Number sense content knowledge and teaching ...

International Journal of Science and Mathematics Education, 15(83), pp.1-19. doi:10.1007/s10763017-9822-6.1

In-service Teachers’ Content and Pedagogical Content Knowledge in Mental Calculations with Rational Numbers Charalampos Lemonidis1 & Helen Tsakiridou1 & Ioanna Meliopoulou1 1 Department of Primary Education, University of Western Macedonia, Kozani, Greece

Abstract The purpose of the present study is to examine content knowledge (CK) and pedagogical content knowledge (PCK) of Greek teachers in number sense and specifically in mental calculations with rational numbers (fractions, decimals, and percentages). Examined within the framework of CK were the type of strategies employed by teachers and the extent of the repertoire of these strategies, which provides an indication of their flexibility. Teachers‟ CK performance in mental calculations with rational numbers was compared with the extent of their strategic repertoire as well as with the PCK they employed when teaching mental calculations with rational numbers. The data revealed that the teachers' high CK performance in mental calculations with rational numbers is positively influenced by the existence of an extensive strategic repertoire. Furthermore, it was found that a high CK performance and an extensive strategic repertoire in mental calculations with rational numbers positively influence the PCK of mental calculations with rational numbers. Keywords Content knowledge. Flexibility. In-service teachers. Mental calculation. Rational numbers

Introduction Fractions, and rational numbers in general, are important mathematical issues in late primary and early secondary education, as they support the development of proportional reasoning and are important for coping with topics in mathematics which follow, including Algebra and Probability (Lamon, 1999). Moreover, in the sense of quantitative numeracy, contemporary curricula demand mathematical knowledge at an advanced level (e.g., Madison & Steen, 2003). Proportional reasoning and calculations are often mentally realized using fractions, decimals, and percentages in various social contexts and life skills (Steen, 2001; Watson, 2004). The subject of fractions and rational numbers in general is considered difficult for many teachers to grasp and implement in their teaching practice (e.g. Ma, 1999; Post, et al., 1993) as well as for many students to learn (e.g. Kerslake, 1986; Nunes, & Bryant., 2009; Stafylidou, & Vosniadou, 2004; Streefland, 1991). Mental calculations are a component of number sense and lead to better understanding of rational numbers (Lemonidis, 2015). The acquisition of number sense has been recognized as a fundamental component of learning mathematics. Number sense, or the capacity to make


sense of numbers and magnitude, promotes the understanding of the decimal system as well as its properties (Reys, 1985). Students tend to use calculation strategies that are based on the analysis and synthesis of numbers (Carraher et al. 1987). Much research (McIntosh et al., 1997; Reys, 1984; Sowder, 1990, 1992; Trafton, 1992), has examined and correlated the students‟ ability in number sense with their flexibility in mental calculations. The results of these studies indicate that number sense is a fundamental condition for the development of students‟ flexibility in mental calculations. Research has demonstrated that teachers play a key role in helping their students develop number sense, with many studies in this regard on pre-service teachers‟ CK and PCK of rational numbers (e.g., Ball, 1990; Behr et al. 1997; Borko et al. 1992; Cramer & Lesh 1988; Depaepe et al. 2015; Newton, 2008; Tirosh, 2000; Turkulku & Yesildere, 2007). Furthermore, a proper learning environment is necessary for students to understand, handle and communicate with numbers and calculations (McIntosh, 2004; Reys, 1994; Siegler & Booth, 2005; Yang & Reys, 2001). It should be noted that according to research, teachers play an important role in helping students develop number sense. However, there have been few studies on in-service teachers‟ CK and PCK of rational numbers (Ma, 1999; Post et al. 1988; Zhou, Peverly, & Xin, 2006). Also, little research has been carried out on in-service teachers‟ number sense performance in mental calculations with rational numbers. While there have been several studies that examine pre-service teachers‟ behaviour in number sense with a focus on rational numbers (e.g. Newton, 2008; Tsao, 2004, 2005; Yang, 2007; Yang, Reys, & Reys, 2009), we found no studies on comparing in-service teachers‟ strategic repertoire with their teaching practices. Unlike the majority of studies which have typically investigated pre-service teachers, this study examined in-service teachers. Our aim was to investigate whether teaching experience diversifies the content and pedagogical knowledge of experienced teachers in terms of number sense of fractions, decimals, and percentages. Moreover, a further aim of the study was to examine the teachers‟ behaviour not only in terms of fractions, but also in terms of decimals and percentages. In the following sections, studies on the teachers‟ content and pedagogical content knowledge as well as mental calculation abilities in rational numbers are presented; the research method is then explained. The main results are thereafter presented and extensively discussed in the conclusion. Teachers’ content and pedagogical knowledge of rational numbers.

The importance of teachers‟ knowledge for teaching has long been established in research literature. The initial work of Shulman (1986, 1987) provides a conceptual framework as well as analytical distinctions between the different kinds of knowledge needed for effective teaching. Shulman (1987) defined seven different categories of teacher knowledge, three of these being content knowledge, curriculum knowledge and pedagogical content knowledge, which comprise the content dimensions of teacher knowledge. Over the past two decades, research into teacher knowledge has focused on two overlapping and interdependent domains: pedagogical content knowledge (PCK) and content knowledge (CK) (e.g. Ball et al., 2008; Depaepe et al., 2015; Fennema & Franke, 1992; Hill et al. 2005; Kleickmann et al. 2013; Krauss et al. 2008; Senk et al. 2012). For an overview of the studies related to CK and PCK of teachers in mathematics see Depaepe et al. (2015). They conclude that: a) the teachers’ CK and PCK are positively related; b) prospective teachers who are more trained in mathematics outperform their less trained colleagues both in terms of CK


and PCK and c) CK and, most importantly, PCK are major determinants of the quality of instruction and, consequently, of students' progress (p. 84). This study examines teachers‟ knowledge in both basic areas of content knowledge (CK) and pedagogical content knowledge (PCK). As presented in „Tasks‟ section, the 24 problems with rational numbers posed in this study belong to the first category of CK, whereas the three questions about teaching practices belong to the second category of PCK. Post et al. (1988) examined the mathematical knowledge of 167 in-service teachers in order to generate profiles of teachers‟ mathematical understanding. Fractions, decimals, percentage problems and interviews were used to investigate the teachers‟ way of thinking along with the procedures followed to solve these mathematical problems, including the way they taught the specific mathematical topics to their students. The average success rate of teachers in problems with rational numbers was 64.75%. Thus, it can be concluded that many teachers simply did not possess adequate mathematics knowledge, while only a minority of those teachers who were able to correctly solve these problems could explain their solutions in a pedagogically acceptable manner. Zhou et al. (2006) found deficits in U.S. in-service teachers‟ knowledge of fractions (CK) and their ability to communicate their fraction knowledge to students (PCK). Such outcomes demonstrate that U.S. in-service teachers lag significantly behind Chinese inservice teachers in CK (concepts, computations, and word problems) and some areas of PCK (e.g., such as identifying important points of teaching the fraction concepts and ensuring students' understanding). The Chinese teachers, on the other hand, performed poorly in comparison to their U.S. counterparts on a test designed to measure General Pedagogical Knowledge (e.g., psychological and educational theories and applications). Olanoff et al. (2014) reviewed 43 research studies on prospective elementary teachers (PTs) mathematical content knowledge in the area of fractions from 1989 to 2013. They indicate that past research was primarily focused on PTs' understanding of fraction operations, predominantly multiplication and division. Recent research also included concepts such as examining PTs' fraction number sense. A general finding of this study is that CK of PTs is relatively strong concerning the execution of procedures, but they generally lack flexibility in moving away from procedures and using "fraction number sense" (e.g. Newton, 2008; Yang et al., 2009) (p. 303). According to the same study (Olanoff et al., 2014), prospective teachers demonstrated the ability to use algorithms to multiply, divide, and compare fractions, but were unable to explain why these procedures worked. For example, most of the prospective teachers were able to perform the procedure of inverting and multiplying for dividing fractions. However, they were not able to explain the significance of this operation (Ball, 1990; Borko et al. 1992; Tirosh, 2000). Most PTs were very dependent on procedures to solve a problem and explained the procedural process when they were asked to justify their strategy (e.g. Caglayan & Olive, 2011; Kajander & Holm, 2011) (pp. 297-298). Several studies indicate that prospective teachers lack CK and PCK regarding rational numbers (see Ball, 1990; Depaepe et al. 2015; Tirosh, 2000; Turkulku & Yesildere, 2007). For example, Depaepe et al. (2015) examined prospective teachers' CK and PCK on rational numbers and the relationship between CK and PCK. They found gaps in prospective teachers' CK and PCK and a positive correlation between them. Furthermore, the authors examined the differences in CK and PCK among prospective elementary teachers (trained as general classroom teachers) and lower secondary teachers (trained as subject-specific classroom teachers) and found better CK but not better PCK on the part of secondary teachers.


Number sense: strategies, flexibility, and strategic repertoire.

As noted in the introduction, mental calculation is an essential component of number sense and leads to better understanding of rational numbers. Important factors that determine the quality of mental calculations and number sense are flexibility and the variety of strategies that can be used. Siegler and his colleagues (Lemaire & Siegler, 1995; Siegler & Lemaire, 1997; Siegler & Shipley, 1995) created the model ASCM (Adaptive Strategy Choice Model), which is a simulation of calculation about making strategic choices and how these change over time due to age and experience. The objective of the ASCM model is to determine how people choose adaptively among strategies, i.e. how to select one strategy among the many available ones in a quick and efficient manner. Strategic repertoire (which refers to the strategies used) is one of the four dimensions of strategic competence and refers to the various strategies employed by a person to solve a number of problems in a given problem area. Studies examining pre-service teachers‟ behaviours in number sense with a focus on rational numbers (Newton, 2008; Tsao, 2004a, 2005; Yang, 2007; Yang, Reys, & Reys, 2009) have shown that teachers‟ performance in number sense as well as their use of number sense strategies and flexibility were low. Newton (2008) found that PTs‟ flexibility was low and did not improve greatly following instruction. He suggests that flexibility was conceptualized as an inclination to choose alternate procedures in contrast to general approaches. For example, 70 out of 85 PTs correctly solved 2/4 – 3/6 on a pre-test by finding a common denominator and subtracting the numerators (e.g., 6/12 – 6/12) and only 7 of the preservice teachers solved the problem by renaming the fractions as ½ (1/2-1/2). Even after instruction the results were similar, 72 out of 85 PTs solved the problem using common denominators whereas 11 used the 1/2 – 1/2 method. Number sense or rule-based strategies for rational numbers A number of studies have been produced on strategies used by students in mental calculations with rational numbers (e.g., Caney & Watson, 2003; Callingham & Watson, 2008; Clarke & Roche, 2009; Lemonidis&Kaiafa, 2014, Post, Cramer, Behr, Lesh, & Harel, 1993; Yang, Reys, & Reys, 2009). McIntosh, De Nardi, and Swan (1994) believe that strategies can be separated into instrumental and conceptual. This division by McIntosh et al. is based on Skemp‟s (1976) terms of instrumental and relational understanding; McIntosh et al. replace the term relational by the term conceptual. According to Skemp, relational understanding is based on the understanding of concepts and their interconnection, so that the student will know what he is doing and why he is doing it without relying simply on the application of rules (rules without reason). In instrumental understanding, the student applies an algorithmic process mechanically. Caney and Watson (2003) distinguish two large categories of instrumental and conceptual strategies for mental operations with rational numbers. These strategies are called instrumental or procedural when students use the techniques learned by heart not accompanied by explanations with a conceptual understanding of the process. The conceptual understanding shows that they grasp the relationships between key structures underlying the numbers and operations. It is possible, of course, that students demonstrate conceptual understanding of a process they use, which is described as a mixed strategy. Yang and colleagues have named subjects‟ strategies as number-sense and rule-based (Yang, 2003, 2005, 2007; Yang et al., 2009). In this study we adopt the terms number-sense and rule-based strategies. Their criterion for distinguishing a strategy as based on number sense


was whether one or more components of number sense are evident in the solution process (Yang, 2003, 2005, 2007). Some examples of number sense strategies are: a. Residual thinking (Behr, Wachsmuth, Post, & Lesh, 1984) wherein the fraction with the smaller residual is taken to be the bigger fraction. When comparing 3/4 and 7/9 a student may think that 3/4 has a residual of 1/4 or 2/8. Consequently the residual for 7/9 (2/9) is less than the residual for 3/4 (2/8). The fraction with the smaller residual is the bigger fraction. b. Benchmarking (or transitive) (Post et al. 1986). When benchmarking, a student may compare a fraction to another well known fraction, such as 1/2, or to a whole number (0 or 1). For example, when comparing 5/8 is bigger than 1/2, and 3/7 is smaller than 1/2, therefore 5/8 is bigger. c. Schematic representation of fractions. Caney & Watson (2003) used a mental picture. For example, in subtraction 1-1/4, a student say “I see 1 as an entire pizza or a clock with four quarters. I remove the 1/4 and 3/4 is left” and d. Conversion of a fraction or a percentage to a decimal before operating. On the other hand, rule-based strategies are based on memorising rules that are not necessarily linked to deep conceptual understanding. Rule-based strategies include: finding equivalent fractions with a common denominator (for adding, subtracting or comparing fractions), cross- multiplying fractions, applying memorized rules as: “In order to divide two fractions, copy the first fraction, invert and multiply the second fraction.” Yang et al. (2009) examined number sense strategies and misconceptions of 280 Taiwanese pre-service elementary teachers who responded to a series of real-life problems. The results of this study show that about one fifth of the pre-service teachers applied number sense-based strategies (such as using benchmarks appropriately or recognizing number magnitude) while the majority of pre-service teachers relied on rule-based methods. This study reports that the performance of pre-service elementary teachers on number sense is low. For example, 95.4% of the pre-service teachers responded correctly, but only 34.3% of them used a number sense method when they were asked to compare the fractions 30/31 and 36/37. Nearly two-thirds of the pre-service teachers relied on writing algorithms to determine the common denominator or change fractions to decimals when comparing fractions and decimals. Such dominant rulebased reasoning also appears in several studies examining PTs‟ ability to compare fractions (Chinnapan, 2000; Domoney, 2002; Whitacre & Nickerson, 2011; Yang, 2007; Yang et al., 2009). The present study.

Initially, the study sets out to answer the following research questions: 1. What are the CK and PCK of in-service teachers in mental calculations of rational numbers (fractions, decimals, and percentages) and their interrelation? 2. Which strategies do teachers know and use in mental calculations with rational numbers? Are these strategies rule-based or are they number sense strategies? What is the extent of the strategic repertoire that teachers use in mental calculations? 3. Do teachers with an extensive strategic repertoire demonstrate better performances in mental calculations?


4. Finally, this study examines teachers‟ performance in CK and the deep-level of knowledge of rational numbers, as well as the breadth of their strategic repertoire with PCK. At this point, we should refer to a number of elements of Greek education concerning mental calculations with rational numbers. The Greek curricula and school books in general refer to mental calculations, underline their significance and give examples together with estimations. Nevertheless, there is no special teaching practice of rational numbers using the logic of number sense, in which, for example, various number sense strategies are presented. Furthermore, teachers are not trained in teaching rational numbers in the logic of number sense. As a rule, in Greece today rational numbers are taught in a traditional environment, in which the focus is more often on teaching using written algorithms than on teaching for conceptual understanding. Method Participants

The participants of this study comprised 70 primary school in-service teachers (22 male, 48 female) from various regions in Greece, who had been teaching fifth and sixth grade classes during the past five years. It should be pointed out that rational numbers are mainly taught in these two grades of primary education. The teachers have on average 6.82 years of teaching experience with a standard deviation of 5.1 years. More specifically, using quartiles points, 25% of the teachers have up to 4 years of teaching experience, 50% of the teachers have up to 6 years of teaching experience and 75% of the teachers have up to 7 years of teaching experience. The youngest teacher has just had two years of teaching experience while the oldest had 33 years of teaching experience. The sample consisted mainly of young teachers so we can say that the results are generally restricted to the young teachers. For this reason, teaching experience was not used as a factor for the statistical analysis. Procedure

All teachers were examined by means of an individual interview comprising 24 questions which aimed to identify the teachers‟ content knowledge in rational numbers (fractions, decimal numbers and percentages) and 3 open questions about their teaching practices concerning mental calculations and rational numbers. They were asked to explain their way of thinking and justify their responses. For every problem, the participants were asked to solve it mentally, mentioning all the possible strategies they could possibly think of. The participants took 16 to 50 minutes to think and answer the questions. Tasks

Table 1 The 19 standard CK problems and 5 deep-level knowledge problems Content Knowledge

Operations

Comparison of fractions

Deep-level knowledge


Fractions

PF1: 1 -

PF6: & (100%),

(100%)*

PF7: & (100%),

PF2: - (98.6%) PF8: & (98.6%), PF9: & , (97.1%), PF3: † PF10: & (95.7%), (88.6%) PF11: & (84.3%), PF12:

&

PF4: Can you find three fractions between 7/8 and 1? (51.4%) PF5: What will happen to the value of the fraction a/b if "a" is increased four times and "b" is halved? (40%)

(67.1%)

PF13: & (58.6%) Decimals

PD1:0.5+0.75 (97.1%) PD2:0.19+0.1 (97.1%)

PD4: Can you find a decimal number between 3.1 and 3.11? (72.9%)

PD3: 1.5 – 0.25

PD5: Estimate the product 47x2.17.

(98.6%)

(38.6%) Percentages

PP1: 10% of 45 (98.6%)

PP6: What percent of 60 is 75? (50%)

PP2: 25% of 80 (97.1%) PP3: 20% of 70 (91.4%) PP4: 25% of 240 (84.3%) PP5: 150% of 12 (80%) *: Rate of accuracy for every problem The 19 CK problems included 11 fraction, 3 decimal, and 5 percentage problems. For example, the 11 fraction CK problems consisted of 3 operations and 8 comparison problems. The 8 fraction comparison problems are the same as those in the research of Clarke and Roche, (2009); the 2 decimal number problems (PD1 and PD2), and 3 of the percentage problems (PP1, PP2 and PP5) are the same as in the research of Caney & Watson, (2003). The 5 deep-level knowledge problems were taken from the research of Post et al. (1988). We use the label „deep-level knowledge‟ for these problems because they require deeper knowledge of rational numbers to solve compared to the previous problems. The three questions that examined PCK were the following: • QA3: How often do you try to develop your students’ mental calculation skills? • QA4: How often do you ask your students to explain the way they thought and the strategy they followed? • QA5: Describe your students’ difficulties in learning rational numbers.


Results Teachers’ Performances, Strategies and Strategic Repertoire in Rational Numbers The teachers' overall performance in fractions, decimals and percentages is presented in Table 2, on a 100-point scale and the results comprise the number of correct responses provided by the teachers to every problem. Table 2 Teachers’ overall, CK and in deep-level knowledge performance on 100 point scale Ν=70

Fractions

Decimals

Percentages

Rational numbers

Overall

M

83.06

80.86

83.56

82.73

Performance

SD

13.53

14.22

17.6

11.46

Performance on CK

Μ

89.96

90.29

97.63

91.14

(19 problems)

SD

11.67

15.13

10.33

9.51

Performance in deep-level knowledge (which are still CK).

M

45.71

55.71

50.00

50.57

SD

38.77

34.62

50.36

30.59

The overall performance in content knowledge, that is the average percent of the success of teachers in the 24 rational number problems (i.e. the average score achieved in all 24 problems out of 100), is M=82.73% with SD= 11.46%. As presented in Table 2, the CK performance in calculations based on the average percentage of teachers' success in the 19 rational number calculations is higher compared to their overall performance (89.96% fractions, 90.29% decimal numbers and 97.63%). The CK (19 problems) performance in rational number calculations shows that the teachers possess the same abilities that they expect their students to have. Concerning their deep-level knowledge performance (which are still CK), teachers attained an average of Μ = 50.57%, but with a wide standard deviation (SD = 30.59%). When it comes to teachers‟ deep-level knowledge of fractions, the average level of their understanding is rather low (Μ = 45.71%) with a wide standard deviation (SD = 38.77%). The teachers appear to have a better understanding of decimal numbers, achieving a mean Μ = 55.71% with a wide standard deviation as well (SD = 34.62%). Their average performance in percentage problems is Μ = 50% with a very wide standard deviation of SD = 50.36%. Because of the large variation regarding their deep-level knowledge performance, the teachers‟ performance in every problem of deep-level knowledge is described separately (Table 1). Therefore, from all the deep-level knowledge problems they were presented with, the one that most teachers (72.9%) were able to successfully solve was: “Can you find a decimal number between 3.1 and 3.11?”. Almost half of the teachers (51.4%) were able to successfully solve the problem “Can you find three fractions between 7/8 and 1?” and precisely 50% of the teachers could successfully solve the problem “What percent of 60 is 75?”. 40% of the teachers were recorded to have been able to successfully solve the problem “What will happen to the value of fraction a/b if "a" is increased four times and "b" is


halved?. Finally, the problem “Estimate the product 47 x 2.17” was successfully solved by 38.6% of the teachers. In conclusion, in-service teachers performed well (> 80%) in simple operations with fractions, decimals, and percentages, as well as in the comparison of fractions. However, they did not perform well in the two last problems comparing fractions, PF12: 5/6 & 7/8 (67.1%) and PF13: 3/4 & 7/9 (58.6%). Finally, the five deep-level CK problems were more difficult for the teachers in the examples considered.

Strategies and strategic repertoire. Table 3 Percentages of teachers’ strategies Questions

Strategy

First Response

Fraction Operations

Rule-based strategy

78.10

Number sense strategy

17.63

Fraction comparisons

Rule-based strategy

37.60


27.60

Rule-based strategy

36.20


59.53

Rule-based strategy

30.30


46.60

1 strat.

2 strat.

3 strat.

50.5

38.56

7.13

47.12

33.93

6.42

30

3.33

28.26

3.14

62.4

Decimals

Percentages

59.72

It should be pointed out that for every problem, teachers were asked to solve it mentally, referring to all the possible strategies they could possibly think of. So every teacher gives one or more responses. The percentages of strategies indicate the number of teachers who successfully solved the task. In the column “First response” of table 3, we present the percentage of the rule-based or number sense strategies used in the teachers‟ first response. In the three operations with fractions, the majority of teachers (78.1%) used rule-based strategies (convert to common denominator for subtractions or invert the second fraction and multiply for division) and a very low percentage of teachers (17.63%) used number sense strategies (convert to decimals or percentage, explain using a representation). When teachers compared the fractions, 37.6% used rule-based strategies and only 27.6% used number sense strategies. For example, for the two comparison problems PF12 (5/6 & 7/8) and PF13 (3/4 & 7/9) with the lowest success rate (67.1% and 58.6%), 41.4% and 34.3% of the teachers respectively, converted the fractions to common denominator, 20% and 8% of them used residual thinking (1/6 > 1/8 and ¼ > 2/9) and 5.7% and 11.4% of them converted them


to decimals. We observe that the number sense strategies, namely the one of residual thinking, which shows a holistic interpretation of fractions and the strategy of convert to decimals, which shows flexibility to change between the symbolic representations of rational numbers, are very seldom used. Nevertheless, the majority of teachers converted fractions to common denominator to compare them. This strategy is characterized as a rule base strategy because firstly, it is based on a very common rule and secondly, teachers use this strategy exclusively (only 5.7% of the teachers answered PF12 and 14.3% PF13 using a second strategy) which shows that they do not know the number sense strategies such as residual thinking. In the 3 operations with decimals, the majority of the teachers (59.53%) used number sense strategies and (36.2%) rule-based strategies. Among the representative number sense strategies used were: a) Converts to natural numbers, e.g., 1.5-0.25 → 150-25=125 → 1.25 (35.7%). b) Dissociation of decimal part, e.g., 0.5+0.75 → 0.75=0.5+0.25 →0.5+0.5+0.25=1.25 (24.3%). Converts to fractions, e.g., 0.5+0.75= ½+3/4 (7.1%). In rulebased strategies, teachers applied mentally the standard algorithm of addition and subtraction of decimals, e.g., 0.19+0.1 teachers say: nine, two, 0.29 (44.3%). In the five percentage problems, the majority of teachers (46.6 %) used number sense strategies while another 30.3% of them used rule-based strategies. The results above show that while teachers more frequently used rule-based strategies than number sense strategies for fractions, they used number sense strategies more for decimals and percentages. To further analyse this result, we compared the strategies used in the three operations of fraction PF1: 1 - , decimal PD3: 1.5-0.25 and percentage PP2: 25% of 80, in which there is the common number

and the operations are similar. With regard to

fraction operation, PF1 teachers overall used rule-based strategies (92.8%) and number sense strategies (84.2%), while regarding decimal operation PD3, they used rule-based strategies (24.3%) and number sense strategies (97.1%), and in percentage operation, PP2 they used rule-based strategies (61.4%) and number sense strategies (98.5%). More specifically, for the operation with fraction 1 – ¼, 92.8% teachers use the rule-based strategy „convert to a common denominator‟. Concerning number sense strategies, 37.1% convert to decimals, 42.8% use representations (e.g. one whole hour minus a quarter in the clock) and 4.3% convert to percentage. For the operation 1.5-0.25 with decimals, 24.3% apply the standard algorithm of vertical subtraction, 37.1% convert to natural numbers (150-25), 38.6% use the strategy „dissociation of decimal part‟ 0.5-0.25 = 0.25 or 0.5 is double 0.25 and finally 21.4% convert to fractions (1.5-0.25=3/2-1/4). To find 25% of 80, 61.4% of teachers use the rule: 25/100 x 80, 74.2% of teachers use number sense strategies and convert 25% to ¼, 18.6% use the benchmark of 50% -50% of 80 is 40, so 25% is 20, 5.7 % use the benchmark of 10% 10% of 80 is 8, and 5% is 4, so 25% is 8+8+4=20. It can be observed that for the simple operations of decimals and percentages, teachers prefer number sense strategies, however, for fraction operations they prefer rule-based strategies. A systematic pattern of teachers‟ behaviour was not found concerning their tendency for errors when they use rule-based or number sense-based strategy, namely if they succeed sometimes with one strategy and fail with another, with the exception of two problems with percentages. When teachers counted 25% of 240 they failed more with a rule-based strategy (6 errors out of 9 responses) in comparison to number sense strategies (4 errors out of 59). Also in the problem 150% to 12 they failed more also with rule-based strategy (7 errors out of 16 responses) in comparison to number sense strategies (2 errors out of 49).


The last three columns in Table 3 show the percent of teachers who used one, two or three strategies effectively in the various categories of operations. It was noted that the strategic repertoire used by teachers is very limited. In operations with fractions, around half of the teachers (50.5%) used only one strategy, which was usually rule-based, 38.56% used two strategies while only 7.13% used three strategies. Only one strategy was used by: (i) 47.12% of the teachers in comparing fraction operations, (ii) 62.4% of the teachers in operations with decimals and (iii) 59.72% of the teachers in percentages. Two strategies for decimal and percentage operations were used by 30% and 28.26% of the teachers respectively, while three strategies were used by only 3.33% and 3.14% of the teachers respectively. It should be emphasised that the teachers' strategic repertoire for dealing with decimals and percentages was even more limited than the one for dealing with fractions. Relation Between Strategic Repertoire and CK Performances As mentioned previously, it was found that the strategic repertoire of teachers concerning rational number calculations was very limited. Certainly, the knowledge of many strategies - a large repertoire - does not automatically imply the existence of flexibility. Although knowledge of many strategies is a prerequisite for the development of flexibility, without the effectiveness of strategy selection in a series of similar problems, such knowledge by itself does not suggest flexibility (Verschaffel et al. 2009). Thus, it is of major importance to examine the relationship of the strategic repertoire with the overall CK performance of teachers; that is to say, their performance in all 24 problems. Two categories were defined with respect to the number of strategies employed by teachers in every operation, namely the "rich repertoire" and "limited repertoire" categories. We used the word „rich‟ to distinguish between the two types of repertoires and without assigning the word „rich‟ its literal meaning. As a criterion for classification, we used the median of the distribution of the number of strategies developed by the teachers, according to which a „rich‟ repertoire was assigned to the teachers who implemented at least two strategies in almost every question. This showed that of the 70 teachers surveyed only 30 had a „rich‟ repertoire while 40 were characterized to possess a rather „limited‟ repertoire. To investigate whether the teachers‟ repertoire affects their performance in rational numbers, the independent samples t-test technique was used. Ttest for independent samples was applied to the dependent variables, the teachers' performance in terms of (a) fractions, (b) decimals, (c) percentages and (d) rational numbers, while the independent variable was the size of the repertoire of strategies of either „rich‟ or „limited‟. The t-test for independent samples showed that the extent of the teachers‟ repertoire had a significant impact on their performance. Specifically, teachers with a „rich‟ repertoire (M = 90.96%, SD = 6.39%) demonstrated a significantly higher performance in rational numbers (t = 7.082, df = 68, p