How Do I Balance Computational and Conceptual Understanding?
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Transcript of How Do I Balance Computational and Conceptual Understanding?
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Dr. Eric MilouRowan University
Department of [email protected]
856-256-4500 x3876
How Do I Balance Computational and Conceptual Understanding?
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Overview
National News in Mathematics EducationNJ mathematics assessmentsConceptual vs. Procedural DebateNumber Sense & Computation Proficiency
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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 11Test Grade
2006 Partially Proficient %
TOTALGEN EdSPEC 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 calculator30 Multiple choice items - 1 pt each 6 Open-ended - 3 pts each25 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 9Geometry 8 10 9 10Algebra 8 10 10 10D/P/D 8 10 10 10Total 33 43 39 39“200” 14 17.5 18 17
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Assessments Points by Cluster
Cluster NJASK7 GEPA HSPANumber 10 12 7Geometry 9 12 12Algebra 10 12 15D/P/D 10 12 14Total 39 48 48“200” 13 25 20.5
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200 Score
Grade3rd 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
816
52
18
46
20
48
19
37
59
001020304050607080
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 procedureAssigns similar problems to students as exercisesHomework assignment
Presents a problem without first demonstrating how to solve itIndividual or group problem solvingCompare and discuss multiple solution methodsSummary, exercises and homework assignment
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We need a BALANCE
Traditional text with conceptual supplementConceptual text (EM, CMP, Core-Plus) with computational supplement
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Conceptual Understanding
24 ÷ 4 = 624 ÷ 3 = 824 ÷ 2 =1224 ÷ 1 = 2424 ÷ 1/2 = ??
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Fractions - Conceptually
€
12
+ 13
=
€
12
+ 13
= More than 1 or Less than 1
Explain your reasoning
€
36
+ 26
= 56
€
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The F word
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Which is larger?
2/3 + 3/4 + 4/5 + 5/6 OR 412.5 x 45 OR 4.5 x 1251/3 + 2/4 + 2/4 + 5/11 OR 2
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Where’s the Point?
2.43 x 5.1 = 123934.85 x 4.954 = 24026921.25 x 1.08 = 22951.25 x 64 = 804.688 x 1.355 = 63522446.88 x 1.355 = 6352244.688 x 135.5 = 63522446.88 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 Mathematicsis a vital
skill
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Computation is Important
Engaging & ActiveLess passive worksheetsCreative!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 GameWipe OutSoftware: 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 xx x x
xx
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 71
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 problemsPlacing mathematical lessons into settingsGiving students a reason to learn the skillMotivating 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
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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 FG H I
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Try Again
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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
CONCEPTUALCONTEXTUALCONSTUCTIVISMCOMPUTATIONTEST-PREP