As recent research has advocated, algebra is among the major sources of confusion to pupils which make them have a negative attitude towards mathematics (Cai & Moyer 2008; Carpenter & Franke, 2001; Carraher & Schliemann, 2007). Any mathematics teacher can easily attest these sentiments. There is also no doubt that pupils can agree to it too. A major cause of this is that most students perceive algebra as too difficult for them to comprehend (Booth, 1988). The question that pops up is why do the students perceive algebra as difficult? Lesly Booth (1988) suggests that to answer this question, we must first find the types of errors that are most common to students who find algebra difficult and try to investigate the causes of the errors.
There are a number of research projects that took this approach. Among the notable ones is a research project conducted in the United Kingdom from the year 1980 to 1983 (Booth, 1988). The students who participated in the research were between the ages of thirteen to sixteen (grades 8 to 10). The students had been studying algebra since they were in grade 7. Though the level of studying was different among the students, all of them had done simple algebra such as simple factorization, solving simple linear equations, simplification of algebraic expressions and substituting in formulas among other algebra. Consequently, the older students had studied more of algebra such as indices, quadratic equations and graphic equalities and inequalities among others.
Despite the age difference and level of study of algebra, the study noted that the students from the different grades had similar errors. Interviews with the students proved that many of the errors could be traced to the ideas that the students had on such aspects as the meaning of letters and variables, types of relationships used in arithmetic, the nature of answers required and the use of notation in algebra (Booth, 1988).
Some student may have a hard time simplifying algebraic expressions such as a+3b. One reason for this may be the how they interpret the operation symbol (Chaiklin, 1989). Usually, students taking arithmetic view symbols as depicting actions that they should perform. For instance, they view the symbol + as requiring them to perform addition and = to give out the answer. This interpretation goes beyond elementary school pupils. Students may not readily appreciate the fact that they can use the addition symbol to signal the action as well as the result of addition and the equal sign signaling a relation rather than requiring an answer (Chaiklin, 1989). However, for a proper understanding of algebra, they need to change their views of such symbols.
Another issue that may be a challenge for students is using notational precision which is critical in algebra when recording statements. Though this precision is also crucial in arithmetic, if the student knows the requirements and performs the operation correctly, then inadequacy results to smaller consequences (Booth, 1988). In arithmetic, for instance, writing 2+5 or 5+2 does not make a big difference so long as the student computes the problem correctly. On the other hand, there is a distinction in algebra between the terms x+y and y+x. More notably is the fact that inadequacy in precision when writing expressions can make students think that division is commutative, just like addition. Such misconceptions can lead to great challenges in later algebra if they are not attended to at an early age.
The notable difference between arithmetic and algebra is the representation of values with letters in the latter. For instance, in arithmetic, the letters g and s may be used to represent grams and seconds respectively rather than representing the values of these quantities as can be done in algebra. Finding ways to this problem may be difficult. For instance, when one is told that 3 m represents 3 meters, they may easily tend to think that 3 p represents three pineapples rather than three times the number of pineapples. Allowing students to choose their letters to represent numerical values as opposed to only taking the initial letter of a given variable may assist them in becoming more familiar with using the letters to represent variables (Booth, 1988).
Different countries may emphasize on different aspects of algebra, and this may have an overall effect on students ability to grasp various algebraic concepts. The Principles and Standards for School Mathematics list four goals related to algebra: (1) Understanding functions, patterns, and relations, (2) analyze mathematical situations using algebraic symbols, (3) use mathematical models to understand quantitative relationships and (4) analyze change which occurs in various contexts (NCTM, 2000). The table below compares the emphasis of five countries curriculum and the emphasis they have put on these goals.
China Russia Singapore South Korea United States
Goal 2 Goal 3
Table 1. Emphasis on goals related to algebra. Adapted from Developing Algebraic Thinking in Earlier Grades: Some insights from International Comparative Studies, by Cai, J., & Moyer, J. (2008).
Professional Development and Preparation of Teachers of Mathematics
Banerjee and Subramaniam, (2012) argue that for students to understand algebra better, there should be professional development and preparation of teachers. Over time, there has been a strong national focus concerning mathematical issues in professional development and preparation of teachers of mathematics and especially at the elementary school level (Carpenter et al., 2005). Unlike in the past, the standards of today have become higher requiring teachers to have competent skills in both teaching and subject knowledge. Each day, new issues arise concerning the individuals responsible for providing professional development and preparing teachers at all levels.
The dominant approach that has been there in the United States for more than 40 years now has been to ensure that teachers are better prepared to teach mathematics in secondary schools has been the requirement of having an undergraduate major in mathematics (Banerjee and Subramaniam, 2012). Usually, prospective teachers have taken courses in linear algebra, calculus, abstract algebra, and analysis. These are the courses that are usually offered for the mathematics major. The assumption is that such a background is quite useful in the teaching of mathematics. The United States investment in mathematics courses is substantial (Banerjee and Subramaniam, 2012). However, there is little knowledge about the effectiveness of these methods of preparation for teaching.
Various discussions exist today about any additional experiences or knowledge about mathematics that may be crucial for teachers who are willing to teach mathematics at secondary schools (Hill, Schilling & Ball, 2004). A lot of effort concentrates on connecting the traditional mathematics present in the undergraduate curriculum to more emphasis on the curricular of the secondary schools that the teachers would eventually be stationed. Many theoretical questions have arisen from researchers concerning the difficulty of separating knowledge and the activities that result in its development (Monk, 1994). The quality of teaching has a long-lasting effect on a students ability to understand various concepts. For this to be the case, a teacher should be effective in the teaching process. The National Commission on Mathematics and Science Teaching (2000) sheds light on the characteristic of effective teachers through this statement:
The most consistent and most powerful predictors of higher student achievement in mathematics and science are: (a) full certification of the teacher and (b) college major in the field being taught. Conversely, the strongest predictors of lower student achievement are new teachers who are uncertified, or who hold less than a minor in their teaching field.
Similar issues emerge in discussions concerning the content and emphasis needed in learning opportunities for teachers (Schilling et al., 2004). There is an increased demand for teacher institutes and professional teacher development experiences that have a strong focus on mathematical content. There is a strong emphasis on the introduction of a wider range of algebra in secondary school curricular and some algebra ideas in middle grades and elementary (Schilling et al., 2004). With this scenario, it makes sense that teachers should have a better understanding and know more algebra ideas. Despite all these, there is still a major question that research has not been able to address well: what algebra ideas do teachers need to understand that will assist them to be effective in the classroom? How can their effectiveness be measured?
Developing Algebraic Thinking of Students at an Early Age
Most students in the United States do not begin to learn algebra until they are in the eighth or ninth grade. As a result, a large portion of them finds it difficult to transit from arithmetic to algebra due them having little experience on the subject (Kilpatrick, Swafford & Findell, 2001). Findings from both national and international assessments indicate that there is an urgent need of development of the algebraic thinking of students in the United States when they are still young (Carraher & Schliemann, 2007). In some countries like China, South Korea, Russia and Singapore, students begin to study algebra formally at an early age. However, it is likely that the educational researchers, curriculum developers and policy makers faced challenges in trying to find the appropriate algebraic expressions for younger children to learn from the United States (Carpenter, Levi, Franke & Zeringue, 2005).
When carrying out a comparison between Davydovs elementary mathematics curriculum with the one that the United States National Council of Mathematics uses in the development of algebra, major differences become clear. The former deviates from viewing algebra in terms of a generalization of the number but rather views that development of the algebraic structure as relationships between various quantities such as weight, volume, area, and length (Cai & Moyer, 2008). The instructional approach emanates from the findings of Luria and Vygotsky that cognitive development is ignited when one overcomes obstacles (Schmittau & Morris, 2004). A study employing Daydovs curriculum which was conducted in a school setting proved that children who used this method were able to obtain solutions to algebraic problems normally (Schmittau & Morris, 2004). Usually, this is not the case in the United States until students reach the secondary level.
Performance of Students on Equation Solving
During the years 1915 1930, researchers made considerable efforts to measure the performance of students on many aspects of the learning of algebra (Kiichemann, 1981). One of the major aspects was the ability of the students to solve algebraic equations. The data the researchers obtained is quite useful as it shows the expectations the researchers made of the students as well as the relative difficulties they experienced in solving various algebraic equations.
Empirical research showed that the greater the step number that students required to arrive at a particular solution, the less likely that they would be able to find a correct solution (Carry, Lewis & Bernard, 1980). Filloy and Rojano (1989) identified that some of the sources of trouble that students may face while solving such algebraic equations might be in combining terms, clearing fractions, and permuting some of the terms across the equal sign. More fundamentally, these troubles might come from the students having a difficulty in understanding the concepts of the equation, variab...
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