Errors in fractions

1. Errors in Rational
Number Operations:
A Case of Preservice
Teachers
Presented by:
SHERWIN E. BALBUENA
Dr. Emilio B. Espinosa Sr. Memorial State College of Agriculture
and Technology (DEBESMSCAT), Masbate, Philippines

2. Contents
 Rationale
 Statement of the Problem
 Objectives
 Significance of the Study
 Methods
 Results and Discussions
 Conclusions
 Recommendations

3. Rationale
 Rational numbers – fractions, mixed
numbers, integers
 Rational numbers have different constructs
(Behr, Lesh, Post, & Silver ,1983)
 Taught as early as Grade III
 Low mastery level of high school graduates,
college entrants
 Preservice teachers’ understanding of
fraction content knowledge is weak (Behr,
Khoury, Harel, Post, & Lesh, 1997; Cramer,
Post, & del Mas, 2002).

4. Rationale
 Preservice teachers find it difficult to
conceptualize fractions (Ball, 1990)
 Can hardly explain fractions to children
and why computation procedures work
(Chinnappan, 2000)
 Cannot operate fractions correctly, even
if they have chosen the correct answer
(Becker & Lin, 2005).
 Future problems posed by this difficulty
 Diagnosis of procedural errors is
imperative

5. Statement of the Problem
 DEBESMSCAT two teacher education programs
 Bachelor in Secondary Education (BSEd)
 Bachelor in Elementary Education (BEEd).
 Admission process
 Enrollees are required to pass the college entrance test
and screenings to ensure that students are highly qualified
to undergo teacher education trainings for four years.
 However, diversity implies that certain learning
difficulties exist among entrants.
 This study is interested about the learning difficulties
exhibited by preservice teachers in understanding
rational numbers.

6. Objectives
 What is the level of performance of
DEBESMSCAT preservice teachers in
operating rational numbers?
 Which of the errors exhibited by
preservice teachers in dealing with
rational number operations are
more prevalent?
 What are the implications for
teaching and learning rational
numbers?

7. Significance of the Study
 Information dissemination of
results to the basic education
teachers
 Diagnosis of the learning
difficulties and research
opportunities for tertiary
educators
 Basis for enhancing preservice
teachers’ procedural skills

8. Methods
 Participants
 38 preservice teachers enrolled in their first of
four-year BEEd program in DEBESMSCAT
 Sampling
 Systematic Random Sampling
 Profile of Participants
 97% are younger than age 25
 76% were female, 24% male
 87% graduated from secondary schools in the
2nd district of Masbate

9. Methods
 Instrument
 Diagnostic pretest with 8 multiple-choice items on
adding and multiplying rational numbers
 Item 1 for identifying errors in adding dissimilar and common
fractions,
 Item 2 in adding dissimilar and uncommon fractions,
 Item 3 in adding a mixed number and a fraction which are
similar and common,
 Item 4 in adding similar fractions,
 Item 5 in adding a mixed number and a fraction which are
dissimilar and uncommon,
 Item 6 in multiplying common fractions,
 Item 7 in multiplying a whole number by a fraction, and
 Item 8 in multiplying a mixed number by a fraction which are
common.

10. Methods
 Instrument (cont’d)
 Example of an item
1/2 + 3/4=
 A. 4/6
 B. 2/3
 C. 10/8
 D. 5/4
 Each distracter has some diagnostic designs to
identify student’s error

11. Results and Discussions

12. Results and Discussions
0
10
20
30
40
50
60
70
80
90
1 2 3 4 5 6 7 8
Percentages of Occurences of Responses in the Item Choices
A B C D
D A A C B B C D

13. Results and Discussions
Total % of errors > Total % of
correct responses
Only 21.05% of the
participants obtained at least
4 marks (50%)
 Very low performance of the
participants in operating rational
numbers

14. Correct vs. Wrong Answers
0
10
20
30
40
50
60
70
80
90
1 2 3 4 5 6 7 8
Item Number
Percentage
of correct
responses
Percentage
of the more
prevalent
wrong answer

15. More Prevalent Errors

16. Procedural Errors
 Item 1 Numerator plus numerator
Denominator plus denominator
Reducing to the lowest term

17. Procedural Errors
 Item 2

18. Procedural Errors
 Item 5
Copying the whole part

19. Procedural Errors
 Item 8
Copying the whole part

20. Conclusions
 Preservice teachers’ knowledge of
rational number operations is very weak
 More prevalent errors were observed in
adding dissimilar fractions and in
multiplying a mixed number by a fraction
 Preservice teachers tend to mix up
memorized fraction rules
 Good at adding similar fractions and
reducing answers to the lowest terms

21. Conclusions
 Lack of sufficient knowledge of equivalence of
mixed numbers and improper fractions
 Confirms that preservice teachers’ procedural
knowledge predominates over their conceptual
knowledge (Forrester & Chinnappan, 2011)
 There is a need for students to gain mastery of
the processes involved in performing operations
on dissimilar fractions
 Preservice teachers are not ready to learn more
advanced topics in mathematics

22. Recommendations
 Improve the quality of teaching and learning
fractions in the elementary and secondary levels
 Enhance students’ retention and conceptual
understanding of fractions
 Give preservice teachers more curative
interventions and trainings in mathematics
 Further studies
 Limitations:
 Small number of participants
 Use of “fixed” questions
 Emphasis on the procedural knowledge

23. 
Thank you very much!
Madamo nga salamat!