Summary

Algebra Transforming by David Pimm

This article examines the effective view of algebra, arguing that algebra is all about transformation. It is highlighted in this chapter that algebraic operations take place between successive written statements, representing not only the results but also the process of change. In algebra, there are counterpart objects-symbols that can be manipulated well as descriptive elements that relate to real-world scenarios. For beginners, however, unifying the two perceptions may prove challenging. In the article, it is argued that as algebraic fluency increases, meaning matters less, and one is more efficient when manipulating symbols as objects instead of always connecting them to real meanings.

To reveal how algebra develops from arithmetic, the chapter gives two lessons from Dave Hewitt. Through these lessons, attention is redirected from determining the value of variables to manipulating equations. The "Think-of-a-Number" lesson introduces algebraic reasoning via verbal and written equation manipulation and complementary inverse operations. The "Rulers" lesson builds algebraic expressions through number line movement. In both lessons, structural knowledge takes over memorization, and students are encouraged to experiment with algebra in action without restricting themselves to symbols.

The chapter continues with a discussion on the role of technology in the teaching of algebra. Computer developments, specifically symbolic manipulation software, may reduce the need for conventional algebra instruction, according to the article. The Logo computer language is cited as a means of introducing students to algebraic notation, enabling them to interpret symbols. Symbolic descriptor software, including computer algebra systems (CAS), is also referred to in the context of its capacity to carry out intricate algebraic manipulations, including graph plotting, equation solving, and differentiation.

Moreover, the article focuses on symbols, going back to concepts of incompleteness or complexity. Symbols are representatives of mathematical objects, to be operated on, but not the objects themselves. This is taken beyond algebra, noting that symbols are important in all branches of mathematics. The chapter ends with a consideration of some of the most significant cultural symbols, including The Vietnam Veterans Memorial and the Holocaust Memorial, to draw an analogy between symbols in society and mathematics as representations of deeper, usually intangible meanings.

Stop 1

What is written in an algebraic demonstration are not the actions but the results of actions: sequences of equations show the results of transformations, not the transformations themselves.(p.g.89)

I paused on this quote because it points our attention to something significant but generally neglected in algebra—the distinction between what we lay out on paper and the process of thought that lies behind it. We document steps and conversions when we're solving equations, but what we document is a representation of what occurs, not the reasoning and choices that led us there. This is connected to the idea in the paper that algebra is a dynamic process and that most of the real work is done in between the steps. It also made me think about how students fixate on algebra sometimes because they're only interested in the printed symbols and not the transformations that bring them there. Like in the case of a science experiment when the results are put into writing but the process is more complex, algebraic proofs write down the results, but the mathematical mind working beneath is invisible.

Stop2

When we attempt to realize a piece of music composed by another person, we do so by illustrating, to ourselves, with a musical instrument of some kind, the composer’s commands. Similarly, if we are to realize a piece of mathematics, we must find a way of illustrating, to ourselves, the commands of the mathematician.(p.g.100)

I paused reading at this quote because it expresses the nature of algebra as demonstrated throughout the paper—algebra is not merely operating with symbols but is about understanding the transformation between steps. Similar to music, algebra is more than what it appears on paper. In the same way that the musician has to decipher and play the piece, the mathematician or student has to work with algebra beyond rule-based behaviour. This quotation also relates to the notion in the paper that algebraic expressions are not inert symbols but are engaged in an active interplay of structure and meaning. It called to mind the manner in which, when teaching algebra, the students should not just be so concerned with solving for x but with being made aware of the change that is occurring, in the same way a musician is more interested in how music progresses rather than the individual notes.

How can we encourage students to view algebra as a dynamic, evolving process rather than a collection of static equations?