Transcript AMTNJ January 07 - Rowan University

How Do I Balance Computational and
Conceptual Understanding?
Dr. Eric Milou
Rowan University
Department of Mathematics
[email protected]
856-256-4500 x3876
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Overview
National News in Mathematics Education
NJ mathematics assessments
Conceptual vs. Procedural Debate
Number Sense & Computation Proficiency
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NCTM Focal Points (9/12/06)
Not Back to Basics at All
Wall Street Journal article 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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2006 State Testing Results
2006 Partially Proficient %
80
70
60
50
40
TOTAL
GEN Ed
30
SPEC Ed
20
10
0
3
4
5
6
Test Grade
7
8
11
5
2006 NJASK 5, 6, 7
30 MC and 3 Open Ended: 39 points
Calculator allowed on ALL questions
NJASK5 JPM was 18/39 (46%)
NJASK 6 JPM was 17/39 (44%)
NJASK 7 JPM was 13/39 (33%)
10 pts per cluster (one cluster with 9 pts)
Sample Items at:
http://www.nj.gov/njded/assessment/ms/5-7/
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2006 NJ GEPA Data
All items allow a calculator
30 Multiple choice items - 1 pt each
6 Open-ended - 3 pts each
25 out of 48 points is a passing score
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Assessments Points by Cluster
Cluster
Number
NJASK
3
9
NJASK NJASK NJASK
4
5
6
13
10
9
Geometry
8
10
9
10
Algebra
8
10
10
10
D/P/D
8
10
10
10
Total
33
43
39
39
“200”
14
17.5
18
17
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Assessments Points by Cluster
Cluster
Number
NJASK7
10
GEPA
12
HSPA
7
Geometry
9
12
12
Algebra
10
12
15
D/P/D
10
12
14
Total
39
48
48
“200”
13
25
20.5
9
200 Score
Grade
Just Proficient Means
3rd 14
4th 17.5
5th
18
6th
17
7th
13
8th
25
11th 20.5
33
43
39
39
39
48
48
42%
41%
46%
44%
33%
52%
43%
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Implications & Inferences
NJ Assessments are rigorous and
conceptual
NJ Math Standards are well aligned
with NJ assessments
Most districts have a well aligned
curriculum
– Then, what’s wrong?
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Compute the following:
4 x 9 x 25
900 - 201
50 ÷ 1/2
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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
20
18
16
19
8
0
St
at
te
d
U
ni
N
et
h
er
la
nd
es
s
n
Ja
K
g
on
H
pa
g
on
ic
bl
R
ep
u
C
ze
ch
A
us
tr
al
ia
0
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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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Lesson Study
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
Traditional text with conceptual
supplement
Conceptual text (EM, CMP, CorePlus) with computational
supplement
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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
– Torture
Big Macs Sell for $1.49, how many Big Macs can
I buy for $10.00?
– 1 is $1.50
– 2 are $3
– 4 are $6
Mental Mathematics
– 6 are $9
is a vital skill
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Computation is Important
Engaging & Active
Less passive worksheets
Creative!
More thinking & reasoning
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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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Patterns
0
9
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
Motivating students
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Example
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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Crossing the River
8 adults and 2 children need to cross a river
and they have one small boat only available.
The boat can hold ONLY:
– One adult
– One or two children
How many one-way trips does it take for all
8 adults and 2 children to cross?
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Fact #1
A
36
Fact #2
B
37
Fact #3
C
38
Fact #4
D
39
Fact #5
E
40
Fact #6
F
41
Fact #7
G
42
Fact #8
H
43
Fact #9
I
44
What is this?
45
What is this?
F
A
C
E
46
What If?
A
B
C
D
E
F
G
H
I
47
Try Again
48
Try Again
DECA D E
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What’s the Point?
Isolated Facts
– Less likely to retain information
Connected Facts, Patterns, Fact in
Context
– More likely to retain information
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Characteristics of a good
mathematics program
CONCEPTUAL
CONTEXTUAL
CONSTUCTIVISM
COMPUTATION
TEST-PREP
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Thank You
Dr. Eric Milou
Rowan University
[email protected]
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