The learning and teaching of mathematics - Essay Example

 COMPARING TEACHING AND LEARNING MATHEMATICS EXPERIENCES WITH EXPERIENCES FROM LITERATURE Introduction Decimals have a significant role in interpretation of ration numbers. Conversely, they are considered as important sources of learning complexity in children. Many children face difficulties in ordering decimals, scale reading and operating with decimals. Investigative studies show that similar problems also exist in teachers who improve perceptions and misconceptions that the subject matter of decimals is hard. This paper develops a critical analysis of different literatures on teaching of mathematics compared to school teaching and learning experiences. To achieve this, this paper will be limited to teaching of decimals to primary school children aged 5/6 to 10/11 years old. Further, in relation to teaching decimals, the paper will analyze decimal notation, denominational thinking, and reciprocal thinking as components in teaching decimals in mathematics. Decimal Notation Decimal notation is a crisis experienced by children in mathematics. I have faced instances where some children are unable to distinguish between small and large values when presented with decimal points. When the subject of decimals is first introduced to primary school children, there is a general belief that when the number is long, the value is high. This means that there is an existing problem in the ability to differentiate the values of digits with decimal points. For example, when I ask children to assign the number with the highest values between 5.555, 5.55 and 5.5, 5.555 is given as the answer. The reasons children give are arguably connected to the way they view whole numbers and denote decimal points. This trend can be explained by a number of misconceptions in decimal notation. There is a general belief that numbers with more digits represent high values than those with small values. 5.555 is believed to have a large value than 5.5 because 5.555 has four digits that make the number look bigger, and 5.5 has two digits that make the number look smaller. While this explanation may be true in some cases, several combination of numbers may not adhere to the misconception. For example, in differentiating between 0.12, 0.111 and 0.1012, children pick 0.1012 as the number with the greatest value because 0.12 has three digits, 0.111 has four and 0.1012 has five, hence, the more the number of the digits, the higher the value of the number. In reality, 0.12 has a higher value compared to 0.1012. My observation leads me to conclude that, children’s brains are conditioned to an assumption that numbers with many digits have more values than those with fewer digits. Detailed research conducted reveals most children fail in exams because of associating small numbers with small values and the vice versa. According to the researchers, the number assumption problem mostly occurred among primary school children because of the underdeveloped reasoning of decimal points in whole numbers (Muir and Sharyn 2014:3-15). Therefore, Muir and Sharyns report confirms the association of digit numbers to values as a result of the cognitive thinking of children. I have also encountered challenges in explaining the existence of decimals in numbers. I have invented a slogan in my classrooms that, ‘decimal points can never represent whole numbers and whole numbers can never represent decimal points’. For example, children view 5.55 as 555, and 5.5 as 55, hence, according to them, 555 is greater than 55. This may be somehow true, but when presented with 0.12, 0.111 and 0.1012 the numbers are being read as 12, 111 and 1012 respectively. So 0.1012, read as 1012, has a greater value than 0.111, read as 111, and 0.12, read as 12. My observation leads me to conclude that this trend is common due to the ignorance created by children on the existence of decimal points in whole numbers. However, it is important to note that they do not premeditate on ignoring the decimal points, but I believe that most of them are just not aware of existing numbers with decimal points. The trend can also be explained by the introduction of whole numbers during the first years of study that familiarises children with whole numbers only. As a result, children get poor grades in exercises that require application of decimal points because they use whole numbers to attempt the exercises instead of using numbers with decimal points as instructed by the teacher. Studies conducted reveal that children develop false impressions of decimals that make them generalise numbers leading to the creation of the decimal point ignored error (DPI) in tests and exercises (Mayson n.d.: par. 4). This studies confirm the teaching and learning trends that I encounter when teaching decimal points. Children create figurative ideas that numbers with decimal points are whole numbers, which, is a false impression. Further, Mayson develops a theoretical outlook on the outcomes of assuming decimal points that presumably explains the failure of children in exercises that require application of numbers with decimal points. I also have observed that children have problems in relating to and arranging numbers that have decimal points. For example, a student would read 5.55 as ‘five point fifty five’ instead of ‘five point five five’, and 5.555 as ‘five point five fifty five’. This results to a number of errors in connection to the size, and value of decimal numbers, where the mention of ‘five fifty five’ gives the connotation that 5.555 has a greater value than 5.55 that is connoted to ‘ point fifty five’. Some may argue that, in reality 5.555 has a high value than 5.55, but this does not occur in all cases. For example, take the case of assigning values to 0.12, 0.111 and 0.1012. The latter is assumed to have the highest value. This is because it is read as ‘zero point one thousand and twelve’ that is greater than ‘zero point twelve’. At this point, I would not deny that almost all children in upper primary are aware of the existence of decimal points, but they unable to relate and arrange the numbers from either least to greatest or vice versa. According to my assessment, habits of reading, arranging and viewing digits after decimal points as whole numbers create this habit. This trend could be explained by confusion of student’s skills on arranging and comparing numbers. Different studies show that, children have mistaken beliefs relating to ordering of numbers (Muir and Sharyn 2014: 8) are caused by the ability to ‘learn how to learn’ solving problems in mathematics and making sense in mathematics. Muir and Sharyn explain the unconscious and habitual predisposition of reading figures after decimal points as numbers I have encountered where, a child denotes 0.1012 as ‘zero point one thousand and twelve’ and picks the meaning of the last words after ‘point’ to identify the value of the number. Denominational Thinking Children view numbers of tenths as having fewer values than numbers of hundredths. For example, children would give the impression that the fraction of 50/400 is greater than 50/50. For example, when told to choose the large value between 50/400 and 50/50, children would pick 50/400. Let us explore the reasons. First, the hundredth digit 400 is perceived to be large than the tenth 50. Secondly, even after dividing the two numbers, the value of 50/400 represented by 0.125 is perceived to have a higher value than the value of 50/50 represented by 1. Thirdly, ‘one twenty five’, where a child ignores the decimal point is perceived to have a higher value than one. This and many other are simply explanations developed to proof children’s stands about determining values of decimal points. However, the facts remain that 50/50 has a large value than 50/100, whether in decimal or fraction form. This trend is explained by; first, the creation of false beliefs that numbers of hundredths have high values than numbers of tenths as they view hundreds as having large values than tenths. Secondly, the creation of the perception of ‘large is larger’; hence, large numbers with decimal points are synonymous to large values and vice versa. Thirdly, mistakes and misinterpretation created during reading of numbers with decimal points where numbers after decimal points are read as whole numbers and not single digits. Lastly, the trend is caused by the assumption that the decimal digit does not exist. All these assumptions rise from denominational thinking that hundredths have higher values than tenths. Lack of children’s understanding in arithmetic procedures also create decimal denotation. From my assessments and observations, most children are used to uncomplicated mathematical operations that involve using whole numbers, and do not use decimals. This creates false impressions among children, who believe only in the existence of simple mathematical expressions using whole numbers. Therefore, when decimal units and introduced in mathematical operations, most children find mathematics difficult, and end up confusing or assuming numbers with decimal points are whole numbers. Early exposure of children to basic multiplication and addition procedures would make children think that, numbers get bigger due to the increase in value of digits. In most cases, children believe that division makes numbers smaller. However, when it comes to dealing with decimal points, the numerous digits presented by operations that include numbers with decimal digits confuse some of the basic rules change and children. According to literatures reviewed, the relationship between fractions and decimals as a ‘coined denominator’ (Muir and Sharyn 2014: 8) signifying that children invent models of thinking that hundredth have a bigger values than tenths leading to a falsified impression that the former has high value than the latter. This is replicated in my observations where children make grave errors in assigning values to fractions. The late introduction of decimals to children is mentioned in a research by Ubuz and Yayan that questions the accuracy of school curriculums in introducing mathematical arithmetics to children (Ubuz and Yayan 2010: 789). According to the research, children identify themselves more with whole numbers than decimal points; hence, explaining the trends created in denominational thinking where students have developed displaced ideas in dealing with fractions. Reciprocal Thinking From my professional observation, I have noted that children view decimals as negative numbers; hence, they read numbers after decimal points as whole numbers. For example, a child would conclude that 0.1012 is larger than 0.12 because when relating to whole numbers, digits 1012 represents a large value than 12. This is as a result of the development of a falsified impression that associates decimals with negativity. Also, there is a rounded mentality of children viewing numbers with short decimal digits as having shorter values and those with many digits and having larger values. Literatures reveal that short decimal numbers are synonymous to large values (Ubuz and Yayan (2010: 791). This develops the notion that ‘short is large’ implying that, in the case of determining the large value between 0.1012 and 0.12, children would identify 0.12 as the number with the highest value. Ubuz’s and Yayan’s studies completely contradict with my experience in teaching and learning. Further, the evidence explained about the decimal denotation and denominational thinking reveal that children believe that ‘large is large’ and ‘short is small’. Therefore, this evidence contradicts the literature reviewed from Ubuz and Yayan about reciprocal thinking among children. Conclusion This paper has so far compared school experiences in learning and teaching decimals in mathematics to various literatures about decimals. From the analysis, it is clear that all the literature reviewed reflect common problems faced when teaching decimal concepts to children. Up to this point, the paper has summarized three basic misconceptions about decimals created by children due to different experiences in learning mathematics. However, it is vital to note that as children advance in education, they accept and gain correct insights about decimals. It should also be the intention of teachers and educators to eradicate misconceptions. It is also fundamental to note that, it may be hard to eradicate student’s misconceptions as it is also hard to teach without creating false impressions. Therefore, many areas in mathematics need extreme understanding as a measure to eradication of hypothesis created behind them. Developing teaching methodologies that eradicate and prevent misconceptions is complex and hard, but achievable with the application of immeasurable knowledge and skills in classrooms. Bibliography MAYSON, S. (n.d.). Teaching Decimals to a year 7 class.http://www.did.stu.mmu.ac.uk/. Retrieved February 28, 2014, from http://www.did.stu.mmu.ac.uk/cme/Studen MUIR, T , AND SHARYN L. "What do they know? A comparison of pre-service teachers and in-service teachers' decimal mathematical content knowledge."http://www.cimt.plymouth.ac.uk/. N.p., n.d. Web. 28 Feb. 2014. Read More