Density of decimals : One of the features distinguishing decimals from whole numbers is the density of decimals. Hiebert et al., (1991) found improving the continuity aspects of decimals’ density was particularly difficult. Working with problems involving continuous models in the written tests and the interviews, such as marking a representation of a decimal number on a number line, or finding a number in between two given decimals such as 0.3 and 0.4 were found to be more challenging than working with discreterepresentation task utilizing MAB models. Analysis for this finding suggested that an extra step in finding the unit of the continuous models explained the lower performance on continuous-representation tasks. In Table 2.1, A2 (money thinking) describes students thinking like this.
Likewise, Merenlouto (2003) found that only a small portion of Finnish students aged 16-17 years old in her study changed their concept of density. She attributed difficulties with grasping density to students’ reference to natural numbers and difficulties in extending their frame of reference to rational or real numbers. Some students relied on the possibility to add decimals in their explanations for recognizing density of decimals. Furthermore, she contended that this kind of explanation was based on an abstraction from natural numbers properties rather than a radical conceptual change from natural to real numbers.
Difficulties with density were also evident in studies involving pre-service teachers. Menon (2004) found only 59% of 142 pre-service teachers recognized the density of decimals. A similar trend was noted by Tsao (2005) who found that of 12 pre-service teachers involved in her study, only the six high ability students demonstrated an understanding of density.
The nature of incorrect responses with regard to the density of decimals is reflected in common misconceptions drawing on analogies between decimals and whole numbers. Clearly density will not make sense to students holding misconceptions identified in Table 2.1 such as money thinking, denominator focussed thinking, reciprocal thinking, and place value number line thinking. In general incorrect answers in recognizing the density of decimals could be classified in two categories. The first category of incorrect answers is identifying no decimal existing in between pairs of decimals. Fuglestad (1996) found that most students in her study of Norwegian students claimed there were no decimals in between two given decimals such as between 3.9 and 4 or between 0.63 and 0.64. Similarly, Bana, Farrell, and McIntosh (1997) reported that the majority of 12 year olds and 14 year olds from Australia, US, Taiwan and Sweden displayed the same problem. Only 62% of 14 year olds from Australia and 78% of 14 year olds from Taiwan showed understanding of decimal density. This evidence reflected incorrect extension of whole number knowledge that there is no whole number in between two consecutive whole numbers such as 63 and 64. Note that students holding money thinking (allocated to A2 code in Table 2.1) also will have difficulty in grasping the density notion of decimals and identify no decimals in between decimals such as 0.63 and 0.64. However, these students might identify 9 decimals in between 0.6 and 0.7 for instance, if they interpret decimals only as a number system for dollar and cents.
The second category of incorrect answer translates knowledge of multiplicative relations between subsequent decimal fractions. For instance, Hart (1981) reported that 22 to 39% students age 12 to 15 year-old thought there were 8, 9, or 10 decimals in between 0.41 and 0.42. Similarly, Tsao (2005) observed the same phenomenon in her study with pre-service teachers. She found that three pre-service teachers from a low ability group believed there were nine decimals in between 1.42 and 1.43 by sequencing only the thousandths: 1.421, 1.422,…, and 1.429. Along with most of the students in L and S groups (see Table 2.1), some students holding A thinking, such as A2 thinking with reference to metric measures (m, cm, mm) might possibly respond in this way.
Likewise, Merenlouto (2003) found that only a small portion of Finnish students aged 16-17 years old in her study changed their concept of density. She attributed difficulties with grasping density to students’ reference to natural numbers and difficulties in extending their frame of reference to rational or real numbers. Some students relied on the possibility to add decimals in their explanations for recognizing density of decimals. Furthermore, she contended that this kind of explanation was based on an abstraction from natural numbers properties rather than a radical conceptual change from natural to real numbers.
Difficulties with density were also evident in studies involving pre-service teachers. Menon (2004) found only 59% of 142 pre-service teachers recognized the density of decimals. A similar trend was noted by Tsao (2005) who found that of 12 pre-service teachers involved in her study, only the six high ability students demonstrated an understanding of density.
The nature of incorrect responses with regard to the density of decimals is reflected in common misconceptions drawing on analogies between decimals and whole numbers. Clearly density will not make sense to students holding misconceptions identified in Table 2.1 such as money thinking, denominator focussed thinking, reciprocal thinking, and place value number line thinking. In general incorrect answers in recognizing the density of decimals could be classified in two categories. The first category of incorrect answers is identifying no decimal existing in between pairs of decimals. Fuglestad (1996) found that most students in her study of Norwegian students claimed there were no decimals in between two given decimals such as between 3.9 and 4 or between 0.63 and 0.64. Similarly, Bana, Farrell, and McIntosh (1997) reported that the majority of 12 year olds and 14 year olds from Australia, US, Taiwan and Sweden displayed the same problem. Only 62% of 14 year olds from Australia and 78% of 14 year olds from Taiwan showed understanding of decimal density. This evidence reflected incorrect extension of whole number knowledge that there is no whole number in between two consecutive whole numbers such as 63 and 64. Note that students holding money thinking (allocated to A2 code in Table 2.1) also will have difficulty in grasping the density notion of decimals and identify no decimals in between decimals such as 0.63 and 0.64. However, these students might identify 9 decimals in between 0.6 and 0.7 for instance, if they interpret decimals only as a number system for dollar and cents.
The second category of incorrect answer translates knowledge of multiplicative relations between subsequent decimal fractions. For instance, Hart (1981) reported that 22 to 39% students age 12 to 15 year-old thought there were 8, 9, or 10 decimals in between 0.41 and 0.42. Similarly, Tsao (2005) observed the same phenomenon in her study with pre-service teachers. She found that three pre-service teachers from a low ability group believed there were nine decimals in between 1.42 and 1.43 by sequencing only the thousandths: 1.421, 1.422,…, and 1.429. Along with most of the students in L and S groups (see Table 2.1), some students holding A thinking, such as A2 thinking with reference to metric measures (m, cm, mm) might possibly respond in this way.
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