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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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middle class suburb in Queensland. Over 20 lesson observations showed that in general the students were cooperative and well behaved although getting some students to complete homework tasks was a constant battle. The school mathematics program included identifying students who were struggling with mathematics and offering them lunch time or after school tutorial sessions. Four of the 54 students were absent on the day the pre-test was administered. The year 10 students had previously completed the unit but since their marks were unsatisfactory they were repeating the unit. The unit included the following procedural tasks; adding and subtracting polynomials, multiplying and dividing polynomials, simplifying, factorising, solving and transforming equations. The subject coordinator had workprogram documents, which related various activities to specific content and process objectives. However, the two teachers responsible for teaching the unit worked from a one page document which listed content material in an abbreviated form such as “Adding and subtracting expressions.” This document also had references to photocopied sheets from various sources. It was observed that the dominant activity source was a traditional textbook, which contained numerous symbolic manipulation exercises of increasing difficulty. Results and Analysis The results are presented as the percentage of students who correctly answered each question. Students who did not attempt to answer a question were given no marks. Incorrect responses are reported, as are the numbers of students who made these responses. This approach to data reporting enables trends in student performance to be examined. The results are described and summarised in Table 1 below. Student responses in each of the major concept areas described and analysed. Table 1: Summary of the results of the arithmetic manipulation test.
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 %

Directed number operations Directed numbers should have been included in the test but were not. However, the students were asked to solve for x in the following equation –4x = 20. This type of problem has been termed operational algebra in that it can be solved through arithmetic operations. Twenty-one students reported that they could not do the problem, one left a space, eight responded with numbers like 16, 24, 6. These responses indicated that arithmetic as well as the directed number concept was problematic for them. Eighteen reported 5 indicating that they were able to solve the problem except for the directed number component part of the operation. Only four students of the 52 had the correct answer of –5. The observation that eighteen reported 5 and eight had other positive numbers is clear evidence that many of the students could not work with directed numbers. The equal concept Almost all students were able to correctly do computations and determine which of the statements like 5 × 4 = 40 ÷ 2 were correct. However, only 10 % could correctly suggest that 56 be added to the left-hand side if it had been added to the right hand side. This failure does not necessarily mean that they did not understand the equal concept although this may well have been the case. Seventy percent of students either did not attempt the answer or reported “I do not know what to do.” The remaining twenty- percent suggested various numerical procedures such as “divide 56 by 43 and add the result to 139;” two suggested “subtract 56 from the other side” and the remainder multiplied 139 by 43 to yield 5977 but were unable to go further or tried seemly unrelated computations. It is possible that most of the students had not been presented with this type of the test of equal sign concept and when faced with an unfamiliar format they did not know what the question was about. However, the responses support the conclusion that many of the students had a computational rather than a sense making approach to the equal concept. Order of Operations Only thirty percent of the students correctly answered 8 for the question 6 + 4 ÷ 2. Thirty-five of the fifty-two students reported that 5 was the answer to 6 + 4 ÷ 2. Clearly these students had done the operations as they occurred. Other incorrect responses included 20 and 4.1, which are difficult to explain. Only two students did not succeed on the question 7 × (2 + 1) = , and their answers of 9 and 12 probably indicate that they may have attempted to add all the numbers and failed to distinguish between the addition and the multiplication symbols. Most students (84%) correctly computed the answer to the problem 20 ÷ (5 –1). One student

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responded with an incorrect answer of 4. This may have indicated that this student had made a computational error. One student responded with 80 suggesting that she multiplied instead of divided. Other answers included 3 ?, which is difficult to fathom, and ? that may have indicated that the student divided 5 by 20 and ignored the subtract one. Clearly the data supports the previously stated finding that all most all the student appreciated the Brackets part of BOMDAS (Brackets of Multiplication Division Addition and Subtraction) must be done first. The Distributive concept. Only 24% of students correctly answered question Two. “Can you work out the answer to (36 + 24) 6 if you are not allowed to add 36 and 24? Explain.” Most of the students who gained no marks simply responded with “no” or “can’t be done,” but ten students made comments similar to, “no because you have to do the brackets first.” Likewise most of the students who gave reasons for not being able to complete 5 × (6 + 7) reported that the brackets needed to be done first. One has to wonder if the use of BOMDAS as a learning hinge or memory prompt has not clouded students’ memory of the distributive law. One conclusion from these observations is that order convention needs to be taught along side the distributive law. Associative Law Twenty six percent of the students correctly answered the question “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?” As in previous responses many students do not elaborate on why the problem could not be solved. However, six students (12 %) responded with numbers which added up to 11 such as 1 + (3 + 7) = 11. Several others said no “because s + n could be any number” and went on to suggest “6 + 5 or 8 + 3 or 10 + 1.” Clearly, many students did not appreciate that the brackets did not have meaning in the case where all the operations were addition. It is possible that the use of brackets and symbols confused students, however the use of such symbols is common in the primary curricula. Fractions as part of a whole Almost all correctly named the two sevenths shaded fraction. However, 16 students could not make up the whole given n n being 1/3. Seven students simply added one square to give a final answer of n n g . This may have indicated that they saw the model as 2/3 to which they added a further square, which was seen to represent 1/3 in order to make up the whole. Alternatively this response may have meant that they focused on the denominator and were preoccupied by the three, thinking that the denominator had to become three. In the following question only 19 students correctly made the 2/5 th rectangle up to a whole. Generally these students carefully measured the rectangle and made up the appropriate length as a horizontal extension. Several used a vertical stacking system. Fifteen students simply did not attempt the question. Of the seventeen who tried and were not able to produce a correct answer, seven simply divided the rectangle into fifths. That is, they appeared to be preoccupied with the denominator and did not appear to understand that a fraction is an expression of a relationship between the numerator and the denominator. Three other students simply treated the 2/5 as 1/5 and added four more

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rectangles. That is the significance of the numerator was not obvious to these students. The remaining students either added three rectangles, nine rectangles or ignored scale factors. In summary, many students were not able to make up a whole given a proportion. These findings indicate that many of the students did not have a good understanding of the nature of fractions. Fraction Operations As with previous concept areas there was considerable consistency among students. For example those students who did poorly tended to do so on all related questions. Only nineteen students correctly completed 2 × 1/8 . Of the remaining 32 students 10 gave the answer as 2/16 indicating that they treated 2 as 2/2 and multiplied both the numerator and denominator by 2. It is likely that these students did not appreciate that 2/2 was 1. Given the number of times this idea is used in algebra in transforming and solving equations this is a significant finding. Two student responded with 16 possibly indicating that they treated 1/8 as 8, in that they simply ignored the fact that the number 8 was the denominator. One student answered with 3/8 suggesting that he simply added the 2 to the 1 and did not account for the denominator. That is, in relation to fractions this student did not distinguish between the operations of addition and multiplication. Two students responded with 16 suggesting that they treated 1/8 as 8/1, which is they ignored the fact that 1/8 was a fraction rather than a whole number. Other answers included 2/6 and 1 ?, which are difficult to relate to possible thinking patterns. The remainder of the students simply did not attempt the problem. The division of fractions problem (12 ÷ 1/3) was correctly answered by only three of the fifty-two students. Five students responded with 4 indicating that they simply divided 12 by 3 and did not treat 1/3 as a fraction, alternatively they may have failed to differentiate between multiplication and division operations in this context. Three students responded with 3, perhaps they had the same misconceptions as their classmates who responded with 4 but also had problems with their number facts. Two students responded with 12/4 and two with 12/3. These students seemed to believe that a division was in order but lacked both the algorithms and conceptual understandings to derive a reasonable answer. Other answers included 6 and 12. The addition of fractions question (15/3 + 24/4 = ) was completed correctly by 12 of the fifty-two students. Six of the students simply added the numerators together to equal 39 and added the denominators together get 7 and expressed the answer as 39/7. Clearly this approach shows little conceptual understanding of the nature of fractions in that the students did not distinguish between thirds and quarters. Two students simply multiplied the numerators to give 132 and the denominators to give 12 and expressed the product as 132/12. One student multiplied the denominators and added the numerators to give 39/12. At least these students seemed to realise that a common denominator was necessary even if they were unable to conserve the fraction by representing each fraction as an equivalent faction. It is difficult to guess what students may have thought when they calculated 4 1/7 and 5 7/12. Squaring Thankfully almost all students squared 3 to give 9. The students who answered this problem incorrectly responded with 6 indicating that they simply multiplied 3 by 2 and did not appreciate the significance of indices.

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Area of a rectangle Surprisingly only 18 of the 52 students correctly reported that the area of a rectangle 3 cm wide and 6 cm long was 18 cm2. Four students answered with 18 cm. We believe it is important for students to use correct units since this type of error may be related to confusion between linear and area measurements. Two other students answered with L × W, but did not substitute. Four students responded with A = (L × W) × 2. This seems to be confounding the formula for area and perimeter. Other confounding errors included 3 + 3 = 9 and 2 + 2 = 4 therefore the area was 9 × 4 = 36 cm2, 3 + 2 × 2 = 10 cm2 and also 3 + 2 × 2 = 6 × 2 = 12. This last response is interesting in that it contains not only confusion with perimeter formula but also errors in order of operation. There were other answers, which are difficult to understand including one response of 6 cm, one with 5 cm2 and finally two with x2. Nine students simply did not attempt to answer the question. Forty two percent of the students correctly responded that the inverse of 5 was 1/5. Five students responded with 5, one with 5/1, one with .02 and one student responded with 10/2. The remaining students simply did not attempt the question or responded with “I don’t know.” It can be argued that this question is related to term knowledge rather than procedural or conceptual understanding. Still it is alarming that many students who were about to begin an algebra course were unable to state the inverse of a number. Discussion and Conclusions Clearly the results indicate that many of the students do not have the necessary prerequisite knowledge of arithmetic to begin the study of algebra. This finding is somewhat surprising in the light of previous studies such as that by Boulton-Lewis et al.,(1998, p. 149) who found that in a similar school located in another middle class suburb, “By grade 9 most students had sufficient understanding of the commutative law to apply this to linear equations, the majority of students also had displayed a satisfactory understanding of inverse procedures and of correct order of operations …most students had satisfactory arithmetic understandings to enable them to apply these principles to algebra.” The findings of this study show that many students had neither operational nor structural understanding of arithmetic and this will almost certainly make it difficult for them to develop operational and structural understandings of algebra concepts. Certainly, it has been argued previously that while an operational understanding of arithmetic may well allow students to succeed at arithmetic such understandings are less so in the case of algebra (Boulton-Lewis & Cooper 1997). The results of this study suggest that prior to beginning formal algebra study teachers might consider determining whether their students are ready to begin formal algebra study. The application of a test similar to the one above, particularly if it is supported by student interviews could provide useful information regarding the types of learning activities that students might be given prior to beginning formal algebra study. Should teachers find that student lack necessary arithmetic understandings there seems to be at least two courses of action. First, the students could take a specific bridging course and the results of the test being used to inform the nature of the bridging course. Second, the arithmetic concepts necessary for algebra study could be taught as they are encountered in algebra study. Clearly the approaches

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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.


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