Algebra reasoning in elementary mathematics ö

Theory and Practice

Algebra Working Group Organizers:

Eugenio Filloy and Teresa Rojano

Department of Mathematics Education,
Cinvestav, MŽxico

The powerful resources nowadays available from hand held calculators and computers offer new ways for the teaching of mathematics. Numerical-based strategies and visual approaches provided by them currently challenge symbolic algebra as means of obtaining the competences desired. These numerical and visual resources allow us to design teaching activities so that students work out mathematical challenges without previously having any formal approach to the mathematics involved. Such an approach implies many theoretical and practical issues.

Recently, much emphasis has been set in the use of computer environments, but less light
(from a theoretical point of view) has been cast on how this may result in an innovative organization of the classroom. This is a beneficiary fact without doubt.Ê Thus it seams relevant to center our discussion in the innovative organization of the classroom environment.

A reflection on the relationship between theory and practice in the context of innovative approaches to the teaching and learning of algebra needs to take into account both the actual classroom practices and curriculum content and the research results from new proposals.

In the present working group sessions, we will focus our attention on the changes that have been produced in the classroom practices with the introduction of innovative techniques based or not in the use of information technology. We will predominately study the changes in the role of the teacher, the role of the students and the role of the media in which the educational act is produced.Ê In particular, we will discuss the following issues:

á New forms of generalization and formalization;

á The difficulties in the translation from natural language to the algebraic one.

á New approaches to analyze problem solving processes;

á Building-up algebraic syntax with graphic calculators and computers;

á Innovative organization of the classroom environment.

In what follows we briefly introduce three possible themes of discussion.

Algebra and Technology: New Semiotic Continuities and Referential Connectivity

ÊThis discussion will deliberately take a somewhat different orientation to algebra than is usually the case, investigating how technology affects basic semiotic assumptions and habits, with a special focus on the algebra of functions ö their definition, manipulation, and use as models.

Our historical applications of technology to help with both the learning and the doing of algebra have passed through several stages. The earliest involved facilitating manipulations of character strings, as was the case in the late 1960âs with MACSYMA being used for complicated symbol manipulations required in General Relativity. In the 1970âs the public increasingly used computer technology to plot coordinate graphs of, and generate numerical data from, algebraic functions of one or more variables. In the 1980âs these notations were increasingly linked to one another so that by the end of the decade one could make changes in one notation and these changes would be almost simultaneously reflected in any of the others.

Two features were common to all the development up to this point. One was the central role played by character-string notations in both the definition and manipulation of the functions-whether they were closed-form or recursively defined functions. The second was the traditional relationships between the algebraically defined mathematical objects as models and the phenomena or situations that they were used to model or represent. Both of these features reflect a deep, but largely tacit view of formalisms as separate and distinct from informal notations or utterances and from the phenomena that they are taken to represent. In particular, algebraic statements are part of the universe of mathematical notations, with separate rules of reference, with syntax distinct from natural languages, and abstract independence from the media in which they happen to be instantiated. This view in turn is intimately integrated with a Platonist and representationalist philosophical orientation that takes:

á Mathematical objects as pre-existing, to act as pre-given reference objects for mathematical notations;

á Language as an inert representational instrument which does not help create mathematics but only enables us (if we are sufficiently skilled in its use) to see and do mathematics; and

á Mathematics as separate from the material and social worlds we inhabit;

All three of these positions are eroding in the face of an increasing flow or technology-enabled semiotic systems that offer:

1. Increased semiotic continuity between mathematical notations and our extra-mathematical methods of manipulating objects in our world, and

2. Increased referential connectivity between inscriptions taken to refer as models to phenomena or situations.

After offering a characterization of the kinds of 21^st century mathematical activity that stimulate the need for new representational forms, the remainder of the discussion will be devoted to illustrating and explicating these two assertions, developed in three sections.

1. Parametrization of Mathematical Objects and Relations, and the Increased Steepness of the 21^st Century Learning Curve.

2. ÊSemiotic Continuity with Ordinary Physical Actions.

3. Referential Connectivity Between Inscriptions and the Phenomena or Situations They Are Taken to Model.

Verbal Arithmetic-Algebraic Problem Solving

We will address the subject of transference of the algebraic operativity, that has been recently learned, to some other contexts, as would be the case of arithmetic-algebraic verbal problem statements.Ê Among the transfer processes of a given algebra operativity to problem contexts, where it could be used for its solution, are those that can identify the procedures for the solution, in which actually such operativity could be applied. These processes of simple recognition are only part of the complex transfer process (which includes, among others, the analytical reading processes of the statement, the production of a strategy and a representation system, as well). When reasoning through a complex problem, it is more than enough to have some kind of distraction for a child to focus on a certain context in which the recognition of what has already been learned and mastered at an operational level could not be applied. The likelihood of experiencing these types of centering phenomena during the development of the procedure for solving the problem is not overlooked and if this is the case, all the procedure could be upset or still, the possibility of solving the problem could be hindered. The solution to these types of obstacles is a level of transfer of the operativity, in which the already, mastered syntax elements could be drawn from the semantics of the context from which the problem is addressed (or solved)

Progress Towards Semantics

In spite of the confidence that is reflected by some students in being able to solve the new equations operationally when these appear in other contexts; to be able to speak of a true transfer of that operational capacity to the solution of problems, still to be considered are the processes that lead to understanding the statement and writing an equation. Among these processes are those of representation of the elements of the problem and this presupposes reading and analyzing the statement that distinguishes between what is given and what must be found and that allows the relevant information to be recovered while leaving aside whatever is not essential. This might also precede (or sometimes follow) the representation, the production of a strategy to attack the problem. The consolidation of the first elements of algebraic syntax is based on their link with a non-algebraic semantics, in this case that of how problems are stated.

When the performance of the students has put into operation the new elements of syntax, there is evident progress in algebraic semantics (as far as its problem-solving use is concerned) that also implies progress in the use of syntax.Ê The opposite is also true: progress in syntax implies progress in the semantics of algebra; this last-mentioned appears to be a fairly generalized opinion, since, in effect a certain level of syntax is always considered to be a factor in helping to solve problems.

Translating from Natural Language to the Algebraic Mathematical Sign System And Viceversa

This section deals with the translation, in both directions, of natural language (NL) into the Mathematical Sign System (MSS₁) generated by previous learning during the arithmetical and pre-algebraic training of the pupils in primary school and the first grades of high school. This translation between NL and MSS₁ is one of the central features of classical teaching strategies for the solution of word problems by means of the Algebraic Mathematical Sign System (MSS₂).

Children at three different levels of performance (high, mŽdium, low) in mathematics were selected for interviews to work with a basic sequence of four blocks of items:

Block 1. The reading of equalities corresponding to geometric formulae, expressed in algebraic symbols, like , , etc.

Block 2. The reading of open algebraic expressions like .

Block 3. The reading of algebraic equivalencies (tautologies) like .

Block 4. The interpretation of sentences expressed in natural language and their translation to mathematical symbols. For example, ãthe double of a,ä ãa increased from two.ä ãa decreased from two.ä Only some children with high and medium performance worked with a fifth Block consisting of systems of simultaneous equations of the type

with a, b, c and d particular whole numbers.

Some Results

I.ÊÊÊÊÊ In Block 1, three levels of the interpretation of the formulae were observed:

A) Textual Reading in NL of the expression without reference to any context.

B) Reading as in A), accompanied also by a verbal reference of the elements of the expression to dimensions of a geometric figure, without specification of the latter by the subject.

C) Reading as in B), accompanied also by the association of a specific geometric figure (circle, square) and of the corresponding attribute (area, perimeter); this was not always done in a correct way. These three interpretative levels appeared both in a partial and in a total manner, depending on the level of pre-algebraic performance of the subject.

II. ÊÊ A) With respect to Block 2, the textual reading in NL of expressions like Êwas accompanied by i) a reference to the dimensions of ãidealä geometric figures (heights, bases); ii) the need to assign specific values to the letters in order to obtain a result and ãcloseä the expression thought up by the subjects themselves; iii) the elaboration of an equation or equality starting from the expression and the numerical substitution for some of the literals.

B) In some case in Block 2, in the numerical substitution, the election of the values by the subject appeared to be arbitrary; however, in expressions such as a-b, identical values for a and b are not immediately accepted, since the association of different values with different letters and viceversa is present. In children with a low pre-algebraic performance, a resistance to assigning a higher numerical value to b than to a was observed, given the imminence of a negative result.

C) Furthermore, within the same Block 2, a tendency to give meanings to the open expressions in the context of word problems was observed. This was found very clearly in the case mid-level case in the following way:

Open Expression ÊÊÊÊÊ ÊÊÊÊÊÊPosing a Problem ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ Formulating an ãEquationä
(the expression is closed) Obtaining a Result.

III.- The interpretation of ãcompositeä expressions like Êand of algebraic tautologies like the development of the squared binomial (Blocks 2 and 3) presented a high level of difficulty and the majority of the subjects did not get beyond the most primitive level of reading in NL. The reading in NL of Êgave the typical error of .