentheses we only need to change the signs’ and she wrote $2m(2p-q)(2m+2)$ (Krutetskii, 1976).

Keeping a respectful distance from the analysis that Krutetskii may have made of the above episode from a psychological perspective and in the context of his research, which was aimed at isolating and characterizing a set of mathematical abilities, it can be said that Sonya's immediate response speaks of the presence of a structure sense with respect to algebraic expressions. A concrete expression of the definition of algebra structure sense (ASS) formulated by Hoch (2007) is perfectly captured in the above episode with Sonya L. According to this author, subjects show that they have ASS if they are able to: (a) recognize a structure in its simplest form; (b) treat a compound term as a single entity and, through appropriate substitution, recognize a familiar structure within a more complex form; (c) choose appropriate transformations to make effective use of the structure. In Sonya's performance, phases (a)–(c) are observed in a single step; by the way, Krutetskii calls this ability abbreviation of thought.

The use of the operational versions of ‘structure sense’ as above, which focus on subject actions, also necessarily involves elements rooted in the form and essence of the object (of knowledge), in the case of symbolic algebra the objects are symbols, expressions, and equalities between algebraic expressions. In the latter respect, some authors refer to intrinsic properties of the object, such as grammar (Esty, 1992), the superficial, and deep structures of expressions (Kirshner, 1989) or to the hierarchy, form, and internal order of expressions (Sfard & Linchevski, 1994). While other authors include elements that belong to the subject's relationship with the algebraic object, and such is the case of Kirshner and Awtry (2004) who, in order to give plausible explanations for certain frequent mistakes that students make in algebra, resort to the attribute of the visual salience of an algebraic expression, an attribute that depends both on the properties of the expression and on ‘whomever reads it’. All of the above shows a set of perspectives, from which one can delve into the notion of ASS, understood broadly as the ability to be aware of the structural properties of an algebraic object. In this chapter, aspects of this notion that have been studied from some of these perspectives are addressed, and elements for an ASS didactics are identified.

The Objects of Symbolic Algebra

Symbolic algebra is a system of signs whose syntactic rules are conventional; they are arbitrary and not natural; however, the system and its rules, as we know them today, are the result of a long and intricate history. With the caveat that in the next chapter L. Puig takes on a more extensive discussion of this topic, in order to provide an idea of the history behind the language of algebra, it is worthwhile to reference the three stages of the evolution of algebraic symbolism stated by G.H.F. Nesselmann in his 1842 work (Cajori, 1928–1929): the rhetorical algebra stage, where the solution to a problem is written as a prose argument; the stage of syncopated algebra, where stenographic abbreviations for quantities, relations, and operations that occur with greater recurrence are used; and the symbolic algebra stage, where solutions to problems are expressed in a synthetic language, composed of symbols that have no apparent relationship with what they represent (Eves, 1983, p. 126). The abacus texts, which include Fibonacci's Liber Abbacci (Sigler (tr.), 2002), are an example of rhetorical algebra. Whereas the Diofanthine algebra, contained in the book La Arithmetica, (Heath 1910), and Jordanus de Nemore's De Numeris Datis (Hughes, 1981) are two variants of the syncopated stage. Finally, according to various mathematics historians, the appearance of The Analytical Art by François Viéte in the 16th century (Witmer (tr.), 1983) marks the birth of symbolic algebra. The following examples from each of these stages show how the relationship between representations and what they represent changes from one stage to another, becoming less explicit as these representations evolve into more symbolic, more abstract forms. However, even in the Vietic syntax, in which the expressions are formed under Zetetic rules¹, the geometric lineage of the expressions can be appreciated.

Example 1: A problem statement and part of its solution expressed in rhetoric algebra (Problem 1 from the abacus book Trattato di Fioretti by M.A. Mazzinghi (1350, Italy) (G. Arrighi, ed., 1967). Pisa, Italy: Domus Galileana) and its translation into current algebraic symbolism (Rojano, 1985, unpublished PhD dissertation, Cinvestav – IPN, Mexico).

Original version in old Italian

Problem 1

Fa’ di 19, 3 parti nella proportionalità chontinua che, multiplichato la prima chontro alla altre 2 e lla sechonda parte multiplichato all'alltre 2 e lla terza parte multiplichante all'altre 2, e quelle 3 somme agunte insieme faccino 228. Adimandasi qualj sono le dette parti.

Translation into modern symbolism

Find three numbers, x, y, z such that:

$$x+y+z=19$$

and that

$$x\text{/}y=y\text{/}z$$

$$x\left(y+z\right)+y\left(x+z\right)+z\left(x+y\right)=228$$

Part of the solution in original old Italian version

Conciosiachosachè Maestro Antonio sottile scriva e’ chasi e’ qualj non ànno asolutione alle quantità numeralj, nientedimeno io mi sforçerò di porre chasi e’ qualj aranno asolutione a quantità che facilmente si potranno provare, cioè che con facilità acciò che lo ‘nteletto possa essere chapacie di tutto.

Ora al nostro chaso, qui è da sapere che se si multiplicha 19 nel doppio de la seconda parte sarà 228, overo si se multiplicha lo droppio del 19, cioè 38, nella seconda parte farà 228. E che questo sia vero il voglio chiarire.

Noi abbiamo proposto che si divida 19 in 3 parti nella proportionalità chontinua che, multiplichata la prima per l'altre 2 e la seconda per l'altre 2 e l'altra, cioè la terça, per l'altre 2 et agunte le dette multiplichatione, insieme faccino 228.

Part of the solution translated into modern symbolism

Solution:

You have to know that

$$\left(19\right)\left(2y\right)=228$$

Which is equal to

$$2\left(19\right)y=228$$

$$38y=228$$

This must be explained.

Example 2: A problem statement and part of its solution expressed in syncopated algebra (Translation of Proposition 2 from Book One of De Numeris Datis by Jordanus de Nemore. Acritical edition. B. Hughes (ed./tr.) (1981). University of California Press).

Proposition 1.2

If a given number is separated into as many parts as desired, whose successive differences are known, then each of the parts can be found. Given is the number a which is divided into w, x, y, and z the least of the parts.

Since the successive differences of all of these are given, each difference can be expressed in terms of the difference of each number with z. Therefore, let $f$ be the difference of w and z, and the sum of g and h be the sum of the differences of x and z with y and z. Now because z makes each of those equal to each of these, it is obvious that thrice z with the sum of $f$, g, and h equals those three. Therefore, four times z with the sum of f, g, and h equals a.

Example 3: An algebraic expression written with the Zetetic's rules (The Analytic Art by François Viéte (1540–1603). Witmer, T.R. (tr.) (1983). Kent, Ohio: The Kent State University Press). Below, the same expression is written with modern algebraic notation.

$$\frac{X^3-3b}{{cY}^2}$$

The aforementioned stages did not follow a chronological order, and, in fact, it is thought that rhetorical algebra was predominant for a long time in the work of Western European algebraists (Eves, 1983, p. 126).

However, circling back to the topic of this chapter, the interest in making a brief reference to the evolution of algebraic language is in pointing out that, despite the fact that the symbols and expressions of this language have their origins in contexts that are loaded with meaning, such as geometric and those coming from word problems, algebraic symbolism, as we know it today, is far from evoking referents to those contexts among current users. In fact, the power of this language is understood to reside in its generality, in the possibility of using it to model situations and problems from a variety of contexts, and that through transformations of the expressions or ‘model’ equations problems can be solved and knowledge of the modeled situations can be deepened. This possibility is precisely due to the fact that the symbols and expressions formed with them are not, from the outset, tied to particular or concrete referents. It is an autonomous, self-contained language, in which the utterance of the theorems and their proofs (Klein, 1968) and the statement of the problems and their solutions (Rojano, 1996) can be entirely expressed.

Along with the recognition of the unequivocal power of the abstract nature and generality of algebraic symbolism, one must acknowledge that teaching and learning said symbolism, its syntax, and its use ...