Transcript define algebra - Rowan University

Concepts vs Skills: A Bogus Dichotomy
NCTM Regional Meeting
Richmond, VA
October 2007
Dr. Eric Milou
Rowan University
Department of Mathematics
[email protected]
856-256-4500 x3876
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Overview
National News in Mathematics
Education
Conceptual vs. Procedural
Debate
Number Sense & Computation
Proficiency
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NCTM Focal Points (9/12/06)
Not Back to Basics at All
Press articles did not represent the substance or
intent of the focal points.
The focal points are not about the basics; they are
about important foundational topics. NCTM has
always supported learning the basics.
Students should learn and be able to recall basic
facts and become computationally fluent, but such
knowledge and skills should be acquired with
understanding.
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Education Week 11/1/06
We cannot afford to waste time on polarization.
What is important is that we pragmatically address
critical target areas to improve mathematics
education. We cannot be distracted from our
primary mission—to match tactical initiatives in
other, newly technological societies that are
snatching our competitive advantage in
innovation—while we bicker over modest
differences in approach. (Jere Confrey)
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Compute the following:
4 x 9 x 25
How many ounces are in a gallon?
50 ÷ 1/2
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Peanuts
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QuickTime™ and a
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QuickTi me™ and a
TIFF (U ncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (U ncompressed) decompressor
are needed to see this picture.
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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
31
20
10
18
16
20
19
8
0
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at
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la
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Ja
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Stated vs Developed
100
90
80
70
60
50
40
30
20
10
0
83
76.9
23.1
78.1
21.9
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Germany
Japan
Stated
United States
Developed
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Math Lessons
Demonstrates a
procedure
Assigns similar
problems to students as
exercises
Homework assignment
Presents a problem without
first demonstrating how to
solve it
Individual or group problem
solving
Compare and discuss
multiple solution methods
Summary, exercises and
homework assignment
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We need a BALANCE
Pedagogical Balance
Content 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 - Conceptually
The 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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Where’s the Point?
2.43 x 5.1 = 12393
4.85 x 4.954 = 240269
21.25 x 1.08 = 2295
1.25 x 64 = 80
4.688 x 1.355 = 635224
46.88 x 1.355 = 635224
4.688 x 135.5 = 635224
46.88 x 13.55 = 635224
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Computational Balance
1000 ÷ 1.49
= 671.1409396
1.49
1000
= Torture!
Big Macs Sell for $1.49, how many Big Macs can
I buy for $10.00?
– 1 is $1.50
– 2 are $3
Mental Mathematics
– 4 are $6
is a vital skill
– 6 are $9
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Computation is Important
Engaging & Active
Less passive worksheets
More thinking & reasoning
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Numbers Are Everywhere
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Name That Number - Computational
Practice
3 8 17 1
3
Target #: 6
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Active Computation
Fifty (1, 2, 3, 4, 5, 6 and addition)
Buzz (3)
Product Game
Wipe Out
Software: Math Arena
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Multiplication Wheels
0
9
4 Facts
1
8
2
7
3
6
5
4
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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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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
x
3
x2
3x
7
7x
21
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Contextual Problem Solving
Not more traditional word problems
Placing mathematical lessons into
settings
Giving students a reason to learn the
skill
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Contextual Problem Solving ≠
More Word Problems
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Q uickTim e™ and a
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Q uickTim e™ and a
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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
– 50 1st time Nike buyers
– 30 1st time Converse buyers
– 20 1st time Reebok buyers
 40% 20% 40% 
 20% 50% 30% 


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: Game Show
4
6
5
9
8
You must select one spinner. Both spinners
above will be spun once.
The spinner with the higher number
showing wins $1,000,000 for that person.
Which spinner will you select?
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Spinner Example
BLUE
4
6
8
4
6
8
ORANGE
5
5
5
9
9
9
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Math Essays?
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Fact #1
A
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Fact #2
B
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Fact #3
C
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Fact #4
D
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Fact #5
E
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Fact #6
F
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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 Facts
– Less likely to retain information
Connected Facts, Patterns, Facts in
Context
– More likely to retain information
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Characteristics of a good
mathematics program
CONCEPTUAL
CONTEXTUAL
CONSTRUCTIVISM
COMPUTATION
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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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