Several researchers (see Booth, 1988; Greeno, 1991; Kieran, 1992; Lins, 1990) suggest that many of the fundamental difficulties experienced by beginning algebra students are due to their failure to identify equivalent forms of an

algebraic expression.

We say that a connected linear

algebraic k-group G is k-split, or split over k, if its unipotent radical [R.

Additionally, when the functions are continuous, the values of x that satisfy f(x) = g(x) represent an

algebraic way of determining the solutions of the inequations f(x) < g(x) and f(x) > g(x).

The neutrosophic

algebraic structures are

algebraic structures based on sets of neutrosophic numbers of the form N = a + bI, where a, b are real (or complex) numbers, and a is called the determinate part on N and bI is called the indeterminate part of N, with ml + ni = (m + n)I, 0-I = 0, I [conjunction] n = I for integer n [greater than or equal to] 1, and I / I = undefined.

The

algebraic students are those who can; understand the functions, relations and patterns; present, analyse mathematical situations and structures; be able to construct and express mathematical model; and have the ability to analyze the variations in different contexts [Principles and Standards for Schools Mathematics, (2000)].

In Section 2, we give a few definitions, mostly illustrating the link between context-free grammars, solutions of positive

algebraic systems and N-

algebraic functions.

Also, in order to establish thoroughly the conceptual links needed for

algebraic (and, in fact, all mathematical) thinking, "multiple representations" of the ideas are needed (Warren, 2003; Sfard, 1991).

By way of example, a typical pedagogical strategy is to suggest various equivalent

algebraic expressions for the general term of a pictorial pattern.

AMS Special Session

Algebraic Methods in Statistics and Probability (2d: 2009: Urbana-Champaign, IL) Ed.

The American Mathematical Society published details on how the

Algebraic Eraser's key agreement protocol for public key cryptography is suitable for low resource devices, such as RFID tags, in their peer-reviewed book

Algebraic Methods in Cryptography.

Research on the development of

algebraic understanding by investigating the difficulties students encountered in solving

algebraic problems has been well documented (Vlassis, 2002; Warren, 2003).

Section I, "The Nature of

Algebraic Thinking," defines

algebraic thinking and provides an argument for its development in the classroom.