How Do I Balance Computational and Conceptual Understanding?

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Dr. Eric MilouRowan University

Department of Mathematicsmilou@rowan.edu

856-256-4500 x3876

How Do I Balance Computational and Conceptual Understanding?

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Overview

National News in Mathematics EducationNJ mathematics assessmentsConceptual vs. Procedural DebateNumber Sense & Computation Proficiency

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NCTM Focal Points (9/12/06)

Not Back to Basics at AllWall 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.

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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)

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2006 State Testing Results

0

10

20

30

40

50

60

70

80

3 4 5 6 7 8 11Test Grade

2006 Partially Proficient %

TOTALGEN EdSPEC Ed

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2006 NJASK 5, 6, 7

30 MC and 3 Open Ended: 39 pointsCalculator allowed on ALL questionsNJASK5 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)Sample Items at: http://www.nj.gov/njded/assessment/ms/5-7/

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2006 NJ GEPA Data

All items allow a calculator30 Multiple choice items - 1 pt each 6 Open-ended - 3 pts each25 out of 48 points is a passing score

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Assessments Points by Cluster

Cluster NJASK3 NJASK 4

NJASK 5

NJASK 6

Number 9 13 10 9Geometry 8 10 9 10Algebra 8 10 10 10D/P/D 8 10 10 10Total 33 43 39 39“200” 14 17.5 18 17

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Assessments Points by Cluster

Cluster NJASK7 GEPA HSPANumber 10 12 7Geometry 9 12 12Algebra 10 12 15D/P/D 10 12 14Total 39 48 48“200” 13 25 20.5

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200 Score

Grade3rd 14 33 42%4th 17.5 43 41%5th 18 39 46%6th 17 39 44%7th 13 39 33%8th 25 48 52%11th 20.5 48 43%

Just Proficient Means

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Implications & Inferences

NJ Assessments are rigorous and conceptualNJ Math Standards are well aligned with NJ assessmentsMost districts have a well aligned curriculum– Then, what’s wrong?

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Compute the following:

4 x 9 x 25

900 - 201

50 ÷ 1/2

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What’s “Typical?” in US

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Third International Math & Science Study (TIMSS)

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816

52

18

46

20

48

19

37

59

001020304050607080

Australia

Czech RepublicHong Kong

Japan

NetherlandsUnited States

Procedures vs. Concepts

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23.1

76.9

17

8378.1

21.9

0102030405060708090

100

Germany Japan United States

Stated Developed

Stated vs Developed

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Lesson Study

Demonstrates a procedureAssigns similar problems to students as exercisesHomework assignment

Presents a problem without first demonstrating how to solve itIndividual or group problem solvingCompare and discuss multiple solution methodsSummary, exercises and homework assignment

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We need a BALANCE

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

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Conceptual Understanding

24 ÷ 4 = 624 ÷ 3 = 824 ÷ 2 =1224 ÷ 1 = 2424 ÷ 1/2 = ??

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Fractions - Conceptually

€

12

+ 13

=

€

12

+ 13

= More than 1 or Less than 1

Explain your reasoning

€

36

+ 26

= 56

€

25

The F word

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Which is larger?

2/3 + 3/4 + 4/5 + 5/6 OR 412.5 x 45 OR 4.5 x 1251/3 + 2/4 + 2/4 + 5/11 OR 2

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Where’s the Point?

2.43 x 5.1 = 123934.85 x 4.954 = 24026921.25 x 1.08 = 22951.25 x 64 = 804.688 x 1.355 = 63522446.88 x 1.355 = 6352244.688 x 135.5 = 63522446.88 x 13.55 = 635224

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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 Mathematicsis a vital

skill

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Computation is Important

Engaging & ActiveLess passive worksheetsCreative!More thinking & reasoning

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Name That Number - Computational Practice

Target #: 6

3 8 17

1 3

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Active Computation

Fifty (1, 2, 3, 4, 5, 6 and addition)Buzz (3)Product GameWipe OutSoftware: Math Arena

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Patterns

01

2

3

456

7

8

9

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Conceptual & Contextual

8+ 7 = ?How do we teach this?

xx xx x x

xx

xx

xx

xx

xxx

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17 - 8 =

1 7- 8//

0 17

2 7

8 --> --> 10 --> --> --> --> --> --> --> 17

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1000 - 279 = ?

1000−279

279 +1 = 280

+ 20 = 300

+700 = 1000

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Multiplication

13 x 17 = ? 1 3

x 1 71

2

9031

2 2 1

-------

-------

10 7

10

3

1 0 0 3 0

7 0

2 1

221

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Conceptual approach leads to ?

x 7

x

3

x2

3x

7x

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Algebra: (x + 3) (x + 7) =

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Contextual Problem Solving

Not more traditional word problemsPlacing mathematical lessons into settingsGiving students a reason to learn the skillMotivating students

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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?

4 6 8 5 9

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Spinner Example

BLUE ORANGE468468

555999

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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?

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Fact #1

A

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Fact #2

B

38

Fact #3

C

39

Fact #4

D

40

Fact #5

E

41

Fact #6

F

42

Fact #7

G

43

Fact #8

H

44

Fact #9

I

45

What is this?

46

What is this?

F A C E

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What If?

A B C

D E FG H I

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Try Again

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Try Again

D E C A D E

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What’s the Point?

Isolated Facts– Less likely to retain information

Connected Facts, Patterns, Fact in Context– More likely to retain information

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Characteristics of a good mathematics program

CONCEPTUALCONTEXTUALCONSTUCTIVISMCOMPUTATIONTEST-PREP

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Thank You

Dr. Eric MilouRowan Universitymilou@rowan.edu