The National Council of Teacher of Mathematics’ [NCTM] Principles and Standards for School Mathematics  states “Effective teaching requires knowing and understanding mathematics, students as learners, and pedagogical strategies” (2000, p. 170). In fact, Ball (1990) and Ma (1999) claimed that teachers’ content knowledge (CK) might be a potential factor affecting classroom instruction. In contrast, Shulman (1986) argued that CK alone is not enough; pedagogical content knowledge (PCK) is also important for making the learning of specific mathematical concepts comprehensible. In the context of learning fractional concepts, it has traditionally been difficult and cumbersome to teach as well as to learn; thus, many have failed to achieve a solid understanding and have negative attitudes towards fractions (Barnett-Clarke, Fisher, Marks, & Ross, 2010; Mewborn, 2003).

Perspectives or Theoretical Framework for the Research

According to von Glasersfeld (1989), individuals acquire knowledge not by absorbing but by constructing it. From this point of view, an individual learns by doing (Dewey, 1938/1997). With regard to preservice teachers (PTs), they must possess profound CK and PCK to facilitate knowledge construction for developing student’s fractional understanding. Subsequently, PTs should cultivate positive attitudes towards fractions that eventually influence their future instructional practices (Timmerman, 2004).

                                                 Methods, Techniques, or Modes of Inquiry for the Research

The present study utilized a partially mixed concurrent dominant status design (Leech & Onwuegbuzie, 2009) in which a quasi-experiment was used for gathering the qualitative data through a convenience sampling of 57 elementary school PTs. The participants were females enrolled in a mathematics methods course during spring 2011 at a public university in Texas. An experienced professor taught the course and also developed a fractional instructional unit based on the Mathematics Texas Essential Knowledge and Skills (TEKS) Connection [MTC Project] module. The class instruction was heavily focused on teaching and learning through hands-on mathematics activities and concrete manipulatives.

The qualitative and quantitative techniques were mixing during the data analysis and data interpretation stages (Nastasi, Hitchcock, & Brown, 2010) to answer the following research questions:

What was the effect of an instructional unit using concrete models on  

(1) the levels of   elementary PTs’ content knowledge and pedagogical content knowledge for teaching fractions?

(2) the attitudes of elementary PTs towards teaching and learning fractions?

                                                           Data Sources or Evidence for the Research

The study involved the use of 10 open-ended tasks that were selected from Sowder, Phillipp, Armstrong, and Schappelle (1998) and Lamon (1999) for measuring PTs’ levels of CK (six items) and PCK (four items). Two mathematics education professors reviewed these tasks to determine the content-related validity (Collins, Onwuegbuzie, & Sutton, 2006). The same tasks were administered during class (CK) and as a class assignment (PCK) before and after the fraction instruction, which lasted three weeks. Additionally, at the end of course PTs were asked to answer an essay question to ascertain whether attitude might or might not have changed after the instruction.

Each written response was assessed based on the degree of accomplishment (1-4 points). For inter-rater agreement, two researchers graded 10% of the written responses and achieved a consistency of 70%. Discussions were conducted to resolve each disagreement before grading the remaining answer scripts. The Statistical Package for Social Science (SPSS) version 17.0 (2008) was used to run the statistical analysis on 53 written responses as four participants did not take one of the assessments. This sample size was adequate to detect differences between two dependent means (paired t tests) at the 5% statistical significance level with .80 power (Faul, Erdfelder, Lang, & Buchner, 2007).

The essay questions were coded using QDA Miner 3.2 (Provalis Research, 2009) based on classical content analysis (Onwuegbuzie & Teddlie, 2003). We read through data, underlined chunks, and coded them into smaller significant parts; then counted the number of codes that were frequently used and might represent important elements of PTs’ attitudes towards fractions. 

Results and/or Conclusion

The t tests results showed the fraction instruction did elicit a statistically significant improvement in the PTs’ level of CK (t= - 4.279, p= 0.000) and PCK (t= - 6.744, p = 0.000). Then, the analysis of individual tasks demonstrated PTs had significant improvement on six fraction tasks of CK and PCK. Specifically, the post assessment measure showed PTs’ level of CK was higher on arithmetic operation, fractional part-whole, and equivalent fractions, a result that was similar to Newton (2008). Similarly, the PCK’s post assessment measure revealed PTs were significantly better on identification of an accurate representation, posing scenarios problem, and analysis of students’ responses.

The analysis of essay questions revealed 18 codes, which emerged from the coding process. Many PTs learned fractions algorithmically (20 PTs) when they were in school and believed the concepts were scary (8 PTs) and confusing (4 PTs). We found that hands-on activities and exploration (38 PTs) might be the most important means that have changed PTs attitudes towards teaching and learning fractions. Meanwhile 63% PTs (34) felt their attitudes have improved, still a number of PTs were concerned (12), needed more time, and practice (14) in what they had learned during the methods course. Sixteen PTs felt confident about teaching fractions after taking the course but 10 PTs worried about explaining the fractional concepts to students. They better understood fractions but were concerned about modeling and teaching the concepts correctly.

In summary, teacher education program is the place for PTs to develop knowledge, teaching skills, beliefs, and awareness for becoming an effective mathematics teacher (Llinares, 2002). Indeed PTs’ prior knowledge and attitudes significantly affect what and how they learn during teacher preparation (Llinares, 2002).

Educational or Scientific Importance of the Research

This study was distinctive because it represents the first study to use mixed analysis techniques to understand the complex phenomena of preservice teachers’ CK, PCK, and attitudes towards fractions. Thus, the present study led to a combination that yielded “complementary strengths and nonoverlapping weaknesses” (Johnson & Turner 2003, p. 299).