Errors and misconceptions are common place in the classroom especially in mathematics. “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. “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. p127). 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. 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 Küchemann (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 “length” and b as the truncated “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 “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...
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