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How Should We Teach Mathematics?

- Dr. Eric Milou
- Rowan University
- Department of Mathematics
- milou_at_rowan.edu
- 856-256-4500 x3876

Overview

- Conceptual vs. Procedural Debate
- Activities and Examples
- NJ mathematics assessments

RhetoricNY Times (5/15/06)

- In traditional math, children learn

multiplication tables and specific techniques for

calculating. - In constructivist math, the process by which

students explore the question can be more

important than getting the right answer, and the

early use of calculators is welcomed.

Motivating Factors for Change

- Societys hate for mathematics that is prevalent

and acceptable - 4 out of 10 adults hate mathematics (twice as

many people said they hated math as said that

about any other subject) - International test scores
- Industry concerns (no problem solving skills)
- National Council of Teachers of Mathematics

(NCTM) Standards

2005 AP-AOL News poll

Compute the following

- 4 x 9 x 25
- 900 - 201
- 50 1/2

Whats Typical? in US

Third International Math Science Study (TIMSS)

- Procedures vs. Concepts

Stated vs Developed

Lesson Study

- 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

What is Prevalent?

- Texts become the curriculum
- Drill oriented
- Mathematics is BORING and does not engage

students - Mathematics Phobia and Anxiety

Standards Based Approach

- Conceptual Understanding
- Contextual Problem Solving
- Constructivist Approach
- Appropriate use of Calculators Technology

We need a BALANCE

- Traditional text with conceptual supplement
- Conceptual text (EM, CMP, Core-Plus) with

computational supplement

Conceptual Understanding

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

Fractions - Conceptually

The F word

More than 1 or Less than 1 Explain your reasoning

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

Wheres 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

- 1000 1.49
- 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
- Creative!
- More thinking reasoning

Name That Number - Computational Practice

- Target 6

3

3

17

1

8

Active Computation

- Fifty
- 1, 2, 3, 4, 5, 6 and addition only

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 ?

279

1 280

20 300

700 1000

Multiplication

10 7

- 13 x 17 ?

2

10 3

1 3 x 1 7

1 0 0

7 0

-------

3 0

2 1

1

9

0

3

1

-------

2 2 1

221

Conceptual approach leads to ?

- Algebra (x 3) (x 7)

x 7

x 3

x2

7x

3x

21

Contextual Problem Solving

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

Example

- 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

4

5

6

5

8

5

4

9

6

9

8

9

Constructivist Approach

- Allow students to develop their own meanings in

mathematics first, then build on those meanings. - ENGAGE students to be active learners with

hands-on cooperative learning activities

Crossing the River

- 8 adults and 2 children need to cross a river and

they have one small boat only available. The

boat can hold ONLY - One adult
- One or two children
- How many one-way trips does it take for all 8

adults and 2 children to cross?

Factor Game

Calculators Technology

- Calculators allowed on 100 of GEPA HSPA
- Calculators allowed on 90 of the points on the

NJASK3 4 - Calculators allowed on 100 of the SAT
- BSI and Special Education should be even more

strongly encouraged to use calculators

NSF funded curriculum projects

- Elementary Everyday Math, Investigations, and

Trailblazers - Middle Connected Math, Math-in-Context,

MathThematics, MathScapes - High School IMP, Core-Plus, SIMMS, Arise, CPM

Research

- USDOE Exemplary Promising mathematics programs

(1999) - Standards-Based School Mathematics Curricula,

edited by Senk Thompson, published by LEA (2003)

Success Factors

- Teachers (what they know, believe and do)
- Teachers Professional Development and Ongoing

Support - Administrative support
- Time on mathematics

Professional Development

- Intensive, Sustained, and Ongoing
- 60 PD hours for new curriculum
- Content knowledge focused
- Pedagogical demonstrations
- Lesson Study

2006 NJ Assessment Data

- NJASK3
- 6 non-calculator items (1/2 pt each)
- 21 MC - calculator allowed - 1 pt each
- 3 Open-ended - 3 pts each
- 14 out of 33 points is a passing score

2006 NJ Assessment Data

- NJASK4
- 8 non-calculator items (1/2 pt each)
- 24 MC - calculator allowed - 1 pt each
- 5 Open-ended - 3 pts each
- 17.5 out of 43 points is a passing score

2006 NJASK 5, 6, 7

- NJASK5 JPM was 18/39 (46)
- NJASK 6 JPM was 17/39 (44)
- NJASK 7 JPM was 13/39 (33)
- 10 pts per cluster (one cluster with 9 pts)

2006 NJ Assessment Data

- GEPA
- All items allow a calculator
- 30 Multiple choice items - 1 pt each
- 6 Open-ended - 3 pts each
- 25 out of 48 points is a passing score

2006 NJ Assessment Data

- HSPA
- All items allow a calculator
- 30 Multiple choice items - 1 pt each
- 6 Open-ended - 3 pts each
- 20.5 out of 48 points is a passing score

Assessments Points by Cluster

Assessments Points by Cluster

Implications Inferences

- NJ Assessments are rigorous and conceptual
- NJ Math Standards are well aligned with NJ

assessments - Most districts have a well aligned curriculum
- Then, whats wrong?

Typical Questions

- Whats wrong with these kids?
- Why wont they buckle down and get serious?
- Why arent they supported at home?
- Why arent the 1st and 2nd grade teachers

preparing them?

Changing the Questions

- What will students likely take away from the

activity? - How is the mathematical idea developed?
- What is the nature of the work of students?
- What is the role of the teacher?

Characteristics of a good mathematics program

- CONCEPTUAL
- CONTEXTUAL
- CONSTUCTIVISM
- COMPUTATION
- TEST-PREP

Thank You

- Dr. Eric Milou
- Rowan University
- milou_at_rowan.edu

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