1 NCSM Annual Meeting March 2007 Dr. Eric Milou Rowan University Department of Mathematics [email protected] 856-256-4500 x3876 Conceptual Understanding.

1

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

mailto:[email protected]

mailto:[email protected]

2

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.

4

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

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.







7

What’s “Typical?” in US

8

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

9

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

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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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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

30

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?



34

Fact #1

A

35

Fact #2

B

36

Fact #3

C

37

Fact #4

D

38

Fact #5

E

39

Fact #6

F

40

Fact #7

G

41

Fact #8

H

42

Fact #9

I

43

What is this?

44

What is this?

F A C E

45

What If?

A B C

D E F

G H I

46

Try Again

47

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

50

Thank You

Dr. Eric Milou

Rowan University

[email protected]

1 NCSM Annual Meeting March 2007 Dr. Eric Milou Rowan University Department of Mathematics [email protected] 856-256-4500 x3876 Conceptual Understanding.

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