"But what about the oneths?" A year 7 student's misconception about decimal place value.[FIGURE 1 OMITTED] Amy: Can you explain to me why you thought the place value of the 8 was tenths? Trissy: Well, because the 7 is in the oneths column, so the 8 must be in the tenths column. The "oneths" column? As revealed in the above dialogue, I recently identified a misconception mis·con·cep·tion n. A mistaken thought, idea, or notion; a misunderstanding: had many misconceptions about the new tax program. about decimal place decimal place n. The position of a digit to the right of a decimal point, usually identified by successive ascending ordinal numbers with the digit immediately to the right of the decimal point being first: value from the examination response of a Year 7 student named Trissy. Trissy and I had been going over a recent decimals examination, when I noticed that Trissy's answers concerning place value were consistently off by one place. After questioning Trissy about the question pictured above, I realised that she had been mistakenly mis·tak·en v. Past participle of mistake. adj. 1. Wrong or incorrect in opinion, understanding, or perception. 2. Based on error; wrong: a mistaken view of the situation. thinking that the first place after the decimal point (character) decimal point - "." ASCII character 46. Common names are: point; dot; ITU-T, USA: period; ITU-T: decimal point. Rare: radix point; UK: full stop; INTERCAL: spot. had the value "oneths". I proceeded to explain to Trissy the structure of decimal notation decimal notation A representation of a fraction or other real number using the base ten and consisting of any of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and a decimal point. (Figure 2): [FIGURE 2 OMITTED] Still confused, Trissy responded "But what about the oneths?" I felt it was important to get to the bottom of this misconception, so I asked Trissy to explain to me why the first place after the decimal point was "oneths." Trissy When you have a whole number, like 346, the 3 is the hundreds column, the 4 is in the tens column, and the 6 is in the ones column, right? So then when you have a decimal Meaning 10. The numbering system used by humans, which is based on 10 digits. In contrast, computers use binary numbers because it is easier to design electronic systems that can maintain two states rather than 10. , the decimal point is like the middle, so you have the same columns on the other side but they go the opposite way and they have "-ths" on the end. Amy: Do you know what the "-ths" mean? Trissy: Yeah, like, if it was 8 hundredths, it means 8 out of 100, and if it was 8 tens, it'd be 8 out of 10, so 8 oneths is 8 out of 1. Amy: Trissy, can you write "8 out of 1" as a fraction for me? Trissy: Isn't 8 over 1 just 8? Amy: So is it a whole number or a part of a number?" Trissy: A whole number ... Ohhhh, I get it. Oops! The mirror metaphor After further investigation, it was interesting to find that other teachers and researchers had also identified this misconception and suggested that it was related to the "symmetry symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. " of the decimal notation (Ball & Bass, 2000; Chinn, 2008; Hiebert, Wearne & Taber, 1991; Ministry of Education, Ontario, 2006; Moskal & Magone, 2000; Resnick, Nesher, Leonard, Magone, Omanson & Peled, 1989). Ball and Bass (2000) suggest that a student's asking, "Where is the oneths place?" emanates from the reasonable expectation that if there is a ones place to the left of the decimal point, and a tens place to the left of that, there should be a symmetry to the right of the decimal point. The decimal point becomes a "mirror," reflecting the place values to the left of the decimal point on to the right side. In their study of students' understandings of decimals, Stacey, Helme and Steinle (2001) describe the confusion between decimals, fractions and negative numbers as a consequence of the so-called "mirror metaphor." However, in their discussion of the applications of this conceptual metaphor In cognitive linguistics, conceptual metaphor refers to the understanding of one conceptual domain in terms of another, for example, understanding time in terms of space (e.g. "time flies"). A conceptual domain can be any coherent organization of experience. , they do not highlight the effect it has on decimal place value. It can be argued that the "oneths" misconception is a prime example of the application of the mirror as a conceptual metaphor. Interestingly, when applied correctly, the mirror metaphor actually explains the structure of decimal place value, however the important distinction to be made is that the symmetry is based around the ones column and not the decimal point. "Oneths": A rational error Talia Ben-Zeev (1998) offers an insight into the psychology of the errors students make in mathematics, describing many of these errors as rational: errors which are "logically consistent and rule based See rules based. rather than being random" (p. 366). She suggests that when faced with an unfamiliar problem, rather than give up, people will construct their own rules or strategies in order to solve it. These strategies make sense to those who created them, however the procedure can lead to erroneous erroneous adj. 1) in error, wrong. 2) not according to established law, particularly in a legal decision or court ruling. solutions (Ben-Zeev, 1998). The strategies constructed in order to assist in solving an unfamiliar problem are usually based on prior knowledge and experience. But as Resnick and colleagues (1989, p. 6) suggest, "prior knowledge of whole numbers and fractions can both support and interfere with construction of a correct concept of decimals." Trissy's application of her understanding of the place value of whole numbers to be mirrored in the place values of fractions of whole numbers is an example. The fact that decimal place value concepts are embedded Inserted into. See embedded system. in a structure that shares key features of place value with whole numbers could suggest to students that the extension of this system is identical to the existing one and lead them to ignore the differences between the two, and this could even be heightened by teachers' attempts to help students use their prior knowledge of whole numbers to facilitate learning the decimal system decimal system [Lat.,=of tenths], numeration system based on powers of 10. A number is written as a row of digits, with each position in the row corresponding to a certain power of 10. (Resnick et al., 1989). Implications for practice When considering student misconceptions Misconceptions is an American sitcom television series for The WB Network for the 2005-2006 season that never aired. It features Jane Leeves, formerly of Frasier, and French Stewart, formerly of 3rd Rock From the Sun. , it is important for teachers to consider what can be incorporated into their lessons in order to address, or avoid, these misconceptions. For example, it might be helpful for teachers to make references to the "endless Base 10 chain" (Steinle & Stacey, 1998, p.418; see Figure 3) when explaining the structure of decimal notation to students. This is one of the important properties of the Base 10 numeration numeration, in mathematics, process of designating Numbers according to any particular system; the number designations are in turn called numerals. In any place value system of numeration, a base number must be specified, and groupings are then made by powers of the system that seems to be overlooked by students with misconceptions (Steinle & Stacey, 1998). [FIGURE 3 OMITTED] By highlighting this chain, emphasis is taken away from the decimal point and is placed on the role of, and relationships between, the individual digits within the decimal. As Chinn (2008) has explained, students tend to focus on the decimal point partly because that is how they are taught e.g., "line up the decimal points; then you can add" (p.19), and partly because it is different. Placing too much emphasis upon the decimal point could increase the likelihood of misconceptions, such as that held by Trissy, developing. Encouraging students to recognise that the symmetry of the decimal revolves around the "ones", and thus highlighting the fact that role of the decimal point is to identify which digit A single character in a numbering system. In decimal, digits are 0 through 9. In binary, digits are 0 and 1. digit - An employee of Digital Equipment Corporation. See also VAX, VMS, PDP-10, TOPS-10, DEChead, double DECkers, field circus. falls into the "ones" position, may help students overcome the desire to label incorrectly the first position after the decimal point. The key to understanding the development of student misconceptions is to ask students to explain their thinking. Time constraints In law, time constraints are placed on certain actions and filings in the interest of speedy justice, and additionally to prevent the evasion of the ends of justice by waiting until a matter is moot. of classroom teaching make it difficult to consult with each and every individual student about their thought processes This is a list of thinking styles, methods of thinking (thinking skills), and types of thought. See also the List of thinking-related topic lists, the List of philosophies and the . . However, when a particular error keeps surfacing, such as that revealed in Trissy's decimals examination, simply marking the response as incorrect will not assist the student. A two-minute conversation, such as that reported at the beginning of the article, can help a student's mathematical understanding. References Ball, D. L. & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83-104). Westport, CT: Ablex. Ben-Zeev, T. (1998). Rational errors and the mathematical mind. Review of General Psychology, 2 (4), 366-383. Chinn, S. (2008). The decimal point and the ths. Mathematics Teaching incorporating Micromath, 208, 19. Ministry of Education, Ontario. (2006). Number sense and numeration, Grades 4 to 6: Decimal numbers. Ontario: Author. Moskal, B. M. & Magone, M. E. (2000). Making sense of what students know: Examining the referents, relationships and modes students displayed in response to a decimal task. Educational Studies in Mathematics, 43 (3), 313-335. Resnick, L. B., Nesher, P., Leonard, F., Magone, M., Omanson, S. & Peled, I. (1989). Conceptual bases of arithmetic errors: The case of decimal fractions. Journal for Research in Mathematics Education, 20 (1), 8-27. Stacey, K., Helme, S. & Steinle, V. (2001). Confusions between decimals, fractions and negative numbers: A consequence of the mirror as a conceptual metaphor in three different ways. In M. van de Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education Vol. 4, (pp. 217-224). Utrecht: PME PME Petites et Moyennes Entreprises PME Professional Military Education PME Pequenas e Médias Empresas (Portugal) PME Petite et Moyenne Entreprise PME Psychology of Mathematics Education PME Pi Mu Epsilon . Steinle, V. & Stacey, K. (1998). Students and decimal notation: Do they see what we see? In J. Gough & J. Mousley (Eds), 35th annual conference of the Mathematics Association of Victoria Vol.1 (pp. 415-422). Melbourne: MAV MAV Micro-Air Vehicle MAV Municipal Association of Victoria (Australia) MAV Mitarbeitervertretungen (German) MAV Magyar Államvasutak (Hungarian State Railways) . Amy MacDonald Amy Macdonald may refer to:
Charles Sturt University Charles Sturt University (CSU) is an Australian multi-campus university in New South Wales and the Australian Capital Territory. It has campuses at Bathurst, Albury-Wodonga, Dubbo, Orange and Wagga Wagga. <amacdonald@csu.edu.au> |
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