QUANTITY

1
QUANTITY

• By
• Matus Muzheve

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Quantity in School Mathematics

• Number concepts and skills form the core of
  school mathematics.
• Many ways to describe quantitative data and
  relationships are taught.
• Operations in number systems are generalized to
  elementary algebra.

3
Influence of Technology

• Technology
• Calculators
• Computer Algebra Systems (CAS)
• Good or bad?

4
Influence of Applications

• Quantitative methods
• Skills needed
• Numbers.
• Calculators.

5
Influence of Psychological Research

• Change in conditions for teaching about quantity
  in school mathematics.

6
Fundamental Concepts

• What quantitative abilities will be fundamental
  in the future of mathematics?
• Build a curriculum that
  
• retracts meandering historical path by which
  numerical techniques have developed.
• Capitalize on structural insights that have
  emerged.

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Numbers and Operations

• Number use involves
• Measuring
• Ordering
• Coding
• School mathematics should emphasize
• ways that different types of number systems serve
  as models of measuring, ordering, and coding
• Ways that standard operations model fundamental
  actions in quantitative situations.

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Variables and Relations

• Variables are not usually significant by
  themselves, but only in a relation to other
  variables.
• Recognition of structural similarities in
  apparently different situations allows
  application of successful reasoning methods to
  new problems.

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Procedures

• Problem solving
• analyze problem to identify number concepts that
  match the problem (conceptual knowledge).
• Infer - get new information that gives new
  insight (procedural knowledge).

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Numerical Representation

• System of symbols that convey mathematical
  information in unambiguous and compact form.
• Representation of ideas aids memory and is a
  medium for communication.
• Representations
• place value system
• variables
• functions and relations

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Graphical Representation

• Identify numbers with points in a geometric line
  or pairs of numbers with points in the plane.
• Be able to interpreting graphic representations
  and understand connections among symbolic,
  graphic, and numerical forms of the same
  information.

12
Computer Representation

• Computer-generated numerical representations of
  algebraic expressions are proving to be a very
  useful tool in practical problem solving.
• Computer curve-fitting tools can be used to find
  symbolic rules that fit patterns in collections
  of numerical data.

13
Algorithms

• Context-independent.
• Current technology undermines any argument that
  students must develop proficiency in executing
  any particular algorithm because they will need
  that skill later in life.

14
Conceptual and Procedural Knowledge

• Routine aspects of both representation and
  manipulation of quantitative information are the
  two key components of procedural knowledge.
• Wise use of calculators can enhance student
  conceptual understanding, problem solving, and
  attitudes toward mathematics without harm to
  acquisition of traditional skills.

15
Number Sense

• It is important to develop student achievement in
  a variety of information aspects of quantitative
  reasoning (number sense)
• Fundamental skills required to test numerical
  results for reasonableness
• Broad knowledge of quantities in the real world.
• Ability to make quick order-of-magnitude
  approximations.

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Symbol Sense

• Skill required to deal with symbolic and
  algebraic operations.
• GOALS for teaching symbol sense
• Ability to scan an algebraic expression to make
  rough estimates of the patterns that would emerge
  in numeric or graphical representation.
• Ability to determine which of several equivalent
  forms might be most appropriate for answering
  particular questions.

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Number Systems
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Number Systems cont.

• Start off with axioms of N and create the number
  systems.
• Natural numbers N
• Integer numbers Z
• Rational numbers Q
• Real numbers R
• Complex numbers C

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Natural Numbers and Integers

• N is the smallest subset of R that includes the
  multiplicative identity and is closed under
  addition
• Z is the smallest ring in R that includes the
  multiplicative identity
• Division Algorithm
• Mathematical induction
• The Fundamental Theorem of Arithmetic

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Rational Numbers

• Q is the smallest subfield of R.
• Q is a dense set
• Q is a countably infinite set

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Real Numbers

• R is the only complete ordered field.
• R is the smallest closed set that contains Q.
• Continuum hypothesis?

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Complex Numbers

• Give solutions to polynomial equations
• A field extension of R.
• Absolute value is euclidean length

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New Number Systems

• Matrices
• Quaternions -- H
• Octonions O

24
Applications

• Enable students to solve problems (quantitative).
• Students must develop the ability to think for
  themselves.

25
Modeling

• Construct mathematical models in order to problem
  solve.
• Real world problem generally are not given on a
  platter.

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Measurement

• A measurement consists of a unit times a number.
• Dimensional analysis.

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Goals

• Develop students ability to reason
• Arithmetic or algebraic algorithms?
• Quantitative reasoning in the modern world.