1 Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876 How Do I Balance Computational and Conceptual Understanding?
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1
Dr. Eric Milou
Rowan University
Department of Mathematics
milou@rowan.edu
856-256-4500 x3876
How Do I Balance Computational and Conceptual Understanding?
2
Overview
National News in Mathematics Education
NJ mathematics assessments
Conceptual vs. Procedural Debate
Number Sense & Computation Proficiency
3
NCTM Focal Points (9/12/06)
Not Back to Basics at AllWall 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
0
10
20
30
40
50
60
70
80
3 4 5 6 7 8 11
Test Grade
2006 Partially Proficient %
TOTAL
GEN Ed
SPEC Ed
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2006 NJASK 5, 6, 7
30 MC and 3 Open Ended: 39 pointsCalculator allowed on ALL questionsNJASK5 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 NJASK3 NJASK 4
NJASK 5
NJASK 6
Number 9 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 NJASK7 GEPA HSPA
Number 10 12 7
Geometry 9 12 12
Algebra 10 12 15
D/P/D 10 12 14
Total 39 48 48
“200” 13 25 20.5
10
200 Score
Grade
3rd 14 33 42%4th 17.5 43 41%5th 18 39 46%6th 17 39 44%7th 13 39 33%8th 25 48 52%11th 20.5 48 43%
Just Proficient Means
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Implications & Inferences
NJ Assessments are rigorous and conceptualNJ Math Standards are well aligned with NJ assessmentsMost 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)
31
8
16
52
18
46
20
48
19
37
59
00
10
20
30
40
50
60
70
80
Australia
Czech RepublicHong Kong
Japan
NetherlandsUnited States
Procedures vs. Concepts
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23.1
76.9
17
8378.1
21.9
0102030405060708090
100
Germany Japan United States
Stated Developed
Stated vs 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, Core-Plus) 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
€
1
2+
1
3=
€
1
2+
1
3= More than 1 or Less
than 1
Explain your reasoning
€
3
6+
2
6=
5
6
€
2
5
The F word
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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– 6 are $9
Mental Mathematics
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
Target #: 6
3 8 17
1 3
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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
01
2
3
456
7
8
9
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Conceptual & Contextual
8+ 7 = ?
How do we teach this?xx x
x x xxx
xx
xx
xx
xxx
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17 - 8 =
1 7
- 8
//0 17
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 = ?
1 3
x 1 7
1
2
9031
2 2 1
-------
-------
10 7
10
3
1 0 0
3 0
7 0
2 1
221
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Conceptual approach leads to ?
x 7
x
3
x2
3x
7x
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Algebra: (x + 3) (x + 7) =
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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
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?
4 6 8 5 9
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Spinner Example
BLUE ORANGE468468
555999
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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
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Fact #2
B
38
Fact #3
C
39
Fact #4
D
40
Fact #5
E
41
Fact #6
F
42
Fact #7
G
43
Fact #8
H
44
Fact #9
I
45
What is this?
46
What is this?
F A C E
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What If?
A B C
D E F
G H I
48
Try Again
49
Try Again
D E C A 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
milou@rowan.edu
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