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2017 Fall ConfChem: Mathematics in Undergraduate Chemistry Instruction

10/26/17 to 11/27/17
Cary Kilner and Eric Nelson

Chemistry is a quantitative science.  For college majors in the sciences and engineering, “general chemistry for science majors” is a required course that is expected to provide students with a foundation in solving scientific calculations: mathematics applied to measurements.  In recent years, many instructors have noted deficits in the background of growing numbers of students seeking to enroll in general chemistry.  In a 2012 survey of nations with highly developed economies, US 16 to 24 year olds had the lowest scores in “numeracy” among the 22 nations tested.

During this four-week conference, each week two papers will be posted for concurrent discussion.  Authors will discuss their interventions to improve student skills in the mathematics used in courses ranging from high school college-preparatory chemistry to college physical chemistry. As the conference organizers, we believe you will find ideas in each paper that can provide substantial “food for thought” in your work to help students learn.

For the two papers posted each week, Monday to Wednesday will be “reading/question days” when readers may question the authors on the content of their papers.  Thursday to Sunday will be “general discussion days” where readers and authors may originate comments and pose questions on issues related to the ideas broached in the papers.

Questions begin for the first two papers on Monday Oct 23th.  Discussion of the last two will end on Sunday, November 19.  Starting Monday, November 20 for one week (including Thanksgiving), comments may be submitted to this the main conference page in a final discussion and wrap-up.  If you have a question or comment on a paper after its week for discussion has passed, this will be the time to present it.  The conference will conclude on Monday, November 27.

Without registration, anyone may read the conference papers, questions, and comments.  However, to ask questions or post comments, you will need to register with CCCE via the instructions at the top left of this page. 

Because registration can take a day or two to activate, if you think you may want to submit questions or comments, we recommend you register as soon as possible.

Additional detail on the conduct of the conference is available at

To begin, read a paper and its discussion by clicking on the title below.

Enjoy the conference!

-- Cary Kilner and Eric Nelson, Conference organizers


Conference Dates

Week Author Title
Week 1: Monday October 23 - Sunday October 29
Paper 1 Penn Estimation – An Empowering Skill for Students
Paper 2 Mason et. al. MUST-Know Pilot—Math Preparation Study from Texas
Week 2: Monday October 30 - Sunday November 5
Paper 3 Ranga Impact of Quick Review of Math Concepts
Paper 4 Leopold Strengthening Math Fluency through Calculator-Free Chemistry
Week 3: Monday November 6 - Sunday November 12
Paper 5 Craig Building Student Confidence With Chemistry Computation
Paper 6 Kilner The Chem-Math Project
Week 4: Monday November 13 - Sunday November 19
Paper 7 Shipman et. al. Applied Mathematics for Chemistry Majors
Paper 8 Nelson Addressing Math Deficits With Cognitive Science
Week 5: Monday Novenber 20 - Sunday November 27
ConfChem Wrap-Up


Conference Articles

Abstracts of Papers:

Estimation – An Empowering Skill for Students in Chemistry and Chemical Engineering

Lynn S. Penn


Today’s students are from the generation that has always used calculators to compute the answers to problems.  This has rendered them with no means to estimate numerical answers and no way to recognize an unrealistic answer.   Lower-level undergraduate students are able to overcome these deficiencies by means of training in piece-by-piece execution of a complex algebraic computation.  By moving to fewer and fewer products and quotients in a single equation, an estimate of the final value of a complicated algebraic equation can be made readily; the students gain a sense of scale when they see that their estimates are within 10% of the exactly computed value.  At first, students resist this training, claiming that they will always have a calculator at hand.  This presentation addresses how to overcome student resistance and how to move step-wise through practice exercises to the goal of rapid and reliable estimation final values.


MUST-Know Pilot Study

Amy Petros, Rebecca Weber, Sue Broadway, Robyn Ford, Cynthia Powell, Kirk Hunter, Vickie Williamson, Deborah Walker, Blain Mamiya, Joselyn Del Pilar, G. Robert Shelton and Diana Mason


Since 2007, the reported SAT (reading + math) scores for the state of Texas have steadily fallen from a high of 999 to an all-time low of 944. Solving this problem requires a multifaceted approach. For our part as instructors of a known gateway course, general chemistry, we chose to focus on the most fundamental crosscutting topic in STEM: arithmetic. Hence, the MUST Know (Mathematics: Underlying Skills and Thinking) study was conceived and implemented. General chemistry is widely considered a gateway course because students' success in general chemistry provides entry into several STEM and some non-STEM careers. Failure to succeed in general chemistry has been linked to students' mathematics fluency that other researchers have attributed to poor algebra skills. However, is it possible that this relationship should really be attributed to students' lack of "must-know" arithmetic skills? In Fall 2016-Spring 2017, a team of 11 chemical educators investigated the relationships between solving simple arithmetic problems and course grades for 2,127 students (60.3% female) enrolled in general chemistry I and II at six post-secondary institutions (3, large public research universities; 2 Hispanic Serving Institutions; and 1, 4-year private university) from varied geographic locations in the heart of the state of Texas overlaying 32,000 square miles. The arithmetic concepts evaluated for this study are introduced to most Texas students starting at the 4th-grade level. The selected concepts include multiplication, division, fractions, scientific notation, exponential notation, logarithms, square roots and balancing chemical equations. Results support that students, without the aid of a calculator, succeeded at the 40%-correct level (Chem I) and 60%-correct level (Chem II). Students' algebra skills might be a better predictor of overall success, but the initiator of the problem we posit starts with lack of automaticity and fluency with basic arithmetic skills. Correlations between final course grades and mathematics fluency ranged from 0.2-0.5 with the Hispanic-serving classes being among the weakest correlations and the research universities exhibiting the strongest. Building a strong profile of a successful general chemistry student is beginning to form from this continuing investigation. Future plans include implementation of High-Impact Practices (HIPs) to increase numeracy followed by dissemination of outcomes and expansion of the study to include other needed success-producing skills like logical thinking, spatial ability, and quantitative reasoning ability.


Impact of Quick Review of Math Concepts in General Chemistry Courses: Engaged Student Learning

Jayashree S. Ranga


Math proficiency is a vital skill for mastering concepts in General Chemistry courses. In this article, the author discusses simple yet powerful, pedagogical interventions implemented in General Chemistry courses to assist students with math. (a) A quick review of math concepts essential for solving chemistry problems has led to positive learning experiences in General Chemistry courses. This includes topics such as rearranging equations, exponents, etc. (b) A major challenge in General Chemistry courses include improper use of calculators. A quick discussion emphasizing the importance of parentheses/various function keys on calculators has led to efficient problem solving sessions. (c) Problem-solving skills are one of the most important skills acquired in these courses. Students learn how to read a problem, identify the given content, and then proceed to solve the problem. To alleviate stress during the problem solving sessions, pedagogies such as pause method were explored.  Sample math review content, tips for using a calculator effectively, and problem solving strategies used in General Chemistry courses will be presented in this article.


Strengthening Students' Math Fluencies through Calculator-Free Chemistry Calculations

Doreen Geller Leopold


In a 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.1  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.  Multiple-choice exams incorporating problems of this type, in which calculators were not allowed, were also administered.  It is hoped that these methods can help motivate students to gain greater intuitive and conceptual insight through solving quantitative chemistry problems, and to become more fluent in expressing science in the language of math.  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.  Finally, we discuss some "broader impacts" of strengthening basic math skills, using as an example a suggested connection between some eligible voters' understanding of percentages and probabilities, and the outcome of the 2016 presidential election.


Building Student Confidence with Chemistry Computation

Peter R. Craig


I work in a liberal arts college as a chemistry professor.  Not educated in the US, I have been welcomed into my adopted culture by being given the opportunity to teach that has allowed me to learn about a system I knew little about beforehand.  At the college level, chemistry can be portrayed as applied algebra.  In my experience algebra takes on a whole new level of difficulty and significance for students when the level they need to understand it underpins the comprehension of a subject it is a prerequisite for.  This appears to be exacerbated by the following factors: the lack of continuity of the offering of high school algebra and chemistry prior to entering college, the increased emphasis of attaining confidence at the cost of learning content at high school, and the numbers of students attaining access to college who don’t know how to study.  This paper looks at attempts to redevelop the robustness of students’ chemistry read-only memory (ROM) – their ability to identify and apply appropriate computational methods to solve problems without much thinking or hesitation.  With the confidence of a reliable ROM, students are better able to learn chemistry at college.


The Chem-Math Project

W. Cary Kilner


Much national effort has been directed at the recruitment of STEM students. However, retention has become an equally important task, especially when many STEM students find the mathematics of chemistry a formidable barrier. In a five-year action-research study, the author has examined this problem closely from the students’ point of view. By anticipating and addressing a myriad of small issues, students’ skills and confidence can be built up and they can become proactive learners.

In this paper, the author shall present some direct quotes from students, along with examples of student work, to show how we may help students overcome their mathematics anxiety and become successful chemistry students. “Chem-math” represents a contradistinction from the “formal-math” to which these students are typically referring. In a chem-math recitation, we will review mathematics fundamentals, show how they are applied to chemistry, and clearly define all words and symbols they will use in their study. We will not allow any sloppiness in execution. Through such rigor we can set students on the road to the successful execution of chemistry exercises and problems.


Applied Mathematics for Chemistry Majors

Rachel Neville, Amber T. Krummel, Nancy E. Levinger, Patrick D. Shipman


The math that chemistry students need is significant.  In physical chemistry, students need to be comfortable with ordinary and partial differential equations and linear operators. These topics are not traditionally taught in the calculus sequence that chemistry students are required to take at Colorado State University, thus mathematics can present a significant barrier to success in physical chemistry courses.  Through the collaboration of the mathematics and chemistry departments, Colorado State University has developed and implemented a two-semester sequence of courses, Applied Mathematics for Chemists (MfC), aimed specifically at providing exposure to the math necessary for chemistry students to succeed in physical chemistry.  The prerequisite for the sequence is a first semester of Calculus for Physical Scientists—that is, a working knowledge of derivatives, integrals and their relation through the Fundamental Theorem of Calculus.  MfC begins with a look at the Fundamental Theorem of Calculus that emphasizes a scientific realization that it provides, namely an understanding of physical phenomena in terms of an initial condition and the rate of change.  This introduces the first topic of MfC, namely first- and then second-order differential equations.  Working with differential equations at the start of the course allows for questions from chemistry to motivate the mathematics throughout the sequence.  Solving the differential equations naturally introduces students to another fundamental mathematical concept for physical chemistry, and another theme of the course, namely linear operators.  The flow of the course allows for topics traditional to second and third semesters of calculus, such as Taylor series and complex numbers, to be motivated by solving chemical problems and leads to some topics, such as Fourier series, which are not part of the standard calculus sequence.  Feedback from students who have taken MfC and then physical chemistry has been positive.


Addressing Math Deficits to Improve Chemistry Success

Eric A. Nelson


The brain solves problems in structures termed “working memory.”  Between 2001 and 2010, cognitive experiments verified that at each step when solving a problem, working memory can hold only a few small elements of knowledge that are not well-memorized.  One implication of this limit is that students must rely almost exclusively on the application of memorized facts and algorithms when solving mathematical or scientific calculations. 

Unfortunately, since 1990, 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 students have deficits in “automaticity” in the recall of math that is necessary for chemistry.  This paper will include evidence that if math fundamentals are moved into memory as preparation for a chemistry topic, student success in first-year chemistry improves substantially.


Hi All,
This is the second test of the ConfChem.  Hopefully this goes out to everyone, and has a link that brings you back to the conference homepage.  On Monday morning we will subscribe papers 1 and 2 to the list, and send out a message to the list, indicating the start of the discussions.  Please feel free to read the articles, but withhold making comments until the list is subscribed to the papers.  That way the authors can be available to respond. 

Hopefully this will be the last test email.


Bob Belford's picture

Hi All, 

Please pardon if this seems a bit in left field, but hopefully it is of relevance to the recent discussion that was carried on the ConfChem list, dealing with the preparation of instructors for the task of instruction.

When my students complain about varied and often contradictory theories I often tell them a story from my grad school days, and the importance of thinking and conditional knowledge. This was the story of a colleague who received his PhD in Russia in the immediate-post Stalin era. He explained to me that it was not “healthy” in the day’s of Stalin to be a scientist who believed in genes, and that there was actually an alternative theory called Lysenkoism,  (Please check it out if you are not familiar – it is an eye opener!)

The point at hand here is that Mendelian genetics were making their way back into the Russian universities when he was a student, and he not only had to know both theories, but guess on who was asking the question, and then determine which theory he should reply with. My point here is that part of the “task of instruction” is to get students to think, and not just parrot what the instructor knows, and things could be worse, a lot worse! 


The goals of assessing and enhancing our chemistry students' pencil-and-paper math skills and abilities to estimate, and decreasing their over-dependence on calculators, were a focus of more than half of the 8 papers at this ConfChem, and of many of the comments.  Some encouraging news, that I was delighted to discover last week, is that the new SAT (Scholastic Assessment Test), which was first administered in March 2016, includes one math section (out of three) that does not allow calculators!  Of the sample questions posted by the College Board, many require manipulating numbers (not just symbols) without a calculator.

Since the SAT is so critical for many high school students in their quest for admission to the colleges of their choice, and sometimes also for their eligibilities for college scholarships, this change may provide a strong impetus for students to strengthen their math fluencies.  This is likely to be an even more compelling motivation for students, as well as for their teachers and parents, than is enhancing their readiness for college chemistry.

Here is an excerpt from the College Board's website under "Calculator Use":

"The Math Test-No Calculator portion of the test makes it easier to assess your fluency in math and your understanding of some math concepts. It also tests well-learned technique and number sense."

The same comment is made for their PSAT/NMSQT and PSAT 10 exams.

If anyone reading this has played a role in implementing these improvements, thank you - you have given us all something more to be thankful for on this Thanksgiving weekend!

You can see 18 sample questions (with answers and detailed explanations that emphasize calculator-free solution methods) from the No-Calculator portion of the new SAT at the following link.  Most of these are multiple-choice, and others require the student to "grid-in" a numerical answer.

Scroll down to the bottom, click on "questions that do not permit the use of a calculator" (in the second-to-last paragraph) and you can download the 58-page Word file, which reiterates on page 5:

"1. The use of a calculator is not permitted."

In addition, in the "Answers and Explanations" section beginning on p. 29, each of the 18 questions is individually labeled "Calculator Usage: No", and the applicability of that question for three types of exams, the SAT, PSAT/NMSQT, and/or PSAT 10, is also specified.

Many of these no-calculator questions require the student to perform calculations that rely on their arithmetic fluency.  For example, Question 3, which is a grid-in question applicable to all three types of exams, requires the student to factor 51.  Question 8, a multiple-choice question, requires one to multiply (or at least estimate the product of) 2-digit numbers.  Also see multiple choice Question 9 (which requires simplifying 27 / 39), and grid-in question 17 (which requires taking the square root of 169), which are also ranked for all three types of College Board exams. 

Khan Academy - free on-line math review:  How can students enhance their math fluency to better prepare for the SAT, without (or in addition to) paying a lot of money for courses or tutors?  Khan Academy has partnered with the College Board to provide practice questions and instruction for free!  Much of this review is also beneficial for enhancing high school students' pencil-and-paper math skills, and better preparing them for college chemistry. 

Here is an excerpt from their website:

"For the first time ever, the creators of the SAT have given Khan Academy exclusive access and advice to build a personalized practice program for anyone, anywhere.

These tools are free and available now for every student to take ownership of their learning and their future."

For example, see their SAT math prep website:

This includes many of the topics in which we would like our college general chemistry students to be fluent.  For example, here is their 3-minute video on solving a basic percentage problem on paper, without a calculator:

Our college general chemistry students can be directed to this set of SAT math prep videos (omitting those that are not as relevant, such as triangle, circle and trig functions under "Additional Topics in Math").  We can also recommend that they review Khan Academy videos on additional relevant topics, such as scientific notation:

ACT:  Many high school students take the ACT (rather than, or in addition to, the SAT) as part of their college admissions process.  Most colleges that require these exams will accept either one.  The ACT's calculator tips say, "You are not required to use a calculator.  All the problems can be solved without a calculator."  But, they do allow calculators (including graphing ones), and I would guess that most American students use them. 

For 60 ACT sample math problems, divided into 5 sets of 12 problems each, see:

It is fun to print out these problems and solve them without a calculator.  After entering the answers to all 12 questions in each set, one can view the answers and detailed explanations.  One notices that a student with a higher level of math fluency, who is comfortable with simplifying expressions and estimating, is likely to solve many of these problems more quickly than would students who are overly calculator-dependent.  Speed is a critical advantage on the ACT, since students have only 60 minutes to solve 60 math problems.

In summary, high school students can likely be motivated to enhance their calculator-free math fluencies to better prepare for both the SAT and ACT college admissions tests, and especially for the new no-calculator math portion of the SAT.  Practice problems and tutorials that model doing pencil-and-paper calculations, available for free from Khan Academy, can also help students be better prepared for their college chemistry and other STEM classes.

Hopefully, the “calculator-free” section of SAT will persuade those who write K-12 state standards to focus on teaching math without a calculator -- always in elementary schools, and often in middle and high schools =-- to avoid embarrassing scores when state students get to the SAT.

However, though encouraging students to review math fundamentals before they are needed for topics in chemistry will help a good number of students, to have maximum impact, state boards of education need to be pushed to focus on automaticity in their K-12 standards starting in first grade. If students were never required to commit some fundamentals to memory, they may take longer to learn those fundamentals than the time available in a “review” course.

State standards determine whether local districts will adopt math curricula that align with how science says math needs to be taught (due to limits that impact human working memory).

Even if the SAT-type pressure does improve state math standards, it will take 12 years before colleges see students who have been pushed to learn math fundamentals since first grade -- where teaching fundamentals needs to start.

Given how long it would take in K-12 to adopt cognitive-science-aligned standards, then adopt curriculum that aligns with science, then buy the textbooks that in K-12 are nearly always required in K-12 financial budgets to be used for at least six years, I suspect efforts by chem departments to improve student calculation skills are likely to be needed in most states for at least the next decade.

-- rick nelson

Cary Kilner's picture

First of all, I would like to thank the seven other authors and all of the discussants for a great online conference.
Now, how do we implement these good ideas to improve instruction and learning for our chemistry students?

Rick and Doreen have expressed the first component in Papers #4 and #8. Students must become numerate (Paulos). This is a long-term component (and commitment by others who are not chemistry educators), since it relies upon the promulgation and application of the following two key attributes. Students must be able to make quick mental calculations in their heads, and they must be able to make an estimation of the reasonability of computed answers.

Obtaining these attributes for our future students relies upon the adoption of suitable mathematics standards in the early years of schooling. It also relies upon lots of practice throughout primary and secondary schooling doing calculations without a calculator. It depends upon the application of current cognitive-science research in designing appropriate pre-service and in-service professional development of teachers at all levels, so they fully understand how their students learn best and thus ensure that students memorize their mathematics and chemistry/physical-science fundamentals.

The second component is expressed in my Paper #6, The Chem-Math Project, consisting of actions that chemistry instructors can take to improve instruction and learning for our current students. Chemistry instructors in high school and general/introductory/GOB courses in college can begin, right now, to harness the mathematics skills that even underprepared students do have by observing some of the points I present from my research, and expressed in the 27 Chem-Math Units of instruction found in my supplementary information. Here is a brief summary of what I found in the Chem-Math Project and expressed in my ConfChem paper.

Class time must be taken to show our students how the formal mathematics they have been taught is used in processing the measurements and applications used in chemistry, aka chem-math.

This simply cannot be done in a large lecture hall. It requires a small-group collaborative-learning format separate from lecture, with rigorous supervision by a pedagogical-content expert, not an untrained and inexperienced graduate-student.

Attention must be paid to a rigorous definition of terms. You and the student must understand each other in order for effective instruction to occur. This includes chemistry terms, mathematics terms, and common conversational English terms used in chemistry such as “substance,” that can be misunderstood.

Students must come to general-chemistry knowing their physical-science fundamentals upon which the subject of chemistry is constructed. If students do not understand these basics, time must be taken to study them and provided some laboratory experience with them.

Instruction in chem-math must utilize the language in which students have been taught their mathematics. This means that instructors must become conversant with that language. They can do this by having conversations with their mathematics colleagues and consulting NCTM research materials, particularly the NCTM Year Books.

Our less well-prepared chemistry students, either from lack of quality secondary instruction in chemistry, mathematics, and physical-science, or simply from an inability to comprehend the subject, will typically come to general-chemistry with great apprehension. They may have no interest in our subject but need it as a prerequisite for their major coursework. Thus addressing the affective domain is very important because these students may not be amenable to our instruction unless we recognize and acknowledge the desultory attitudes they may bring to our class. We need to show them that chemistry IS learnable, and that our mission is to provide a tenable route to that learning provided they do the hard work, in and out of class, that we advise.

Students’ previous mathematics instructors will have used specific vocabulary (mathematics nomenclature) in their instruction. It is important that we can use the same nomenclature in order to access that previous mathematics instruction when we teach chem-math.

Students will typically have practiced using proofs and derivations in a geometry class, and possibly in their algebra class. This can be a very effective learning tool when we can link it to the chemistry concepts we are teaching, and so our students can see how our chem-math equations come about.

There is much published recent research regarding on the use of multiple representations for teaching chemistry concepts. Johnstone’s macroscopic, submicroscopic, and symbolic representations provide one such schema. Jensen recommends the use of molar, molecular, and electronic representations. The NCTM recommends integrating graphical, numeric, algebraic, and verbal representations into their instruction, and we should use these in teaching chem-math.

As we teach chem-math, it is important to consistently use and demonstrate one clear method of setting up and solving chemistry exercises and problems. Only in this way can we extirpate sloppy pencil-and-paper habits that will have become intransigent, and that will perjure all attempts to teach students chem-math if we ignore them.

Finally, we know that all the good work we do will not result in student success unless they do the necessary hard work outside of class. Whether we use the flipped-classroom design or a traditional lecture, they must practice on their own. We cannot rely on students finding what they need, but we must recommend or assign quality online homework. Some good online resources are the Khan Academy, ALEKS, and authored by my co-moderator, Eric Nelson.

Hopefully, these suggestions from my work might be welcomed by chemistry instructors interested in a more global look at chem-math instruction.