Report Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
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ConfChem Conference on Mathematics in Undergraduate Chemistry Instruction: Strengthening Students’ Math Fluencies through Calculator-Free Chemistry Calculations Doreen Geller Leopold* Chemistry Department, University of Minnesota, Minneapolis, Minnesota 55455, United States
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S Supporting Information *
ABSTRACT: In our previous study of students in second-semester general chemistry classes at the University of Minnesota, higher scores on a calculator-free math assessment, administered at the start of the semester, were found to correlate with higher grades in this course, despite the use of calculators during exams. The present paper describes some methods subsequently used to enhance students’ math fluency through solving numerical problems using pencil-and-paper math, without the use of a calculator. When doing such problems in class, the instructor can efficiently interleave reminders of basic algebraic methods to simplify expressions, to work with common and natural logs, and to estimate results to one or two significant figures. Multiplechoice exams incorporating problems of this type, in which calculators were not allowed, were also administered. Examples of such problems and their pencil-and-paper solution methods are presented in the areas of chemical equilibrium, acid−base reactions, buffers, and titrations. Students’ evaluations of this pedagogical approach are also discussed. It is hoped that these methods can help motivate students to gain greater intuitive and conceptual insight into the chemistry through solving quantitative problems, and to become more fluent in expressing science in the language of math. This report summarizes one of the invited papers for the ConfChem online conference on Mathematics in Undergraduate Chemistry Instruction, 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, Calculator-Based Learning, Curriculum, Mathematics/Symbolic Mathematics, pH
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BACKGROUND
includes the common logs of integers up to 10 and some basic log relationships. This report summarizes one of the invited papers5 for the ConfChem online conference on Mathematics in Undergraduate Chemistry Instruction,6 held from October 23 to November 27, 2017, and hosted by the ACS DivCHED Committee on Computers in Chemical Education (CCCE). The Supporting Information includes the ConfChem paper with the accompanying discussion, and an Appendix with 38 quantitative multiple-choice questions selected from calculatorfree exams and descriptions of the pencil-and-paper methods that can be used to solve them conveniently without a calculator. Several problems involving pH and buffer calculations are highlighted in the paper to illustrate how one can “remind” students of the reasons behind fundamental relationships involving logarithms and how to use them, without distracting too much from the chemistry content. Also included in the Supporting Information are the following resources: a math review for calculator-free problem solving (32 pages); a collection of 165 additional multiple-choice exam problems (including descriptions of calculator-free solution methods) that cover all of the quantitative topics that we typically include in second-semester general chemistry; and a
In 2008, we reported a study of the math fluencies of secondsemester general chemistry students at the University of Minnesota (Twin Cities), based on a math assessment for which calculators were not allowed.1 Although these scores did not count toward the course grades, and calculators were allowed on all exams, students’ scores on this math assessment were found to correlate with their grades in this course. Many readers are probably amply convinced of the importance of improving the math fluencies of our chemistry students at both the high school and college levels, and are familiar with some of the rich literature on this subject.2−4 For example, as pointed out by Hartman and Nelson:2 “Arithmetic facts and fundamental algorithms” must be “thoroughly mastered, and indeed, overlearned” to avoid the bottleneck in novel working memory. In the intervening decade, we have found that the best way to motivate students to master the basic math techniques used in general chemistry is to give at least one exam that counts toward the course grade, and that includes many numerical problems, for which the use of calculators is not permitted. We have done this during several semesters for the exam that covers chemical equilibrium and acid−base chemistry, including buffers and titrations. To enable students to do these calculations on paper, we use paper-and-pencil mathfriendly numerical values in the problems, and allow extra time to do the exam. We also provide an equation sheet that © XXXX American Chemical Society and Division of Chemical Education, Inc.
Received: February 14, 2018 Revised: May 28, 2018
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DOI: 10.1021/acs.jchemed.8b00113 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Report
chemistry (Appendix to the original paper), with worked solutions showing pencil-and-paper calculation methods (PDF, DOCX) Multiple-choice numerical problems (136 pages, 165 problems) for these and other topics in second-semester general chemistry, with worked solutions (PDF, DOCX) Math practice problems (51 pages, 123 problems) based on the above chemistry problems, with worked solutions (PDF, DOCX) Math review for calculator-free problem solving, 32 pages (PDF, DOCX)
packet of the corresponding math problems with the chemistry content removed to provide practice with just the math.
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ADVANTAGES OF THIS APPROACH The choice of simple numbers in calculator-free numerical problems can allow the chemistry concepts to be communicated more clearly. Thus, these types of problems are useful for introducing new content in lectures. The deliberate choice of pencil-and-paper math-friendly numerical values, and estimating results to one or two significant figures, can reduce the extent to which the chemistry is obscured by the math. Of course, in laboratories, the need for more precise values often requires the use of calculators or computational programs. The potential advantages of encouraging students to train themselves to perform calculations on paper, rather than going right from the question to the calculator as is often done, are numerous. On paper, students may be more likely to include units with the numerical values in their equations. They are more likely to use algebra to solve for an unknown, and to simplify products or ratios of expressions, saving themselves work while also helping to highlight the underlying relationships among the quantities. Most importantly, the most efficient way to check that one is avoiding numerical errors, in the absence of a calculator, is to think about the meaning of the numbers. Frequently checking whether their calculations make sense, and making rough estimates, can reinforce students’ conceptual understanding of the chemistry.
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*E-mail:
[email protected]. ORCID
Doreen Geller Leopold: 0000-0001-5739-6550 Notes
The author declares no competing financial interest.
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REFERENCES
(1) Leopold, D. G.; Edgar, B. Degree of Mathematics Fluency and Success in Second-Semester Introductory Chemistry. J. Chem. Educ. 2008, 85 (5), 724−731. (2) Hartman, J. A. R.; Nelson, E. A. Automaticity in Computation and Student Success in Introductory Physical Science Courses. arXiv:1608.05006v2 [physics.ed-ph] https://arxiv.org/abs/1608. 05006 (accessed May 2018). (3) Watters, D. J.; Watters, J. J. Student Understanding of pH: “I Don’t Know What the Log Actually Is, I Only Know Where the Button Is on My Calculator. Biochem. Mol. Biol. Educ. 2006, 34, 278− 284. (4) National Mathematics Advisory Panel. Foundations for Success: The Final Report of the National Mathematics Advisory Panel; U.S. Department of Education: Washington, DC, 2008. https://www2.ed. gov/about/bdscomm/list/mathpanel/report/final-report.pdf (accessed May 2018). See the Panel’s recommendations on pages xxiv, 24, 25, 50, 78, and 89 regarding the use of calculators. (5) The ConfChem paper, Strengthening Students’ Math Fluencies through Calculator-Free Chemistry Calculations, and associated appendix and discussion are available at https://confchem.ccce. divched.org/content/2017fallconfchemp4 (accessed May 2018). (6) 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 May 2018). (7) College Board. SAT (Scholastic Assessment Test), Math Test.https://collegereadiness.collegeboard.org/sat/inside-the-test/ math Math Sample Questions. https://collegereadiness.collegeboard. org/sample-questions/math (both accessed May 2018).
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CONCLUSIONS Students’ evaluations of this pedagogical approach were mixed. While some viewed it favorably, others found it unnecessary: for example, “I will probably never be in a situation where I won’t have access to a calculator.” In the broader impacts section of the paper, it is argued, on the contrary, that encouraging students to develop a deeper level of mathematical fluency can better inform not only their future academic and career activities, but also the feelings and intuitions that will influence some of their most consequential decisions. During conference participants’ subsequent discussions of this and related papers, the encouraging news emerged that the new SAT (Scholastic Assessment Test), first administered in March 2016, includes one math section (out of three) that does not allow calculators.7 This improvement is consistent with some of the recommendations of the 2008 National Mathematics Advisory Panel.4 Of the sample questions for this section posted by the College Board for the SAT, as well as for the PSAT/NMSQT and PSAT 10 exams, many require manipulating numbers (not just symbols).7 It is hoped that this change will provide an additional impetus for high school students to strengthen their calculator-free math skills, enabling them to undertake their college chemistry and other STEM classes more fluent in the language of math.
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AUTHOR INFORMATION
Corresponding Author
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00113. Full text of the original paper (pp 1−29) with associated discussions (pp 30−57) from the ConfChem Conference (PDF) Multiple-choice numerical problems (42 pages, 38 problems) on chemical equilibrium and acid−base B
DOI: 10.1021/acs.jchemed.8b00113 J. Chem. Educ. XXXX, XXX, XXX−XXX
