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Concepts vs Skills: A Bogus DichotomyPowerPoint Presentation

Concepts vs Skills: A Bogus Dichotomy

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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

856-256-4500 x3876

Overview

- National News in Mathematics Education
- Conceptual vs. Procedural Debate
- Number Sense & Computation Proficiency

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

- 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)

Third International Math & Science Study (TIMSS)

Proceduresvs. Concepts

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

- Pedagogical Balance
- Content Balance
- Conceptual Understanding
- Algorithmic Proficiency

- These are NOT Dichotomous

Conceptual Understanding

- 24 ÷ 4 = 6
- 24 ÷ 3 = 8
- 24 ÷ 2 =12
- 24 ÷ 1 = 24
- 24 ÷ 1/2 = ??

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?

- 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

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

- Engaging & Active
- Less passive worksheets
- More thinking & reasoning

Active Computation

- Fifty (1, 2, 3, 4, 5, 6 and addition)
- Buzz (3)
- Product Game
- Wipe Out
- Software: Math Arena

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

Contextual Problem Solving

- Not more traditional word problems
- Placing mathematical lessons into settings
- Giving students a reason to learn the skill

100 Students How many would buy Nike the second time? 50 x .4 + 30 x .2 + 20 x .1 = 28

- 50 1st time Nike buyers
- 30 1st time Converse buyers
- 20 1st time Reebok buyers

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?

Fact #1

A

Fact #2

B

Fact #3

C

Fact #4

D

Fact #5

E

Fact #6

F

Fact #7

G

Fact #8

H

Fact #9

I

What is this?

F A C E

Try Again

D E C A D E

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

- CONCEPTUAL
- CONTEXTUAL
- CONSTRUCTIVISM
- COMPUTATION

Thank You

Dr. Eric Milou

Rowan University

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

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