Report Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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ConfChem Conference on Mathematics in Undergraduate Chemistry Instruction: EstimationAn Empowering Skill for Students in Chemistry and Chemical Engineering Lynn S. Penn* Department of Chemistry, Drexel University, Philadelphia, Pennsylvania 19104, United States
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S Supporting Information *
ABSTRACT: Today’s students have always used calculators to compute answers to problems, and as a result, they have a poor sense of the expected size of a physical quantity and cannot tell if their computed value is realistic or unrealistic. Students place great faith in the correctness of calculators, but they fail to acknowledge their mistakes in typing numerical values or in entering mathematical operations. These factors together not only contribute to poorer grades on exams and problem sets in their coursework, but also lead to embarrassing mistakes on group projects and in the professional workplace. The anodyne to this is a workshop approach to developing estimation skills. This report summarizes paper 1 of the 8 invited papers to the ConfChem online conference held from October 23 to November 27, 2017, and hosted by the ACS DivCHED Committee on Computers in Chemical Education (CCCE). KEYWORDS: First-Year Undergraduate/General, Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Problem Solving/Decision Making, Testing/Assessment, Enrichment/Review Materials, Mathematics/Symbolic Mathematics
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INTRODUCTION TO THE ESTIMATION APPROACH As a rule, many of today’s students cannot tell whether a value computed on their calculator is realistic or preposterous. This report will summarize a paper prepared for the recent ConfChem Conference on Mathematics in Undergraduate Chemistry Instruction,1 which describes a mathematics estimation strategy that I have taught to students in physical chemistry for a number of years. This approach has been observed to improve students’ ability to check calculations and make computations correctly, and also enhances their confidence and speed in working problems. Details and the online discussion are provided in the Supporting Information. An annual workshop offered by the author takes only 1 h and is composed of training in estimation skills and in step-bystep presentation of the computation. The approach to estimation consists of five successive exercises, all done by the students with pencil and paper and without calculators. Calculators may be used, at the instructor’s discretion, after each exercise for comparison of calculator values with estimated values. The first exercise focuses on overcoming the students’ assumption of invincibility in making calculations. A good way to make them realize their vulnerability is to give them a nontrivial equation in class and to ask them all take out their calculators and compute the answer. A useful example is the equation for enthalpy change for the process of heating a mole of copper at constant pressure from 500 to 1000 K. The heat capacity coefficients and the formula are supplied to students. Typically, 15−20% of the students get answers that differ from the correct answer by significant amounts. Students seem to have no trouble doing the integration; where they stumble is on the arithmetic operations required after plugging in the © XXXX American Chemical Society and Division of Chemical Education, Inc.
temperature limits. They are taken aback when they state their answers orally and there are so many different answers, some of them orders of magnitude different from others. This is enough to convince them that there is indeed a problem. After this, the students must work without calculators. In the second exercise, the students reduce simple quotients to a single value expressed in scientific notation. The numerators and denominators are selected so that students can readily divide mentally without mistake; students find it easy to do mental division when the numerator is larger than the denominator. The third exercise is to have the students round the numerators and denominators of realistic quotients to simple numbers, use powers of 10 to convert the numerator to a larger number than the denominator, and mentally compute the final answer. Although the final answers to these quotients are estimates, typically they are well within 10% of the answer one would obtain with a calculator. After half a page of this type of exercise, students are usually proficient. In the fourth exercise students are given more complicated initial expressions and are asked to show their work in multiple steps. They are instructed to do no more than two arithmetic operations per step. Clearly, there is room for individual differences in rounding the numbers as well as in choosing which powers of 10 to use. The closeness of the estimated values, usually within 10% of the calculator values, boosts the students’ confidence. Received: May 16, 2018 Revised: June 28, 2018
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DOI: 10.1021/acs.jchemed.8b00363 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Report
The fifth and final exercise is to estimate the final values for some typical physical science equations, given the constants and the values of the variables.2 This exercise, like the previous ones, must be done without a calculator. When the students get to the point where they can estimate the final values for realistic physical science equations without calculators, they are ready to do problems on an exam without calculators. For such an exam, this instructor counts their answers correct if the final values are within 15% of the value correctly obtained by computer or calculator. Students taking this type of exam typically do well and do not require extra time.
available at https://confchem.ccce.divched.org/2017fallconfchemp1 (accessed Jun 2018). (2) Gaskell, D. R.; Laughlin, D. E. Introduction to the Thermodynamics of Materials, 3rd ed.; Taylor & Francis: Washington, DC, 1984. (3) Leopold, D. G.; Edgar, B. Degree of Mathematics Fluency and Success in Second-Semester Introductory Chemistry. J. Chem. Educ. 2008, 85 (5), 724−731. (4) Willingham, D. T. How Knowledge Helps. Am. Educator 2006, 30, 30−37.
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DISCUSSION SUMMARY Over 35 questions, answers, and comments (included in the Supporting Information) were generated in response to this paper. Discussion included difficulties observed by instructors in simple mental arithmetic, including recall of the multiples of 12 and simplification of numeric fractions by students, even those with round numbers, when the denominator was larger than the numerator. It was also noted that practice with estimation is a way to strengthen mental arithmetic skills. Previous studies have found that the ability of students to solve simple numeric calculations without a calculator was among the best predictors of success in introductory chemistry.3,4 Also discussed was the importance of estimation in fields that require introductory chemistry, including health and engineering where, for reasons of human safety, answers found with the assistance of technology must be checked.
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CONCLUSIONS In conclusion, undergraduate and graduate students who have been taught to practice estimation and step-by-step display of calculations exhibit better performance on exams, and better technical communication, and they make fewer computational mistakes. This estimation workshop has been well-received by students and offers a practical model for emulation in other quantitative science courses.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00363. Full text of the original paper with associated discussions from the ConfChem Conference (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Lynn S. Penn: 0000-0002-8579-7359 Notes
The author declares no competing financial interest.
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REFERENCES
(1) American Chemical Society, Division of Chemical Education, Committee on Computers in Chemical Education. 2017 Fall ConfChem: Mathematics in Undergraduate Chemistry Instruction. https://confchem.ccce.divched.org/2017FallConfChem (accessed Jun 2018). The EstimationAn Empowering Skill for Students in Chemistry and Chemical Engineering paper and discussion are B
DOI: 10.1021/acs.jchemed.8b00363 J. Chem. Educ. XXXX, XXX, XXX−XXX
