doing what they want instead of what they are taught

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DOING WHAT THEY WANT INSTEAD OF WHAT. THEY ARE TAUGHT: METHODS OF DIVISION. USED BY FOURTH AND FIFTH GRADE. STUDENTS.
DOING WHAT THEY WANT INSTEAD OF WHAT THEY ARE TAUGHT:

METHODS OF DIVISION

USED BY FOURTH AND FIFTH GRADE STUDENTS Cheryll L. Fitzpatrick, Darcy Hallett, Nicole Pelley, & Kyle Morrissey

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

I. 

INTRODUCTION/BACKGROUND

Pre-formal assessment (e.g., Correa, Nunes, & Bryant, 1998; Squire & Bryant, 2002) - illustrative (Kornilaki & Nunes, 2005; Matalliotaki, 2012) - toys (Correa et al., 1998)

Demonstrates conceptual understanding of dividend, divisor, quotient (Squire & Bryant, 2002)

Equal Sharing typically has highest accuracy (Correa et al., 1998; Kornilaki & Nunes, 2005)

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

II. CURRENT STUDY Examine methods of division used by grade 4’s and 5’s after they have received formal assessment. Open-ended questions

-  Newfoundland curriculum -  Introduces 3 core methods

-  Equal Sharing -  Equal Grouping

Equal Repeated Grouping Inverse Equal Sharing Subtraction Multiplication

-  Repeated Subtraction -  Inverse Multiplication

Newfoundland Department of Education, 2010 RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

III. METHODOLOGY Ps: 76 4th graders (30 boys, 46 girls), M(age) = 9.6 yrs 37 5th graders (16 boys, 21 girls), M(age) = 10.5 yrs 20 ÷ 5 =

Measures: 10 division computation problems (pencil-paper) Easy

»  DV: Accuracy and Method type (open-ended) »  5 Easy and 5 Hard

54 ÷ 9 =

Hard

20 multiplication computation problems (computer) »  DV: Accuracy »  10 Easy and 10 Hard

5x4 Easy

9x6 Hard

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

III. METHODOLOGY Inverse Multiple Multiple Multiplication and Additionby by Addition Decomposition Quotient Divisor

Coding: Recall Eq. Sharing à Equal Sharing Eq. Grouping à Equal Grouping RepSub à Repeated Subtraction InvMult à Inverse Multiplication MultAdd-D à Multiple Addition by Divisor MultAdd-Q à Multiple Addition by Quotient IM-Dec – Inverse Multiplication and Decomposition Other - Oth

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 1.  What methods are Gr. 4s using and what are they using most often? Methods Used by Grade 4 40

Horizontal Axis

Percentage Used out of 10

35

1.  2.  3.  4.  5.  6.  7.  8.  9. 

30 25 20

53.3%

15 10 5 0 1

2

3

4 5 6 Type of Method

7

8

Recall Eq. Sharing Eq. Grouping RepSub InvMult MultAdd-D MultAdd-Q IM-Dec Other

9

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 1.  What methods are Gr. 5s using and what are they using most often? Methods Used by Grade 5 40

Horizontal Axis

Percentage Used out of 10

35

1.  2.  3.  4.  5.  6.  7.  8.  9. 

30 25 20

40%

15 10 5 0 1

2

3 4 5 Type of Method

6

7

8

Recall Eq. Sharing Eq. Grouping RepSub InvMult MultAdd-D MultAdd-Q IM-Dec Other

9

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS Inverse Multiplication vs The Rest - Grade 4 70 60

Percentage Used out of 10

Horizontal Axis 50

58%

40

1.  2.  3.  4.  5.  6. 

30 20

Recall Eq. Sharing Eq. Grouping RepSub Other All InvMult

10 0 1

2

3 4 Type of Method

5

6

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS Inverse Multiplication vs The Rest - Grade 5 80

Percentage Used out of 10

70 60

74%

50

Horizontal Axis 1.  2.  3.  4.  5.  6. 

40 30 20

Recall Eq. Sharing Eq. Grouping RepSub Other All InvMult

10 0 1

2

3 4 Type of Method

5

6

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 2. Which grade uses All InvMult most often? All IM Use on Matched Items 5

Number of Questions Answered

4.5 4 **

3.5 3 2.5 2 1.5 1 0.5 0

** p < .01

Grade 4

Grade 5

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 2.1 How many different methods are Gr. 4s using? Number of Methods Used - Grade 4 35

Percentage of Students

30

MNum = 2.76

25 20 15 10 5 0 1

2

3

4 Number of Methods

5

6

7

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 2.1 How many different methods are Gr. 5s using? Number of Methods Used - Grade 5 35

Percentage of Students

30

MNum = 2.78

25 20 15 10 5 0 1

2

3 Number of Methods

4

5

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 2.1 Classifying Gr. 4s as belonging to a dominant strategy.  

Dominant Strategy Used - Grade 4 30

Percentage of Use

25

Horizontal Axis 1.  2.  3.  4.  5.  6.  7.  8. 

20 15 10 5

Recall Eq. Sharing Eq. Grouping InvMult MultAdd-D MultAdd-Q IM-Dec Other

0 1

2

3

4 5 Type of Method

6

7

8

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 2.1 Classifying Gr. 5s as belonging to a dominant strategy. Dominant Strategy Used - Grade 5 35

Horizontal Axis

Percentage of Use

30

1.  2.  3.  4.  5.  6. 

25 20 15 10

Eq. Sharing Eq. Grouping InvMult MultAdd-D MultAdd-Q IM-Dec

5 0 1

2

3 4 Type of Method

5

6

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 2.2 Method Use Gr. 4: Easy vs. Hard Methods used in Easy vs. Hard Questions - Grade 4 20

Percentage Used out of 5

18

Horizontal Axis

16

***

14

*

**

12

Easy

10

Hard

8 6

** **

4 *

2

1.  2.  3.  4.  5.  6.  7.  8.  9. 

Recall Eq. Sharing Eq. Grouping RepSub InvMult MultAdd-D MultAdd-Q IM-Dec Other

0 1

2

* p < .05; ** p < .01; p < .001

3

4 5 6 Type of Method

7

8

9

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 2.2 Method Use Gr. 5: Easy vs. Hard Methods used in Easy vs. Hard Questions - Grade 5 20 18

Percentage Used out of 5

16

Horizontal Axis

***

14 12

Easy

10

Hard

*

8 6 4 2

1.  2.  3.  4.  5.  6.  7.  8.  9. 

Recall Eq. Sharing Eq. Grouping RepSub InvMult MultAdd-D MultAdd-Q IM-Dec Other

0 1

2

3

* p < .05; ** p < .01; *** p < .001

4 5 6 Type of Method

7

8

9

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS

Grouping dominant strategies

- Taught Methods: Eq. Sharing, Eq. Grouping, RepSub -  Pseudo Recall: Recall, InvMult, MultAdd-Q -  Untaught IM: MultAdd-D, IM-Dec

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 3. Are the Gr. 4s correct ? Percent Correct on Division Problems - Grade 4 100 **

90 80

Accuracy - %

70 60 50 40 30 20 10 0 Taught Proc ** p < .01

Pseudo Recall Strategy Group

Untaught IM Proc

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 3. Are the Gr. 5s correct ? Percent Correct on Division Problems - Grade 5 100 90 80 Accuracy - %

70 60 50 40 30 20 10 0 Taught Proc

Pseudo Recall Strategy Group

Untaught IM Proc

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 4.  Multiplication Facts Is performance correlated with performance on division problems?

Gr 4: No, r(46) = .265, p = .076 Gr 5: No, r(37) = .213, p = .207 Is performance correlated with division problems for those with a dominant strategy?

Gr 4: YES, r(37) = .367, p = .025 Gr 5: YES, r(26) = .413, p = .036

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 4.  Multiplication Facts – Gr. 4 Percent Correct on Multiplication Problems - Grade 4 100 **

90

**

80

Accuarcy - %

70 60 50 40 30 20 10 0 Taught Proc ** p < .01

Pseudo Recall Strategy Group

Untaught IM Proc

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

IV. RESULTS 4. Multiplication Facts? Percent Correct on Multiplication Problems - Grade 5 100 90 80

Accuracy - %

70 60 50 40 30 20 10 0 Taught Proc

Pseudo Recall Strategy Group

Untaught IM Proc

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

V. DISCUSSION -- Taught methods are being used, BUT only half of the time, less for Gr. 5s -- Strategy development occurs with experience - Consistent with Siegler’s Model of Strategy Choice - Untaught strategy tends towards inversion methods, especially for Gr. 5 students -  Seems to contradict difficulty children have with inversion questions (Robinson & Dube, 2009)

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

V. DISCUSSION

-- Method use varied by difficulty level - retrieval method used most for easy problems (e.g., Imbo & Vandierendonck, 2008; Lemaire & Siegler, 1995)

-- Division and Multiplication Fact accuracy differs based on dominant strategy classification – BUT only for the Gr. 4’s

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

VI. CONCLUSION -- Children approach division as the inverse of multiplication -- Siegler’s Model of Strategic Choice -  strategy that produces correct answers -  less variety of strategy with age and cognitive ability

- - Multiplication knowledge is important for furthering mathematical concepts in division.

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca

Thank you RCDMC Lab Members Nicole Pelley

Funding Agency

Martha Ennis Kyle Morrissey Supervisor: Darcy Hallett Newfoundland English School Board Schools/Principals/Students

RESEARCH CENTRE FOR THE DEVELOPMENT OF MATHEMATICAL COGNITION www.mun.ca