Conceptual Understanding and Basic Skill: Discussion Points for Teachers and to Parents
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NCSM Annual MeetingMarch 2007
Dr. Eric MilouRowan University
Department of [email protected]
856-256-4500 x3876
Conceptual Understanding and Basic Skill:Discussion Points for Teachers and to Parents
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Overview
National News in Mathematics EducationConceptual vs. Procedural DebateNumber Sense & Computation Proficiency
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NCTM Focal Points (9/12/06)
Not Back to Basics at AllPress 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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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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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
€
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= 671.1409396
= 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
More thinking & reasoning
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Numbers Are Everywhere
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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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Multiplication Wheels
01
2
3
456
7
8
94 Facts
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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
21
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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Contextual Problem Solving ≠ More Word Problems
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Example 1: Sneakers
40% 20% 40%
20% 50% 30%
10% 30% 60%
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
SECOND purchase
Nike Converse Reebok
Nike
Converse
Reebok
F
I
R
S
T
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100 Students
– 50 1st time Nike buyers– 30 1st time Converse buyers– 20 1st time Reebok buyers
How many would buy Nike the second time?
50 x .4 + 30 x .2 + 20 x .1 = 28
40% 20% 40%
20% 50% 30%
10% 30% 60%
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
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Example 2: Game Show
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 ORANGE
4
684
68
5
559
99
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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
D E C A 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