Misconceptions

Misconceptions in teaching and learning mathematics Within mathematics if a student gives an incorrect answer to a question or problem, it will be one of two things. The answer can either be a mistake; (Almeida, 2010) “The pupil understands an algorithm but there is a computational error due to carelessness. A mistake is normally a one-off phenomenon.” Or a misconception; “The pupil has misleading idea or misapplies concepts or algorithms. A misconception is frequently observed.” A misconception can be a natural stage of conceptual development. I.e. the idea “multiplication gives bigger numbers” is a valid generalisation when applied to natural numbers; however the idea fails when it is applied to the rational numbers. (Swan, 2001, p. 154) “Although I do not believe it is possible to prevent misconceptions arising, a skilled classroom teacher will plan ahead when giving explanations so that he or she does not actively encourage pupils to believe that learning mathematics is about following an extensive set of unrelated, arbitrary rules. Indeed, it is wor4thwhile offering ‘rules’ to pupils and asking them to discuss their domain of validity.” (Swan, 2001, p. 154) One of the main misconceptions made by secondary school mathematics teachers is by assuming that the year 7 pupils have no prior knowledge of mathematics and so must be taught from basics. However this is not the case as year 7 pupils have been building their mathematics knowledge since the age of 4, so by the time students are in year 7 at secondary school they have been practising mathematics for the past 7 years. It is vital that mathematics teachers consider the misconceptions that may arise within the lessons. Children construct meanings internally by accommodating new concepts within their existing mental frameworks. Therefore unless there is intervention, there is likelihood that the pupil’s conception may deviate from the intended one. Pupils are also known to misapply previously learnt algorithms and rules in domains where they are inapplicable. A large proportion of pupils generally share the same misconceptions. (Almeida, 2010) For a pupil to learn new ideas and skills, they have to actively participate in the construction of their own mathematical knowledge. This involved the reception of new ideas and the interactions of these with the pupil’s extant ideas. The ideal teaching style to use to combat misconceptions within the classroom is diagnostic teaching. This teaching style depends on the students taking much more responsibility for their own understanding, being willing and able to articulate their own lines of thought and to discuss them in the classroom. However a teacher cannot correct a misconception unless they understand the reasoning behind the train of thought that lead to the answer gained. Via discussion with the students the teacher has to diagnose the faulty connection between the student’s extant ideas and the new conception being learnt. Once this faulty connection has been identified the teacher can then challenge or counteract the misconception with the faithful conception. It has been shown that a pupil is more likely to not only adopt the faithful conception but also retain the correct understanding by the approach. Within the United States, the United Kingdom, and some other countries, multiplication is sometimes denoted by either a period or a middle dot, i.e. 5 . 2 or 5.2 However the form of multiplication that uses a period dot is more commonly known as a decimal place rather than multiplication. For example, if the question 4+5.2 was posed to students, two possible answers would be gained. If the 5.2 was considered as a decimal then the calculation would look as follows: 4+5.2=9.2 If the 5.2 was considered as a multiplication then the calculation would look as follows: 4+5.2=4+5*2=4+10=14 These two answers are vastly different and so could cause some confusion as to which of the answers was correctly as both methods are viable. Within the national curriculum for Key Stage 3 mathematics there are many different misconceptions made by pupils in each topic. I have recently taught the first three lessons, out of 6, of the topic fractions, decimals and percentages to a year 8 set 3 class. During the teacher exposition and individual practice work I have observed the students make many of the common misconceptions found in this topic. One of the misconceptions that was observed when teaching the student’s how to convert between fractions, decimals and percentages were as follows: * Thinking that a single digit percentage, i.e. 3%, has an equivalent over 1000, i.e. 31000 instead of the correct answer being over 100, i.e. 3100. In the above slide used as a starter in the opening lesson of fractions decimals and percentages, the students were asked to match up the equivalent fractions, decimals and percentages. Some students got confused with the equivalent fraction linked to the percentage 3% as they could not distinguish which answer was correct between 31000 and 3100 as they are both of a similar form. To combat this misconception the students were asked to take out the calculators as type in the following 3÷100 which gave the answer 0.03 and also 3÷1000 which gave the answer 0.003. The students were then asked to make a connection between the number of 0’s in the fractions and the number of decimal places in the decimal answer. Most students were able to identify that when the fraction was over 100, the answer had two digits after the decimal point and three digits after the decimal place when the fraction was over 1000. One of the main design principles of a diagnostic teacher is: (Swan, 2001, p. 158) 1. Before teaching, assess pupils existing conceptual frameworks “... We attempt to assess pupil’s intuitive interpretations and methods before teaching. This can be quite short, involving just the posing of a few critical questions...” On the above slide the values shown were specifically chosen to identify if any of the students would make the misconception of making the following equivalent pair: 3%=31000 One of the misconceptions observed when teaching the students how to add and subtract fractions were as follows: (Siegler, 2010) * Believing that the numerator and denominator of a fraction can be treated as separate whole numbers. When the students were asked to answer the question 19+69, one of the answers gained was 718 with half of the students in the lesson believing that that was the answer to the questions. When asked what method they used to get to that answer they told me that they had added the numerators and added the denominators i.e. 19+69=1+69+9=718 When the students were asked why they had used that method they told me that that was the method that they had been taught in previous years on how to multiply fractions and so they had then applied this method to the current question. Without wanting to tell the students bluntly that they were wrong I asked them to use the same method on the following question 12+12 to which the students answered 24 with a puzzled expression. I asked them to explain why they had a puzzled expression and was told that they knew that the answer should have come out as 1 not 24. I then showed them my way of working out the fraction by firstly making sure that the denominator is the same on both fractions, adding the numerators of the fractions together and putting the common denominator of the two fractions being added as the denominator of the answer. The students then saw that this gave the answer 22 which is the same as 1 which is the answer they were expecting. Having accepted that the method they were using did not work on all addition questions, the adopted to use the method I had just showed them on a few questions that had been set as exercises within the class and found that they gained the right answer for all questions. Another of the main design principles of a diagnostic teacher is: (Swan, 2001, p. 159) 5. Consolidate learning by using the new concepts and methods through problem-solving “New learning is utilised and consolidated by: * Offering further problems to consider.” This was done in the above situation where the student was corrected on their misconception and asked to practice the new concept to make sure that the new concept works on all questions. Word count: 1386 Bibliography Almeida, D. (2010, March 26). Misconceptions in mathematics and diagnosic teaching. Retrieved November 10, 2011, from Academia: http://exeter.academia.edu/DennisAlmeida/Talks/31014/Misconceptions_in_mathematics_and_diagnostic_teachinghttp://exeter.academia.edu/DennisAlmeida/Talks/31014/Misconceptions_in_mathematics_and_diagnostic_teaching Siegler, R. (2010). Developing effective fractions instruction for Kindergarten through 8th grade. National Center for Education Statistics. Swan, M. (2001). Dealing with Misconception in Mathematics. In P. Gates, Issues in mathematics teaching (pp. 147-165). Oxon: Routledge.