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How Should We Teach Mathematics?. Dr. Eric Milou Rowan University Department of Mathematics [email protected] 856-256-4500 x3876. Overview. Conceptual vs. Procedural Debate Activities and Examples NJ mathematics assessments. Rhetoric NY Times (5/15/06).

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how should we teach mathematics
How Should We Teach Mathematics?

Dr. Eric Milou

Rowan University

Department of Mathematics

[email protected]

856-256-4500 x3876

overview
Overview
  • Conceptual vs. Procedural Debate
  • Activities and Examples
  • NJ mathematics assessments
rhetoric ny times 5 15 06
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
Motivating Factors for Change
  • Society’s 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
Compute the following:

4 x 9 x 25

900 - 201

50 ÷ 1/2

lesson study
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
What is Prevalent?
  • Texts become the curriculum
  • Drill oriented
  • Mathematics is BORING and does not engage students
  • Mathematics Phobia and Anxiety
standards based approach
Standards Based Approach
  • Conceptual Understanding
  • Contextual Problem Solving
  • Constructivist Approach
  • Appropriate use of Calculators & Technology
we need a balance
We need a BALANCE
  • Traditional text with conceptual supplement
  • Conceptual text (EM, CMP, Core-Plus) with computational supplement
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
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
Computation is Important
  • Engaging & Active
  • Less passive worksheets
  • Creative!
  • More thinking & reasoning
active computation
Active Computation
  • Fifty
    • 1, 2, 3, 4, 5, 6 and addition only
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

slide22
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
  • Motivating students
example
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
Spinner Example

4

5

6

5

8

5

4

9

6

9

8

9

constructivist approach
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
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?
calculators technology
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
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
Research
  • USDOE Exemplary & Promising mathematics programs (1999)
  • Standards-Based School Mathematics Curricula, edited by Senk & Thompson, published by LEA (2003)
success factors
Success Factors
  • Teachers (what they know, believe and do)
  • Teachers Professional Development and Ongoing Support
  • Administrative support
  • Time on mathematics
professional development
Professional Development
  • Intensive, Sustained, and Ongoing
    • 60 PD hours for new curriculum
  • Content knowledge focused
  • Pedagogical demonstrations
    • Lesson Study
2006 nj assessment data
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 data38
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
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 data40
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 data41
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
implications inferences
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, what’s wrong?
typical questions
Typical Questions
  • What’s wrong with these kids?
  • Why won’t they buckle down and get serious?
  • Why aren’t they supported at home?
  • Why aren’t the 1st and 2nd grade teachers preparing them?
changing the questions
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
Characteristics of a good mathematics program
  • CONCEPTUAL
  • CONTEXTUAL
  • CONSTUCTIVISM
  • COMPUTATION
  • TEST-PREP
thank you
Thank You

Dr. Eric Milou

Rowan University

[email protected]

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