Evaluation of Number Sense on the Subject of Decimal Numbers of the Secondary Stage Students in Turkey
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Abstract
The aim of this study is to examine number sense on the subject of decimal numbers of the secondary stage students in Turkey. Number sense, which can be described as well-understanding of numbers and calculations, is one of the main foci of mathematical education. When students improve their number sense, which are ability to estimate, mental computations and analytical thinking that will increase their use of mathematics in daily life. So as to measure students’ number senses about decimal numbers, a test called Number Sense Test about Decimal Numbers which was prepared by the researchers was applied to 573 students (Secondary-stage students: 6th, 7th and 8th grades.) in 6 schools from the different regions in Turkey. Later on, in order to find out the possible solutions, 9 students were interviewed. As a result of the study, students’ number senses on decimal numbers turn out to be very low. Root causes of these can be considered as the use of rule-based solution techniques, imperfect knowledge and misinformation about decimal numbers. This type of sense has a good correlation with students’ ages, however, it turns out that there is no correlation between genders of the students and their number senses. In addition to these, medium level correlation was found out between mathematical achievement and number sense on decimal numbers. It is believed that after the importance of number sense is acknowledged, the problems of failure will be prevented.
Keywords
Abstract
The aim of this study is to examine number sense on the subject of decimal numbers of the secondary stage students in Turkey. Number sense, which can be described as well-understanding of numbers and calculations, is one of the main foci of mathematical education. When students improve their number sense, which are ability to estimate, mental computations and analytical thinking that will increase their use of mathematics in daily life. So as to measure students’ number senses about decimal numbers, a test called Number Sense Test about Decimal Numbers which was prepared by the researchers was applied to 573 students (Secondary-stage students: 6th, 7th and 8th grades.) in 6 schools from the different regions in Turkey. Later on, in order to find out the possible solutions, 9 students were interviewed. As a result of the study, students’ number senses on decimal numbers turn out to be very low. Root causes of these can be considered as the use of rule-based solution techniques, imperfect knowledge and misinformation about decimal numbers. This type of sense has a good correlation with students’ ages, however, it turns out that there is no correlation between genders of the students and their number senses. In addition to these, medium level correlation was found out between mathematical achievement and number sense on decimal numbers. It is believed that after the importance of number sense is acknowledged, the problems of failure will be prevented.
Keywords
- There are lots of international resources on number senses. In these studies, from different grades, students’ number senses are examined in different countries (Alajmi & Reys, 2010; Howell & Kemp, 2006; Mohamed & Johnny, 2010; Yang, 2005b; Yang & Li, 2008; Yang, Reys, & Reys, 2009). Results showed that levels of students’ number senses are not sufficient. In one of studies, 808 3rd grade students’ number sense from Taiwan was examined (Yang & Li, 2008). Approximately, 34% of accuracy rate was obtained in each component of the number sense and this shows that students in Taiwan do not have good performance on number sense. Another study was carried also in Taiwan on 5th grade students to examine their success on number sense (Yang, Li, & Li, 2008). According to the study results, the students performed best on “recognizing relative number size” and performed worst on “judging the reasoning of estimates of computed results”. Yang’s (2005b) study showed that rather than number sense strategies, students tend to use rule based methods and written standard algorithm. Similar results were obtained on a study done with pre-service teachers (Yang, et al., 2009). According to the result of these studies, preservice teachers could not obtain good performance on number sense while they preferred to use rule-based strategies.
- In addition to researches about students’ number sense, in the literature, there also exist some researches about how different teaching methods affect students’ number sense development (Griffin, 2004; Irwin & Britt, 2005; Kaminski, 2002; Markovits & Sowder, 1994; Yang, 2003, 2005a; Yang, Hsu, & Huang, 2004; Yang & Wu, 2010). Researchers showed that use of realistic number sense activities has a positive effect on number sense improvement.
- In other studies on number sense, its relation with other abilities is examined (Louange & Bana, 2010; Pike & Forrester, 1997; Tsao, 2004; Yang & Huang, 2004). In these studies, it was observed that there is a correlation between number sense and estimation, mental computations and problem solving.
- Number of studies on number sense in Turkey is limited with thesis. In the first study, 95 6th grade students were tested. According to results of this study, only 9% of students used number sense (Harç, 2010). Another study was carried with secondary stage of primary school students so as to see their number senses’ changes according to their grade, gender and mathematics performance. Results of this study showed that their number senses are seriously low (Kayhan Altay, 2010). The results also revealed that most commonly used method was rule-based methods on each component of number sense questions. Aim of the Study
- One component of number sense, “Understanding of the general meaning of numbers” underlies the importance of 10-number system which consists of whole numbers, fractions, and decimals. Decimal numbers, which are used in daily life as much as in science, have been recognized for some time to be a significant source of learning and teaching difficulties (Stacey et al., 2001). According to these studies, misconceptions of decimal numbers can be listed as below (Resnick et al., 1989; Widjaja, Stacey, & Steinle, 2011; Yılmaz, 2007)
- Sowder and Markovits (1989) stated that meaningful understanding of the size of fractions and decimal numbers can help students in developing number sense in general. When students’ misconceptions are taken into consideration, it can be stated that students’ number senses are not sufficient. For instance, a student whose number sense is not improved can normally think that 0.897 is greater than 0.9 just by looking at their number of decimal digits. On the other hand, a student whose number sense is high can easily say that the second number is greater than the first one. Therefore, as long as students develop their number senses, misconceptions that they have about decimal numbers will be solved as a result.
- After the revision made in 2005 by the Turkish Ministry of National Education (MEB), some changes were made. After these changes take place, although the term of number sense is not included in the curriculum, its effect can be realized. Using abilities like estimation, mental computation, and flexible thinking effectively is one of the aims of this program (MEB, 2009). Although number sense’s importance was emphasized with this aim, it wouldn’t be wrong to state that necessary actions were not taken so as to make students to have number sense (Umay, Akkuş, & Paksu, 2006)
- Decimal numbers are included in the curriculum starting from 4th grade. Acquisitions about this subject happen mostly on 6th grade. For example, it is expected from 6th grade students to have the ability of comparing decimal numbers in terms of their size, estimation of the solution of computations with decimal numbers and knowing how to round a decimal number (MEB, 2009).
- Methodology Study is designed with a survey method which aims to describe a situation in the past or current as it is (Karasar, 2000). Quantitative data of the study were obtained via Number Sense Test about Decimal Numbers (NSTDN) and qualitative data were obtained via interviews. Sample 573 students from 6th, 7th and 8th grade were participated in the study (284 girls and 289 boys). Study was carried in different regions of Turkey at 19 classes in 6 different schools. These schools are settled on different regions with respect to economical status. Distribution of the students according to their grades and gender can be seen in Table 1. Table 1. Distribution of students according to their grades and gender
- Study was done in the spring semester of 2010-2011 using the improved version of the test that included 16 questions. It was given in a 40-minutes lecture hour. Before the test started, the teacher informed the students about the instructions of the test. The students were told that they shouldn’t use any pen or pencil to make written calculations; they should just do mental computations and estimate the result. Also by stating “without detailed calculations”, it was tried to prevent written calculations in the test. It was forbidden to pass a question or move into the next page until the previous page was completed. Thus, it is tried to provide that the students spend equal time for the questions and answer accordingly.
- In addition to above, additional to students grades, gender etc., it was also asked that their mathematic achievement, because in the study, one of the main aims was to investigate the relation between mathematical achievement and the level of number sense about decimal numbers. To specify students’ achievement, 2010-2011 Fall semesters’ math grade marks were used. Marks were in the scale of 1 to 5.
- 2. The insufficiency of the conceptual knowledge of students in terms of decimal numbers can be shown as a second reason. As also mentioned in performed studies (Arslan & Ubuz, 2009; Bilgin & Akbayır, 2002; Resnick et al., 1989), the students are hard put to understand the decimal numbers subject and have misconceptions about the subject. For instance, success ratio of the question given in Table 8 also shows that students could not totally get the idea of decimal numbers and they have some misconceptions about decimal numbers. And, for instance, the correct answer ratio for the question in Table 8 (How many different decimal numbers between 1.52 and 1.53?) reveals that the students were not able to configure exactly the meaning of decimal numbers and that there are deficits in their knowledge.
- 3. Another reason of the obtained results is that teachers may have deficiencies about number sense. Yang, Reys and Reys (2009) found that teachers in Taiwan are not capable of using number sense and they tend to use written methods. In our country, since there is no study about the topic, it is impossible to reach a definite conclusion like this. However, since teachers are educated in a similar environment, it can be deduced that they also tend to use rule-based strategies. For a teacher who uses rule-based strategies and thinks that number sense is not required it is impossible to teach number sense to his/her students. Teachers who will implement the program should be motivated and should be taught how to prepare an environment for the students so that they can be successful during education of number sense.
- The least successful component was “being able to judge the reasoning of a computational result that includes decimal numbers” in 6th, 7th and 8th grade students’ study. This result is also parallel to Yang & Li (2008) and Yang, Li & Lin (2008)’s results. The researchers from Taiwan emphasize this failure. On the other hand, “Being able to use benchmark appropriately” component is the one that students obtained the best success. This result is consistent with Harç’ (2010) study. According to Harç, the main reason why students obtained the best result from this component is that they are kind of visual questions. Estimating the objects’ weights and sizes are enough to solve these questions. Not using any algorithm or rule may be considered as a reason of this success.
- According to research results, it was observed that age has a positive effect on number sense of decimal numbers. This result also shows consistence with other results (Aunio, Ee, Lim, Hautamäki, & Van Luit, 2004; Aunio et al., 2006; Yang, 1995) and inconsistence with other studies (Kayhan Altay, 2010). As algorithmic written and rule based methods are learned, students internalize them more (Markovits & Sowder, 1994). This situation reduces the use of number sense based strategies of students.
- Research shows that gender does not have a significant effect on number sense. This result also shows consistence with other results in the literature (Aunio, et al., 2004; Aunio, et al., 2006; Harç, 2010; Kayhan Altay, 2010; Yang, 1995; Yang & Li, 2008). When NSTDN grades of students and their mathematical achievements are compared, moderate level correlation was found. Similar studies can be found in the literature (Kayhan Altay, 2010; Yang, et al., 2008). This situation is a demonstration of significant relation between mathematical success and number sense. This moderate correlation indicates that even the student in this study having high mathematics achievement is not able to sufficiently use number sense base strategies at problem solving. It can be said that number sense education is in adequate.
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