Systematic Errors In Arithmetic Of Some College Students (journal Paper)

Systematic Errors in Arithmetic of Some College Students36 Joel R. Noche Department of Mathematics and Natural Sciences Abstract Knowledge of students’ systematic errors can improve instruction and assessment. In this study, a basic number skills test created by Somerset was administered to 125 college students in four intact sections during the first day of their basic algebra class. The test contains mostly supplyitem questions involving: arithmetic on positive integers, decimals, and fractions; conversions from fractions to decimals; estimation of metric quantities; and number problems presented verbally with a context. The results of the test are summarized and the most common systematic errors are identified.

Introduction [F]rom the preschool years onward, children learn abstract mathematical concepts and principles, as well as procedures and facts. Fairly often, however, they either fail to grasp the concepts and principles that underlie procedures or they grasp relevant concepts and principles but cannot connect them to the procedures. Either way, children who lack such understanding frequently generate flawed procedures that generate systematic patterns of errors. Depending on how one looks at it, these systematic errors can be seen as either a problem or an opportunity. They are a problem in that they indicate that children do not know what we have tried to teach them. On the other hand, they are an opportunity, in that their systematic quality points to the source of the problem, and thus indicates the specific misunderstanding that needs to be overcome. (Siegler, 2003, p. 221)

We first define some terms commonly used in the mathematics education literature. In this study, if a teacher gives a learner a task to perform, we assume that there is only one expected correct response.37 An error is a learner’s response that differs from 36

Presented at the 2009 Bicol Mathematics Conference held at the Ateneo de Naga University 37 More precisely, there is a known set of equivalent correct responses.

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the teacher’s expected correct response. Errors may be due to the learner (not using a correct procedure, not having the correct concepts, not understanding the task, and so on) or the teacher. In this study, we focus on systematic errors—errors resulting from the learner’s use of an incorrect procedure—in written arithmetic tasks done without using notes or calculators.38 For example, let the task be answering the question “What is 19 ÷ 2 equal to?” The correct response is “9.5” (or “9 ½,” or “9 with a remainder of 1,” and so on). Responses like “9.1,” “95,” “17,” “8,” “1,” “quotient,” and no response are errors. We may consider the first three of these as systematic errors; they can be explained as results of incorrect procedures (using the decimal point to represent the word remainder, misplacing the decimal point, using an incorrect operation). The fourth error (“8”) does not seem to be due to an incorrect procedure. It seems to be an error in recalling a basic number fact. (The learner has probably incorrectly recalled “18 ÷ 2 = 9”.) Errors such as this are sometimes called “careless” (Somerset, 2002, p. 23) or “unintentional mistakes” (VanLehn, 1986, p. 134). Somerset (2002, p. 23) calls them calculation errors. The fifth error (“1”) does not seem to be a systematic error or a calculation error. Errors such as this (like the last two errors in the list) are called basic errors (Somerset. 2002, p. 23). There have been many studies on arithmetic systematic errors (see, for example, the references cited in Riccomini (2005), VanLehn (1986), and VanLehn (1990)). Of special interest to us is Somerset’s (2002) survey of high school students in the Central Visayas Region of the Philippines. A sample of 567 students (360 second year and 207 fourth year) from fifteen high schools (eleven regular public schools, three private schools, and one science-augmented high school) was obtained using random list sampling. Of the four diagnostic tests given, we concern ourselves here only with the basic number skills test. The test contains mostly supply-item questions involving: arithmetic on positive integers, decimals, and fractions; conversions 38

Some researchers use the term bug to indicate a small change to a correct procedure (for example, VanLehn (1986, p. 134)). Thus, a systematic error is the result of a correct procedure with one or more bugs.

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from fractions to decimals; estimation of metric quantities; and number problems presented verbally with a context. Somerset analyzed the responses for each question and classified them into as many as ten groups—correct response, calculation error, no response, other basic errors, and different kinds of systematic errors.

Figure 1.. Percentage of correct responses for each question for the four sections in this study

We administered Somerset’s test to some college students (mostly freshmen) in one university in the Bicol Region of the Philippines. We analyze the results and identify the most common systematic errors. Methodology This study involved a convenience sample of four sections of basic algebra taught by the researcher during the first semester of school year 2008–2009. Section A had 39 respondents, section B had 31, section D had 27, and section N had 28. To assure the subjects that the test was diagnostic in nature and that their performance on it would not affect their grades in the course, age and gender were the only personal information requested on the test instrument. The subjects’ ages ranged from 15 to 34 years old, with a mean of 16.9. (A majority were first-year students.) Of the 125 subjects, 59 were males. The test Somerset actually used in his study included two selection-item questions (the occupations of the father and the mother of the subject) that we did not include in the test for the present study. Our test (see the Appendix) was given during the first day of class and the participants were given the whole class period (one and a half hours) to complete it. The actual instrument was printed back-to-back on one 8.5” × 13” sheet of paper, with spaces in between the questions where the participants were to show their solutions. Participants were told not to use notes or calculators during the test. The researcher classified the responses for each question using the response coding scheme provided by Somerset (2002, Annex B). In cases where a response fell into more than one category (for example, the response included calculation and systematic errors), it was classified under the error that occurred first.39 (This is consistent with Somerset’s coding procedure (Somerset, 2002, pp. 25, 33).)

39

Admittedly, the coding is sometimes subjective.

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Figure 2.. Percentage of correct responses for each question for all the students in this study, the fourth-year year high school students in Somerset’s study, and the second-year year high school students in Somerset’s study

Results A few highlights of the results are presented here.40 Figure 1 shows the percentage of correct responses for each question for the four sections in this study. Note that the percentages differ significantly for questions 3a, 3b, 3c, 4a, 4b, and 4c, effectively creating two subgroups, with sections A and D outperforming sections B and N. Figure 2 shows the percentage of correct responses for each question for all the students in this study, the fourth-year high school students in Somerset’s study, and the second-year high school students in Somerset’s study. (Somerset (2002) provides no data for the responses to question 7b.) In general, the first-year college students in this study performed better than the fourth-year high school students in Somerset’s study, and the latter outperformed the second-year high school students. For all three groups, the three lowest percentages are for questions 5c, 5h, and 11b. For the mechanical arithmetic on positive integers and decimals part of the test (questions 1a to 1f), the three most common errors were for questions 1g, 1e, and 1i. For the question “812 ÷ 4 =,” 44 students (35.2%) gave the response “23,” a split-dividend error. (The students split the dividend 812 into two numbers, 8 and 12, and carried out division on each term separately (Somerset, 2002, p. 26).) For the question “47.1 − 0.65 =,” 40 students (32%) gave the response “46.55.” Somerset (2002, p. 71) classifies this as a calculation error, although Standiford, Klein, and Tatsuoka (1982) classify this as a bug. (The student “‘brings down’ numbers in the subtrahend if there are no corresponding digits in the minuend. This may be a carry over from addition thinking that there is ‘nothing to do.’” (Standiford, et al., 1982, p. 4)) For the question “2.05 × 0.52 =,” 24 students (19.2%) gave a response with the correct digits but with the decimal point misplaced (for example, “106.6”). For mechanical arithmetic on fractions (questions 2a to 2c), the three most common errors were for questions 2b and 2c. For the question “2/5 + 1/4 =,” 33 students (26.4%) gave the response “3/9” 40

Interested readers may contact the author ([email protected]) for complete details.

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(or “1/3”), ignoring the fraction line and adding the numerators and the denominators as if they were separate whole numbers (Somerset, 2002, p. 28).

For the question “Arrange 0.07, 0.23, and 0.1 in order, from the smallest to the largest,” 18 students (14.4%) gave the response “0.07, 0.23, 0.1.” This response deserves further study.

For the question “2/5 × 1/4 =,” 29 students (23.2%) gave the response “8/5” (or “5/8”) by “cross-multiplying” (or by inverting the second term, then multiplying) while 11 students (8.8%) gave the response “3/20” (or “13/20”) by calculating the least common denominator, then adding the numerators without (or with) conversion.

For estimation of metric measures (questions 6 to 7b), 32 students (25.6%) estimated the weight of a pair of shoes to be 6 kg while 22 students (17.6%) estimated it to be 6 g. The height of a classroom door was estimated to be 20 m by 26 students (20.8%).

For decimals on a number line (questions 3a to 3d), a rather large percentage of the responses were basic errors (16% for 3a, 16.8% for 3b, 23.2% for 3c, and 27.2% for 3d). For question 3b, 27 students (21.6%) gave the response “5.2,” counting single decimal units for every tick mark from the left. For question 3c, 27 students (21.6%) gave a response that was correct except for the second decimal digit (“0.9,” “0.99,” or “0.09”). For conversion from fractions to decimals (questions 4a to 4c), a large percentage had no response (14.4% for 4a, 14.4% for 4b, and 18.4% for 4c). For the question “Convert 2/100 to a decimal,” 27 students (21.6%) gave a response with the correct digits but with the decimal point misplaced (for example, “00.2”). For the question “Convert 4/5 to a decimal,” 12% of the responses were basic errors. Many participants divided the denominator by the numerator (11.2% for 4a, 10.4% for 4b, 12% for 4c); these include those who made calculation errors (for example, a response of “1.1” for question 4c). For ordering positive integers and decimals (questions 5a to 5h), a large number of responses were decimal-point-ignored (DPI) errors (ignoring the decimal point and treating the decimals as whole numbers). For example, for question 5h (“Arrange 0.438, 0.4, and 0.44 in order, from the smallest to the largest”), 56 students (44.8%) gave the response “0.4, 0.44, 0.438.” Many students also made DPI errors for question 5c (39.2%) and 5g (16.8%). A significant number of responses were reverse-order-ofdecimals (ROD) errors (ordering in the reverse direction to the whole number order). For example, for question 5h, 30 students (24%) gave the response “0.438, 0.44, 0.4.” The percentages of ROD errors for the other questions are 21.6% for 5e, 19.2% for 5c, 16.8% for 5b, and 10.4% for 5g.

For the number application problems (questions 8 to 12), the most common errors involved questions 8, 11, and 12. For the money problem (question 8), 35 students (28%) made calculation errors (but used the correct procedure). Common calculation errors were 19 ÷ 2 = 8 and 19 ÷ 2 = 9.1. (The latter was counted as a calculation error in this instance to follow Somerset’s coding procedure.) For the land measure problems (questions 11a and 11b), a large percentage of students had no response (17.6% for 11a and 21.6% for 11b). In answering the area problem (question 11b), 28 students (22.4%) used incorrect procedures involving the multiplication of two lengths (for example, A = 2L × 2W). For the percentage problem (question 12), 21 students (16.8%) calculated 20% of the initial weight, giving the response “30 kg.” Discussion The results of this study are quite similar to those of Somerset’s study. Questions that had a large percentage of correct responses and questions that had a large percentage of errors were, in general, the same for both groups. The kinds of systematic errors most commonly found in both studies were also quite similar.41 The three questions with the lowest percentages of correct responses were “Arrange 0.55, 0.8, and 0.14 in order, from the smallest to the largest,” “Arrange 0.438, 0.4, and 0.44 in order, from the smallest to the largest,” and “What is the area of the [irregularly shaped] land for building the high school?” Although the participants in this study had significantly better performance in the ordering questions than the participants in Somerset’s study, their performance in the area problem was unusually low. This implies that the solution of non-standard problems is not being well taught to these students.

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The interested reader is referred to Somerset (2002) for more details.

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Very noticeable in Figures 1 and 2 are the group differences in correct response percentages for questions 3 and 4. This implies that problems involving the labeling of points on a number line and those involving conversion of fractions to decimals are good at discriminating between those who have good basic number skills and those who don’t. Conclusion The results of this study have implications for improving instruction and assessment. Teachers should emphasize why certain procedures work so that students will know under what conditions the procedures are valid and under what conditions the procedures are not. Students should be taught how to answer problems that cannot be solved using ready-made formulas. In order to see if students fully understand the concepts, a wide variety of tasks should be given. The labeling of points on a number line is a particularly good arithmetic-related task to give students as it involves the concepts (and not just the procedures) of addition, subtraction, multiplication, and division in one problem. The test used in this study covers only basic concepts and skills. It does not involve, for example, negative numbers and mixed (non-decimal) numbers. The division operation is also not given much emphasis. One incorrect procedure involving division that was prevalent but not sufficiently analyzed in this study is the use of the decimal point to represent the word remainder. Future studies should address the identification of systematic errors in more complicated operations. We end by quoting Riccomini (2005, p. 234): “Educators typically analyze students’ mathematical errors with the intent to improve instruction and correct misconceptions. [...] Identification and analysis of students’ arithmetic errors has the potential to improve instructional planning and, ultimately, student performance. [...] Although identification of errors in mathematics is an important first step for remedial or corrective instruction, there is little evidence to suggest that teachers are able to perform systematic error analysis of students’ work.” We hope that by presenting some common systematic errors in arithmetic of some college students, we are able to help teachers identify and correct them.

References Riccomini, P. (2005). Identification and remediation of systematic error patterns in subtraction. Learning Disability Quarterly, 28(3):233– 242. (ERIC Document Reproduction Service No. EJ725675) Retrieved January 30, 2009, from ERIC database. Siegler, R. (2003). Implications of cognitive science research for mathematics education. In J. Kilpatrick, W. Martin, and D. Schifter, editors, A Research Companion to Principles and Standards for School Mathematics, chapter 20, pages 219–233. National Council of Teachers of Mathematics, Reston, VA, USA. Standiford, S., Klein, M., & Tatsuoka, K. (1982). Decimal fraction arithmetic: Logical error analysis and its validation. Technical report, Illinois University, Urbana. (ERIC Document Reproduction Service No. ED215907) Retrieved February 2, 2009, from ERIC database. Somerset, A. (2002). Basic Number Skills: Why Students Fail in Math: A Diagnostic Survey of Fifteen High Schools in Central Visayas. Quezon City, Philippines: National Institute for Science and Mathematics Education Development, University of the Philippines. VanLehn, K. (1986). Arithmetic procedures are induced from examples. In J. Hiebert, editor, Conceptual and Procedural Knowledge: The Case of Mathematics, chapter 6, pages 133–179. Hillsdale, NJ: Erlbaum. VanLehn, K. (1990). Mind Bugs: The Origins Misconceptions. Cambridge, MA: MIT Press.

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Procedural

Appendix: Basic number skills test42 June

, 2008 • Age: _____ •

Male

Female

Answer the questions below. Please show your solution in full on the test paper. Do not use other pieces of paper to calculate your answers. 1.

42

82

of

Calculate: a.

23 + 9 + 168 =

b.

5.07 + 1.3 =

c.

4138 − 753 =

d.

6.25 − 4 =

e.

47.1 − 0.65 =

The spaces between questions have been removed.

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2.

f.

42.13 − 6.7 =

g.

812 ÷ 4 =

h.

360 × 105 =

i.

2.05 × 0.52 =

c. 5.

Examples:

Smallest

Largest

Calculate:

2

3

1

1

2

3

a.

9

2

7

2

7

9

Questions:

b. c. 3.

Arrange each group of three numbers in order, from the smallest to the largest.

In the box write the decimal number indicated by the arrow.

6.

Smallest

a.

6

9

5

b.

0.3

0.1

0.6

c.

0.55

0.8

0.14

d.

168

97

201

e.

0.37

0.1

0.23

f.

0.65

19

8.7

g.

0.07

0.23

0.1

h.

0.438

0.4

0.44

The height of a classroom door is about 20 cm

7.

200 cm

20 m

200 m

6g

600 g

The weight of a pair of shoes is about 6 kg

8.

Largest

60 kg

The temperature in a classroom is usually about 10 °C

30 °C

50 °C

70 °C

Answer these number problems in the space below each question. Show your solution in full. Do not use other pieces of paper to calculate your answers. 9.

4.

Convert the following fractions to decimals. a. b.

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Here are the prices of some goods in Manang Tonya’s sari-sari store: Tea Soap Salt Toothpaste Sugar

P 6.00 per pack P 8.50 per bar P 2.00 per pack P 9.50 per tube P 19.00 per kg

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Maria wants to buy three bars of soap and ½ kg of sugar. What is the total amount that Maria must pay? 10. On Sunday I left home to walk to the ricefield at 6:45 AM. I arrived at the ricefield after walking for 25 minutes. At what time did I arrive at the ricefield? 11. A doctor gave a patient 30 tablets, and told her to take 2 tablets, 3 times a day. After how many days will the tablets be all consumed? 12. Here is the plan of the land for building a new high school.

a.

What is the perimeter of the land for building the high school?

b.

What is the area of the land for building the high school?

13. Mang Doming has just completed harvesting. He wishes to dry 150 kg of paddy rice. When he dries the rice, it loses 20 % of its original weight. How manyy kilos of dry rice does Mang Doming get?

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