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Concepts vs Skills: A Bogus Dichotomy. NCTM Regional Meeting Richmond, VA October 2007 Dr. Eric Milou Rowan University Department of Mathematics [email protected] 856-256-4500 x3876. Overview. National News in Mathematics Education Conceptual vs. Procedural Debate

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Concepts vs skills a bogus dichotomy
Concepts vs Skills: A Bogus Dichotomy

NCTM Regional Meeting

Richmond, VA

October 2007

Dr. Eric Milou

Rowan University

Department of Mathematics

[email protected]

856-256-4500 x3876


Overview
Overview

  • National News in Mathematics Education

  • Conceptual vs. Procedural Debate

  • Number Sense & Computation Proficiency


Nctm focal points 9 12 06
NCTM Focal Points (9/12/06)

  • Not Back to Basics at All

  • Press 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.


Education week 11 1 06
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)


Compute the following
Compute the following:

4 x 9 x 25

How many ounces are in a gallon?

50 ÷ 1/2






Math lessons
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


We need a balance
We need a BALANCE

  • Pedagogical Balance

  • Content Balance

    • Conceptual Understanding

    • Algorithmic Proficiency

  • These are NOT Dichotomous


Conceptual understanding
Conceptual Understanding

  • 24 ÷ 4 = 6

  • 24 ÷ 3 = 8

  • 24 ÷ 2 =12

  • 24 ÷ 1 = 24

  • 24 ÷ 1/2 = ??


Fractions conceptually
Fractions - Conceptually

The F word

More than 1 or Less than 1

Explain your reasoning


Which is larger
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


Where s the point
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


Computational balance

1.49

1000

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


Computation is important
Computation is Important

  • Engaging & Active

  • Less passive worksheets

  • More thinking & reasoning




Active computation
Active Computation

  • Fifty (1, 2, 3, 4, 5, 6 and addition)

  • Buzz (3)

  • Product Game

  • Wipe Out

  • Software: Math Arena


Multiplication wheels

0

9

1

8

2

7

3

6

4

5

Multiplication Wheels

4 Facts


Conceptual contextual
Conceptual & Contextual

  • 8+ 7 = ?

  • How do we teach this?

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x


17 - 8 =

0

17

/

/

1 7

- 8

2 7

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


1000 279
1000 - 279 = ?

279

+1 = 280

+ 20 = 300

+700 = 1000


Multiplication
Multiplication

  • 13 x 17 = ?

10 7

2

10

3

1 3

x 1 7

1 0 0

7 0

-------

3 0

2 1

9

1

1

3

0

-------

2 2 1

221


Conceptual approach leads to
Conceptual approach leads to ?

  • Algebra: (x + 3) (x + 7) =

x 7

x

3

x2

7x

3x

21


Contextual problem solving
Contextual Problem Solving

  • Not more traditional word problems

  • Placing mathematical lessons into settings

  • Giving students a reason to learn the skill



Example 1 sneakers
Example 1: Sneakers

SECOND purchase

Nike Converse Reebok

F

I

R

S

T

Nike

Converse

Reebok


100 students
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


  • Example 2 game show
    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?


    Spinner example
    Spinner Example

    4

    5

    6

    5

    8

    5

    4

    9

    6

    9

    8

    9













    What is this1
    What is this?

    F A C E


    What if
    What If?

    C

    A

    B

    F

    D

    E

    I

    G

    H



    Try again1
    Try Again

    D E C A D E


    What s the point

    Isolated Facts

    Less likely to retain information

    Connected Facts, Patterns, Facts in Context

    More likely to retain information

    What’s the Point?


    Characteristics of a good mathematics program
    Characteristics of a good mathematics program

    • CONCEPTUAL

    • CONTEXTUAL

    • CONSTRUCTIVISM

    • COMPUTATION


    Thank you
    Thank You

    Dr. Eric Milou

    Rowan University

    [email protected]

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


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