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Applied Mathematics for Chemistry Majors


Rachel Neville1, Amber T. Krummel2, Nancy E. Levinger2, Patrick D. Shipman3

1 University of Arizona, Department of Mathematics, Tucson, Arizona, United States
2 Colorado State University, Department of Chemistry, Fort Collins, Colorado, United States
3 Colorado State University, Department of Mathematics, Fort Collins, Colorado, United States

11/15/17 to 11/23/17

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.



The depth and breadth of mathematical skills that chemists need is significant. Like most American college and university chemistry curricula leading to the BA or BS degree, Colorado State University (CSU) has previously required students complete three semesters of calculus. This more than fulfills the requirements for the ACS approved chemistry degree (ACS, 2015). However, these calculus courses omit mathematical topics such as differential equations and linear operators that are imperative for understanding physical chemistry.  Similarly, the traditional calculus courses like those at CSU, cover content such as a broad range of integration techniques that are not of immediate use in physical chemistry.  From the instructors' perspective, the chemistry major would require students to take significantly more mathematics, including linear algebra and differential equations, prior to taking physical chemistry.  However, requiring these math courses would add credits to the chemistry major that already requires a lot of classes, making the curriculum less flexible and potentially decreasing the number of students majoring in chemistry. 

To provide chemistry students with appropriate mathematical background, and to refresh topics that students may have forgotten since their last math course, some CSU chemistry instructors have offered a "just-in-time math review" as an addendum to the Physical Chemistry 1 course. Because it is optional, not all students enrolled in the math review, reducing its potential impact.  To address the mismatch between the required calculus courses and to provide a math curriculum more aligned with the needs of chemistry courses, we have developed a two-semester math sequence, Applied Math for Chemists I and II (MATH 271 and 272) at CSU. 


Motivation and Background

Among students at CSU and elsewhere, physical chemistry has the reputation of being a very challenging course. Derrick and Derrick studied success of students at Valdosta State University and suggest that the “formidable perception'' of physical chemistry is due to the mathematical and conceptual difficulty rather than the chemistry itself (Derrick & Derrick, 2002). Early attempts to identify students who would struggle in a physical chemistry course resulted in a diagnostic quiz that tests students’ background in mathematical concepts deemed necessary for physical chemistry (Porile, 1976).

Prior success in math courses significantly impacts a student's success in physical chemistry. For instance, Hahn and Polik showed that student success in physical chemistry correlate significantly both with the amount of mathematics that a student has taken and grades earned in these mathematics courses (Hahn & Polik, 2004). Instructors at CSU have observed the same trend. In another study surveying instructors of physical chemistry courses across several hundred universities, 61% of instructors indicated that students struggle because they lack the necessary mathematical background and a third of instructors reported that students do not make connections between physical chemistry concepts and the mathematics on which those concepts are based (Fox & Roehring, 2015). This suggests that not only the mathematical concepts, but also their connections to chemistry are important to student success. In fact, after lengthy conversations with colleagues, one professor concluded “College students in the sciences often grasp the operations of mathematics but miss the connection between mathematical operations and the physical systems they describe.'' (DeSieno, 1975) Given these observations, it seems that we could provide a better math background to help our students succeed in physical chemistry.  

In 2000, the Mathematical Association of America (MAA) organized a series of Curricular Foundations Workshops to seek input on mathematics curriculum from chemists, biologists, physicists, and engineers whose students rely on a strong foundation in mathematics (Craig, 2001). Various working groups developed recommendations regarding the mathematical skills necessary for students in specific fields.  A working group composed of chemistry and mathematics faculty from different institutions gave a thorough recommendation of the content and conceptual principles that students should be taught and a recommendation for the division of responsibility (see table in the appendix of (Craig, 2001)).

Several topics were given high priority for mathematics competence of students in the chemical sciences, namely multivariate calculus, creating and interpreting graphs, spatial representations and linear algebra. Nearly all relations that students will encounter in chemistry contexts are multivariate. Therefore, students should be comfortable with handling multivariate problems, thinking of variables as more than merely a spatial extent or time. Due to large variations in physical scale of problems, students should be able to decide if solutions are reasonable with estimation techniques and order of magnitude calculations. There should also be an emphasis in visualizing structures in three dimensions.

The course sequence at Colorado State University was initiated by a request by faculty members in the Department in Chemistry who were seeking ways to improve student performance in the two-semester, upper-division undergraduate course in physical chemistry.  These faculty members believed that deficiency in mathematical preparedness presented a significant barrier to student success, both in terms of the mathematical topics covered in the prerequisite courses (a standard three-semester calculus sequence covering topics through multivariate calculus and targeting students in physical sciences and engineering) and student ability to apply the mathematical topics covered in those courses in their chemistry courses.  Faculty members from the Departments of Chemistry and Mathematics collaborated to design the sequence of two 4-credit, semester-long courses, called Applied Mathematics for Chemists (MfC).  The sequence was taught as an experimental course in the academic years 2014-2015 and 2015-2016 (with temporary course numbers, standard at CSU) and was accepted into curriculum of the Mathematics Department and as a prerequisite for the physical chemistry sequence in 2016 (course numbers MATH 271 and MATH 272).


Course Content

MfC has a prerequisite of Calculus for Physical Scientists 1 (derivatives and integrals) and serves as the mathematics prerequisite for the physical chemistry course. While there is some necessary mathematical background required for other chemistry courses, physical chemistry has the highest mathematical demands. The goal of the MfC courses is to provide students with a working proficiency of the mathematics so that they can focus on learning and understanding the chemistry.

Two texts are used for MfC, namely Enrich Steiner's The Chemistry Maths Book (Steiner, 2007), and Donald McQuarrie's Mathematics for Physical Chemistry (McQuarrie, 2008).  Both books focus specifically on mathematical topics relevant to chemists. These texts take a practical, straightforward approach, with less emphasis on theory or proofs of theorems and more emphasis on developing a student's mathematical tools applied to practical problems. The texts cover similar material, but the Steiner book is more complete mathematically, whereas the McQuarrie book has more detail on connections with physical chemistry. Students appreciated the full solutions freely available on the publisher’s website for The Chemistry Maths Book, as it offered quick feedback and an opportunity for individual practice. Mathematics for Physical Chemistry is written by the same author as the text that is used in the physical chemistry course at CSU and expands on math review sections that are included in the chemistry text (McQuarrie, 2008).   

Clear recommendations for mathematics courses for chemistry majors were given in the MAA Curricular Foundations Workshops (Craig, 2001), specific to the chemistry context. The expectation is set that math courses should develop 14 conceptual principles, nearly all of which are addressed in MfC. The exceptions are an extensive discussion of numerical methods, representation of information as analog or digital, statistics and curve fitting. Statistics and regression are covered in a statistics course that chemistry students are also required to take. All principles are marked in two categories, (1) they should be developed by mathematicians, and (2) the teaching of the mathematical concept in the specific context of chemistry is particularly effective. The material covered in this course is substantial, though necessary for the future success of chemistry students.

The course is topically divided into five parts. Parts 1 (differential equations, series, and complex variables) and 2 (linear algebra), are covered in the first semester.  The second semester covers parts 3 (inner product spaces and Fourier series), 4 (multivariable calculus), and 5 (partial differential equations).

The highlight in the prerequisite course (one semester of The Calculus) is the Fundamental Theorem of Calculus (FTC), typically written as 

Students see two interpretations of this relation.  With s equal to a spatial variable x, the FTC gives an area underneath the graph of f ' (x) in the domain a ≤ x ≤ b . With s equal to time t, the FTC gives the total change in f over the time interval a ≤ t ≤ b.  But, honestly, why calculate the total change f(b) - f(a) by some complicated integral?  MfC opens with a slight but tremendously revealing rewriting of the FTC;

Any differentiable function f(t) can be written in terms of an initial condition f(0) and a rate of change f '(t).  This mathematical insight also opens up a whole new way of thinking scientifically and leads into the first part of MfC, namely ordinary differential equations.   We cover chemical basic first- and second-order linear homogeneous and inhomogeneous differential equations and solution methods such as separation of variables, integrating factors, and the method of undetermined coefficients. Applications in chemical kinetics, the harmonic oscillator and a first look at Schrödinger's equations for a particle in a box motivate each class of equations. Complex numbers and series are taught as necessary theory for working with more complex systems. The grand finale of the unit on ordinary differential equations is the method of using power series to solve differential equations. Chemistry students are typically not exposed to these mathematical topics because they comprise topics in an ordinary differential equations course, which is not required for chemistry majors.

Part 2 covers linear algebra. Students are introduced to vectors and are encouraged to think of vectors as coordinates in physical space as well as holding variables that are not necessarily distance. There is an emphasis on what insights determinants and eigenvalues give when modeling a physical system. Symmetries and group axioms are taught primarily through linear transformations, with some discussion on finding group representations. Compelling examples come from symmetries of planar molecules (Hückel molecular orbital method) and distributions of electrons in p-orbitals. Several students reported this application as being the most compelling example from the entire course.

The second semester and Part 3 of MfC begins with the notion of a vector space and a basis. As inner product spaces are introduced, parallels are drawn between finite-dimensional vector spaces and infinite-dimensional inner product spaces. This gives students a concrete footing in a topic that they find very theoretical.  Orthogonal polynomials (including special sets of polynomials) are introduced. Rather than emphasizing the (often fairly involved) derivation of these polynomials, students are challenged to understand them as a basis for modeling specific physical systems. This notion is initiated here and developed further in the end-of-the-year project. Finally, students learn Fourier series and work with Fourier transforms and their interpretation in a mini-Matlab project. This section is generally the most challenging for students.

Part 4 returns to material that is usually covered in a standard course on multivariate calculus (third semester of a traditional calculus sequence). By this point in the course, students have become comfortable with working with expressions in multiple variables. Visualization in three dimensions is taught, as well as partial derivatives and multiple integrals. There is an emphasis on physical interpretation of these quantities. However, the level of coverage is not as extensive as a typical third semester calculus course.  For example, a topic from a typical course in multivariable calculus that is not covered in MfC is Stoke's Theorem.

The concluding part, the shorter Part 5 is a basic introduction to partial differential equations.  Students are introduced to separation of variables and the method is applied to solve the heat equation and the classical wave equations. Boundary conditions and initial conditions are discussed, again with an emphasis on modeling a physical system. This is a topic that students would not encounter until a course in partial differential equations after a course in ordinary differential equations, a course that very few chemistry students take.  We considered taking more time in Part IV and omitting Part V, but an advantage of covering Part V is that many concepts from the course come together when solving partial differential equations.  Indeed, this topic allows students to combine their knowledge of ordinary differential equation boundary value problems, partial derivatives, and Fourier series. Another advantage is that students are likely to see the wave equation near the start of a physical chemistry course, and we want them to feel like they are mathematically prepared from the beginning of the course.

Near the end of MfC, students are assigned a group project, applying separation of variables. This project is discussed further in Section 4.

To allow for this material to be covered in a year-long course, some sacrifices from the traditional sequence clearly need to be made.  This includes some integration techniques and theorems on convergence of sequences and series as well as Stoke's theorem.  Although the topics covered in MfC range from differential equations to linear algebra to understanding multivariable relationships, the fact that they are tied together by a theme of linear operators helps to unite the course and allows for the reinforcement of previously learned topics throughout.  

The focus of this course is on developing students' mathematical dexterity and reasoning skills with motivation coming from chemistry.  One challenge is that some of the most compelling examples require a good deal of chemistry to understand.   For example, students were assigned a project on Nuclear Magnetic Resonance (NMR). This is a compelling application of Fourier transforms. However, the theory on molecular structure and NMR is taught in Organic chemistry. The students who had taken organic chemistry (i.e. had seen NMR in a classroom setting) thought the application was neat, though oversimplified. The students who had not had an organic chemistry course, could perform the transform but were at a loss when it came to connecting the output signal to the molecular structure, even with an (oversimplified) explanation in the project description.

At the end of the second semester, students were given a final group project. Students are guided through the analytic solution to the Schrödinger equation for the hydrogen atom. This project pulls together concepts from operators, correct handling of multiple variables, partial derivatives, techniques of solving differential equations and partial differential equations, visualizing in three dimensions, and the postulates of quantum mechanics in an example that is very compelling for chemistry students.


Impact in Physical Chemistry Course

The difference in students completing the calculus sequence versus the MfC sequence to fulfill their mathematics requirements for chemistry is dramatic.  The difference in students’ daily engagement in the Physical Chemistry course is different between the two student populations.  For example, in Physical Chemistry 1, students that have taken the MfC course sequence have already been exposed to the concept of a differential equation so they do not have to grasp what a differential equation is before striving to understand the interpretation of the solutions they generate for the Schrödinger equation.  Instead, students that have taken the MfC sequence are confident in their practical knowledge of finding solutions to ordinary differential equations.  Thus, they have the capacity and are free to begin thinking about the interpretation of solutions to the Schrödinger equation, rather than being stuck on mathematical mechanics associated with solving differential equations.  Likewise, students having completed MfC approach the Maxwell relations in thermodynamics without trepidation having already manipulated partial differential equations.  These are only two of many examples that speak to the divide that MfC bridges by producing a course that nests the mathematics required as a chemistry practitioner in chemical applications. 

The feedback from students who have taken MfC and then physical chemistry has been positive.  These students have encouraged their colleagues to take MfC rather than the traditional Calculus sequence, noting that students having taken the traditional Calculus sequence struggle more in Physical Chemistry than students having taken the MfC course sequence. Even students who struggled in MfC have remarked how familiar they found the math in physical chemistry, which improved their outlook about the traditionally dreaded physical chemistry course. 

Finally, MfC offered at CSU does not require additional credit hours of math for our chemistry majors.  Instead, we have tailored the mathematics and the application of the mathematics to be aligned with the needs of a chemistry practitioner. To accommodate transfer students and students changing majors, we still allow chemistry majors to take the traditional three semesters of Calculus for Physical Scientists, but strongly urge our majors to take MfC. 



The authors would like to thank Francis Motta for his contribution to developing materials for this course.



ACS. (2015). Guidelines and Evaluation Procedures for Bachelor's Degree Programs. Washington DC: American Chemical Society Committee of Professional Training.

Bressoud, D. (2002, Aug./Sept. ). The Curriculum Foundations Workshop on Chemistry. FOCUS, 22(6). Washington D.C.: Mathematical Association of America.

Course Catalog. (2015-2016). Colorado State University. Retrieved from

Craig, N. (2001). Chemistry Report: MAA-CUPM Curriculum Foundations Workshop in Biology and Chemistry. Journal of Chemical Education, 78, 582-6.

Derrick, M., & Derrick, F. (2002). Predictors of Success in Physical Chemistry. Journal of Chemical Education, 79(8), 1013-1016.

DeSieno, R. (1975). How Do You Know Where to Begin? Journal of Chemical Education, 52(12), 783.

Fox, L., & Roehring, G. (2015). Nationwide Survey of the Undergraduate Physical Chemistry Course. Journal of Chemical Education, 92, 1456-1465.

Hahn, K., & Polik, W. (2004). Factors Influencing Success in Physical Chemistry. Journal of Chemical Eduaction, 81(4), 567-572.

McQuarrie, D. (2008). Mathematics for Physical Chemistry: Opening Doors. University Science Books.

N. Craig, D. B. (2000). CRAFTY Curriculum Foundations Project: Chemistry.

Porile, N. (1976). Diagnostic quiz to identify failing students in physical chemistry. Journal of Chemical Education, 2(53), 109.

Prussel, D. (2009). Enhancing Interdisciplinary, Mathematics, and Physical Science in an Undergraduate Life Science Program through Physical Chemistry. CBE- Life Sciences Education, 8(1), 15-28.

Steiner, E. (2007). The Chemistry Maths Book. Oxford UP.



Rich Messeder's picture

I'd like to start the week off by saying how much I appreciate the time and effort that went into preparing these papers. These threads are a most valuable resource to me, and made more useful by the comprehensive nature of the comments. I am primarily research- and engineeering-oriented, but I value the intrinsic worth of each student. I hope that the threads are available after the conference, because I intend to mine them for ideas.

Rich --

My understanding is that if you go to the main ConfChem site, the "Useful Links" will be posted on the left as they are on this page, and the "temporal article list" will have all of the articles and threads.

-- rick nelson

Hi All,

We actually  have three nonredundant backups of the ConfChem discussion archive. 

First, there are the actual papers, which as Rick states, can be found through the temporal article list, but also through the sortable article list (just pick the ConfChem you want to see), and note you can tag the papers if that helps your research (but not the comments, although those can be tagged with an annotation service like

Second is the actual Confchem List archive at UALR, .  Just choose the month, and there are several sort options (subject line, date...).

Third, after the ConfChem is over the authors have the option to submit these to the Journal of Chemical Education as a series of bundled communications.  Attached to each communication as "Supporting Information" is the actual ConfChem paper with the discussions, and so if the CCCE website goes down, and the UALR list goes down, you still have the discussions archived in the supporting Information of the JCE communications. I should add, that we remove personal identifiers like the names and images of the people making comments in the JCE supporting information, but what they say is preserved.



Rich Messeder's picture

Info captured for the future.

Dear CSU Team,

If I understand correctly, what you have done at CSU is to give chemistry majors a choice of the traditional CSU sequence of 3 semesters of 4-credit Calculus for Physical Scientists, or the new sequence of one semester of Calculus for Physical Scientists followed by 2 semesters of Applied Math for Chemists (taught by the math department).

My suspicion would be there were obstacles that needed to be overcome to achieve these offerings of “Calculus Customized for Chemistry.” For instructors who would like to gain the same type of sequence on their campuses, would there be advice you might be able to give on what bottlenecks to anticipate and how they might address them?

-- rick nelson

Yes, you understand correctly.

The main obstacle may be that for a math department to make a sequence of courses that is ideal for evey major would make for a lot of different course sequences and it gets unwieldy!  Plus, it takes special grad students like Rachel to teach the course--they need to be willing to learn some p-chem, perhaps even to learn some maths that they never learned (self-adjoint operators, for example) so this course can't be taught in the normal factory method of teaching calculus.    So, chemistry needs to really have enough students to put into the course, and maths faculty need to realize that this is a fun course to teach.

Maths faculty are used to the traditional sequence, and it can be hard to fathom some variation on it--differential equations are supposed to be a topic after Calc III, and we do them right at the beginning of the MfC sequence! 


First of all, I congratulate Dr. Nelson and his co-organiser (whose name momentarily escapes me, apologies) for their commendable organisation and moderation of this conference on internet.  This occasion is the first in which I have participated in such a meeting of minds, and I am pleased to read the various points of view and the dedication to improve the teaching and learning of chemistry.

The fact that students who are admitted to tertiary institutions are poorly prepared in mathematics is clearly not confined to USA, but is likely worse there consistent with the standing of USA in comparison with other developed countries.  One topic, to which I here respond, is the provision of courses by departments of mathematics for students of other departments.  Some years ago, a responsible senior professor of mathematics informed me that the policy of his department was to respond to any such request from another department, but within standard courses of mathematics there was no attempt to include examples or problems in any applied area, because although that content might be of interest to a fraction of students in the common course it would be boring and a distraction for the other students.  He emphasised that mathematicians were prepared by their academic experience to teach mathematics, not chemistry nor physics nor .... 

One approach that is more common in European universities than in Canadian and USA institutions is to have a course such as 'mathematics for chemistry' taught by either an instructor of mathematics devoted to a particular department, such as was the practice in Danish Technical University in the past, or a chemistry instructor who is suitably prepared to undertake such tasks.  In University of York UK two distinguished instructors, both active in fields of quantum chemistry, taught such courses, and even published a textbook Mathematics for Chemistry, by G. Doggett and B. T. Sutcliffe.  On examining that book, I ventured to express the opinion that I considered it to be at a rather low level, but Dr. Doggett replied that it was actually deemed to be beyond many students in British universities; for that reason he was asked to write two other short books at an even lower level, "Maths for Chemists (Tutorial Chemistry Texts)" published by Royal Society of Chemistry.  The problem addressed in this conference has two parts:  the mathematical preparation of students entering general chemistry, most discussed, and the mathematical requirements for succeeding courses in chemistry.  The fundamental solution to the preceding case is to improve the teaching and learning of arithmetic and mathematics in school, according to all the aspects described here, including estimation; in lieu of that solution for the present students, remedial courses at the tertiary level must be arranged, whether organised by chemists or mathematicians.  For higher courses in chemistry, if what departments of mathematics offer is deemed unsatisfactory or insufficient, chemistry instructors can offer their own courses, based on such a textbook as I specified above, for example, or other comparable books.  My interactive electronic textbook Mathematics for Chemistry is an alternative approach in which the objective is to teach, with advanced mathematical software (Maple), the concepts and principles of all pertinent mathematical topics and aspects, from arithmetic to group theory and graph theory, and then to encourage the students to apply their knowledge of the use of that software to solve chemical problems.  I despair of that book being of significant utility if students lack the basic arithmetical and symbolic skills that participants in this conference have decried.

When I was an undergraduate in a Canadian university, I was required to complete five year courses in mathematics (equivalent to ten semester courses) as a requisite of an honours degree in chemistry.  The average requirement of mathematics for chemistry in Canadian universities is now three semester courses, although the nominal mathematical level of content of chemistry courses has risen significantly in the interim.  I have noticed that standard textbooks of general physics, which might in many cases be a corequisite with general chemistry, have mathematical content at an increasingly high level.  For instance, Schroedinger's partial differential equation is introduced before its solutions and their properties.  How can students of chemistry cope with such content when we have read that even multiplication of small numbers challenges the capability of many such students?

Much of the discussion within this conference addresses a complaint that students entering general chemistry are unprepared mathematically.  It is ironic that the instructors who complain so vociferously are themselves unprepared mathematically to teach even general chemistry.  I refer to the content of all textbooks of general chemistry that I have seen that includes orbitals, electronic configurations of atoms and analogous material based ultinately on quantum mechanics, which is now recognised to be not a chemical theory, not even a physical theory, but a collection of mathematical methods, or algorithms, that one might apply to systems on an atomic scale -- which is far from the laboratory experience of general chemistry.  The authors of such textbooks, and the instructors who duly prescribe and teach the textbook in the fashion of a parrot, so act not because they understand the underlying mathematics and its consequences but because they do not so understand.  How many of you are aware that there is not just one set of orbitals for the hydrogen atom but four sets, not just one set of quantum numbers associated with those orbitals but one set of quantum numbers for each set of orbitals?  [The descriptions of these orbitals are freely available from 1709.04759, 1709.04338, 1612.05098, 1603.00839] You are "the credulous masses [or their successsive generations] -- that sad benighted chemistry professoriate -- dazzled with beguiling simplifications" by Pauling, "a master salesman and showman" [A. Valiunas, The man who thought of everything, The New Atlantis, No. 45, 60 - 98, 2015;, J. S. Rigden, Review of Linus Pauling -- a man and his science, by A. Serafini, Physics Today,  43 (5), 81 - 82, 1990].  Some years ago, I wondered whether a really intelligent student who encountered this incomprensible and indigestible rubbish about orbitals in general chemistry might decide that, because he could not understand what he was taught but would be forced merely to memorize the material quasi-religiously, the fault was his, so that he transferred to some other subject such as computer science that he could genuinely understand, leaving the mediocre students of general chemistry to progress onward in chemistry and eventually to become the next generation of professors to perpetuate the charade.

How many of you who have taught, in any shape or form, orbitals, which are indisputably solutions to the Schroedinger equation for the hydrogen atom, have actually read Schroedinger's papers?  They are available in authorised English translation; within them you can learn about a second set of orbitals and quantum numbers, beyond the first set for spherical polar coordinates with quantum numbers k,l,m.  Pauling never admitted the existence of this second solution, which would accordingly undermine his proffered ideas.  Your libraries might contain this book at QC172.....

Next time you feel like complaining about the quality of mathematical preparation of students entering chemistry, please reflect that your students have absolutely the same right to complain about the quality of mathematical preparation and understanding of their instructors -- only the ignorance of the students precludes such complaint, just as the mathematical ignorance of instructors of chemistry -- "that sad benighted chemistry professoriate" -- perpetuates the current paradigm of teaching chemistry.

Rich Messeder's picture

Hailing from a sub-culture in the US noted for its often too-frank speech, I appreciate John's frank discourse, however unsettling I found its direct criticisms. His comments directed toward "that sad benighted chemistry professoriate" apply equally to other fields, I opine. I see two root problems in academia (and perhaps beyond) that seemingly contradict one another. The first is that there seems to be a great deal of insecurity among academics, in my experience, and this relates to an unwillingness to be different, lest one not be "accepted". The second also relates to insecurity: arrogance. I think that this varies by institution, but I have seen it in all academic quarters over the past 4 decades. It is a bad enough example among academic peers, but it is especially destructive when faculty "talk down to students". We need outspoken leaders, yes. I am the captain of my ship, and my students know that as well as did those serving under me years ago in the military. Students look for confidence and leadership, but shy from arrogance.

"dazzled with beguiling simplifications"
I have lately coined the term "simplication" to mean just this. Why? Because I have seen too many instances of complicated material "simplified" in discussion to the point of irrelevance. For example, when working on a major physics research project recently, we were tasked with implementing an algorithm that was kicked around for years in popular terms that over-simplified its real complexity and impact on research. It was not until several frustrating years of increasing pressure from approaching deadlines that the issue was put on the table for open discussion. I recall that at one meeting the PI in charge of the research was stunned that he did not truly understand the algorithm that he been advocating all along. No one did, because they had all (PhDs) been kicking around a "simplicated" version of the algorithm. The beginning of the algorithm was replaced with an equivalent, much simpler, statistical model. There are places for both appropriate simplification ("as simple as possible, but no simpler"), and "the harrowing complexity of honest science."

I might be accused of being frank, as Frank is my middle name!

Having passed some years as a visiting professor or equivalent in significant departments of mathematics and physics, I have some intimate knowledge of those two fields, accumulated long after my undergraduate degree of which the programme was formally described as combined honours in physics and chemistry, with a healthy component of mathematics (equivalent to ten semester courses, as I mentioned) plus a course in mathematical physics as part of the physics sequence.  On the basis of that direct experience, I find it implausible that those two subjects suffer from the same systemic rot to the same extent as chemistry arising from orbitals and related rubbish -- even though physicists might be prone to attribute experimental observations directly to mythical 'hybridisation', clearly an infection from chemistry.  I have no doubt of the value of quantum-chemical calculations -- they were invaluable in our identification of two new boron hydrides, B2H4 and B3H3, for instance -- although for molecules containing elements other than boron perhaps 'molecular mechanics' would have produced similar results. One must, however, distinguish between orbitals, which pertain only to the hydrogen (or one-electron) atom, and members of a basis set that might be applied in quantum-chemical calculations.  Even the latter are superfluous because density-functional theory without an orbital basis set is a practical alternative.

What is an orbital?  It is incontestably an algebraic function.  An ignorance by chemistry instructors of the mathematical basis of such concepts is just as reprehensible as students admitted to general chemistry being incapable of undertaking basic arithmetical and mathematical operations for the solution of chemical problems.

The sources of the quotations cited in my preceding comment can provide an ample basis for the recognition of the deficiencies that must be rectified.

Yes, one of the challenges of teaching new things is eliminating students' conflicting misconceptions, some of which have been installed by well-meaning teachers seeking to help students feel that science "makes sense".  On the subject of atomic and molecular orbitals, I think these are some (among many) pedagogically useful articles (and two of the authors are also named "Frank"!):

"4s is Always Above 3d! or, How to Tell the Orbitals from the Wavefunctions," Frank Pilar, J. Chem. Educ., vol. 55 #1, Jan. 1978, pp. 2-6.

"Tomographic Imaging of Molecular Orbitals," D. M. Villeneuve and coworkers, Nature, vol. 432, 16 Dec. 2004, pp. 867-871 (includes a tomographic reconstruction of the HOMO of N2)

This image also appeared in C&E News, vol. 82 # 51, 20 Dec. 2004,  p. 10

"The Covalent Bond Examined Using the Virial Theorem," Frank Rioux, Chem. Educator, vol. 8, 2003, pp. 10-12.

   George Box pointed out many years ago that all models are wrong, some are useful.  Anybody who talks to an organic chemist knows the truth of this.

The standard sequence of teaching general chemistry proceeds through any number of simple models for chemical bonding and reaction following the historical development of the science. The atomic/molecular orbital model neatly summarizes these and also explains much about their limits of applicability.  Thus it is both useful and teachable at a beginners level.  Indeed, as far as structure and reactivity on the atomic level is concerned there is very little mathematics in the first semester of GChem and a whole lot of visualization and memorization for which atomic orbitals are sufficient.  If the correct description of electron density is 95% pz and 5% whatever, for teaching a general chemistry student does this make a difference?

I have found that the "atoms first" sequence has the advantage of making it easier to justify earlier models to the students, such as valence, oxidation number, octet rule, etc.  each of which, as well as atomic & molecular orbitals, have something useful to say about bonding and reactions as well as about the "real" nature of molecules as shown in the image referenced by Doreen (thanks:)

and others that are increasing appearing showing bonds, defined as regions of high electron density between atoms similar to what a good GChem student would write on an exam using orbitals.

Orbitals are, of course, algebraic functions, but they are not unconstrained by physical limits and they do convey useful and generally correct information about the shapes and reactivity of atoms and molecules which is what we are trying to teach. 

The "atoms first" sequence has the natural advantage that the students in the laboratory for general chemistry work directly with single atoms and molecules, and the students can directly see and measure the orbitals -- is that not the case? -- so that there is a direct connection between the lecture material and the laboratory material, which is a primary pedagogical objective.

Which orbitals do you use for your explanations?  There are four sets of orbitals for the hydrogen (or one-electron) atom, and each set has its individual shapes and set of quantum numbers -- but of course you understand that if you have been teaching about orbitals.  Unfortunately, all those orbitals apply strictly to the hydrogen atom; the corresponding algebraic functions of the helium atom -- yes, they are known -- have quite distinct and complicated algebraic forms.  You would not commit the logical fallacy of extrapolation from a point, fron H to any other atom, would you, Dr. Halpern?

The orbitals of H are presented both algebraically and pictorially in these four items freely available from

1709.04759,  1709.04338,  1612.05098,   1603.00899

Instructors who teach about orbitals might wish also to read "The nature of the chemical bond, 1990 -- there is no such thing as orbital", Journal of Chemical Education, 67, 280 - 289 (1990), republished, by request of the editors, with additional material in Conceptual Trends in Quantum Chemistry, Kluwer, 1994

First, IMHO, Dr. Ogilvie starts from a bad place.  It is not necessary that students use algebraic functions to describe hydrogen like orbitals for hydrogen or other atoms.  There are wonderful visualization tools that provide these images, including 3-D JMol, versions etc.  For discussion of atomic and molecular orbitals at the GChem level these are all that is needed and used.  Atomic and molecular structure in GChem are a visual, not a mathematical exercise.  A good site for these images is

Moving on what kind of experiments could one do in a gchem lab starting with atoms first?  Now obviously what follows can be refined and improved, but I believe it is a place to start.  YMMV

My general approach would be to have the students make a measurement and then interpret the results using concepts taught in class.  Availability of on line apps and data bases makes this much simpler than in the past.  I have a few suggestions, and others will hopefully chime in (pun intended).  Of course dry labs are also possible but really not what Dr. Ogilvie or I would want, at least in part.

For example, a mass spectrometer with unit resolution measuring the isotopes in a simple sulfur compound like SO2, could demonstrate isotopes and results and interpretation could be done using the NIST webbook.  

A simple limiting reagant experiment could be used to explain the mole concept and its relationship to atomic number and atomic weight.  The emphasis would be on the mole concept and not the stoichiometry

Both of these are taught at the beginning of an atoms first course as well as in the historical sequence.  

Moving on to atomic structure students could measure the hydrogen Balmer band spectra and relate that to the Bohr formula.  You could then measure the sodium atom spectrum and see how it does not exactly fit the Bohr formula, and use that to motivate the discussion of how the hydrogen orbitals are not quite the ones for sodium.  One could compare the hydrogen orbitals to the hydrogen like ones for more complex atoms.  See for example 

The web site I mentioned above also discusses this as well as showing the more complex shapes of the hydrogen like orbitals

There are worksheets at VIPEr that can be used in conjunction

  The NIST atomic spectra data base would help the students assign lines to transitions between hydrogen like orbitals. 

Since these spectra are relatively sparse and in the visible region, small spectrometers such as those sold by ocean optics would be fine.  If you are interested in He I, then the NIST atomic spectra data base  which generates Grotrian diagrams would be key to assigning the orbitals involved in the one electron transitions.

Molecular bonding could be demonstrated by selecting a small molecule.  The students would then describe bonding in the molecule using (the dreaded hybrid) orbitals.  Then a simple ab inito program would be used to calculate electron density maps and IR spectra.  The calculated spectra would be compared to the measured ones (either directly, or using the NIST Webbook or similar). The electron density maps would be compared to the initial prediction.  The calculation would, be all black box, Gaussian abuse as it were, but the students would start the calculations and measure the spectra as well as starting with the prediction of the shape and orbitals involved.

FWIW folks, might enjoy playing with this app

on their phones and comparing the orbitals to Dr. Ogilvie's

And so, good night to all :)

Dr. Halpern has raised some stimulating points, on only a few of which I comment here.

Despite the fact that the knowledge of orbitals existing in four distinct sets, each with its individual shapes and set of quantum numbers, has been available since at least 1976, I have no doubt that Dr. Halpern, like almost all other instructors of chemistry who teach orbitals, is blissfully ignorant of this fact.  "Where ignorance is bliss, 'tis folly to be wise."  [Thomas Gray, 1742]  If Dr. Halpern were so aware, he might have difficulty justifying the selected particular set of orbitals, in spherical polar coordinates, for the purpose of his teaching, but blissful ignorance precludes such a tiresome chore.  Both the formulae and the pictures (plots) of orbitals in all four sets are freely available at

Dr. Halpern suggests the use of a "simple ab initio program to calculate electron density maps and IR spectra".  The problem is that a truly "ab initio program" will calculate no such results.  If the atomic nuclei and electrons are treated in an equitable manner, there is NO molecular structure (apart from trivial cases such as diatomic molecules); cf. M. Cafiero, L. Adamowicz, Molecular structure in non-Born-Oppenheimer quantum mechanics,  Chemical Physics Letters, 387 (1), 136 - 141, 2004.  Dr. Halpern seems confused between truly "ab initio" quantum-chemical calculations and semi-empirical calculations, in which a structure is included in the input with some chosen 'canned' basis set.

I applaud Dr. Halpern for having arranged the use of a mass spectrometer, even with resolution only unit mass (dalton), for the direct use -- "hands on" -- of each of his students in his laboratory for general chemistry.  Perhaps other expensive instruments might preclude the necessity for students to learn to titrate a base with an acid, but in any case beginning with "atoms first" might require several years of courses to reach the level of practical chemistry in other than the most superficial manner.

When the senior author. P. Corkum, of this paper by Itatani, Villeneuve and others presented a lecture on this topic, I challenged him to define an orbital, but he demurred.  The authors of that paper understood what they claimed to measure neither during the experiments nor afterward.  Anybody who takes seriously the claim of these authors to have recorded an image of a molecular orbital (not molecular orbitals, plural) has only the most superficial understanding of the experiment and its interpretation, which is replete with errors.  cf Foundations of Chemistry, 13, 87 - 91 (2011); DOI 10.1007/s10698-011-9113-1

Cary Kilner's picture

A (hopefully) balanced response from the co-moderator (I respectfully forgive the slight):

John’s vociferous post (and not meant as a derogatory description) provokes the following response. He makes many good points in his diatribe, but we need to get back to the masses we are trying to educate. Of course, we seek the excellence he wishes for the upper-level courses for majors. But the fact remains that we must focus on our service-course clientele, who will be our future health-care professionals and who are the most challenged students in the physical-sciences and the least prepared mathematically. While he decries instructor misunderstandings in teaching MO theory, I decry instructors who take mathematical competence for granted and still ply the sink-or-swim perspective. If students’ primary and secondary education is not sufficient for their study of chemistry, we simply must take up the baton ourselves—hence this ConfChem.

His point regarding mathematics teachers teaching “formal math” and neglecting applications seems to be a fault of mathematics instructors more interested in their own egos than in any collective effort to prepare students for a meaningful career that requires the use of mathematics in any capacity. After all, mathematic instruction represents even more of a service course than our own gen-chem, serving a greater number of students and more diverse majors. We might address this issue through pleas to the NCTM, who seem to be at the forefront of mathematics education.

And I agree with his argument, which I shift slightly here, that the student might rightly complain of the chemistry instructor being unable to understand and address the neophyte students’ troubles with formal mathematics and with its translation into chem-math -- which, of course, is why we presented this ConfChem on mathematics in the teaching of chemistry—for the edification of interested and concerned chemistry instructors.

I really don’t care if these students know an orbital from an orbit. I want them to understand how to make a serial dilution, how to calculate the volume of gas at a given temperature and pressure from a given reaction, how to determine if a given reactant is limiting or in-excess, how to perform a successful titration, how to use Beer’s law and do UV-vis spectroscopy, how to conduct a meaningful calorimetry experiment. Some instructors might feel that these calculations are too abstract for the life-science majors. But I believe that you simply cannot teach chemistry meaningfully without showing how the science developed from an engineering perspective, i.e. in the service of solving practical problems. I want them to know that a chloride salt is NOT a “pale green gas,” and that carbon has several allotropic forms, and the difference in behavior between concentrated sulphuric acid and concentrated nitric acid. In other words, I want them to know some descriptive chemistry with its associated chem-math measurements and calculations.

An understanding of a need for chemical calculations (“chem-math”) has to arise from a need to understand interesting chemical and physical phenomena, either presented via provocative demonstrations or carefully-developed wet-chemistry activities or formal experiments. For instance, in seventh-grade I was reading about the shock sensitivity of potassium chlorate. Of course my local mentor and pharmacist sold me some of this salt, since back then pharmacists WERE, in fact, “chemists.” I also obtained chromates and dichromates and potassium permanganate and iodine crystals. My father allowed me to take sodium hydroxide pellets from the 55-gallon barrels in his shop, where I avoided inhaling the aggravating dust and observed the pellets immediately take on water from the humid air. My grandfather helped me obtain the concentrated acids I needed for my home basement laboratory.

Back to my KClO3 story; since I understood stoichiometry as a way to DO chemistry, I was able to balance the equation for its reaction with table sugar, that I knew to be a disaccharide, and to calculate how much sugar to mix with my one gram of KClO3. Unfortunately this was too large a mixture. As I ground it on the cement floor of the basement using a lead plate I had melted down, it detonated with a huge BANG, instantly filling the basement with a fog. My parents cheerfully called down the back stairs, “Everything O-K down there?” To which I responded in a cold sweat, “Yeah, it’s all good.”

The main point in my paper for this ConfChem is that to address difficulties in mathematics, you must first have the student PRESENT – not on his/her smart-phone, not downloading a power-point, not sitting in the back scribbling inchoate notes, not practicing Educational Darwinism and merely passing with a D- to get the credit, but actually engaged with the material. Otherwise, how else can they learn? And why else are they there? The long-lost lecture-demonstration pedagogy, with a formally-hired and designated demonstrator/demonstration coordinator was the way in the past that we have been able to engage a large lecture hall of students—not as entertainment but to show how concepts are related to phenomena, with concomitant measurements.

This speaks to the value of the flipped classroom and of POGIL as a way to engage students. Nevertheless, however uniquely interested and energetic instructors have tried to implement these initiatives into 100+ classrooms, it is really the small classroom that enables these practices to work well, where the instructor can get in the face of EACH student to ensure he or she is engaged, and to ferret out issues preventing engagement. I’m speaking, of course, from 23 years of high-school teaching with the luxury of 15-25 student classes. And small liberal-arts colleges have this luxury as well. It’s up to chemistry educators to continue to research ways to effectively engage students so they are actively THERE in the large lecture halls of our large public universities.

Finally getting back to mathematics, in my doctoral research I examined thirty-five pamphlets, booklets, paperbacks, small books and textbook chapters and appendices, to see how chem-math was being addressed by other concerned instructors. Of all these I found "Maths for Chemistry; A chemistry’s toolkit of calculations,” by Paul Monk and Lindsey Munro (Oxford U. Press, 2nd edition) to be outstanding; written very clearly and the best of the bunch. John and some other participants in this conference have cited various British publications, so I wonder if he is familiar with this fine book. It may not have quite the depth he requests, but it is very thorough. Besides the dimensional-analysis, algebra (and graphing) review typical of most chem-math primers, it provides three chapters on powers and logs, two on statistics, one on trig, six on differentiation, four on integration, and one each on matrices (including group-theory), vectors, and complex-numbers. So it seems to me this chem-math text would serve most physical-chemistry instructors well.

(my apology again to Dr. Kilner)

The objective of my attention to mathematics for chemistry has been mathematics for chemists, i.e. students proceeding to an academic degree with chemistry as major subject.  I had been unaware, and am somewhat astonished, at the severity of the mathematical incapability of students of general chemistry for most of whom the ultimate interests lie elsewhere than in chemistry.  The latter problem evidently requires concerted attention, such as remedial courses for present students and reform of school curriculum for future students, and this conference has been addressed mainly to this concern.

"His point regarding mathematics teachers teaching “formal math” and neglecting applications seems to be a fault of mathematics instructors more interested in their own egos than in any collective effort to prepare students for ..."  How other than "interest in their own egos" can one explain the propensity of instructors of general chemistry, following the authors of their selected textbooks, to teach orbitals and electron configurations to students of biology, nursing ... within common courses of general chemistry?  I persist in maintaining that, if those instructors, and the authors, understood the mathematics, they would not teach that material because it is nonsense and irrelevant for chemistry.  Furthermore, is that fact that chemistry teachers teach "formal chem", such as electron configuartions, and neglect applications in nursing or biology not the same fault of which mathematics teachers are accused?  Both mathematics and chemistry are academic disciplines in their own rights, and chemistry is a science with an associated chemical industry. 

Since I discovered in 1971 the existence of practical 'computer algebra' (IBM Formac), I have devoted efforts first to do my own extensive mathematical calculations for chemical or physical applications with software, and then, as that software developed into its present advanced form, to teach mathematics with that software (progressively Mumath, Reduce, Derive, Maple ... through more than three decades).  For me, mathematics consists not merely of reading a book and scribbling separate calculations with pen on nearby paper but of reading a large computer screen that describes, with sufficient profundity, the concepts, principles and practice, and then that reader implements the appropriate operations with the same software on the same screen.  That scheme underlies my interactive electronic textbook Mathematics for Chemistry, now in its fifth edition, and I respectfully suggest that an analogous design of teaching arithmetic to algebra, with interactive testing built into the content of the lessons, would be an effective pedagogical approach, provided that the students were sufficiently prepared to cope with that software.  Is 'computer-aided instruction' really so novel in year 2017?  The ratio of students to instructor becomes then not 15 or 25 to 1 but 1 to 1.  This approach would seem to be applicable for remedial purposes of the students of general chemistry.  For further chemistry my electronic textbook might be brought to bear. Dr. Kilner mentioned a book by Monk and Munro that includes various topics; my own electronic textbook includes all those topics and more, with rotatable plots in three dimensions and other pedagogical devices beyond the printed page. I have made no effort to become acquainted with various printed textbooks of mathematics for chemistry; I mentioned that by Sutcliffe and Doggett merely in relation to the discussion of the varied level of mathematics.  I consider the entire concept of the traditional printed static textbook to be obsolescent, although when I read for pleasure I greatly prefer a book in my hands to staring at a computer screen, especially a small one.

Cary raises key points.  The most important thing we could do is to agree on the way forward.  Allow some suggestions, starting with the easiest one

First, the issue with chemistry majors might be best met by a Mathematics for Chemistry course as the terminal math course for chemistry majors with dropping of Diff Eq or maybe even Calc III.  This is the path that physics and engineering have taken.  It, IMHO, should be some combination of differential equations, linear algebra, statistical analysis and computation.  It could be team taught with maybe the analytical chemists taking the lead on statistics.  I would strongly recommend that it be centered around a symbolic computation system such as Mathematica, Maple, or shudder, MathCAD, the later because of its ubiquity in engineering, the other two depending on the local license situation.  If we can reach some agreement on this it is something to be brought to the Committee on Professional Training.

Second, the more difficult question is the mathematical preparation for GChem.  Much of the discussion has been about identifying those students who need help.  We might start with a list of tools that have been suggested and perhaps then survey ourselves about which the majority feel are the most useful.  An open discussion as we have seen can be scattered.

If we can come to agreement on what students need to know and generally how to identify the students who need help, that alone will be useful in discussing remediation with our colleagues (maybe not so much), our chairs, deans and so forth because it is no longer simply a personal or local opinion but something broader.  Perhaps then the moderators could draft a short paper for J Chem Ed.

How to remediate is a much more difficult problem. As Cary says you go to class with the students you have, not the students you want to have.  A point that recently came up on Twitter is that the first thing a new Assistant Professor needs to know is that they were not the typical student in their GChem class.  As we have heard there is no magic bullet, although, again, I agree with Cary small classes or recitation sections are key.

In closing, thanks to all for their constructive work.

Excellents points were raised during these discussions.   I agree on "As Cary says you go to class with the students you have, not the students you want to have" . If I may add based on my experiences, most of the students are aware of their limitations and are willing to do the extra work to get on the same page with math. A bit of guidance on math is much appreciated by them (especially with commuter/returning students).

Perhaps a free online "math for general chemistry" course with short videos on math related to general chemistry topics (maybe on ACS or elsewhere). Students from diverse math backgrounds can watch these short videos and bring their math upto speed for Gen Chem classes. Every instructor can direct students to the same place.

Thank you all for sharing your great pedagogies/ideas.


scerri's picture

Orbitals may not have 'real physical significance' and may indeed be unobservable.

Yet they are very useful in rationalizing many aspects of spectroscopy and in chemical education.  


Eric Scerri, UCLA Department of Chemistry


Both atomic and molecular orbitals have been observed.  See the recent work of Wilson Ho at UC Irvine.  An earlier perspective is found in Dinse and Pratt, "Orbital Rotation", JACS 104, 2036 (1982).

Rich Messeder's picture

Many ideas in this conference address how to move forward, and I think that they should all be given serious consideration in the immediate future. My perspective is one of finding a core approach that is useful for STEM students, and have that core modified for specific fields (chem, physics, etc.).

I iterate the recommendation for using computers to relieve faculty of the press of supporting many students in low-level reviews. For example, ALL entering STEM students could be enrolled in a computer class that "teaches" those concepts that we want memorized. Students would be //required// to meet minimum performance criteria; for example, be able to enter the answer to multiplication tables through 12s, randomly presented, in some reasonable time (constrained, calculators not permitted), with, say, 95% accuracy. Students who meet the requirement on the first pass have effectively complete the course. To ensure that "the cramming effect" is not active, students who did not pass everything on the first pass would be required to log in and retake the drill|exam periodically (weekly? monthly?) to provide the repetition that we have discussed here. These remarks are a just a starting place. I first saw computer-aided teaching at the UI/Urbana campus in the 1970s. Students were very excited to get time on the systems, which were advanced for the day, but primitive by today's standards. Nonetheless, I have been surprised by how little computers have been used for teaching over the decades. Amateur radio operators (I'm one) have used computers for decades to help them commit to memory material necessary to pass the different FCC license level exams. There are many resources available, and perhaps one of the tasks of academia is to sift through those resources and find the best of them to recommend to students. This would be an ongoing process, and it would be nice to have a sort of "clearinghouse" for all of it. This suggestion has the risk that the task will become overwhelming. Faculty from different institutions must be ready to "tolerate" recommendations from various sources. At any rate, for the fundamental material that we would like to see our students demonstrate mastery, this approach may be useful. The benefit of this approach is that it is easily absorbed by secondary schools, relieving them of the same time-burden, and could shorten the time that it takes to raise the math capabilities of our students. This approach also lends itself to "teaching" basic math tools, such as spreadsheets and basic MATLAB, Maple, etc., programming.

Cary Kilner's picture

Kudos to you and your team for showing how the problem of understanding mathematics in upper-level coursework can be addressed! My original degree was chemical engineering, so I did study much of this mathematics myself (not that I remember it). Here are some questions for you.

I have seen little discussion of this issue in teaching P-Chem, although the problem must exist for many programs and instructors. Why do you suppose it has received so little attention, despite the importance for the conceptual understanding of this material for chem-majors in this very important course? (I know the chem-majors are certainly a very small subset of the gen-chem students we have been discussing in this ConfChem.)

Do you think it likely that we have seen a decline in mathematics facility and understanding with even the stronger mathematics students found in the major’s track in the past few decades? Is this the result of changes in the way upper-level mathematics is being taught? And do you feel this could also be a reflection of recent changes in earlier mathematics education?
Thank you!

An obstacle to giving attention to maths for p-chem may be simply that the usual sequence works OK for other STEM disciplines, but a slightly different set of material (more understanding of linear operators, self-adjoint operators, orthogonal functions,  for example) is needed for p-chem, and there are usually not enough chemistry majors for math departments to worry about them.


Regarding your question on decline in understanding of math majors, I can only give my impressions.  One hears growing frustrations with decline in students' ability in proof, particularly what we call "analysis"--proving the theorems of calculus rigorously.  However, I don't think that that is true for the stronger mathematics students--they seem as strong as ever to me. 

Rich Messeder's picture

I looked online at the course texts you referenced, examining the table of contents, and browsing pages where it was permitted. It seemed to me that the scope is such that it would cover 90%+ of applied math for several STEM majors. In a chem-specific context, examples use the vocabulary and semantics of chemistry. But it seems to me that most often college-level intro math courses spend a great deal of time on theory, sacrificing application, so that students walk away from the class able to write proofs, but less able to actually use the math. Much of the math course development that I see represented in these papers seems to be oriented toward practical application, which seems appropriate to me (even for theoretical physics). Did you consult with other STEM departments at your institution? Do you think that your course can be used by other departments with little modification? Physicists and engineers, for example, might want more numerical methods. I noted especially the comment about 3D visualization, which I think is an important element often overlooked at the undergrad level. What aspects of 3D visualization are covered? And +1 for Dr Nelson's question: I am surprised at the fast track to course acceptance. To what do you attribute this success?

It is fairly understood that prior knowledge is the most predictive variable towards success in any course.  I would love to see you develop your anecdotal observations into a research study and attempt to discover where students' level of proficiency is: are they proficient in algebra, pre-cal, cal?  We might all be surprised to see where the problems stem from, and it might not be limited to just mathematics skills.  

Discovering levels of proficiency of entering students would be a good starting point. One of my other questions is how do you get students to participate in the drill, practice, rest, revisit idea without using marks as a reward?  I also believe that getting students to get into the habit of doing practice before entering college or university will help students in being successful beyond secondary or high school. 

Rich Messeder's picture

No. And, you might have guessed, I won't be surprised to see research supports this answer.

I have tried to refrain from writing pages of replies, in order that I not seem too pushy. My experiences in the private sector, especially as an engineering supervisor, and my further experiences teaching at HS and university, suggest strongly to me that the same principles and goals that are appropriate for success in the private sector apply to academia. I have taken management courses over the years that address the psychology of managers:employees::faculty:students, and paid close attention over the years to what works and what does not, to what adds or detracts from me as a leader and as a teacher, in an effort to continually improve myself. Regarding the suggestion that I might turn my experiences in to a research project: There are reasons why that is not likely to occur, though a related project could apply. Why? Time is of the essence. We are wasting our students' time, and doing both them and their employers an injustice, as well as hugely impacting scientific and engineering progress. It is time for well-thought action. Some of the research here that I find so useful and relevant is a decade or more old. I admire those institutions that have stepped up to the plate and changed for the better; the majority, it seems to me, have not, and it shows in the quality of students entering STEM courses in university.

For example: In class, I regularly emphasize collaboration on studying and problem-solving, followed by individual writing...and relate specific factual anecdotes from my experiences that reflect dramatic improvements in performance. I roll these out occasionally during the course...Why occasionally? Because it seems to me that students are not up to speed on either of these points (collab & literacy), as evidenced in part by comments from the private sector regarding grads entering the workforce, and, just as we have been saying about repetition in math, repetition in all things results in internalization. I grade my STEM students on literacy, and tell them so the first day of class.

An example of comments from the private sector: I have sat in on many industry "panels" at the undergraduate level. These panels are ostensibly to share industry perspectives, but I often think of them as recruiting opportunities (which is OK, too). Almost invariably, at the end of a panel discussion, some student will ask what the panelists opine that students might take from their undergrad experiences other than strictly academic work (grades, papers, etc.), and almost invariably the panelists reply with "collaboration skills and literacy".

Part of my research: I have personal knowledge of a series of events (circa mid-1980s) at a huge private facility where the consequences of certain kinds of equipment failures posed significant risks to inhabitants of local communities. One day, the computers that monitored all that equipment made a mistake, and declared that something was wrong (everything BUT the computers was working just fine, it later proved). Nonetheless, this spurious fault condition triggered an attempt to activate several safety systems. One very important system did not activate. Post-game analysis showed that it had a design flaw, and that design flaw was identified by an engineer earlier. It seemed that the engineer was reviewing systems (for reasons I never knew), and decided that his calculations indicated a design flaw. He wrote it up and passed it to his immediate lead engineer. Well, that certainly should have gotten some attention, eh? But the document was so poorly worded that the lead engineer didn't get the point, and then, not realizing what the problem was, didn't follow up with the author. (Two significant problems: literacy and leadership.) When all this surfaced, the VP of engineering of this very large engineering staff had all the engineers take a literacy exam. All those who failed were tasked with taking remedial classes of the VP's choosing --- on their own time --- with the understanding that those who failed the first exam would be tested ~6 months later. Those who failed a 2nd time would be fired. Scientific research? Not exactly. Message received? Absolutely. Yet, when I mentioned this to a university physics faculty member, he said that literacy was not his concern...that's what the English department is for. But, I opine, standards /there/ are as poor as they are in mathematics, and anyway writing technical papers is very different from writing a paper criticising a novel. (Prior to the event mentioned above, I had already informed my engineers that literacy would be part of their annual review.)

I opine that most US students entering college or university do not meet the reading, writing, and math skills of their forerunners of a few decades ago. They struggle to read challenging material, they struggle to write with any degree of literacy appropriate to their level of education, and they struggle to manage conceptual material in STEM classes because of the issues addressed in this conference.

Side note: Sorry if I have already mentioned this: Compare the user manuals for the HP-11C and 15C (on the web) with that for the TI-89. This comment is NOT about the devices themselves, but about user manual content then and now. Sorry, I don't have a reference for older TI calculators.

I find the research here, and referenced here, quite valuable, because it gives me something substantial to add to my anecdotes as I continue to work toward improving the quality of US academic life.


A common criticism of courses of mathematics is that the theory is emphasized at the expense of the practice.  The same criticism might be made of general chemistry in that orbitals and other baggage eventually traceable to fraudulent quantum-mechanical bases are emphasized at the expense of the real basis of chemistry as a practical science; in the latter case, the instructors of general chemistry, merely teaching ill chosen textbooks, teach that material not because they understand it but because they fail to understand it.  I find nearly impossible to believe that instructors of mathematics at any level in general exhibit the analogous ignorance or that the textbooks of mathematics contain a similar proportion of rubbish, because mathematics is much more readily intrinsically testable.  The problem of poor mathematical preparation for general chemistry seems to be ultimately attributable to the failure in learning arithmetic -- multiplication tables et cetera, long before algebra and geometry, let alone calculus, are confronted.  To any strategies within the environment of college or university applied to entering students to redress the accumulated arithmetical or mathematical deficiencies, I admire the recognition of the need and the practice to remedy the deficiencies, as discussed in this conference.

At the level above general chemistry, and under the assumption that the deficiencies noted above have been resolved for the students who advance therefrom, one can then apply various courses of mathematics within chemistry departments, to avoid the excessive emphasis on 'theory' -- theorems, corollaries, lemmas -- in courses taught by mathematicians who have little interest or knowledge of chemical or other applications.  The use of advanced mathematical software in the latter circumstances can be greatly beneficial, but is of no use if the students lack the fundamental skills of arithmetic.

I do agree that this course (with some modification) could be opened up to other departments as an applied math sequence. In fact, CSU does have a Calculus for Biological Scientists sequence as well, which uses in part the text by Erich Steiner. However, that course sequence is two semesters in total, the second of which is not a required course for the major. My biggest concern might be for students wishing to go further in math. Since topics come from a variety of traditional math courses, it is a little bit unclear what other math classes they might take if they chose to continue with math. For example, students have some differential equations, but not a whole course worth.  Numerical methods, for example, may be more important to other fields, but adding this in would cut other topics in an already full course. 

In my opinion, part of the joy of teaching this course was having the chance to engage with students mathematically on topics that they already cared about/felt were valuable to think about. Because all of my students were chemists, I think that keeping chemical examples central helped with student buy-in. Instead of a set of rules to be memorized, math was shown to be useful for thinking about physical systems that interested them.  I know not every school is large enough to be able to support these types of "flavored" math classes. I wonder if some of that feeling of relevance would have disappeared if the applications were varied. 

To answer the question on 3D visualization, we spent time learning how to sketch surfaces in 3D,  set up and evaluate volume integral and some work with plotting in matlab.  

I guess it was kind of fast.  We ran the course two years as an experimental course, and in the second year the process was put through to make it a regular course.  I think that we can officially run a course three years as an experimental course.  It was important that the Maths department chair and undergrad director were supportive and that faculty from chemistry were involved with the course design and wanted the course to continue running.  Positive feedback from students was important too.

Since we introduced the course, faculty from physics, computer science, and chemical engineering have all expressed interest in switching to this sequence.  The possible obstable to having physics and computer science join is that we want to keep the focus on p-chem applications.  Also, physics and chemical engineering majors will need to take a differential equations course as well, and that makes for a lot of overlap with the first course in the Maths for Chemists sequence.  I have suggested for chemical engineering that they do the first semester of Maths for Chemists, and then switch back to the normal Calc. III course.

I think that even for maths majors a sequence of i) Calculus I (up to fundamental theorem of calculus, ii) differential equations-based course, much like the first semester of Maths for Chemists, iii) Calculus III

would be a good sequence.  The Calc II coordinator here is interested in taking some of the ideas from the MfC sequence and giving more of a focus on differential equations in Calc. II.

Your paper states one of your goals as: “students should be able to decide if solutions are reasonable with estimation techniques and order of magnitude calculations.” That topic was addressed previously in this conference (in Paper #1) for a course in physical chemistry. How is this done in your program? Are some numeric calculations on graded assignments expected to be done without a calculator?

-- rick nelson

I appreciate the careful thought and methodology of teaching estimation in the first paper. In practice, in our program, estimation was a recurring theme throughout the semester, not a topic taught on its own. It was often the response to "Does this answer make sense?" or "Is this what we expected?". I would run through estimations on the board or verbally after computations (often with a calculator) were completed. The accuracy of estimation would depend on the problem, sometimes it would be pretty rough--even just an order of magnitude argument. Hopefully, these checks became part of the routine of problem-solving. 

I did not require students to do computations without a calculator. I can see how this forces students to sharpen arithmetic skills.  However, I hope that using estimation to check answers was communicating that even when using a calculator you need to do a "gut check" at the end. 

I was the managing author and 2nd listed author of the book The Unified Learning Model (Shell et al.) referred to elsewhere in this conference.

Willingham's book (Why Don't Students Like School?) is an excellent, readable summary of where educational psychologists are today in terms of their views on learning. IF that book has a shortcoming, it is that reading it will suggest to you that students actually DO like school. Of course, that would not be the title of a best seller.

Two of my colleagues have joined me in writing a newer book rooted in information theory. This book is available at:

The new book is edited periodically. A new round of edits will be posted before the end of the year. The edits are maintained such that readers of earlier editions are directed to the new changes. 

The book has four sections. First is the general theory. Next are the applications. Third are elements of the basic underpinning science (such as EEG or studies of snails). Last are the enumerated edits. The book was first posted in August 2015.

The advantages of a Web-based book include the opportunity to edit based on new information and the ability to link directly to multimedia. For example, check out:



Interesting--concentrating on other things makes the details hard to see! 

Dr. Brooks –

It is rare to find a single source on the “science of learning for educators” that is both comprehensive and up-to-date. Your “Minds, Models, and Mentors,” at the link you provided, I think is Number One.

For instructors in “science-major chemistry” with its focus on well-structured problem solving, my personal sequence of “recommended reading,” from short and simple to more comprehensive, would include:

1. Four pages on fundamentals of how the brain solves problems in the section “The Human Brain – Learning 101,”on pages 8-11 at

2. Eight pages on problems in the physical sciences and math on pages 4-2 to 4-10 of The Report of the Task Group on Learning Processes in the Final Report of the National Mathematics Advisory Panel (NMAP) at
The section on “automaticity” is especially important in helping students solve scientific calculations.

3. The book Make It Stick by Brown, Roediger, and McDaniel (2014) describing specific study strategies such as retrieval and interleaved practice, summary sheets, and elaboration.

4. Your “Minds, Models, and Mentors” as a comprehensive summary of the brain’s structure and its impact on learning.

-- rick nelson

Cary Kilner's picture

Thank you for your contribution to the ConfChem. You may recall that we met at Princeton in 1984 at the Woodrow Wilson Dreyfus Master Teachers Institute, where our charge was periodicity and descriptive chemistry. That one month was an outstanding experience for me; it kick-started my career as a chemistry teacher (I had been a professional musician after college), and reinvigorated my love of DOING chemistry, and not just talking about it. It encouraged me to continue to develop demonstrations, which I eventually used to help teach chem-math to apprehensive students.

As I recall you were editor of an ACS publication, and came for a week to participate as a leader. I don’t believe it was Chem-Matters, but maybe so. Please refresh my memory. I ordered a class set and used it for 20 years. My students looked forward to it coming every few months and loved reading it. I had them read back issues as well as the four that came each year.

Were you interested in cognitive science at that time? I will certainly check out your references.