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NCTM 2008

Dr. Eric MilouRowan University

Department of Mathematicsmilou@rowan.edu

856-256-4500 x3876

Basic Skill and Conceptual Understanding: Not Dichotomous at All

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Overview

National Math Panel Recommendations

Conceptual vs. Procedural Debate

Number Sense & Computation Proficiency

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National Math Panel (NMP)

A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.

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NMP

Instructional practice should be informed by high-quality research, when available, and by the best professional judgment and experience of accomplished classroom teachers. High-quality research does not support the contention that instruction should be either entirely “student centered” or “teacher directed.”

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NMP

A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions).

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Response (Gary Stager)

The Report of the National Mathematics Advisory Panel does not dispute that teachers spend lots of time teaching fractions. The report merely urges that teachers do even more of the same while hoping for a different result. A definition of insanity comes to mind.

It would be bad enough if wasted time was the only consequence of the fanatical fraction focus, but too many students get the idea that they can’t do math. This damages their inclination towards learning other forms of mathematics. Given the importance of mathematics and the widespread mathphobia sweeping the land, students can ill afford to a diminution in their self-image as capable mathematicians.

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Response (Peanuts)

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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?

4 x 4 ÷ 4 x 4

30 ÷ 3/4

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What’s “Typical?” in US

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Third International Math & Science Study (TIMSS)

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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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We need a BALANCE

BalanceDirect InstructionConstructivism

BalanceConceptual UnderstandingAlgorithmic Proficiency

These are NOT Dichotomous

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Conceptual Understanding

24 ÷ 4 = 624 ÷ 3 = 824 ÷ 2 =1224 ÷ 1 = 2424 ÷ 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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Conceptual Fraction task

Kim’s teacher asked her class to design a flag using four colors, dividing a square into parts, and to color the parts as follows:1/2 is colored red1/4 is colored blue1/8 is colored greenAny other part is to be left white

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Flags

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Harder Task

A chocolate bar is separated into several equal pieces.If Laura eats 1/4 of the pieces; andPaul eats 1/2 of the remaining pieces;There are six pieces left overInto how many pieces was the original bar

divided?

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Chocolate Bar

LAURAP A U L

16 pieces

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Decimals

1000 ÷ 1.49= 671.1409396

= Torture!Big Macs Sell for $1.49, how many Big Macs can

I buy for $10.00?1 is $1.502 are $34 are $66 are $9

Mental Mathematics

is a vital skill

1.49 1000

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Computation is Important

Engaging & ActiveLess passive worksheetsMore thinking & reasoning

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Numbers Are Everywhere

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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

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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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8 + 7 = ?

8 + 7 = ?

2 5

10 + 5 = 15

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17 - 8 =

1 7

- 8

//0 17

2 7

8 --> --> 10 --> --> --> --> --> --> --> 17

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How Many Circles?

50

30

1000 - 279 = ?

1000

−279

279 +1 = 280 + 20 = 300 +700 = 1000

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1000 - 279 = ?

1000 - 1 = 999

999-278

721

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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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Fact #1

A

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Fact #2

B

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Fact #3

C

37

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

42

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 FactsLess likely to retain information

Connected Facts, Patterns, Facts in ContextMore likely to retain information

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Characteristics of a good mathematics program

CONCEPTUALCONTEXTUALCONSTRUCTIVISMCOMPUTATION

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Thank You

Dr. Eric Milou

Rowan University

milou@rowan.edu

http://www.rowan.edu/colleges/las/departments/math/facultystaff/milou/eric.html