## On Algebra and Logic

Regarding the position of algebra in today’s math education, some educators state that the procedures of algebra run counter to those of natural thought.  This viewpoint can be understood if we take into account that natural thought tends to be forward looking, in contrast with the retrospective nature of formalization. Furthermore, educators holding this position believe that a mechanical approach to the solution of problems using algebraic procedures makes it unnecessary for the student to reach true understanding of the situations being involved. In fact, they argue that algebra is less concerned with how thought does proceed and more with how it should proceed to reach new mathematical findings. However, both algebraic and logic procedures are the result of the separation of form from content, the most important feature of mathematics and the driven force behind its amazing achievements. This becomes very apparent in an examination of the individual’s mental development and the development of algebra and logic in the history of mathematics. The evolution of mathematics shows that it moves backwards in a retrospective reflective abstraction, and also forward, generalizing the structures so constructed. We cannot deprive the students of the power of mathematics by refraining them from either a true understanding of the problems in hand, or the retrospective look of reflective abstraction contained in algebraic and logic procedures. Therefore, our work as mathematics teachers must be that of engaging the students in activities that keep them deeply focused on what they are doing, as well as in activities that make them think of the current and previous procedures in place to try to find general features in them.

In terms of the individual’s mental development, Jean Piaget (1896-1980), without a doubt one of the greatest researchers in this field, constructed a model which identifies four stages and the processes by which children progress through them. The four stages are: a) Sensorimotor stage (birth – 2 year-old)–The child, through physical interaction with his environment, builds a set of concepts about reality. Because he lacks the ability to hold an image in his mind, a child in this stage does not know that physical objects remain in existence even when out of sight. b) Preoperational stage (ages 2-7)–The child now develops mental representation and can use symbols. However, he is not yet able to conceptualize abstractly and needs concrete physical situations. c)  Concrete operations (ages 7-11)–As physical experience accumulates, the child starts to conceptualize, creating logical structures that explain his physical experiences. But, at this point, the child must still perform these operations within the context of concrete situations. d) Formal operations (beginning at ages 11-15)–By this point, the child’s cognitive structures are like those of an adult and include conceptual reasoning.

According to Piaget’s model of mental development, there are several levels of knowledge, the first of them being determined by just acting upon the objects. Following this model, we can see that the sequence of levels of knowledge has a hierarchical order, the highest of which is scientific knowledge. In the intermediate levels, the subject gradually forms what Piaget defines as “operation,” coordinating elements of previous levels.  Abstraction is the transition from one hierarchical level to another level, from the level of action to the level of operation, for example. In conformity with Piaget’s theory of knowledge, there are two possible ways we can get knowledge: a) Simple abstraction, which happens when we act upon an object and our knowledge is derived from the object itself. For example, this take place when we determine the weight of an object, as occurs in the case of experimental or empirical knowledge for the most part. b) Reflective abstraction, which takes place when the knowledge is derived from the actions carried out on (or with) the objects, not from the object that is acted upon. For example, when we count “five” objects, it does not matter what kind of objects we are counting—whether marbles, or pencils, or textbooks—the knowledge is drawn from our actions when counting. This is the basis of logical and mathematical abstraction. Furthermore, when passing from a less organized to a better organized level of knowledge given by the abstraction taking place, a mental reorganization also takes place. Piaget refers to this mental reorganization when using the word reflective in the expression reflective abstraction.

One important feature in Piaget’s model is the differentiation between form and content taking place in the different stages of his model. Thus, the sensori-motor structures toward the end of the first developmental stage are forms in relation to the previous simple movements they coordinate; however, these structures are content in relation to the interiorized actions of the following level, which become forms for these structures. Also, ‘concrete’ operations are forms in relation to those interiorized actions, but content with respect to the formal operations of eleven to fifteen years; and these again are only content in relation to the operations applying to them at later levels. This cognitive development model also shows that the separation of form from content happens naturally at every stage of the individuals’ mental growth.

The evolution of logic and algebra in the history of mathematics shows that it has been determined by the separation of form from content too, and in the creation of new forms by reflective abstraction starting from those of a lower level. As a matter of fact, in the early nineteenth century, algebra was regarded simply as symbolized arithmetic. This started with the attempt of George Peacock (1791-1858) to give arithmetic a logical structure (as it was given to geometry in Euclid’s Elements). That is, instead of working with specific numbers as is done in arithmetic, he used letters to represent these numbers. It was thus possible to apply the statements referred to these symbols to some set of elements other than the set of positive integers initially considered. For example, if S is the set of natural numbers to which the operations of addition and multiplication refer, we could change S to be the set of integers, or the set of all rational numbers, or the set of all real numbers, or something else (polynomials, functions, ordered pairs, etc), as long as the corresponding binary operations involved were clear. August De Morgan (1806-1871) went even further and left without meaning not only the letters that he used to represent numbers, but also the symbols of operation. Thus, he insisted that “with the only exception of the equal sign, no word or sign of arithmetic or algebra has one atom of meaning.” Later, George Boole (1815 – 1864) used letters x, y, z,… to represent set of objects (numbers, points, ideas, etc.) selected from a universal set of objects U, and defined formally the operations of union and intersection of sets. Boole’s ideas, a formal extension of De Morgan’s previous ideas, were published in 1847 with the title The Mathematical Analysis of Logic. This book, along with The Laws of Thought, also authored by Boole and published in 1854, were the culmination at that time of a process of separation of form from content that led to the discovery of the nature of pure mathematics.

These notes presented on Piaget’s mental development model, as well as those on the development of logic and algebra in the history of mathematics, reveal that the formalization contained in algebraic and logic procedures is the result of the natural tendency of human beings to look backwards in trying to learn from past experiences. Furthermore, the mental reorganization that takes place in doing so also provides elements that can be used for generalizations that show the power of mathematics. Since the separation of form from content that is at the core of this activity happens independently of the subject’s awareness of it, mathematics teachers should strive to make this separation a conscious one. However, even more important is the awareness of the need of the separation of form from content for the evolution of mathematics. The exclusive focus on either the current mathematical activity, without the retrospective look of the reflective abstraction, or the mechanical use of the results of the latter, deprives us of the power needed to take mathematics—and ourselves—to upper levels of development.

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