Pupils Errors and Misconceptions free essay sample

Errors and misconceptions are common place in the classroom especially in mathematics. â€Å"It is important to establish a distinction between an error and a misconception† (Spooner, 2002, p3). An error can be due to a number of different factors, such as lack of concentration, carelessness and misreading a question. On the other hand, a misconception is generally when a student misinterprets the correct procedure or method. â€Å"Students often misunderstand or develop their own rules for deciding how something should be done. This is part of normal development. † (Overall et al. 2003. 127). Whilst many of these invented rules are correct, they may only work under certain circumstances. It is important, when teaching, that error patterns and misconceptions are eradicated and corrected when pupils are learning and that they use procedures and algorithms correctly to obtain the right answer. In this report I am going to focus on the basic errors and misconceptions made by pupils studying algebra, specifically within key stage 3. We will write a custom essay sample on Pupils Errors and Misconceptions or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page Algebra is the generalisation of arithmetic, containing a wealth of symbolic notation, in which students have not previously met. It is not surprising that students find the basic concepts hard to grasp, resulting in many errors and misconceptions. I am going to cover ‘what does the equal sign mean? ’ and students’ understanding of algebraic letters. These are the two fundamental concepts in which pupils need to be fully competent in, in order to be successful in working with algebra. Pupils Understanding of Letters in Algebraic Expressions In order for students to be confident in working with algebra they first need to be able to understand algebraic expressions and variables. In a study by Kuchemann (1981), he found less than half the children, in his study, seemed able to use a letter as a numerical entity in its own right, instead the letter was ‘evaluated’ or regarded as an ‘object’. Children can interpret letters in a number of different ways. The first is that the student may refer to the letter as an object. Letters in algebraic expressions are frequently thought of as representing an object. For example when measuring a length a pupil may refer to the side they are measuring as x, rather than the measurement. The idea of seeing letters as labels (truncated words) rather than as a variable might stem from the use of the letters l and b in relation for the area enclosed by a rectangle. l is seen as truncated â€Å"length† and b as the truncated â€Å"breadth†, but l and b are representing the measurements i. e. number of length units and NOT the object (the sides). (Kesianye, 2001, p16). The ‘fruit salad’ approach to teaching algebra can often lead pupils to believe that a letter stands for an object, reinforcing these misconceptions. When explaining what 3a + 2b means, teachers will often say three apples and two bananas. This is especially common when collecting terms; 5c + 2c means 5 cow plus 2 cows which results in 7 cows. However, the ‘fruit salad’ approach is flawed when questions such as: If a = 2 and 6a = 4b find b, arise. If taught the ‘fruit salad’ method, pupils’ immediate thoughts would be â€Å"6 apples don’t equal 4 bananas†. They take 6a to mean 6 things, or objects, rather than 6 multiplied by a value. Issues like this can also be seen in questions such as; if x = 2 what does 3x equal? Students may answer this question with 32 rather than the correct answer, 6. A remedy for this approach would be to consider the letter as the cost of the object, thus the question could be phrased differently; the cost of 6 apples is equal to the cost of 4 bananas. When teaching algebra it is extremely important to emphasise that the letters represent numbers and not objects. Another misconception can be found when students are asked to evaluate a letter. When asked to solve for x in 4x + 25 = 73, a student literally inserted x=8 into the equation, resulting in 48 + 25 = 73. This student has understood the property of the equivalence as he pasted the correct number to make the equivalence work, although he did not follow the normal equation solving procedures†. (Egodawatte, 2011, p95). This misconception can be stemmed back to the ‘fruit salad’ approach where pupils do not recognise that 4 is multiplied by the x. Many children will try and avoid having to solve pr oblems with a specific unknown. Instead they will give the unknown a value. Kuchemann (1981) presented a group of children with the question â€Å"what can you say about a if a+5=8? 92 percent of the group answered the question correctly however he found that most of them relied on their knowledge of a â€Å"familiar number-bond or counting from 5 until they reach 8†. In classes I have observed I have also found that this is the case. When a class was presented with the question 50m=100, they automatically knew that 50 x 2 = 100. They did not think about the algebraic process in order to determine the correct answer. The procedures pupils learn may be correct or they may be full of misconceptions. It is important or teachers to ensure pupils understand the correct mathematical procedures and algorithms for solving such equations. What Does the Equals Sign Mean? Another misconception I have come across, whilst observing in the classroom, was the misconceptions students make a bout the meaning of the equals sign. Does it mean ‘equivalent to’ or ‘the answer is’? Students interpret equals as an instruction to do something to determine a result rather than as a symbol that indicates the equivalence of two expressions. This arises in a natural way through the use of equals in numerical calculations. It is also encouraged by the presence of a key labelled with an equals sign on many calculators (French 2002, pp 13-14). â€Å"When two algebraic expressions are combined together with an equals sign, it is called an equation† (Egodawatte, 2011). In an equation the equals sign is used to express the equivalence between two sides of an equation whereas in arithmetic, normally students are given an operation to act upon on the left side of the equals sign and they are to write their answer on the right hand side of the equals sign. When presented with a question such as simplify x + x + 3 students may be able to collect the x terms together resulting in 2x+3 however they become baffled as to what to do next. Many students will over simplify this and write x + x + 3 = 2x + 3 = 5x as they will recognise the equals sign as a symbol asking them to compute something instead of a relationship. This is because in arithmetic the equals sign is the symbol to announce a result containing no operational signs. â€Å"The presence of the operator symbol, +, makes the ‘answer’ appear unfinished† (Lovell, n. . p13) therefore students are reluctant to accept 2x + 3 as their final answer because the expression seems incomplete. This can be very confusing for students who have learned that the equals sign means ‘the answer is’. The idea of a balanced scale can be introduced to students to help them understand the meaning of the equals sign when it is used in equations. â€Å"Students can connect representations of a balanced scale with operations that preserve equalities in an equation. The equals sign is synonymous with the centre of the scale† (Foster 2007, p166). If a weight is added to one side of the balanced scales, then it must be added to the other to maintain the balance. Foster illustrates the question â€Å"solve 3x + 5 = 11 for x† through the use of a diagram, shown below. On the left side of the scales are three boxes each representing the unknown value x and 5 marbles. On the right hand side of the scales are eleven marbles. To get the unknown value by itself 5 marbles may be removed from the left side of the scales. To keep the scale balanced, 5 marbles must be removed from the right side. This illustrates the process of subtracting equal quantities from each side of an equation. Now we are left with the three unknown values, which are equal to each other, on the left side of the scales and 6 marbles remain on the right side of the scales. We may separate the blocks so we can see the 3 separate values. We can also share the 6 marbles into three groups. This can be shown in the illustration below. It is easy to see from the diagram that one block is equal to 2 marbles. Thus x = 2. Depicting a set of scales is only a useful tool for students if they understand that an equation works in the same way as a set of balanced scales, where both sides are equal to each other, i. . have the same value. Students must understand the correspondence between the arithmetic operations and their scale counterparts of adding objects, removing objects, or partitioning objects. With these understandings, students can solidify meanings of solving equation. (Foster, 2007. p166). The scales demonstrate that if you add or subtract the same value from both sides of an equation then eq uality is conserved. This representation allows students to understand that the equals sign means ‘equivalent to’ as well as ‘the answer is’. Conclusion My research has identified a number of different meanings that can be given to the letters in algebra and to the equals sign. This can often be very confusing and cause misunderstanding when pupils are problem solving in algebra. It is therefore extremely important for teachers to explain these varied definitions to prevent misconceptions from developing. It is necessary for students to grasp these different meanings as algebra appears in different parts of the key stage 3 and 4 curriculums; in mathematics and science. Further review is needed to cultivate a clear assessment of what factors help students to fully grasp all aspects of algebra. We already know that even very basic mathematical concepts such as addition of whole numbers involve complicated cognitive processes. Since teachers are already familiar with those basic concepts, this leads them to ignore or underestimate the complexity by taking a naive approach to teaching those concepts (Schoenfeld, 1985; Edgodawatte, 2011. p22). Regardless of the previous knowledge students have from studying general processes in arithmetic; they will still find it bewildering when coming across symbols for the first time. If students are unfamiliar with algebraic expressions, notation and symbols then the students understanding and method may not be what the teacher intends. Research on student errors and misconceptions can provide support for both teachers and students. Whilst it is not always possible to teach in a way that eradicates all errors and misconceptions, it will allow teachers to come up with methods of overcoming these problems. Teachers must be able to apprehend pupils thought processes and their understandings in order to provide lessons which reduce pupils’ errors and misconceptions and support them in their learning.