Chapter 5b Summary

1
Chapter 5b Summary

• The Nature of Contemporary Algebra

2
What is Algebra?

• Generalized arithmetic
• Has algorithmic nature
• Use in abstract systems (fields groups)
• Most associate at the very basic level with
  variables expressed by symbols

3
Four fundamental components

• Understanding patterns, relations, and functions
• Representing and analyzing mathematical
  situations and structures using algebraic symbols
• Using mathematical models to represent and
  understand quantitative relationships
• Analyzing change in various contexts

4

• Algebra has its historical roots in the study of
  general methods for solving equations. The
  Algebra Standard emphasizes relationships among
  quantities, including functions, ways of
  representing mathematics relationships, and
  analysis of change
• (Principles and Standards for School Mathematics,
  p. 37)

5
Change in School Algebra

• Mostly during last two decades
• Prepare students in elementary school for the
  concept of function
• Through systematic experiences with patterns
• Use of technology
• Move among numeric, symbolic, tabular and graphic
  representation of data

6
Historic Development of Algebra

• Began as a set of tools for solving practical
  problems
• Verbal recipes for the Babylonians
• Geometric representations for the Greeks
• Letters and symbols for the people of the
  Renaissance
• Algebra is from the Latin translation of the
  Arabic title of al-Khwarizmis book on algebra
• A.D. 825

7
Historic Development of Algebra

• Gerolamo Cardano (1501-1576)
• Organized negative numbers (thought of as debts)
  use and rules
• René Descartes (1596-1650)
• First used imaginary and real as they are used
  today
• Francois Viete (1540-1603)
• Introduced the systemic use of letters for
  constants
• Carried further by Descartes in algebraic
  representation of geometrical quantities
• Made algebra into an organized body of knowledge

8
Historic Development of Algebra

• Now a tool for representing numbers and numerical
  relationships by symbols, letters, and coordinate
  graphs
• The development of the real-number system, its
  subsets (natural numbers, whole numbers, etc.),
  and its properties provides the basis for the
  structure of algebra

9
Historic Development of Algebra

• The algebra taught today has the same content as
  that subject prior to 1824
• 1824 marked a radical change algebra that cannot
  be reduced to geometry
• For example, matrix theory is a part of modern
  algebra beyond the ability of the Euclidean tools
• Neil Henrik Abel (1802-1829)
• In 1824 proved that it is impossible to solve the
  general equation above the fourth degree
• ax5 bx4 cx3 dx2 ex f 0
• Cannot be solved by a finite number or arithmetic
  steps

10
Todays School Algebra

• Considered one of three types
• Generative activities
• Such as solving equations with an unknown and
  finding patterns and numerical relationships
• Transformational
• Such as finding equivalent equations
• Global
• Problem solving, modeling, proving theorems,
  predicting new relationships

11
Todays School Algebra

• Researchers are working with classroom teachers
  to study mathematical activities that require
  student to generalize quantitative relationships
  other than with symbols
• Algebraic thinking is the processes involved in
  this expanded notion of algebra
• Should be considered as another way of looking at
  how students do and understand mathematics

12
Algebraic Thinking Example

• At the middle school level, the main algebra
  emphasis is linear relationships
• When working with the tables students notice that
  the y column has an increase of three
• They then count up by 3s until the 10th term is
  reached
• Students need to show how the recursive pattern
  can help them write an equation

13
What is Algebra?

• Depending on the viewpoint, this cannot be
  answered simply
• However there are two commonalities
• Algebra is not a discrete subject
• Technology can greatly assist students in
  accessing an updated view of algebra

14
Technology for Problem Solving

• Technology includes computers, probeware,
  graphing calculators, scientific calculators,
  four-function calculators, and even manipulatives
• The use of technology allows student new methods
  of representing relationships that may not
  require letter or symbolic form

15
Example of Multiple ways to solve a problem

• You have finally been offered a paying job as a
  rock band drummer. Two different groups want you
  to play with them, but their contract offers are
  drastically different. Group 1 will pay a flat
  fee of 1000 per concert. Group 2 will pay only
  100 per concert but will double this amount for
  each additional concert day played. Which is the
  better deal?

16
Example of Multiple ways to solve a problem

• Method 1 Table
• Method 2 Spreadsheet

17
What is Algebra for all?

• 1988 NCTM Yearbook The Ideas of Algebra K-12
• Peggy House identifies two influencing forces on
  teaching algebra
• Growth and availability of computer technology
  for the secondary school classroom
• Social demands on school to be accountable to
  provide a better educated workforce

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What is algebra for all?

• Students entering technology-intensive careers
  (such as engineering) are not the only careers
  needing technology
• Also students entering vocational careers, etc.
• The owner of a technologically advanced body shop
  stated,
• Were finding that we cant accept the people
  who were forced to go to the automotive side of
  high school because they couldnt conform to the
  scholastic side. They need the algebra and the
  math to run the mechanics

19
What is algebra for all?

• NCTM (1994) declares,
• First year algebra in its present form is not
  the algebra for everyone. In fact, it is not the
  algebra for most high-school graduates today

20
What is algebra for all?

• New curricula are designed to present algebra
  topics in ways that give more students access to
  their understanding and use
• Includes problems that can be approached in a
  variety of ways from a variety of levels
• Teachers find their own understanding of
  mathematics to be enhanced and challenged when
  teaching these new materials to diverse student
  populations

21
In Conclusion

• Since the formation of algebra in the 9th
  century, equation solving has played a major
  role. However, in todays classrooms technology
  allows students deeper insight about the concept
  of functions as they can move easily from tables,
  to graphs, to equations and see how changing a
  feature of one of these representations affects
  the others. Imagine what the curriculum will
  look like 30 years from now.