Embed Size (px)
Citation preview
1
NCSM 2008
Dr. Eric MilouRowan University
Department of [email protected]
856-256-4500 x3876
Conceptual Understanding and Basic Skills: Discussion Points for Teachers and Parents
2
Overview
National Math Panel Recommendations
Conceptual vs. Procedural Debate
Number Sense & Computation Proficiency
3
National Math Panel (NMP)
A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.
4
NMP
Instructional practice should be informed by high-quality research, when available, and by the best professional judgment and experience of accomplished classroom teachers. High-quality research does not support the contention that instruction should be either entirely “student centered” or “teacher directed.”
5
NMP
A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions).
6
Response (Gary Stager)
The Report of the National Mathematics Advisory Panel does not dispute that teachers spend lots of time teaching fractions. The report merely urges that teachers do even more of the same while hoping for a different result. A definition of insanity comes to mind.
It would be bad enough if wasted time was the only consequence of the fanatical fraction focus, but too many students get the idea that they can’t do math. This damages their inclination towards learning other forms of mathematics. Given the importance of mathematics and the widespread mathphobia sweeping the land, students can ill afford to a diminution in their self-image as capable mathematicians.
7
Response (Peanuts)
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
8
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)
9
Compute the following:
4 x 9 x 25
How many ounces are in a gallon?
4 x 4 ÷ 4 x 4
30 ÷ 3/4
10
What’s “Typical?” in US
11
Third International Math & Science Study (TIMSS)
31
8
16
52
18
46
20
48
19
37
59
00
10
20
30
40
50
60
70
80
Australia
Czech RepublicHong Kong
Japan
NetherlandsUnited States
Procedures vs. Concepts
12
23.1
76.9
17
8378.1
21.9
0102030405060708090
100
Germany Japan United States
Stated Developed
Stated vs Developed
13
We need a BALANCE
BalanceDirect InstructionConstructivism
BalanceConceptual UnderstandingAlgorithmic Proficiency
These are NOT Dichotomous
14
Conceptual Understanding
24 ÷ 4 = 624 ÷ 3 = 824 ÷ 2 =1224 ÷ 1 = 2424 ÷ 1/2 = ??
15
Fractions - Conceptually
€
1
2+
1
3=
€
1
2+
1
3= More than 1 or Less than 1
Explain your reasoning€
3
6+
2
6=
5
6
€
2
5
The F word
16
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
17
Conceptual Fraction task
Kim’s teacher asked her class to design a flag using four colors, dividing a square into parts, and to color the parts as follows:1/2 is colored red1/4 is colored blue1/8 is colored greenAny other part is to be left white
18
Flags
19
Harder Task
A chocolate bar is separated into several equal pieces.If Laura eats 1/4 of the pieces; andPaul eats 1/2 of the remaining pieces;There are six pieces left overInto how many pieces was the original bar
divided?
20
Chocolate Bar
LAURAP A U L
16 pieces
21
Decimals
1000 ÷ 1.49= 671.1409396
= Torture!Big Macs Sell for $1.49, how many Big Macs can
I buy for $10.00?1 is $1.502 are $34 are $66 are $9
Mental Mathematics
is a vital skill
1.49 1000
22
Computation is Important
Engaging & ActiveLess passive worksheetsMore thinking & reasoning
23
Numbers Are Everywhere
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
24
Computational Practice
Target #: 6
3 8 17 1 3
25
Active Computation
Fifty (1, 2, 3, 4, 5, 6 and addition)Buzz (3)Product Game
26
Conceptual & Contextual
8 + 7 = ?How do we teach this?
xx xx x x
xx
xx
xx
xx
xxx
27
8 + 7 = ?
8 + 7 = ?
2 5
10 + 5 = 15
28
17 - 8 =
1 7
- 8
//0 17
2 7
8 --> --> 10 --> --> --> --> --> --> --> 17
29
How Many Circles?
50
30
1000 - 279 = ?
1000
−279
279 +1 = 280 + 20 = 300 +700 = 1000
31
1000 - 279 = ?
1000 - 1 = 999
999-278
721
32
Multiplication
13 x 17 = ?
1 3
x 1 7
1
2
9031
2 2 1
-------
-------
10 7 10
3
1 0 0
3 0
7 0
2 1
221
33
Conceptual approach leads to ?
x 7 x
3
x2
3x
7x
21
Algebra: (x + 3) (x + 7) =
34
Contextual Problem Solving ≠ More Word Problems
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
35
Example 1: Sneakers
40% 20% 40%
20% 50% 30%
10% 30% 60%
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
SECOND purchase
Nike Converse Reebok
Nike
Converse
Reebok
F
I
R
S
T
36
100 Students
50 1st time Nike buyers30 1st time Converse buyers20 1st time Reebok buyers
How many would buy Nike the second time?
50 x .4 + 30 x .2 + 20 x .1 = 28
40% 20% 40%
20% 50% 30%
10% 30% 60%
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
N C RN
C
R
37
Example 2: Drug Steady State
10 mg Zrytec daily50% of Zrytec is eliminated dailyWhat % is in the body
after 7 days10 daysn days?
38
10mg Daily of Zrytec
Day Mg. of Zyrtec
1
234
56
10
1517.518.75
19.37519.6875
39
Fact #1
A
40
Fact #2
B
41
Fact #3
C
42
Fact #4
D
43
Fact #5
E
44
Fact #6
F
45
Fact #7
G
46
Fact #8
H
47
Fact #9
I
48
What is this?
49
What is this?
F A C E
50
What If?
A B C
D E F
G H I
51
Try Again
52
Try Again
D E C A D E
53
What’s the Point?
Isolated FactsLess likely to retain information
Connected Facts, Patterns, Facts in ContextMore likely to retain information
54
Characteristics of a good mathematics program
CONCEPTUALCONTEXTUALCONSTRUCTIVISMCOMPUTATION
55
Thank You
Dr. Eric Milou
Rowan University
http://www.rowan.edu/colleges/las/departments/math/facultystaff/milou/eric.html
