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ConfChem Conference on Mathematics in Undergraduate Chemistry Instruction: Addressing Math Deficits with Cognitive Science Eric A. Nelson*,† Fairfax County Public Schools, Falls Church, Virginia 22042, United States
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S Supporting Information *
ABSTRACT: During the past decade, cognitive scientists have reached a consensus that when solving problems in mathematics and the physical sciences, the brain must rely almost entirely on the application of facts and algorithms that have previously been overlearned (thoroughly memorized). Since 1990, however, K−12 math standards in most U.S. states assumed that, with access to calculators and computers, memorization in math could be de-emphasized. As a result, many current students have extensive deficits in math that is prerequisite for chemistry. Evidence indicates, however, that if math fundamentals are moved into memory just before they are needed in chemistry, student success improves substantially. This report summarizes one of the invited papers to 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). The entire paper and discussion are provided in the Supporting Information. KEYWORDS: First-Year Undergraduate/General, High School/Introductory Chemistry, Interdisciplinary/Multidisciplinary, Curriculum, Problem Solving/Decision Making, Learning Theories, Mathematics/Symbolic Mathematics, Mnemonics/Rote Learning
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TRENDS AMONG SCIENCE-MAJOR GRADUATES As chemistry instructors, our mission to prepare students for scientific careers has become increasingly difficult. Since 1966, the percentage of college students who graduate as chemistry majors is down by about 60% (Table 1). Table 1. U.S. Bachelor’s Degrees and Chemistry Majors, 1966−2012a
Figure 1. A fraction with four denominators.
Count of Degrees Awarded by Year (% of Bachelor’s Degrees) U.S. Bachelor’s Degrees Awarded, by Major In all fields Chemistry majors
1966
1978
1990
524,008 930,201 1,062,151 9,735 11,474 8,289 (1.86%) (1.23%) (0.78%)
(solving problems with mathematical content), U.S. 16−24 year olds scored 22nd among 22 developed-world nations.2,3 Evidence suggests that U.S. math deficits are due in substantial part to changes in how K−12 mathematics has been taught.
2012 1,810,647 13,714 (0.76%)
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DECREASED ATTENTION TO MATH IN MEMORY For millennia, of necessity, students learned math by memorizing facts and procedures. With the arrival of electronic calculators in 1970, instruction changed. By 2002, most U.S. state K−12 standards required calculator use in third grade.4 Most state standards after 1990 also required “decreased attention” to “memorizing rules and algorithms”, “manipulating symbols”, and “rote practice”.5−7 These changes assumed that during problem solving, the brain could utilize non-
a
National Science Foundation, Science and Engineering Degrees: 1966−2012. Data from ref 1.
More recently, between 1984 and 2012, the percentage of graduates who majored in chemistry fell by over 30%, and the percentage majoring in engineering, for which chemistry is a foundation, fell by 40%.1 What factors may contribute to these declines? First-year chemistry is where science and engineering majors learn the fundamentals of solving scientific calculations: applying math to numbers with units attached (Figure 1). As preparation for science, knowing the math of numbers is essential, but in 2012 international testing in numeracy © XXXX American Chemical Society and Division of Chemical Education, Inc.
Received: February 7, 2018 Revised: May 13, 2018
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DOI: 10.1021/acs.jchemed.8b00085 J. Chem. Educ. XXXX, XXX, XXX−XXX
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are likely to have math deficits. Can we improve their rates of chemistry success? In recent experiments, the author and a collaborator designed and assigned homework tutorials which integrated relevant math review and topics in general chemistry. Evidence of a positive impact on chemistry achievement was seen for students with math backgrounds average or above those of students enrolled in first-year college chemistry. Discussion of this paper noted17 that as students applied the estimation strategies described by other articles in this ConfChem, they also practiced the overlearning advocated by cognitive research. Participants also suggested sharing with students the findings of research on learning.
memorized information as easily as it utilizes memorized information. Unfortunately, that assumption has proven to be mistaken.
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IMPORTANCE OF KNOWLEDGE IN MEMORY In the terminology of cognitive science, to solve a problem, the brain processes information in structures termed working memory. Working memory can acquire two types of knowledge: Nonmemorized information can be supplied by the senses or determined during steps of processing, and memorized information can be recalled from prior storage in long-term memory. In the past decade, in the case of well-structured problems (problems with answers experts agree upon, including math and science calculations), scientists who study the brain have reached the following consensus: During problem solving, working memory retains well-memorized information with ease but is exceptionally limited in retaining information that is not well-memorized.8−12 If an answer must be looked up or calculated, it occupies space in working memory that is limited. In contrast, for information quickly recallable from long-term memory, space in working memory is essentially unlimited. During the steps of problem solving, if more than a few elements of information are needed that cannot be recalled from long-term memory, some nonmemorized information drops out of working memory, and confusion tends to result.9−12
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CONCLUSION For instructors, how the brain solves problems is “central science”. Forums such as the Commentary papers in this Journal, designed to probe a variety of viewpoints, can assist in exploring the implications for chemistry instruction of the new paradigms in cognitive science. When our guidance on learning aligns with science, our students and society benefit.
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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.8b00085.
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IMPLICATIONS OF COGNITIVE ARCHITECTURE In a 2008 report of a U.S. Presidential Commission on K−12 mathematics instruction, five of the nation’s leading cognitive experts wrote the following (ref 13, p 4−6): [I]n support of complex problem solving, arithmetic facts and fundamental algorithms should be thoroughly mastered, and indeed, over-learned, rather than merely learned to a moderate degree of proficiency. To overlearn means to practice recall beyond mastery, using retrieval practice such as flashcards, repeated over multiple days. The need for overlearning also applies when students solve well-structured problems in chemistry.12,14 When recallable knowledge is utilized during problem solving, the connections of conceptual understanding grow physiologically between the neurons of long-term memory. However, this wiring cannot form until after knowledge has been stored in neurons.15 For the exact (verbatim) relationships that characterize math and science, long-term storage generally requires overlearning.13
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Full text of the original paper with associated discussions from the ConfChem Conference (PDF)
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Eric A. Nelson: 0000-0003-2909-8892 Notes
The author declares the following competing financial interest(s): The author has co-authored textbooks in chemistry. † Retired.
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ACKNOWLEDGMENTS This paper is an extension of cognitive literature reviews and publications by the author and JudithAnn R. Hartman, Department of Chemistry, U.S. Naval Academy (retired).
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DISCUSSION During the initial study of a topic, students must thoroughly memorize fundamental definitions and relationships. The Supporting Information materials provide details: In 2018, among cognitive experts, this is uncontested science. However, between 1975 and 2014, K−12 math curricula in most states gradually de-emphasized memorization. Since 2014, in response to cognitive research, most states have implemented revised standards that restored in part an emphasis on recalling math fundamentals from memory,16 but change in K−12 takes time. For the next 8−10 years, most U.S. students entering college will have spent at least some years with textbooks that discouraged memorization. Those students
REFERENCES
(1) National Science Foundation. Science and Engineering Degrees: 1966−2012, 2015 Tables 5, 37; https://www.nsf.gov/statistics/2015/ nsf15326/ (accessed May 2018). (2) Goodman, M.; Sands, A.; Coley, R. America’s Skills Challenge: Millennials and the Future; Educational Testing Service: Princeton, NJ, 2015. (3) Organisation for Economic Cooperation and Development. Literacy, Numeracy, and Problem Solving in Technology-Rich EnvironmentsFramework for the OECD Survey of Adult Skills; OECD Publishing: Paris, 2013. (4) Klein, D.; Braams, B. J.; Parker, T.; Quirk, W.; Schmid, W.; Wilson, W. S. The State of the State Math Standards 2005; Fordham Foundation: Washington, DC, 2005.
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DOI: 10.1021/acs.jchemed.8b00085 J. Chem. Educ. XXXX, XXX, XXX−XXX
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(5) National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics; NCTM Publishing: Reston, VA, 1989. (6) Raimi, R.; Braden, L. State Mathematics Standards; Fordham Foundation: Washington, DC, 1998; https://files.eric.ed.gov/fulltext/ ED421513.pdf (accessed May 2018). (7) Loveless, T. The Regulation of Teaching and Learning. In Stability and Change in American Education: Structure, Process, and Outcomes; Hallinan, M., Gamoran, A., Kubitschek, W., Loveless, T., Eds.; Eliot Werner Publications: Clinton Corners, NY, 2003; pp 171− 192. (8) Ericsson, K. A.; Kintsch, W. Long-Term Working Memory. Psychological Review 1995, 102, 211−245. (9) Cowan, N. The Magical Number 4 in Short-Term Memory: A Reconsideration of Mental Storage Capacity. Behavioral and Brain Sciences 2001, 24, 87−114. (10) Cowan, N. The Magical Mystery Four: How Is Working Memory Capacity Limited, and Why? Current Directions in Psychological Science 2010, 19 (1), 51−57. (11) Willingham, D. T. How Knowledge Helps. Am. Educator 2006, 30, 30−37. (12) Clark, R.; Sweller, J.; Kirschner, P. Putting Students on the Path to Learning: The Case for Fully Guided Instruction. Am. Educator 2012, Spring, 6−11. (13) Geary, D. C.; Boykin, A. W.; Embretson, S.; Reyna, V.; Siegler, R.; Berch, D. B.; Graban, J. The Report of the Task Group on Learning Processes; U.S. Department of Education: Washington, DC, 2008; pp 4-2 to 4-10. http://www2.ed.gov/about/bdscomm/list/mathpanel/ report/learning-processes.pdf (accessed May 2018). This report contributed to a subsequent work: National Mathematics Advisory Panel. Foundations for Success: The Final Report of the National Mathematics Advisory Panel; U.S. Department of Education: Washington, DC, 2008; http://www2.ed.gov/about/bdscomm/list/ mathpanel/report/final-report.pdf (accessed May 2018). (14) Willingham, D. T. Practice Makes Perfect-But Only If You Practice Beyond the Point of Perfection. Am. Educator 2004, 28 (1), 31−33. (15) Hebb, D. The Organization of Behavior; Wiley: New York, 1949. (16) Nelson, E. Cognitive Science and the Common Core Mathematics Standards. Nonpartisan Educ. Rev. 2017, 13 (3). http://nonpartisaneducation.org/Review/Articles/v13n3.pdf (accessed May 2018). (17) 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). The Addressing Math Deficits with Cognitive Science paper and discussion are available at https://confchem.ccce.divched. org/content/2017fallconfchemp8 (accessed May 2018).
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DOI: 10.1021/acs.jchemed.8b00085 J. Chem. Educ. XXXX, XXX, XXX−XXX
