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Journal of Education for Business

ISSN: 0883-2323 (Print) 1940-3356 (Online) Journal homepage: http://www.tandfonline.com/loi/vjeb20

Order of Operations: Do Business Students

Understand the Correct Order?

Ed Pappanastos , Marc A. Hall & Ava S. Honan

To cite this article: Ed Pappanastos , Marc A. Hall & Ava S. Honan (2002) Order of Operations: Do Business Students Understand the Correct Order?, Journal of Education for Business, 78:2, 81-84, DOI: 10.1080/08832320209599702

To link to this article: http://dx.doi.org/10.1080/08832320209599702

Published online: 31 Mar 2010.

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Order

of

Operations: Do

Business Students

Understand

the Correct Order?

ED PAPPANASTOS

MARC

A.

HALL

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Troy State University

AVA S. HONAN

Troy, Alabama

Auburn University at Montgomery

Montgomery, Alabama

any professors who teach basic

M

computer programming and quantitative methods have noted an appreciable number of students each term who must be retaught the mathematical order of operations. Given the computer-driven world in which we live, the implications of this erroneous understanding could have a potentially devastating impact on businesses whose employees are required to use spreadsheet programs. Formulas that are entered incorrectly in spreadsheets and databases obviously produce erroneous results. To ensure correct outcomes from spreadsheet and database operations, businesses must make sure that their employees know the order of operations required for use of these tools.

When asked the correct order of operations, students often will recall the elementary school acronym “Please Excuse My Dear Aunt Sally,” or PEM- DAS, which stands for Parentheses,

Exponents, Multiplication

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& Division

(left to right), Addition & Subtraction (left to right). This acronym has been used as a teaching tool by textbook authors and teachers for many years. In fact, a quick Internet search easily reveals scores of tutorial Web pages, mostly authored and maintained by elementary and secondary school teachers, relating to the reteaching and reinforce- ment of such concepts. In this study, we

ABSTRACT. Many university students enrolled in quantitative methods

and basic computer programming courses do not understand the proper order of precedence that should be applied to mathematical operations. Thus, many professors are compelled to reteach the proper order of opera-

tions before embarking on the initial coursework. In this study, the researchers surveyed over 300 primar-

ily nonfreshman business school stu-

dents at two regional universities to

determine the extent of this lack of understanding. Results show that one third of the respondents incorrectly applied the order of operations.

sought to ascertain the extent to which business school students suffer from a lack of understanding, and perhaps mis- understanding, of the most basic order of operations.

Most previous research on order of operations has focused on elementary and secondary textbook and curriculum development (Lee & Messner, 2000), elementary pedagogy (Schneider &

Thompson, 2000), elementary educa- tional technology in the classroom, and the role and/or propriety of mathematical remediation in the college curriculum (Waycaster, 2001). Very little research seems to have focused on the seriousness of college students’ lack of understanding of the most basic conven- tions necessary for success in quantitative methods and computer program-

ming coursework. Although Lee and Messner (2000) investigated concatena- tion and order of operations content in 6th through 9th grade textbooks, they acknowledged anecdotally that college professors are frustrated by the fundamental algebraic mistakes made by cal- culus students (p. 178). Additionally, they suggested anecdotally that college students become confused about the proper hierarchy when they are present- ed with both negation and exponentiation, as in the following question: Does -3* refer to 9 (incorrect response) or -9 (correct response)?

It is noteworthy, however, to point out that this confusion or disagreement over the proper hierarchy of precedence between negation and exponentiation exists even within the computer spreadsheet programming industry. In fact, it

has been openly observed and reported

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as early as 1987 that spreadsheet programs are inconsistent (Ecker, 1987). Currently, Microsoft’s Excel spreadsheet program, for example, places negation above exponentiation in the hierarchy, a clear departure from the generally accepted convention. The other market leader, Lotus 1-2-3, has applied the convention consistently.

Schwartzman (1996) conducted a simple study with trigonometry students in a community college setting in

which he examined a limited area of

November/December 2002 81

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the order of operations convention. Specifically, Schwartzman tested whether the grouping symbols (parentheses, brackets, and braces) “must be used in a specific order” “when nest- ed.” In this rather unscientific study, students were asked to evaluate whether or not parentheses, brackets,

and braces had been used correctly in

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1 1 different mathematical expressions.

Only one out of 21 students correctly responded that order was not relevant so long as each symbol was used consistently within a particular expression

(Schwartzman, 1996).

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Method

To measure the degree of college students’ lack of understanding of the generally accepted “correct” order of operations, we asked business school students at two regional public universities to complete a survey instrument. The business school programs at both universities are professionally accredited. One of them is a metropolitan campus of a land- grant institution accredited by the Amer- ican Assembly of Collegiate Schools of Business (AACSB) with a total university enrollment of approximately 5,500, students, of which 1,500 are declared business majors. The average ACT score there is just over 19. The other university is a traditional campus institution with a total university enrollment of approximately 4,300, of which approximately 1,100 are declared business majors. This business school is accredited by the Association of Collegiate Business Schools and Programs (ACBSP), and the average ACT score is approximately 21.5. We received a total of 306 survey responses from students enrolled in sophomore and junior level business

classes.

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In addition to asking students to

evaluate several arithmetic expressions, we asked them a variety of demographic questions. The arithmetic expressions were open-ended questions and, as such, no choice of answers was provided.

Results

Student responses to four of the expressions warrant discussion. In two of these arithmetic expressions, we tested the students’ understanding of the

82

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Journal

of Education for Business

order of operations for the basic mathematical operators: multiplication, division, addition, and subtraction. These expressions were 6/3*2 and 10+5/5.

Because multiplication and division should be performed from left to right, the correct answer for the first expression would be 4. A student believing that multiplication takes precedence over division would incorrectly arrive at a final answer of 1. Similarly, a student regarding the division operator as a fraction bar, thereby “clearing the fraction” first, would incorrectly arrive at 1. In our evaluation of the percentage of correct responses for this expression and all subsequent expressions, we ignored incorrect responses that resulted from a reason other than misapplication of the order of operations. (For example, one student evaluated this expression as equaling 9, which required a rather cre- ative application of the order of operations. As such, this student’s response was removed from consideration.) Out of 332 respondents, 14.8% evaluated the first expression incorrectly.

Following the order of operations, we would perform the division before the addition on the second expression (10+5/5) and arrive at the correct answer, 11. A student simply choosing to work from left to right or to “clear the fraction” would arrive at a final answer of 3. Out of 312 respondents, 33.3% evaluated the second expression incorrectly.

[image:3.612.232.564.382.738.2]

These results provide substantial evidence confirming the hypothesis that college students lack an understanding of the most basic rules incor- porated in the order of operations. It is interesting to note that 50% of the students who evaluated the first expression incorrectly also evaluated the second one incorrectly. Incorrect evaluation of both expressions suggests that those students had regarded the division symbol as a fraction bar. A student working from left to right and not considering the order of operations would evaluate exactly one of the expressions correctly. The only other explanation for an incorrect evaluation of both expressions would be that the

TABLE 1. Description of Demographic Variables

Demographic variable

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%

State of majority of high school education (n = 324)

Alabama Other U.S. state Abroad Female Male Sophomore Junior Senior ACT SAT

Both ACT & SAT

Other

Princeton’s Review Barron’s Review

Live course

Material from Internet

Major

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(n = 326)

Computers & math Quantitative business (accounting & finance) Other business

1997 & before

1998 & after Gender (n = 240)

Classification (n = 227)

Admission test ( n = 316)

Preparation material for admission test (n =114)

High school graduation year ( n = 333)

80.2 11.5 8.3 43.8 56.2 18.9 46.3 34.8 74.1 10.7 7.6 7.6 12.3 56.1 22.8 8.8 26.4 26.1 47.5 39.0 61.0

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respondent believed that both multiplication and addition take precedence over division.

We created two additional arithmetic

expressions,

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-52 and

8-5*, to test stu-

dents’ understanding of the precedence relationship between negation and exponentiation. Recall that the program Lotus 1-2-3, using the traditionally accepted order of operations, evaluates exponentiation before negation, whereas the Microsoft Excel program evaluates negation first. Out of 332 respondents,

88.9% incorrectly stated that the expres-

sion -5*equaIs 25 (the same incorrect answer arrived at through the Microsoft Excel spreadsheet program). Out of 205 respondents, 20% incorrectly believed that the expression 8-52 equals 33 (the same answer arrived at through the Excel program).

These results suggest that students are not completely sure how to apply the order of operations. Though our study lacks the kind of data that could indicate the root cause of students’ lack of understanding, it does indicate possible rela-

tionships between student understanding and the demographic variables that we obtained. In Table 1, we provide information on the respondents’ demographic variables.

We collected data on gender and classification (sophomore, junior, or senior) for only 205 of the 306 respondents. All of the respondents were completing the survey in a sophomore or junior level quantitative methods course.

We performed chi-square tests of independence for each combination of demographic variable and arithmetic expression. For the “state of high school education” and “preparation materials” variables, we found no statistically sig-

nificant differences (using

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a =

0.10) for

any of the arithmetic expressions. We found statistically significant dif- ferences with regard to gender. A larger percentage of female students incorrectly answered the basic expressions. However, a larger percentage of male students evaluated one of the more diffi- cult expressions (-5*) incorrectly, by a statistically significant margin.

We obtained statistically significant results regarding student classification and the first basic expression, 10+5/5.

This expression was incorrectly evaluated much more frequently by seniors than by sophomores and juniors. This result could indicate that those seniors taking the sophomore and junior level quantitative courses postponed these courses because of a pre-existing weak- ness in math. It is interesting to note that juniors were more frequently correct than sophomores in their responses to both expressions. This might suggest, unfortunately, that students are not gaining a better understanding of the basic order of operations until their later years in college. We found that seniors gave incorrect responses more frequently for the second basic expression also, but this difference was not statistically significant.

We found statistically significant differences with regard to the variable “admissions test” for only one of the basic expressions. Students who took

both the ACT and the SAT incorrectly

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TABLE 2. Summary of Percentage of Incorrect Answers and Chi-square Results for Tests of Independence That

Were Statistically Significant (Using a

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= 0.10)

Statistically significant differences for each mathematical expression

Demographic variable

I0+5/5 6/3*2 -52 8-5*

State of majority of high school Gender

education Classification Admission test Preparation material for

Major

High school graduation year admission test

No differences Female: 39%

Male: 28%

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x2

= 3.062, df=

1, p = .080

Sophomore: 29% Junior: 26% Senior: 42%

xZ

= 4.940, df = 2,

p = .085 ACT 45% SAT 34% Other: 36% ACT & SAT 9%

x2

= 7.670, df = 3,

p = ,053 No differences No differences 1997 or earlier: 46% 1998 or later: 27%

x2

= 11.825, df = 1 ,

p = .QQ1

No differences No differences No differences Female: 19% Female: 88% No differences Male: 8% Male: 95%

x2

= 7.285, df = 1, p = .007

No differences No differences No differences

xZ

= 3.199, df= 1 ,

p = .074 No differences

No differences

No differences No differences

No differences No differences No differences No differences

No differences No differences

No differences No differences November/December 2002

83

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answered the expression 10+5/5 only 9% of the time, whereas students who took only one of those admissions tests or another admissions test incorrectly

answered this expression over

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36% of

the time. Possible explanations for this difference might be that the students who took two tests may have been more motivated to learn the correct order or operations, or that repetition is the key to learning.

We expected that students majoring in computer- and math-related fields would answer correctly more frequently. Surprisingly, we found no statistically significant differences related to major.

We found only one statistically significant difference related to graduation year. Forty-seven percent of the students who graduated from high school in 1997 or earlier incorrectly evaluated the first expression (10+5/5), whereas only 28% of the students who graduated in 1998 or later did so. This difference may be attributable to greater recall of precedence relationships among students who graduated from high school more recently.

In Table 2, we provide a summary of

the statistically significant differences

that we found.

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Recommendations

Our results suggest that business students lack the fundamental mathematical knowledge necessary for success in business-related spreadsheet applica- tions programming. The problem may be acute, but the solution is simple. We suggest that instructors implement a refresher course on the order of operations convention, which is as follows:

1. Symbols first (brackets, braces, parentheses, etc.), starting with the innermost ones

2. Exponents

3. Negation

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4.

Multiplication and division from

5. Addition and subtraction from left left to right

to right.

Although it is unfortunate that the two dominant spreadsheet programs lack consistency in their relative treat- ments of the order of operations, specifically negation and exponentiation, it is essential that persons using spreadsheets and databases be apprised of both

the proper convention and the underly- ing difference between the two programs’ treatment of them. Without this knowledge, errors could certainly be expected that would adversely affect the quality of users’ results. Future research could focus on the degree to which the community of spreadsheet users is aware of the inconsistencies and the possible consequences of importing and exporting calculations between and among various spreadsheet programs.

REFERENCES

Ecker, M. (1987). Mathemagical computing:

Order of operations and new software.

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Journal

of Computers in Mathematics and Science Teaching, 8(2), 103-105.

Lee, M. A.,

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

& Messner,

S . J. (2000). Analysis of

concatenations and order of operations in wnt- ten mathematics. School Science and Mathe- matics, 100(4), 173-1 80.

Schneider, S. B. , & Thompson, C. S. (2000). Incredible equations develop incredible number sense. Teaching Children Mathemutics, 7(3),

146- 154.

Schwartzman, S. (1996). Some common algebra- ic misconceptions. Mathematics and Computer Education, 30(2), 164173.

Waycaster, P. (2001). Factors impacting success in community college developmental mathematics courses and subsequent courses. Community College Journal of Research & Practice, 7(3), 403-417.

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Journal of Education for Business