Teaching and Learning Algebra - Essay Example

Teaching and Learning Algebra Most find it difficult to understand and apply algebra, especially when first introduced to the This paper examines some of the reasons for this and focuses specifically on student difficulties with three major algebraic issues. Students’ Persistent Algebra Difficulties The National Curriculum has for many years set forth standards for teaching algebra to students from ages younger than 16 years through 19 years. An extensive analysis of the impact of these standards was conducted by the Joint Mathematical Council and The Royal Society during 1994-1997. It concluded that the standards were not achieving adequate results from students, and made many recommendations for improvement (Joint, 1997, pp.28-32). Despite this attention from some of the U.K.’s highest scholars, algebra students of all ages continue to struggle, and there is a consensus that it is best to introduce students to algebra after they have learned arithmetic, thus beginning in pre-16 schools. Introducing Algebra When introducing something new to anyone, it is often helpful to begin by defining major terms, thus briefly explaining the nature of the subject and providing some context for what is to come. Unfortunately for students, there does not seem to be a single, readily understandable definition of algebra, as shown by these representative examples: The branch of mathematics in which letters and other symbols are used to represent numbers and quantities in formulae and equations. (Compact, 2010, n.p.) The science of equation solving. (Kieran, 2004, p.139) The mathematics of working with variables. (Mathwords, 2008, n.p.) The abstract study of number systems and operations within them. (Weisstein, 2010, n.p.) The mathematics of generalized arithmetical operations. (Wordnet, 2010, n.p.) If an instructor uses a version of some of these examples, students may be denied a useful frame of reference from the very start of their algebra studies. However, the last example does relate algebra to arithmetic, a subject which most students will have studied, may know reasonably well and may actually use in their daily lives. But even though algebra and arithmetic both involve signs, equations, numbers and computation, approaching algebra with an arithmetic mindset can cause students major problems. As noted by Kieran (2007) among others, these problems arise from the different roles of the equals sign (=) and the minus (–) sign in algebra compared to arithmetic. The Equals Sign As Filloy and Sutherland (1996, p.147) explain, in arithmetic: “the lefthand side of the equation corresponds to a sequence of operations performed on numbers… the righthand side represents the consequence of having performed such operation.” As a result, when students first are presented with an algebraic equation they tend to immediately try to compute an answer. Kieran (2007, n.p.) points out that this causes confusion because a student “might be able to correctly interpret algebraic equations such as 2x + 3 = 7 but not equations such as 2x + 3 = x + 4.” Thus, when introduced to algebra, students must reorient their thinking and recognize that “the expressions on both sides of the equals sign are of the same nature (or structure), and that there are actions which give meaning to the equality of the expressions (for instance, the action of substituting a numerical value for the unknown)”. (Filloy and Sutherland, 1996, p.147) The Minus Sign The second major arithmetic-related problem is that students need to interpret the minus sign in additional new ways. As summarized by Kieran (2007, n.p.), the minus sign in algebra “has to be extended from a sign denoting subtraction to include negative something as in – 3, and the opposite of as in – x.” This is often explained by showing that a negative something is simply another way of saying that one is adding the opposite of that something; for example, in 5x – 3 = 5x + (– 3), – 3 on the left side should be thought of as the adding the opposite of + 3 on the right side. But students’ difficulty with the minus sign is exacerbated when terms grouped within parentheses are preceded by a minus sign, for example, 5x – (3 – x). In these cases, instructors emphasize that the opposite includes everything within the parentheses, or in our example 3 and – x, so the sign of every term inside the parentheses has to be changed. A common technique for doing this is for students to think of the expression as having an implied factor of a negative one preceding the parentheses, so that when simplifying the expression, the student’s thought process might look like this: 5x – (3 – x) = 5x + (– 1) [3 + (– x)] =  5x + (–1) (3) + (–1)(–x) = 5x – 3 + x = 6x – 3 Brennan (2010, n.p.) offers a very practical way to handle the minus sign in algebra: “It is helpful to always think of minus signs as being ‘stuck’ to the term directly to their right. That way…the ‘adding the opposite’ business will be taken care of because the minus signs will go with whatever was to their right.” The process to handle the minus sign before parentheses results in simplifying the expression. Thus, in the above example, 5x – (3 – x) becomes 6x – 3, a more concise and clearer way of restating the original expression without changing its value. In algebra, there is another rule for simplifying expressions, known as combining like terms, which is the third major issue confounding algebra students. Combining Like Terms Unlike the issues of the equals sign and the minus sign, solving the issue of combining like terms is a matter of following concrete rules rather than grasping concepts. Like terms are those that contain the same variable raised to the same power, regardless of their coefficients. To illustrate, consider simplifying the expression 5(4x – 3) – (3x + 2) + 2y – x2 First, eliminate any parentheses: 20x – 15 – 3x – 2 + 2y – x2 Next, identify constants whether a term is a constant or a variable: constants are – 15 and – 2, variables are 20x, 3x, 2y and x2 Identify those variables that have the same letter raised to the same power: includes 20x and 3x but excludes 2y and x2 Add all like terms together: 20x + (– 3x) = 20x – 3x = 17x Array all variables: 17x + 2y – x2 Add constants together: – 15 + (– 2) = – 17 Array all variables and constants: 17x – x2 + 2y – 17 Compared to the conceptual challenge of adjusting from an arithmetic mindset to an algebraic one, this should be the least difficult of the three issues discussed in this paper. References Brennan, James W. (2010). JamesBrennan.org [online]. [Accessed 02 May 2010]. Available from: . Compact Oxford English Dictionary [online]. (2010). [Accessed 2 May 2010]. Available from: . Filloy, E. & Sutherland, R. (1996). Designing Curricula for Teaching and Learning Algebra. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde, (Eds.) International Handbook of Mathematics Education. Dordrecht: Kluwer Academic Publishers. Joint Mathematical Council and The Royal Society (1996? 01 July 1997). Teaching and Learning Algebra pre-19. London: Authors Kieran, Carolyn (2004). Algebraic Thinking in the Early Grades: What Is It? The Mathematics Educator. 8, pp.139-151. Kieran, Carolyn (2007). What Do Students Struggle with When First Introduced to Algebraic Symbols?. National Council of Teachers of Mathematics; Research Brief [on line]. none, [Accessed 02 May 2010], Available from: . Mathwords: Algebra [online]. (2008). [Accessed 02 May 2010]. Available from: . Renz, John and Weisstein, Eric W. (2010). Wolfram Mathworld: Algebra [online]. [Accessed 02 May 2010]. Available from: . Wordnet: A Lexical Database For English [online]. (2010). [Accessed 02 May 2010]. Available from: . Read More