This document discusses using fuzzy clustering algorithms to group students for differentiated instruction. It describes how fuzzy c-means clustering was used to analyze student responses to math problems and assign each student percentages of belonging to different groups based on their response patterns. Three groups were identified: students who answered all problems correctly, students who answered some problems correctly, and students who answered all problems incorrectly or partially. The document concludes that fuzzy clustering shows potential for identifying student groups but has limitations and requires teacher support to be useful for differentiation. Future applications in intelligent tutoring systems are suggested.

1. Embrace the Fuzz of
Differentiated Instruction
Saad Chahine, PhD
June 2, 2013
CERA | Victoria, BC
13-04-30

2. Differentiated Instruction
• Huge push for teachers to provide Differentiated
Instruction (DI)
• Many publications are oriented to different ways of
attempting to “do” DI in the classroom
• There is a great deal of speculation on the ways in
which you do “DI” in the classroom
• In practice, the attempt to be more differentiated is
often intuitive rather than evidence-based

3. Purpose
- It is almost impossible for teachers to provide students
with individualized attention for prolonged periods
during the day
- It is possible to create smaller groups of student from
a pedagogical perspective
Big Questions:
Can we use mathematical algorithms to identify groups
from students response patterns?
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4. Fuzzy Logic
• Introduced in 1965 by Lotfi A. Zadeh
• Questions the crisp boundaries that
we form that may be artificial
• Is becoming more widely used in
engineering, computer science and
machine learning etc…
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5. Some Interesting Applications
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6. Algorithm
1. K “means” are randomly generated
based on the data
2. Clusters are created with data points
closest to these means
3. The centroid of each cluster
becomes the new mean
4. Repeat steps 2 & 3 until convergence
FUZZY C-Means:
For each point, calculate the
Coefficient of being in the cluster
http://home.deib.polimi.it/matteucc/Clustering/tutorial_html/cmeans.html
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7. Traditional Grouping
Group 1
Sally
Robin
Students
Bob
Sally
Jim
Robin
Group 2
Bob
Jim

8. Fuzzy Grouping
Group 1
Bob
Sally
Jim
Robin
Students
Bob
Sally
Jim
Robin
Group 2
Bob
Sally
Jim
Robin
40% 60%
20%80%
25%
80%
75%
20%

9. Methods
• TIMSS 2011 Math Number -
Reasoning Items - Book 1
• Random selection of 30 students
• Items coded:
– “2” for correct
– “1” for partially correct
– “0” for incorrect
• Analysis conducted using R
“fannyx” ππ99

10. Trading Card Items
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11. Trading Cards Item 1
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12. Trading Cards Item 2
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13. Trading Cards Item 3
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14. Soccer Tournament Item 4
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15. Results
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Group 1 Group 2 Group 3
Student 1 43 40 17
Student 2 20 71 8
Student 3 44 33 23
Student 4 20 71 8
Student 5 20 71 8
Student 6 43 41 16
Student 7 43 41 16
Student 8 22 69 9
Student 9 22 69 9
Student 10 22 69 9
Student 11 45 35 20
Student 12 45 35 20
Student 13 34 47 19
Student 14 43 22 35
Student 15 35 18 47
Student 16 35 18 47
Student 17 42 33 24
Student 18 35 18 47
Student 19 2 1 97
Student 20 2 1 97
Student 21 2 1 97
Student 22 47 23 30
Student 23 42 36 21
Student 24 43 22 35
Student 25 39 30 30
Student 26 2 1 97
Student 27 2 1 97
Student 28 2 1 97
Student 29 2 1 97
Student 30 2 1 97

16. Results
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Group 1 Group 2 Group 3
Student 1 43 40 17
Student 2 20 71 8
Student 3 44 33 23
Student 4 20 71 8
Student 5 20 71 8
Student 6 43 41 16
Student 7 43 41 16
Student 8 22 69 9
Student 9 22 69 9
Student 10 22 69 9
Student 11 45 35 20
Student 12 45 35 20
Student 13 34 47 19
Student 14 43 22 35
Student 15 35 18 47
Student 16 35 18 47
Student 17 42 33 24
Student 18 35 18 47
Student 19 2 1 97
Student 20 2 1 97
Student 21 2 1 97
Student 22 47 23 30
Student 23 42 36 21
Student 24 43 22 35
Student 25 2 1 97
Student 26 39 39 30
Student 27 2 1 97
Student 28 2 1 97
Student 29 2 1 97

17. ππ1717

18. Response Patterns
• Really good at identifying Groups
2 & 3
• Difficulty with Group 1
• Percentages are more important
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19. ππ1919
Group 1 Group 2 Group 3
Student 1 43 40 17
Student 2 20 71 8
Student 3 44 33 23
Student 4 20 71 8
Student 5 20 71 8
Student 6 43 41 16
Student 7 43 41 16
Student 8 22 69 9
Student 9 22 69 9
Student 10 22 69 9
Student 11 45 35 20
Student 12 45 35 20
Student 13 34 47 19
Student 14 43 22 35
Student 15 35 18 47
Student 16 35 18 47
Student 17 42 33 24
Student 18 35 18 47
Student 19 2 1 97
Student 20 2 1 97
Student 21 2 1 97
Student 22 47 23 30
Student 23 42 36 21
Student 24 43 22 35
Student 25 2 1 97
Student 26 39 39 30
Student 27 2 1 97
Student 28 2 1 97
Student 29 2 1 97
Group 3:
-Answered all items wrong
or partially correct on
item 2
Group 2:
-Answered items 1, 2, & 4
correctly (or partially on
Item 2)
Group 1:
-Answered all items
correct
-Answered items 1 & 2
correctly
-Answered item 3 correct
-Answered item 4 correct

20. ππ2020
Group 1 Group 2 Group 3
Student 1 43 40 17
Student 2 20 71 8
Student 3 44 33 23
Student 4 20 71 8
Student 5 20 71 8
Student 6 43 41 16
Student 7 43 41 16
Student 8 22 69 9
Student 9 22 69 9
Student 10 22 69 9
Student 11 45 35 20
Student 12 45 35 20
Student 13 34 47 19
Student 14 43 22 35
Student 15 35 18 47
Student 16 35 18 47
Student 17 42 33 24
Student 18 35 18 47
Student 19 2 1 97
Student 20 2 1 97
Student 21 2 1 97
Student 22 47 23 30
Student 23 42 36 21
Student 24 43 22 35
Student 25 2 1 97
Student 26 39 39 30
Student 27 2 1 97
Student 28 2 1 97
Student 29 2 1 97

21. Fuzzy Clustering for DI
- May be useful in identifying
response patterns for students
- Is not fully informative on its own
- Needs support of educator
- Current format of analysis is not
user friendly
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22. Future
• Intelligent Tutoring/Testing
programs
• Possible alternative to stats
methods that are
computationally heavy
• FCA can easily be programed
into a software program for
educators’ use
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23. Thank You
saad.chahine@msvu.ca
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