If Denham could have seen how visibly books of

algebraic symbols, pages all speckled with dots and dashes and twisted bars, came before her eyes as they trod the Embankment, his secret joy in her attention might have been dispersed.

However, an

algebraic symbol (letter) that stands for an unknown in a linear relationship was yet an external representation that had not been internalized and integrated into the students' prior mental structure, and thus it was rejected in favor of the lower level of representation (number symbol) which had been a tool of their mathematical communication mastered to date (Javier, 1987; Lesh, et.

For these students an

algebraic symbol (letter) that stands for an unknown in a linear relationship was an external representation that had not been internalized and integrated into the students' prior mental structures, and thus it was rejected in favor of the lower level of representation, number symbols, which had already been a tool of their mathematical communication (Javier, 1987; Lesh, Post, & Baker, 1987).

The second part looks at the history of

algebraic symbols, their origins in Ancient Greece and their evolution and proliferation through the centuries, intertwining the threads of many cultures, and culminating in the work of "The Last of the Magicians"--Isaac Newton.

Thus, the reader is gently encouraged to make the effort to comprehend

algebraic symbols and their rules of operation.

Hopkins argues that Husserl provides only incomplete analyses of three major issues essential for understanding the nature of modern symbolic mathematics: 1) the source and possibility of formalization itself, 2) the unity of a number, and 3) the basis upon which

algebraic symbols function as they do.

Single, double, triple and quadruple elements are referred to, together with various

algebraic symbols and cross-headings such as Type x + (fx).

Kuchemann (1981) described the students' views of

algebraic symbols in terms of hierarchical levels, namely, letter ignored, letter is evaluated, letter as object, letter as specific unknown, letter as generalized number and letter as variable.

In the process, Wolfram came up with new ways of manipulating the

algebraic symbols that those diagrams represent.

The great mathematical power of a CAS calculator or software program, especially its ability to work with

algebraic symbols, means that examination processes must change if CAS is permitted.

But, they also studied how their own students connected number sense to a variety of algebra-based representations, such as blocks, pictures of blocks, number patterns, coordinate graphs, tables,

algebraic symbols, and real-world situations.

Tricks and extension activities vary in difficulty to challenge all students as they learn opposite functions,

algebraic symbols, multiplication tables and more.