Transcript General Recommendations for Elementary

General Recommendations for
Elementary Education Teachers
(Math 14001, Summer I - 2008)
NCTM: National Council of Teachers of
Mathematics
MAA: Mathematical Association of America
An Example
53 – 36
= (50-30) + (3-6)
= 20 + (-3)
= 17
General Recommendations
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Prospective teachers need mathematics courses
that develop understanding of the mathematics
that they will teach.
All courses should develop careful reasoning and
teach both mathematical reasoning and
“mathematical common sense.”
There needs to be partnerships between school
teachers and mathematics faculty.
Teachers should have opportunities for professional
development.
Middle School mathematics should be taught by
mathematics specialists.
Specific Recommendations for
Elementary Teachers
NUMBERS AND OPERATIONS
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Addition, subtraction, multiplication, division
Place value
Algorithms and mental mathematics
Integer, rational, and real numbers
ALGEBRA & FUNCTIONS
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Arithmetic to algebra
Notation
Laws
Functions
Specific Recommendations for
Elementary Teachers
GEOMETRY AND MEASUREMENTS
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Visualization skills
Basic shapes and their properties
Communicating geometrical ideas
Measurements
DATA ANALYSIS, STATISTICS & PROBABILITY
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“Data” at the heart!
Describing data
Drawing conclusions
Probability
Polya’s Quotations
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The traditional mathematics professor of the popular legend is
absentminded. He usually appears in public with a lost umbrella in
each hand. He prefers to face a blackboard and to turn his back on
the class. He writes a, he says b, he means c, but it should be d.
Some of his sayings are handed down from generation to generation.
Geometry is the science of correct reasoning on incorrect figures.
My method to overcome a difficulty is to go round it.
Even fairly good students, when they have obtained the solution of
the problem and written down neatly the argument, shut their books
and look for something else. Doing so, they miss an important and
instructive phase of the work. ... A good teacher should understand
and impress on his students the view that no problem whatever is
completely exhausted.
One of the first and foremost duties of the teacher is not to give his
students the impression that mathematical problems have little
connection with each other, and no connection at all with anything
else. We have a natural opportunity to investigate the connections of
a problem when looking back at its solution.
Polya’s Quotations
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In order to translate a sentence from English into French two
things are necessary. First, we must understand thoroughly the
English sentence. Second, we must be familiar with the forms of
expression peculiar to the French language. The situation is very
similar when we attempt to express in mathematical symbols a
condition proposed in words. First, we must understand thoroughly
the condition. Second, we must be familiar with the forms of
mathematical expression.
If there is a problem you can't solve, then there is an easier problem
you can't solve: find it.
A GREAT discovery solves a great problem, but there is a grain of
discovery in the solution of any problem. Your problem may be
modest, but if it challenges your curiosity and brings into play your
inventive faculties, and if you solve it by your own means, you may
experience the tension and enjoy the triumph of discovery.
If you have to prove a theorem, do not rush. First of all, understand
fully what the theorem says, try to see clearly what it means. Then
check the theorem; it could be false. Examine the consequences,
verify as many particular instances as are needed to convince
yourself of the truth. When you have satisfied yourself that the
theorem is true, you can start proving it.
All the above are from How to Solve It (Princeton 1945).
Polya’s Quotations
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Mathematics consists of proving the most obvious thing in the least
obvious way.
Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C
1988).
Mathematics is the cheapest science. Unlike physics or chemistry, it
does not require any expensive equipment. All one needs for
mathematics is a pencil and paper.
Quoted in D J Albers, G L Alexanderson and C Reid, Mathematical
People (Boston 1985).
When introduced at the wrong time or place, good logic may be the
worst enemy of good teaching.
The American Mathematical Monthly 100 (3).
A mathematician who can only generalize is like a monkey who can
only climb up a tree, and a mathematician who can only specialize is
like a monkey who can only climb down a tree. In fact neither the up
monkey nor the down monkey is a viable creature. A real monkey
must find food and escape his enemies and so must be able to
incessantly climb up and down. A real mathematician must be able
to generalize and specialize.
Quoted in D MacHale, Comic Sections (Dublin 1993)
Chapter 1 – Problem Solving
(1.1): Exploring with Patterns
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An Example: What is 3x4?
Jack has 3 marbles. Jill has 4 marbles. They want to pick
two marbles of different colors. How many different ways
are there to pick two such marbles?
Inductive Reasoning: Reasoning based on examining
several data sets or patterns. Inductive reasoning leads to
conjectures.
Counter Examples: These are specific examples that
prove a conjecture is wrong.
Using logic to show that an idea is true, is deductive
reasoning.
There are three important mathematical ideas in this
section. Arithmetic Sequences, Geometric Sequences, and
sequences that are neither arithmetic nor geometric.
(1.2): Problem Solving Process
George Pólya introduced four steps for solving
mathematical problems.
STEP 1: UNDERSTAND THE PROBLEM.
STEP 2: DEVISE A PLAN
(1.2): Problem Solving Process (cont.)
STEP 3: CARRY OUT THE PLAN
STEP 4: LOOK BACK
(1.3): Algebraic Thinking
Can the calculator be used as a tool in
problem solving?
Scientific calculator: log, exp, yx, sin, cos, …
(grades 5-8, 9-12)
Graphing calculator: graphing and algebra
(grades 9-12)
Do you know scientific notation?
(1.3): Algebraic Thinking (cont.)
The technique of writing an equation is an important
way of introducing algebraic thinking.
Addition Property of Equations
If A = B then A ± C = B ± C
Multiplication Property of Equations
If A = B then A•C = B•C and A/C = B/C (C≠0)
Applied Problem
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Interpretation
Math model
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Math solution