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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

slide23
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 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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