Transcript define algebra - Rowan University

The Mathematics Debate:
Conceptual Understanding and Basic Skills
Wake County Public Schools
September 2008
Dr. Eric Milou
Rowan University
Department of Mathematics
[email protected]
856-256-4500 x3876
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Content Overview
National Math Panel
Recommendations
Conceptual vs. Procedural
Debate
Number Sense & Computation
Proficiency
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Issues
 Society’s hate for mathematics that is prevalent
and acceptable
 4 out of 10 adults hate mathematics* (twice as many people said
they hated math as said that about any other subject)
 International test scores
 Industry concerns (no problem solving skills)
 National Council of Teachers of Mathematics
(NCTM) Standards
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National Math Panel (NMP)
 A focused, coherent progression of
mathematics learning, with an emphasis on
proficiency with key topics, should become
the norm in elementary and middle school
mathematics curricula. Any approach that
continually revisits topics year after year
without closure is to be avoided.
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NMP
 Instructional practice should be informed by
high-quality research, when available, and
by the best professional judgment and
experience of accomplished classroom
teachers. High-quality research does not
support the contention that instruction
should be either entirely “student
centered” or “teacher directed.”
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NMP
A major goal for K–8 mathematics
education should be proficiency with
fractions (including decimals,
percents, and negative fractions).
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Response by Peppermint Patty
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Your Test!
4 x 9 x 25
How many ounces are in a gallon?
50 ÷ 1/2
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Algebra
If x + y = 28 and xy = 7,
then
1 1
 
x y
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x + y = 28 and xy = 7
1 1 y  x 28
 

4
x y
xy
7
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Find x
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Define “ALGEBRA”
 The intensive study of the last three letters
of the alphabet.
 using variables to replace something
significant in a problem
 equations, formulas, numbers, headaches
 using letters such as x to represent an
unknown number and solving for that letter
 The class between prealgebra and geometry
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Traditional Definitions!
Dolciani Verbs
Solve
Simplify
Factor
Rationalize
Expand
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Expand
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Balanced Algebra Curriculum
Algebraic Symbolic Manipulation &
Patterns & Algebraic Thinking
Connections to Geometry
Connections to Probability, Data
Analysis, and Discrete Mathematics
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What’s “Typical?” in US
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Third International Math & Science
Study (TIMSS)
Procedures vs. Concepts
80
70
59
60
50
40
30
52
48
46
37
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20
10
20
18
16
19
8
0
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on
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STATED vs. DEVELOPED
Teaching Gap,
Stigler and
Hiebert, 1999
We need a BALANCE
Balance
Direct Instruction
Constructivism
Balance
Conceptual Understanding
Algorithmic Proficiency
These are NOT Dichotomous
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Conceptual Understanding
24 ÷ 4 = 6
24 ÷ 3 = 8
24 ÷ 2 =12
24 ÷ 1 = 24
24 ÷ 1/2 = ??
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Fractions The
- Conceptually
F word
1 1
 
2 3
1 1
 
2 3
3
2 2 5
 
6
5 6 6
More than 1 or Less than 1
Explain your reasoning
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Which is larger?
 (2/3 + 3/4 + 4/5 + 5/6) OR 4
 12.5 x 45 OR 4.5 x 125
 (1/3 + 2/4 + 2/4 + 5/11) OR 2
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Computation is Important
Engaging & Active
Less passive worksheets
More thinking & reasoning
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Computation
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Active Computation
Fifty
(1, 2, 3, 4, 5, 6 and addition)
Buzz (3)
Product Game
 http://illuminations.nctm.org/ActivityDetail.aspx?id=29
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Web Applets
 NCTM Illuminations
 http://illuminations.nctm.org
 National Library of Virtual
Manipulatives (NLVM)
 http://www.nlvm.usu.edu/
 Shodor Interactivate
 http://www.shodor.org/interactivate/activities/
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Conceptual & Contextual
8 + 7 = ?
How do we teach this?
x x x x x
x x x x x
x x x x
x x x
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8+7=?
8 + 7=?
2
5
10 + 5 = 15
28
17 - 8 =
0 17
1/ 7/
-8
2
7
8 --> --> 10 --> --> --> --> --> --> --> 17
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1000 - 279 = ?
1000
279
279 +1 = 280 + 20 = 300 +700 = 1000
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Multiplication
13 x 17 = ?
2
13
x17
------91
130
------221
10
3
10
7
100
70
30
21
221
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Conceptual approach leads to ?
Algebra: (x + 3) (x + 7) =
x
7
x
3
x2
3x
7x
21
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Contextual Problem Solving ≠
More Word Problems
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Example 1: Sneakers
SECOND purchase
F
I
R
S
T
Nike
Converse
Reebok
 40% 20% 40% 


Converse 20% 50% 30%


Reebok 10% 30% 60% 
Nike
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100 Students
N
C
R
N  40% 20% 40% 
50 1st time Nike buyers
 20% 50% 30% 
C
30 1st time Converse buyers 

20 1st time Reebok buyers R 10% 30% 60% 
 How many would buy Nike the second
time?
 50 x .4 + 30 x .2 + 20 x .1 = 28
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Example 2: Drug Steady State
 10 mg Zrytec daily
 50% of Zrytec is eliminated daily
 What % is in the body
after 7 days
10 days
n days?
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10mg Daily of Zrytec
Day
1
2
3
4
5
6
Mg. of Zyrtec
10
15
17.5
18.75
19.375
19.6875
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Example 3: Jeopardy
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Homework
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Fact #1
A
40
Fact #2
B
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Fact #3
C
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Fact #4
D
43
Fact #5
E
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Fact #6
F
45
Fact #7
G
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Fact #8
H
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Fact #9
I
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What is this?
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What is this?
F
A
C
E
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What If?
A
B
C
D
E
F
G
H
I
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Try Again
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Try Again
DECA D E
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What’s the Point?
 Isolated Basic Skills
Less likely to retain information
 Connected Facts, Use of Patterns,
Facts in Context
More likely to retain information
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Characteristics of a good
mathematics program
CONCEPTUAL
and COMPUTATION
CONTEXTUAL
CONSTRUCTIVISM
and direct instruction
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Thank You
Dr. Eric Milou
Rowan University
[email protected]
http://www.rowan.edu/colleges/las/departments/math/facultystaff/milou/eric.html
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