Preparing for the NJ Math Assessments
in the Middle Grades
Dr. Eric Milou
Rowan University
Department of Mathematics
milou@rowan.edu
856-256-4500 x3876
1
Overview
Conceptual vs. Procedural Debate
– National Math Panel
Number Sense & Computation Proficiency
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
NCTM Focal Points (9/12/06)
September 12 Wall Street Journal article 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.
4
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)
5
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
6
*2005 AP-AOL News poll
Compute the following:
4 x 9 x 25
900 - 201
50 ÷ 1/2
7
What’s “Typical?” in US
8
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
0
St
at
te
d
U
ni
N
et
h
er
la
nd
es
s
n
Ja
K
g
on
H
pa
g
on
ic
bl
R
ep
u
C
ze
ch
A
us
tr
al
ia
0
9
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
10
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
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)
Buzz (3)
Product Game
Wipe Out
Software: Math Arena
20
Patterns
0
9
1
8
2
7
3
6
5
4
21
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
22
17 - 8 =
0 17
1/ 7/
-8
2
7
8 --> --> 10 --> --> --> --> --> --> --> 17
23
1000 - 279 = ?
1000
279
279 +1 = 280 + 20 = 300 +700 = 1000
24
Multiplication
13 x 17 = ?
2
13
x17
------91
130
------221
10
3
10
7
100
70
30
21
221
25
Conceptual approach leads to ?
Algebra: (x + 3) (x + 7) =
x
x
3
x2
3x
7
7x
21
26
Contextual Problem Solving
Not more traditional word problems
Placing mathematical lessons into
settings
Giving students a reason to learn the
skill
Motivating students
27
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?
28
Spinner Example
BLUE
4
6
8
4
6
8
ORANGE
5
5
5
9
9
9
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
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
31
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
32
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)
33
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
34
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
35
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
36
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
37
200 Score
Grade
Just Proficient Means
3rd 14
4th 17.5
5th
18
6th
17
7th
13
8th
25
11th 20.5
33
43
39
39
39
48
48
42%
41%
46%
44%
33%
52%
43%
38
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?
39
Algebra Placement
Districts should not encourage all students
to take Algebra I in grade 8; students should
be taking Algebra I in grade 8 only if they
are highly motivated, have a strong
foundation in middle school mathematics,
receive high grades in previous courses,
intend to study calculus in high school, and
only if the Algebra I courses are taught by
teachers with mathematics certification.
40
Fact #1
A
41
Fact #2
B
42
Fact #3
C
43
Fact #4
D
44
Fact #5
E
45
Fact #6
F
46
Fact #7
G
47
Fact #8
H
48
Fact #9
I
49
What is this?
50
What is this?
F
A
C
E
51
What If?
A
B
C
D
E
F
G
H
I
52
Try Again
53
Try Again
DECA D E
54
What’s the Point?
Isolated Facts
– Less likely to retain information
Connected Facts, Patterns, Fact in
Context
– More likely to retain information
55
Characteristics of a good
mathematics program
CONCEPTUAL
CONTEXTUAL
CONSTUCTIVISM
COMPUTATION
TEST-PREP
56
Thank You
Dr. Eric Milou
Rowan University
milou@rowan.edu
57