Transcript 8 + 7 - Rowan University

Basic Skill and Conceptual Understanding:
Not Dichotomous at All
NCTM 2008
Dr. Eric Milou
Rowan University
Department of Mathematics
[email protected]
856-256-4500 x3876
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Overview
National Math Panel
Recommendations
Conceptual vs. Procedural
Debate
Number Sense & Computation
Proficiency
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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 (Gary Stager)
 The Report of the National Mathematics Advisory Panel
does not dispute that teachers spend lots of time teaching
fractions. The report merely urges that teachers do even
more of the same while hoping for a different result. A
definition of insanity comes to mind.
 It would be bad enough if wasted time was the only
consequence of the fanatical fraction focus, but too many
students get the idea that they can’t do math. This damages
their inclination towards learning other forms of
mathematics. Given the importance of mathematics and the
widespread mathphobia sweeping the land, students can ill
afford to a diminution in their self-image as capable
mathematicians.
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Response (Peanuts)
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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?
4x4÷4x4
30 ÷ 3/4
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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
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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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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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Conceptual Fraction task
 Kim’s teacher asked her class to design a
flag using four colors, dividing a square into
parts, and to color the parts as follows:
1/2 is colored red
1/4 is colored blue
1/8 is colored green
Any other part is to be left white
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Flags
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Harder Task
 A chocolate bar is separated into several
equal pieces.
If Laura eats 1/4 of the pieces; and
Paul eats 1/2 of the remaining pieces;
There are six pieces left over
Into how many pieces was the original bar
divided?
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Chocolate Bar
LAURA
P A U L
16 pieces
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Decimals
 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
QuickTime™ and a
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are needed to see this picture.
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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
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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
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17 - 8 =
0 17
1/ 7/
-8
2
7
8 --> --> 10 --> --> --> --> --> --> --> 17
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How Many Circles?
50
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1000 - 279 = ?
1000
279
279 +1 = 280 + 20 = 300 +700 = 1000
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1000 - 279 = ?
1000 - 1 = 999
999
-278
721
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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
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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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