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# Do our students really have the arithmetic knowledge to start algebra Analysing misconcepti

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Do our students really have the arithmetic knowledge to start algebra? Analysing misconceptions. Stephen J. Norton QUT Tom J. Cooper QUT

Abstract
The study of algebra has traditionally been one of the most problematic areas of study for secondary students. This paper examines arithmetic prerequisites for algebra study. A sample of secondary students who were about to begin a unit of algebra was tested for their ability to do the prerequisite arithmetic. Their responses were analysed and revealed that most of them were poorly equipped for algebra study. The patterns of their responses can be used to inform teaching strategies.

Introduction Sfard, (1995) described algebra as a science of generalised computations. Difficulties in learning algebra have been well-documented (Herscovics & Linchevski, 1994). Algebra processes have been divided into operations, which refers to processes, algorithms and actions and structural conceptions. Structural conception refers to thinking in terms of abstractions (Sfard, 1991). Usually in Queensland it is introduced in late year 8 or early year 9. The research literature indicates a number of concepts need to be understood before students can begin algebra study they include: ? The concept of equal, that is, both sides are equivalent and that information can be processed in either direction in a symmetrical fashion (Linchevski, 1995). It has been noted that some students understand equal to be a place where something should be written (Filloy & Rojano, 1989), or as “makes or gives” (Stacey & MacGregor, 1997). ? The concept of variable has also been problematic for students (Cooper, BoultonLewis, Atweh, Willss & Mutch, 1997). For example Kieran (1992) reported that when students were given a rectangle with sides of 7 and f + 3 and were asked to calculate number only a few 13 to 15 year old students were able to consider the letter as a generalised number and even fewer were able to interpret the letter as a variable. The majority of students treated the letter as a concrete object or ignored it. But even before looking at algebra proficiency an understanding of basic operational skills is essential. For example take the task of solving for x in the expression below which is found in the Canadian text The Learning Equation Mathematics 9 (Ministries for Education Alberta British Columbia Manitoba, Saskatchewan the Northwest & Yukon Territories, 1998). It should be noted that

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similar problems are found texts in commonly used Queensland schools, including Mathematics 9 (Priddle, Davies & Pitman, 1991). 2 /5(x – 1) = - ? (2 – x) In order to solve this equation the following conceptual understandings are necessary: ? Equal concept ? Directed numbers ? Fractions ? Order convention ? Variable concept. If any one of these understandings is lacking the solving is almost impossible. For example misconceptions related to the equal sign might result in a student not doing the same operation to both sides of the equation. Students who do not understand directed numbers are likely not to change the sign of 2 and – x when they multiply through. Lack of understanding of fractions will probably result in students not seeking appropriate ways to convert the denominators to 1. Misconceptions in order convention (and the distributive law) will probably lead to problems related to the sequence in which the students try to solve the problem. And finally if the students do not understand the variable concept it is likely that even if they do solve the equation their solution will probably lack meaning. That is the students will see the exercise as an abstract activity involving the application of arbitrary rules. The question as to whether many students begin algebra study without prerequisite arithmetic skills has been examined previously. For example, BoultonLewis, Cooper, Atewh, Pillay, & Willss, L. (1998) studied 33 students over three years from grade 7 to 9 using interview techniques. Their study probed students’ understanding of commutative and inverse laws of operations, meaning of equal, meaning of unknown, variable concept and solutions of linear equations. Their results indicated that by Year 9 most students had sufficient understanding of these concepts to operate operationally on algebra problems. That is they were able to use arithmetic operations to gain closure. However, they noted that about half the students still did not understand equal in the algebraic sense to need to do the same operations to both sides to maintain equivalence. Further, these authors concluded that pre-algebra instruction should include focus on operational laws, equality as equality of sides leading to equivalence; inverse procedures and use of letters to represent unknowns. This paper follows on from their study in exploring the nature of student preparedness to begin algebra. Method The study uses descriptive statistics and qualitative analysis of a pencil and paper test to report on students’ ability to do arithmetic tasks considered being necessary for the study of algebra. While pencil and paper tests are more limited in their ability to probe students’ thought the authors believe that useful understandings can be derived from analysis of students’ procedures. This test was designed by the authors to test students’ abilities to do arithmetic that research has indicated necessary for beginning symbolic algebra study. The students in this study were 45 Year 9 students and 9 Year 10 students who were about to begin a 35-hour unit on algebra. All these students had passed Year 8 prerequisite subjects which included the arithmetic operations included in this study. These students were enrolled in a middle sized coeducational school located in a

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Item If I add 56 to the right hand side how do I keep the equation equal? 139 × 43 = 5977 Can you work out the answer to (36 + 24) 6 if you are not allowed to add 36 and 24? Explain. Could you work out the answer to the problem 5 × (6 + 7) if you were not allowed to add 6 and 7? Explain. The triangle s and the square n represent unknown numbers. Can you calculate the answer to s + (n + 7) if you know s + n = 11? Write the fraction that has been shaded Concept Equal concept % correct Response 10%

Distributive law

24 %

Distributive law

8%

Associative law and operations

26 %

Fractions

90 %

This is 1/3 , add extra counters to make up the whole. n n If is 2/5 make up a box to represent the whole 6+4÷2=

Fractions Fractions

68% 38%

Order of operations

30%

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7 × (2 + 1) = 20 ÷ (5 –1) 2 × 1/8 12 ÷ 1/3
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/3 + 24/4 =

32 Find the area of a rectangle length 6 cm and width 3cm Write the inverse of 5/1

Order of operations Order of operations Multiplication of fractions Division by fractions Addition of fractions Squaring Area Inverse

98 % 84 % 36 % 6% 24 % 88 % 36 % 42 %

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6

7

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differ radically. It would be interesting to probe the effect of teaching arithmetic concepts while students are engaged in algebra study. Previous research on cognitive load in relation to the use of concrete representations when teaching algebra suggest that the teaching of arithmetic within an algebraic may be problematic (BoultonLewis et al., 1997). It would be interesting to know the proportion of Queensland schools which routinely test for prerequisite arithmetic knowledge before beginning algebra study. The findings of this study indicate that wider research needs to be undertaken to determine if poor understanding of pre-request arithmetic is more extensive in Queensland schools that has been indicated by this study. For example, it could be argued that the results are distinctive to the study school. However, the concepts examined are usually taught in primary school as well as Year 8. This suggests that there may be merit in secondary schools cooperating with their primary feeder schools to design and implement programs that help students to become prepared for the study of algebra.
Copyright declaration The author(s) assign(s) to the Queensland Association of Mathematics Teachers and other educational and non-profit institutions a non-exclusive license to use this document (and any included computer files) for personal use and in courses of instruction, provided that the article is used in full and this copyright statement is reproduced. The authors also grant a non-exclusive license to the Queensland Association of Mathematics Teachers to publish this document in full on the World Wide Web. Any other usage is prohibited without the express permission of the authors. QAMT Inc. ? 1999.

References Boulton-Lewis, G. M., & Cooper, T. J., Atweh, B. Pillay, H. Wilss, L & Mutch. (1997). The transition from arithmetic to algebra: A cognitive perspective. In E. Pehkonen (Ed.), Proceedings of the International Group for the Psychology of Mathematics Education. (pp.2: 144-151). Helsinki: University of Helsinki. Boulton-Lewis, G. M., Cooper, T., Atewh, B., Pillay, H., & Willss, L. (1998). Pre-Algebra: A cognitive perspective. In A. Oliver & K. Newstead (Ed.), Proceedings of the 22nd conference of the International Group for the Psychology of Mathematics Education. (pp.2: 185-192). Stellenbosch, South Africa: University of Stellenbosch . Cooper, T., Boulton-Lewis, G., Atweh, B., Willss, L., & Mutch, S. (1997). The transition from arithmetic to algebra: Initial understandings of equals, operations and variable. In Erkki Pehkonen (Ed.), Proceedings of the International Group for the Psychology of Mathematics Education.(pp. 2: 89-96). Helsinki: University of Helsinki. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19-25. Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27, 59-78. Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390-419). New York: National Council of Teachers of Mathematics.

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Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra. Journal of Mathematical Behavior, 14, 113-120. Ministries of Education for Alberta, British Colombia, Manitoba and Saskatchewan, the Northwest and Yukon Territories and ITP Nelson (1997) The Learning Equation Mathematics 9. Toronto: ITP Nelson. Priddle, A., Davies, T., & Pitman, P. (1991). Mathematics 9. Brisbane: Jacaranda Press. Sfard, A. (1991). On the dual nature of mathematics conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36. Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspective. Journal of Mathematical Behavior, 14, 15-39. Stacey, K., & MacGregor, M. (1997). Ideas about symbolism that students bring to algebra. The Mathematics Teacher, 90(2), 110-113.