How Should We Teach Mathematics?
Dr. Eric Milou
Rowan University
Department of Mathematics
milou@rowan.edu
856-256-4500 x3876
1
Overview
Conceptual vs. Procedural
Debate
Activities and Examples
NJ mathematics assessments
2
Rhetoric
NY 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.
3
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
4
*2005 AP-AOL News poll
Compute the following:
4 x 9 x 25
900 - 201
50 ÷ 1/2
5
What’s “Typical?” in US
6
Third International Math & Science
Study (TIMSS)
Procedures vs. Concepts
80
70
59
60
50
40
30
52
48
46
37
31
20
10
20
18
16
19
8
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7
Stated vs Developed
100
90
80
70
60
50
40
30
20
10
0
83
76.9
23.1
78.1
21.9
17
Germany
Japan
Stated
United States
Developed
8
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
9
What is Prevalent?
Texts become the curriculum
Drill oriented
Mathematics is BORING and does not
engage students
Mathematics Phobia and Anxiety
10
Standards Based Approach
Conceptual Understanding
Contextual Problem Solving
Constructivist Approach
Appropriate use of Calculators &
Technology
11
We need a BALANCE
Traditional text with conceptual
supplement
Conceptual text (EM, CMP, CorePlus) with computational
supplement
12
Conceptual Understanding
24 ÷ 4 = 6
24 ÷ 3 = 8
24 ÷ 2 =12
24 ÷ 1 = 24
24 ÷ 1/2 = ??
13
Fractions - Conceptually
The F word
1 1
 
2 3
1 1
 
2 3
3
2 2 5
 
6
5 6 6
More than 1 or Less than 1
Explain your reasoning
14
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
15
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
16
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
Mental Mathematics
– 4 are $6
is a vital skill
– 6 are $9
17
Computation is Important
Engaging & Active
Less passive worksheets
Creative!
More thinking & reasoning
18
Name That Number - Computational
Practice
3 8 17 1
3
Target #: 6
19
Active Computation
Fifty
–1, 2, 3, 4, 5, 6 and addition only
20
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
21
17 - 8 =
0 17
1/ 7/
-8
2
7
8 --> --> 10 --> --> --> --> --> --> --> 17
22
1000 - 279 = ?
1000
279
279 +1 = 280 + 20 = 300 +700 = 1000
23
Multiplication
13 x 17 = ?
2
13
x17
------91
130
------221
10
3
10
7
100
70
30
21
221
24
Conceptual approach leads to ?
Algebra: (x + 3) (x + 7) =
x
x
3
x2
3x
7
7x
21
25
Contextual Problem Solving
Not more traditional word problems
Placing mathematical lessons into
settings
Giving students a reason to learn the
skill
Motivating students
26
Example
4
6
5
9
8
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?
27
Spinner Example
BLUE
4
6
8
4
6
8
ORANGE
5
5
5
9
9
9
28
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
29
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?
30
Factor Game
31
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
32
NSF funded curriculum projects
Elementary: Everyday Math,
Investigations, and Trailblazers
Middle: Connected Math, Math-inContext, MathThematics, &
MathScapes
High School: IMP, Core-Plus,
SIMMS, Arise, & CPM
33
Research
USDOE Exemplary & Promising
mathematics programs (1999)
Standards-Based School
Mathematics Curricula, edited by
Senk & Thompson, published by
LEA (2003)
34
Success Factors
Teachers (what they know, believe
and do)
Teachers Professional Development
and Ongoing Support
Administrative support
Time on mathematics
35
Professional Development
Intensive, Sustained, and Ongoing
– 60 PD hours for new curriculum
Content knowledge focused
Pedagogical demonstrations
–Lesson Study
36
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
37
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
38
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)
39
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
40
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
41
Assessments Points by Cluster
Cluster
Number
NJASK
3
9
NJASK NJASK NJASK
4
5
6
13
10
9
Geometry
8
10
9
10
Algebra
8
10
10
10
D/P/D
8
10
10
10
Total
33
43
39
39
“200”
14
17.5
18
17
42
Assessments Points by Cluster
Cluster
Number
NJASK7
10
GEPA
12
HSPA
7
Geometry
9
12
12
Algebra
10
12
15
D/P/D
10
12
14
Total
39
48
48
“200”
13
25
20.5
43
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?
44
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?
45
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?
46
Characteristics of a good
mathematics program
CONCEPTUAL
CONTEXTUAL
CONSTUCTIVISM
COMPUTATION
TEST-PREP
47
Thank You
Dr. Eric Milou
Rowan University
milou@rowan.edu
48