Algebraic Notation
Algebraic notation is a system for representing mathematical expressions and equations using symbols and letters. It is commonly used in algebra to express relationships and operations in a concise and standardized way. In algebraic notation, variables are often used to represent unknown quantities, and operations are denoted by specific symbols such as +, -, *, and /.
7 Key excerpts on "Algebraic Notation"
eBook - ePubMathematical Reasoning
Patterns, Problems, Conjectures, and Proofs
- Raymond Nickerson(Author)
- 2011(Publication Date)
- Psychology Press(Publisher)
...It is precisely this requirement to write and hence to think in terms of symbols which makes mathematics a difficult subject in the classroom today unless attention is paid to this particular intellectual demand. (p. 224) The invention of symbolic algebra represented a very significant step forward, not only for mathematics, but for the history of thought. An algebraic equation provides a means of packing a large amount of information into a few symbols. Jourdain (1913/1956) puts it this way: “By means of algebraic formulae, rules for the reconstruction of great numbers—sometimes an infinity—of facts of nature may be expressed very concisely or even embodied in a single expression. The essence of the formula is that it is an expression of a constant rule among variable quantities” (p. 38). The wide adoption of a standard notational scheme greatly facilitated communication among mathematicians and the building of any given mathematician on the work of others. A standard symbology was also a practical necessity for the printing of mathematical works, and the emergence of print technology was an impetus to the development of one. Algebraic Notation not only makes possible great economies of expression, but it also facilitates computation. Indeed, one may see the history of improvements in algebraic symbolism as a shifting of an ever-greater portion of the burden of computation and inference from the person to the symbolism. The symbolism encoded much of what its developers had learned about mathematical inference and preserved that knowledge so that it would not have to be rediscovered anew each time it was needed. Inferences that would be very difficult to make without the use of this, or some comparable, notation may become, with its use, matters of straightforward mechanical symbol manipulation. In many cases the need to make inferences was replaced with the ability to apply an algorithmic procedure...
eBook - ePubPlanting the Seeds of Algebra, PreK–2
Explorations for the Early Grades
- Monica M. Neagoy(Author)
- 2012(Publication Date)
- Corwin(Publisher)
...11 Final Thoughts Algebraic thinking is a major area of school mathematics that is crucial for students to learn but challenging for teachers to teach. National Council of Teachers of Mathematics (2000) T his volume, along with the next one titled, Planting the Seeds of Algebra, Grades 3-5: Explorations for the Intermediate Grades, empowers elementary school teachers to meet the goal articulated by the National Council of Teachers of Mathematics (NCTM) to “help students build a solid foundation of understanding and experience as a preparation for more sophisticated work in algebra in the middle grades and high school” (NCTM, 2000). “Algebraic” Topics By working through Explorations I through IV, students and teachers will engage with foundational concepts of algebra: Numbers and operations Notation Equality Unknowns and variables Patterns and functions The particularity of the Explorations is the algebraic lens through which these topics are framed: Properties of numbers (for example, being odd or even) are examined with an eye on structure, and properties of operations (for example, being commutative or not) are examined with an eye on relationship. In both cases, properties are identified, discussed, represented, and generalized. Algebraic Notation (for example, ABBC to describe a pattern, y = x + 4 to denote a function, or x + y = y + x to express a property of addition) is an efficient, shorthand representation used to express a mathematical idea. Only after students first interact with multiple, grade-level appropriate representations—kinesthetic, concrete, verbal, pictorial, numeric, and graphical—can they make sense, when the time comes, of the most abstract of all representations: symbolic representation. Students and teachers know that the equals sign signifies equality, but in the minds of many children it is solely associated with the action of carrying out one or more computations to find a numerical answer (as in 3 + 4 = ?)...
eBook - ePubStatistics: An Introduction: Teach Yourself
The Easy Way to Learn Stats
- Alan Graham(Author)
- 2017(Publication Date)
- Teach Yourself(Publisher)
...2 Some basic maths In this chapter you will learn: • the basics of algebra, including the meaning of symbols and some useful definitions • how to draw and interpret graphs • some important statistical notation, including the symbols used for ‘sum of’ and ‘mean’. If you flick through the rest of this book, you will find that complex mathematical formulas and notations have been kept to a minimum. However, it is hard to avoid such things entirely in a book on statistics, and if your maths is a bit rusty you should find it worthwhile spending some time working through this chapter. The topics covered are algebra, simple graphs and statistical notation and they offer a helpful mathematical foundation, particularly for Chapters 5, 11 and 12. Algebra Algebraic symbols and notation crop up quite a lot in statistics. For example, statistical formulas will usually be expressed in symbols. Also, in Chapter 11 when you come to find the ‘best fit’ line through a set of points on a graph, it is usual to express this line algebraically, as an equation. Clearly it would be neither possible nor sensible to attempt to cover all the main principles and conventions of algebra in this short section. Instead, your attention is drawn to two important aspects of algebra which are sometimes overlooked in textbooks. The first of these is very basic – why do we use letters at all in algebra and what do they represent? The second issue concerns algebraic logic – the agreed conventions about the order in which we should perform the operations of +, −, × and ÷. WHY LETTERS? Can you remember how to convert temperatures from degrees Celsius (formerly known as centigrade) to degrees Fahrenheit? If you were able to remember, the chances are that your explanation would go something like this: Suppose the original temperature was, say, 20°C...
eBook - ePub- James J. Kaput, David W. Carraher, Maria L. Blanton, James J. Kaput, David W. Carraher, Maria L. Blanton(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
...Both representations, the one in algebraic script as well as the one in natural language, are symbolic. Both express generalizations. Without a doubt, algebraic script has some distinct virtues over spoken language—it is succinct and well suited to further analysis and derivations. However, the expression in natural language has its own merits: It conveys information about the thermal context that is missing in the algebraic expression. In fact, natural language typically serves as an important starting point for children to learn algebra because it allows them to begin to make sense of and describe algebraic concepts using a known language. The early algebra researcher is also likely to pay special attention to two other systems of symbolic representation: tabular representations and graphs. To be fair, all four of these symbolic systems (natural language, algebraic script, tabular representations, and graphs) are recognized as legitimate among research mathematicians as well as mathematics educators. Nonetheless, there are sometimes striking differences in the relative weight given to the systems. Early algebra researchers sometimes introduce algebraic script only after students have become well versed in using the other representational systems. Whereas some experts may see this as ill advised or even wrong, the underlying goal of early algebra is for children to learn to see and express generality in mathematics. Initially, the language of generalization is often broadly defined and evolves toward more specialized forms such as algebraic script and Cartesian graphs. Is Algebra the same as Generalization? If So, Isn’t It Everywhere? People have sometimes criticized inclusive views of algebraic reasoning on the grounds that it becomes difficult to distinguish thinking algebraically from thinking mathematically or (just plain) thinking...
eBook - ePubBringing Out the Algebraic Character of Arithmetic
From Children's Ideas To Classroom Practice
- Analucia D. Schliemann, David W. Carraher, Barbara M. Brizuela(Authors)
- 2006(Publication Date)
- Routledge(Publisher)
...Accordingly, we view the introduction of algebraic activities in elementary school as a move from thinking about relations among particular numbers and measures toward thinking about relations among sets of numbers and measures, from computing numerical answers to describing relations among variables. Children need to be aware that, as Schoenfeld and Arcavi (1988) emphasized, “a variable varies” (p. 421). This requires providing a series of problems to students, so that they can begin to note and articulate the general patterns they see among variables. Tables play a crucial role in this process because they allow one to systematically register diverse outcomes (one per row) and look for patterns in the results. Algebraic Notation, even at the early grades, is also fundamental as a tool to represent multiple possible values and to understanding relationships between two sets of variables. We hope that the set of interview and classroom data discussed in the following chapters will support our claim that algebra can become part of the elementary mathematics curriculum and that the many difficulties students have with algebra are exacerbated by the restrictive approach to arithmetic presently practiced in most schools. Furthermore, we believe that mathematical understanding is an individual construction that is transformed and expanded through social interaction, experience in meaningful contexts, and access to cultural systems and cultural tools. When psychologists evaluate the “development” of children who have already entered school, they are not dealing directly with cognitive universals. In attempting to fully understand the development of mathematical reasoning, we need analyses of how children learn as they (a) participate in cultural practices, (b) interact with teachers and peers in the classroom, (c) become familiar with mathematical symbols and tools, and (d) deal with mathematics across a variety of situations...
eBook - ePub- Wayne A. Wickelgren(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
...10 Topics in Mathematical Representation As stated in Chapter 2, problems contain information concerning givens, actions, and goals. The first and most basic step in problem solving is to represent this information in either symbolic or diagrammatic form. Symbolic form refers to the expression of information in words, letters, numbers, mathematical symbols, symbolic logic notation, and so on. Diagrammatic form refers to the expression of information by a collection of points, lines, angles, figures, directed lines (vectors), matrices, plots of functions, graphs, and the like. Often the same information should be represented using a variety of symbolic or diagrammatic notations. In fact, diagrammatic representation is generally labeled; for example, points, lines, and cells in a matrix have symbols attached to them in the diagram. Of course, problems are stated originally in some form, often relying heavily upon verbal language. The first step in solving such a problem is to translate from the representation given explicitly or implicitly in the original statement of the problem to a more adequate representation. This chapter is concerned with selected topics in the mathematical or precise representation of information in problems. Although precise representation of the information in a problem is the first step to take in trying to solve a problem, I deferred discussing this important topic to this late chapter of the book for two reasons. First, although some general statements can be made about the representation of information in a large variety of problems, most of the principles of representation are specific to particular problem areas. Effective representation for problems from some area of mathematics, science, or engineering depends upon knowing centuries of conceptual development in the relevant areas of mathematics, science, and engineering...
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...C HAPTER 7 Algebra Some key concepts An algebraic symbol, such as x, can be used in different ways. It can represent a specific, unknown value, or it can represent a variable which can be given a range of alternative values. The equals sign means ‘the total value of everything on one side of the ‘=’ is equal to the total value of everything on the other side’. It does not mean ‘Work this out and find the answer’! An equation must be kept balanced. Whatever we do on one side, we must do the same thing on the other. An algebraic expression represents something – for example, the area of a shape or the number of counters in the n th member of a sequence. a) Using Symbols Algebra is full of symbols. The quintessential algebraic symbol for most people – adults as well as children – is x. x crops up all over the place, with different meanings and different values in different situations. This can be very confusing. x can represent one or more specific values in an equation. These values are (at least to begin with) unknown, but it may be possible to work out what they are – so in 4 + x = 10, for example, x is 6, but in x 2 = 9 it is 3 or - 3. x has different values in different equations, but only one value, or a particular set of values, in any one equation. But x can also represent the variable in a function. You can choose different input values for x, and these will produce different outputs. So in the function y = x + 3, for example, y is 4 when x is 1, but y is 96 when x is 93. These two uses of an algebraic symbol such as x, in an equation where it has a specific, unknown value, and in a function where it serves as a variable that can take different values, need to be understood. So for early work in algebra the first thing we need is a symbol that indicates clearly, in itself, the range of meanings and values that x can have. Some textbooks use a box for the unknown when equations are introduced, with 4 + = 10, for example, or – 7 = 2...
