Turkish Pre-service Middle Level Mathematics Teachers’ Knowledge for Teaching Fractions

Figures & data

Table 1 The Components of the Pre-service Teachers’ Mathematics Knowledge for Teaching Framework (Li & Kulm, 2008, p. 836)

Components	Specifications
1. Common content knowledge	The type of mathematics content knowledge that is commonly required in school mathematics for all students. For this type of knowledge, pre-service teachers need to know no less than in-service teachers.
2. Specialized content knowledge	The type of mathematics content knowledge that is more specialized, and is at a higher and more abstract level than what school students need to learn. It also includes the expectations for pre-service teachers to know not only what and how, but also why for specific content topics. Because pre-service teachers are still in their college program studies, they are in a position to learn more about this type of mathematics content knowledge.
3. Knowledge of content and students	The knowledge about students’ learning difficulties and misconceptions related to specific mathematics content topics. Because of their limited teaching experience, pre-service teachers may have very limited exposure to this type of knowledge.
4. Knowledge of content and teaching	The type of knowledge that is used in the practice of teaching, such as the use of various representations for teaching specific mathematics content topics, the use of problem solving for teaching mathematics and in assessment. Given pre-service teachers’ inexperience in teaching, we will not expect pre-service teachers to be proficient in this type of knowledge.
5. Knowledge of mathematics curriculum	Part of Pedagogical Content Knowledge in Shulman’s categorization (Shulman, 1986), but not part of MKT. As curriculum knowledge is an important part of what pre-service teachers need to learn through their program of study, we include this component as well. It is the type of knowledge about state mathematics framework and the process of curriculum transformation from the intended to the implemented and to the achieved. However, we will restrict this type of knowledge requirement as not including the details about topic sequence and specific learning requirements.

Table 2 The Distribution of Participants with respect to Their Background Characteristics

Pre-service middle level mathematics teachers’ background characteristics
Gender	f (%)	Cumulative grade point averages	f (%)
Male	12 (24)	Between 2.00 and 2.49	15 (29)
Female	39 (76)	Between 2.50 and 2.99	24 (47)
Enrollment in the fractions course		Between 3.00 and 3.49	9 (18)
Yes	13 (25)	Between 3.50 and 4.00	3 (6)
No	38 (75)	Methods of Teaching Mathematics course scores
Age		Between 50 and 59	2 (4)
Twenty-two	29 (57)	Between 60 and 69	25 (49)
Twenty-three	16 (31)	Between 70 and 79	14 (27)
Twenty-four	6 (12)	Between 80 and 89	9 (18)
		Between 90 and 100	1 (2)

Table 3 Pre-service Teachers’ Knowledge for Teaching Fractions by Components and Tasks

Components and tasks	M	SD
Common content knowledge	0.82	0.24
Task 1	0.88	0.31
Task 2	0.66	0.45
Task 3	0.90	0.26
Specialized content knowledge	0.35	0.15
Task 4	0.31	0.28
Task 5	0.33	0.26
Task 6	0.23	0.24
Task 7	0.53	0.22
Knowledge of content and students	0.76	0.12
Task 8	0.46	0.14
Task 9	0.90	0.25
Task 10	0.92	0.16
Knowledge of content and teaching	0.50	0.15
Task 11	0.83	0.27
Task 12	0.59	0.28
Task 13	0.30	0.17
Task 14	0.26	0.43
Knowledge of mathematics curriculum	0.66	0.14
Task 15	0.91	0.15
Task 16	0.69	0.32
Task 17	0.40	0.24
Overall mean score	0.62	0.08

Figure 1. Sample participant responses to Task 2.

Figure 2. Sample participant responses to Task 5a.

Figure 3. Sample participant responses to Task 8.

Figure 4. Participant 23’s response to Task 14.

Table

8. A teacher asked his/her students to shade one-third of a triangle. Ayşe shaded the triangle as given on the right. Is Ayşe’s answer wrong? If yes, what is her mistake? If you were Ayşe’s teacher, what would you do to have Ayşe notice her mistake (adapted from Johnson, 1998)?

Table

Fraction topics	Yes	No
a. Ordering fractions	()	()
b. Locating fractions on the number line	()	()
c. Multiplication and division with fractions	()	()

Table

Statements	True	False
a. Comparison of unit fractions is taught in grade 5.	()	()
b. The common denominator algorithm should be the first strategy to introduce when teaching addition and subtraction of fractions.	()	()
c. The 7th grade mathematics curriculum gives more weight to the teaching of rational number concepts when compared to the teaching of rational number operations.	()	()
d. The method for converting repeating decimals into rational numbers in the form of a/b is first taught in grade 7.	()	()

Li, Y., & Kulm, G. (2008). Knowledge and confidence of pre-service mathematics teachers: The case of fraction division. ZDM, 40(5), 833–843.

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Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

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Johnson, N. R. (1998). A descriptive study of number sense and related misconceptions about selected rational number concepts exhibited by prospective elementary teachers ( Unpublished doctoral dissertation). University of South Florida, Florida.

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