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How can we effectively improve the mathematical capabilities of students of chemistry?


Erick Castellón,  John F. Ogilvie


We describe our extensive collective experience in the conduct of courses based on the use of advanced mathematical software to enhance the capabilities of students of chemistry to solve problems with a mathematical component. These courses have been based on an interactive electronic textbook Mathematics for Chemistry with Symbolic Computation that has been concurrently developed and expanded.  The admirable performance of the students who have benefited from our courses leaves no doubt that mathematical software is an invaluable tool for the teaching, learning and practice of mathematics in a chemical context.

To assist students of chemistry with the necessary mathematical knowledge and capability, during the past 18 years, one author has presented short courses, of duration up to four weeks, and full semester courses, and the two authors together have presented semester courses on other occasions; in each case these courses are based on an interactive electronic textbook, of title Mathematics for Chemistry with Symbolic Computation, that has been in concurrent development and is now in its fifth edition [1].  The objective of the course, and of the underlying textbook, is to provide students of chemistry with both an understanding of essential mathematical concepts and principles and a practical ability to solve mathematical problems over the entire range of chemistry -- general, analytical, inorganic, organic, physical and theoretical -- that a student might confront in an undergraduate programme.  Both the textbook and the courses are based on advanced mathematical software, Maple [2], that contains 5000 years of mathematical knowledge, that has a less steep learning curve than other mathematical software and that is available in many universities. Initiated in 1980, Maple was originally developed in University of Waterloo, Canada, for the purpose of assisting students of science and engineering to undertake mathematical calculations, analogously as previous compilers WATFOR and WATFIV, which were based on Fortran and are now obsolete, had been generated to assist calculations involving only arithmetic operations.  These three programs became eventually released as commercial products that have been duly installed in universities around the world.

            Of two parts of the textbook Mathematics for Chemistry, the first is designed to encompass all mathematics that an instructor of chemistry might wish that his students would have learned in courses typically, but not exclusively, offered in departments of mathematics as a service to science students; the second part comprises chapters each devoted to a particular area of mathematics, with its chemical applications, that might be taught within advanced courses in chemistry, including group theory, graph theory and dynamic chemical equilibrium.  After a summary of various useful Maple commands, Part I hence has two chapters before the introduction of calculus, covering arithmetic in various forms and scientific notation, elementary functions, algebra, plotting, geometry, trigonometry, series and introductory complex analysis.  Differential and integral infinitesimal calculus of one variable precedes multivariate calculus, linear algebra including matrices, vectors, eigenvalues, vector calculus and tensors, before differential and integral equations, and statistical aspects including probability, distribution and univariate statistics for the treatment of laboratory data, linear and non-linear regression and optimization. 

            The recommended duration of a course to present all that material in Part I within the sole instruction in mathematics to undergraduate students of chemistry is at least three semesters, but our courses so far have occupied only one semester, based on prerequisite differential and integral calculus taught elsewhere. Under the latter conditions, the first half of the course, before multivariate calculus, serves as a review of mathematics that students are supposed to have learned, which serves as a valuable vehicle for learning the Maple language, which in turn serves as a foundation to enable a rapid progress through the succeeding material.  Each such course consists of lecture demonstrations of the particular concepts and principles of mathematics and their implementation with the Maple program, and supervised practical sessions, of total duration per week consistent with the length of the course, the course load of the students and availability of appropriate space and resources.

            As explained with examples in a preceding report [3], the objective of our courses is to enable the students to solve problems with a mathematical component using their mathematical knowledge to direct their effort and their acquaintance with the computer program to implement the required calculations.  When one has acquired an overview of the entire content of mathematics deemed useful for students of chemistry, one appreciates that the actual extent of mathematical knowledge of a particular topic is small:  for instance, differential calculus is based nearly entirely on a definition of a derivative as a limit of a ratio of two quantities. A combination of an explanation in words, an algebraic formulation, some numerical examples and the use of an animation that demonstrates how a secant to a curve becomes a tangent in the limit illustrates this concept far better than is practicable on even many printed pages or with drawings and scribbling on a blackboard.  The mathematical program hence presents a basis to display text in an appropriate and conventional notation, an interactive algebraic derivation, a few appropriately selected critical numerical examples and, especially, powerful graphical capabilities to produce diagrams in two or three dimensions and with practical animation.  Once a student understands that principle, its application is mere detail, but the instruction about the topic naturally includes not only a few examples, based on the algebraic properties of the selected formulae, but also pertinent exceptions such as what occurs at a discontinuity, again illustrated with appropriate live plots.  In contrast, a typical textbook exhibits more or less crude and static plots with much verbiage in an attempt to convey the dynamic aspect of impracticable animation and hundreds of exercises that a student is supposed to undertake in a mechanical manner.  Einstein's advice was "never memorize anything that you can look up" [4]. With a few judiciously chosen exercises that a student in our course is expected to undertake to illustrate the principles, that student is ready, with direct access to that software, to differentiate any formula, anytime and anywhere. 

            Likewise for integral calculus, our explanation of the principle comprises a definition in words accompanied by an animated display of the area under a curve as the number of subdivisions of the axis increases; the total area of the rectangles is simultaneously exhibited at each stage.  When we authors were students before access to mathematical software, the advice given to us in the course on integral calculus was to learn standard methods, such as change of variables, integration by parts and trigonometric substitutions, and to memorize a list of common integrations; if such manual methods failed to yield a convincing answer in a particular application, we should find a similar expression in a table of integrals, but then we had to undertake whatever transformation was required into the particular variables in the problem at hand. With powerful mathematical software such as Maple that contains more knowledge about integrals than any single printed table ever, one need not be concerned about typographical or other errors that have been prevalent in static printed tables; in any case one should directly differentiate the result of an unfamiliar indefinite integration to prove its correctness.  The emphasis in our course is to encourage the student to think about the solution of a problem, and then to let the software, according to the subset of total commands and instructions included in the course, undertake the tedious algebraic manipulations or prepare an illuminating graph.  Some 70 years ago in advanced countries, pupils in schools were trained how to extract manually a square root of a number, with no explanation of the principle (actually a binomial expansion, but that was never stated); the flood of pocket calculators more than 40 years ago terminated that drudgery, and now even most professors of mathematics are unable to extract a square root of a number by hand. The use of computer software for symbolic mathematics is a natural progression that is consistent with the use of computers for manifold other routine or boring purposes. To replace the learning of mechanical sequences of manual operations, an enhanced understanding of the principles is a valuable dividend of symbolic computation, as incorporated in the design of our interactive electronic textbook and as proven in tests over nearly two decades.

            How effective have these courses and their underlying interactive electronic textbook been in improving the ability of students of chemistry to solve problems with a mathematical component? One measure of the efficacy arises from a direct comparison of the performance of  students in a given course, for instance physical chemistry that has a significant mathematical element, as we describe below.  Another measure is what the students have accomplished that could not have been done without this capability acquired through such a course as we have presented.  What a student feels, on completing this course, is a tremendous power to attack chemical problems with a strong mathematical component.  This power has resulted in its spontaneous application to undertake original calculations for some purpose or other, or perhaps for no particular purpose at all -- just curiosity to prove that a calculation is feasible.  Several cases of such calculations have resulted in submissions to the Maple Application Centre [5], of which we note the following instances.  A procedure for optimization with sequential simplex was originally developed separately, but became incorporated in Mathematics for Chemistry with the permission of the author [6].  Likewise some admirable examples of chemical equilibrium involved in titration curves [7] now constitute a major part of chapter 9 in Mathematics for Chemistry.  Two other former students, completely independently of us instructors of their courses, submitted programs to treat the equations of states of non-ideal gases [8, 9] and the density of probability of an electron near a nucleus [10] and the magnetic field and inductance of a torus [11].  Other students have contributed valued content, particularly animated plots to illuminate some aspect of the content of Mathematics for Chemistry at a germane point, that is duly acknowledged within the present fifth edition.  Such material was clearly devised by imaginative and superior students in our courses, but even more pedestrian applications of their mathematical knowledge and capability acquired in the same manner have contributed to the academic experience and performance of many other students beyond the particular course for which they received credit.  The fraction of submissions to the chemistry category in the Maple Application Centre contributed from students in our courses is astonishing in view of the limited number, less than 100 in total, who have completed our semester courses over the years.

            The following observations about our past students enable a comparison of performance in courses of physical chemistry of both lecture and laboratory type between the students who enrolled in our course and other students.  The major advantages that students gain on learning to use mathematical software for symbolic and numeric computation are rapid solving of assigned problems, improved presentation of information in producing plots of high quality, animations, three-dimensional plots -- most students are amazed when they apply rotations to an object in a 3D-plot, and those who are able to produce such a plot are certainly proud of that achievement -- and the ability to perform non-linear regression on experimental data.

            Among the benefits acquired by students with skills in mathematical computation is notably the loss of fear about facing elaborate calculations. In many situations a complicated physical or chemical model is based on a statement of systems of equations. Each of those equations is typically not difficult to set based on a knowledge of the underlying physics and chemistry; the difficult part of solving many problems is finding the solution for the mathematical system (algebraic or differential), because that task is tedious and commonly difficult to perform with merely paper and pencil. In other cases a problem can be correctly stated equation by equation, but the knowledge to solve the problem lies beyond the expected mathematical preparation of a student of chemistry. That barrier disappears on learning how to use software for symbolic and numeric computation: the student does the thinking and makes the statement, and the computer does the slave labour that brings in the answer. In this respect we recall the superb performance of a student in the physical-chemistry laboratory who undertook a special experimental project on the chemical kinetics of oscillating reactions; this student was able to model his experimental data on solving the system of differential equations that described the variation of concentration of each involved substance with time; the demonstration of his results inspired awe in the other students present. Another instance was solving the Schroedinger equation for the hydrogen atom, which can be readily effected in several coordinate systems simply on translating the laplacian operator from one system to another.

            Breaking the mathematical barrier in solving end-of-chapter problems and modeling experimental data using symbolic and numerical computation have opened a new knowledge perspective for some students who embraced the enterprise of learning about programming, to create their own algorithms. In this regard, the built-in help documentation of software, such as in Maple, provides effective guidance to assist self-learning. In this way, some students applied powerful utilities such as programming routines and image analysis for their post-graduate studies.

            In practical terms, an instructor of a course in chemistry requires a textbook for each course, which he or she follows to a lesser or greater extent.  Even before 1950, printed textbooks were published with the stated aim to assist students of chemistry to undertake various calculations.  Different in concept from any printed textbook, Mathematics for Chemistry with Symbolic Computation [1] is an interactive electronic textbook that is operated on a computer with the Maple program simply on executing the computer files that it comprises; it is interactive in that all calculations contained and discussed within the text are directly executed at the reader's instigation, simply on depressing the key to enter a selected command, which can be modified at will.  The approach to the preparation of the composition of this textbook has been holistic: although its chapters might have names reminiscent of conventional courses, the total content has been designed to include material that is both valuable for chemical applications and lacking from conventional courses of constrained content, and the entire Part I of the textbook is the objective of fulfillment, not merely the individual chapters thereof.  This textbook is intended to be sufficient for all purposes involving the included topics, but traditional printed textbooks or material from internet might usefully supplement the descriptive content and explanation, no matter how comprehensive is that content or its exposition.  As an alternative to a traditional course such as we present in computer laboratories or classrooms with either computers provided or facilities for students to use their own portable computers, Mathematics for Chemistry is, by its nature, highly appropriate for self instruction.  For the purpose of preparing to offer a course based on this textbook, a present instructor of chemistry can readily and rapidly progress through this textbook, learning to use Maple in the process as appropriate, to remind himself or herself of the mathematics that he once studied in traditional courses -- or that was lacking from those courses because of inadequate scope.  Despite the extent and nature of the explanation and the explicit examples in this textbook, after a dozen years of school in one form or another, most university students are inured to more of the same style of teaching and learning; after our lecture demonstrations, we have found that the supervised practical sessions, in which the students are encouraged to ask questions and to request hints for the solution of assigned exercises, play a major role in the acquisition of competence by the students.  Our students have naturally had, and availed themselves of the opportunities, to extend their practice of Maple and the concerned mathematics beyond those supervised sessions, as desired and appropriate in each individual case, with the option to explore unresolved questions in the next practical session. 

            Many instructors of chemistry have developed and tested myriad devices and techniques to try to assist their students to cope with some mathematical component of chemistry in their courses.  Our experience indicates that a truly successful achievement of the desired results necessitates a completely fresh and inclusive approach, namely to embrace a thorough application of symbolic computation in the teaching and practice of mathematics for students of chemistry. The availability, gratis, of textbook Mathematics for Chemistry with Symbolic Computation combined with powerful and easily learned Maple software can cause that successful achievement to materialize.  Once a student has learned to use one particular program for symbolic computation, it is not difficult to learn the language of an alternative program.  Free software for computer algebra, such as Maxima or Reduce each of which is commendable, lacks, however, the pedagogical aids that are incorporated in Maple, and that are applied to great advantage in Mathematics for Chemistry; as the free software is naturally being developed less rapidly than commercial software, it is much less powerful for various aspects of higher mathematics, such as group theory or the solution of differential equations.  For these reasons Maple is particularly pedagogically valuable for teaching mathematics for chemistry, even though some packages in Maple for higher mathematics, such as for general relativity in physics, are unlikely ever to be engaged for chemical purposes.

            Our answer to the question in the title of this essay is, not astonishingly, to teach students of chemistry more mathematics, specifically in the form of a course based on a particular textbook with a broad coverage.  When one of us was an undergraduate in a Canadian university, a student of chemistry there was required to enroll in five year courses, equivalent to ten semester courses, in mathematics to complete the degree.  The average requirement in Canadian universities is now three semester courses, despite the increasingly mathematical nature of at least analytical and physical chemistry in recent decades. In other countries the situation might be even worse. What matters, however, is not just more mathematics but a greater capability in mathematics, which naturally arises, as we have endeavoured to demonstrate above, from instructing the students to apply symbolic computation with powerful mathematical software. Lord Kelvin stated that the human mind is never performing its highest function when it is doing the work of a calculating machine.  With even just three semester courses each based on symbolic computation or a single semester course extending traditional courses, the results can be outstanding. With such resources at their fingertips on a computer keyboard, the mathematical achievement of chemistry students is nearly unlimited. For chemistry, mathematics is no end in itself but a tool to be applied in attacking chemical problems.  The rate-determining step in implementing the appropriate courses that offer and implement the requisite tools is the rate at which instructors of chemistry themselves make the effort to learn to use symbolic computation; instructors should not expect a student to do what they have not done.


[1]       Mathematics for Chemistry with Symbolic Computation (accessed on 2017 June 15).

[2]       Maple: software for symbolic computation. (accessed on 2017 June 15).

[3]       J. F. Ogilvie, M. B. Monagan. Journal of Chemical Education, 2007, 84 (5), 889-896.

[4]       “What you can learn from Einstein’s quirky habits” (accessed on 2017 June 15).

[5]       Maple Application Centre. (accessed on 2017 June 15).

[6]       E. Romero-Blanco. Optimization with sequential simplex of variable size. (accessed on 2017 June 15).

[7]       R. Hidalgo. Chemical equilibrium. (accessed on 2017 June 15).

[8]       C. Viales-Montero. van der Waals equation of state (I). (accessed on 2017 June 15).        

[9]       C. Viales-Montero. Analysis of basic equations of state (II). (accessed on 2017 June 15).

[10]     D. A. Hercules. Density of probability of an electron near the nucleus. (accessed on 2017 June 15).

[11]     D. A. Hercules. Toroids: Magnetic Field and Inductance. (accessed on 2017 June 15).

05/10/18 to 05/12/18


Here I amplify my comments on a preceding contribution to this conference that described the use of R for statistical applications in chemistry.  The current paper being discussed encourages the use of a free interactive electronic textbook of title Mathematics for Chemistry with Symbolic Computation, which requires Maple software for its operation. 

Chapter 8 of this textbook is devoted to statistical topics that are applicable in chemistry, including probability, combination and permutation, distribution and univariate statistics, linear and non-linear regression and optimization under linear and non-linear conditions.  The sections on distributions and univariate statistics include all topics that I found in a few consulted textbooks of analytical chemistry and instrumental analysis, which are explained, with examples and exercises, in a formal pedagogical manner, whereas the underlying software Maple has an enormous range of statistical capabilities including graphical depictions that supplement that basis to whatever desired extent an instructor of chemistry might require.

When a student learns, and operates, the mathematics of the preceding chapters, on arithmetic, algebra, elementary functions, construction of plots in two and three dimensions, geometry, trigonometry, complex analysis, differential and integral calculus of single and multiple variables (with explicit thermodynamic applications), linear algebra, differential (with explicit applications to chemical kinetics) and integral equations, the same knowledge of the operation of Maple is readily applied to the statistical applications that are essential in chemistry.

The same basis of mathematics that is the standard requirement for chemical applications is then applicable to group theory, graph theory, Fourier analysis, quantum mechanics ... in the latter stages of an undergraduate education in chemistry or in post-graduate conditions.  There is no need for a separate language, such as R, for a particular aspect of mathematics relevant for chemistry, because Maple covers all possibililties. 

If a university site licence does not provide student access to Maple, an individual student licence costs as little as usa$79, or $99, depending on the terms, which is about half the cost of some (exorbitantly priced) printed textbooks typically prescribed for courses in chemistry.

In considering how to teach mathematics for chemistry students with the aid of computers, one should adopt an overall or holistic view, not merely patching one particular aspect with software of one type and another aspect with other type.

Cary Kilner's picture

I have nothing in particular to add, other than to agree completely with John's last paragraph above, and supporting it as a univeral pedagogical admonition. In teaching "chem-math," the holistic view IS to always teach the mathematics in context. Note, this is not to denigrate individual creative efforts, such as the development of R for statistical apps, that may serve an individual instructor better for his/her students.

If I follow your reasoning to its logical conclusion I believe you are saying that we should be looking for a general tool that can be used across the curriculum for the least cost.
If my interpretation is correct, then I believe the logical conclusion is to teach with something open-source (usually meaning negligible cost to the students) that is generally applicable for mathematical manipulations. If that is the case then among the options I am aware of SageMath/CoCalc ( or is probably the best option as it has the generality of Maple and the open source advantages of R.
That said, I also find that trying to use one tool for all tasks can lead to problems. Some examples:
1) I would not use SageMath/Maple/Mathematica for data analysis and plotting of significant data sets that need publication quality output. There are much better tools for that (SciDavis, IgorPro, SigmaPlot, Origin with IgorPro being my favorite). A side note, someone put IgorPro in the same category as Excel. There is no comparison. IgorPro generates publication quality plots by default. It contains a full-fledged C-like programming language and well documented versions of all the tools described in _Numerical_Recipies:_the_art_of_scientific_computing_. By comparison Excel is a toy.
2) I use LibreOffice/OpenOffice and under duress MSOffice for wordprocessing, but I would never write computer code in either one. I even use different tools for different programming tasks: Eclipse for java and C; Jupyter for simple python scripts; JEdit for most of my html/javascript.
3) I use Jmol for most molecular visualization tasks, but use other more adept software (wxMacMolPlt and Avogadro) for generating initial guesses at molecular structures because they have better molecule building interfaces.
Anyway, I think it is important to choose the right tool for the task, while keeping in mind that we have to avoid cognitive overload for our students. I am constantly learning to use new tools and we do need to help our students learn how to do that. We cannot predict what they may need to be able to do or learn in the future.

My understanding of the purpose of this discussion is that the intention is to find ways and practical means of assisting students of chemistry to improve their mathematical knowledge and its application for purposes that arise in various courses in chemistry.

1) Your reasoning about the output of publication quality in relation to the use of mathematical tools to assist students of chemistry is distinctly unhelpful, if not ludicrous.  Do you require your students to submit plots of "publication quality" for your assigned exercises?  With regard to the quality of plots generated with Maple, in the past three years I have published seven papers that included graphs prepared with Maple with no further refinement; no editor nor reviewer objected to their quality.  In one case, Molecular Physics, a reviewer asked for more papers that contained such plots, which I considered a strange request at the time.

2)  A student might undertake a report or the solution of exercises completely from Maple, as pdf for instance, that might include calculations, explanatory text, graphs and plots of data.

3)  If you are not an undergraduate student of chemistry, what you use is faintly irrelevant in relation to the present discussion. 

I agree that "it is important to choose the right tool for the task"; the advantage of a general mathematical computing environment, such as Maple, is that it provides the correct tools for almost any task with a mathematical component in a chemical framework.  My book, Mathematics for Chemistry, is designed to assist the teaching, learning and doing of mathematics for students of chemistry; it comprises all the mathematics that I, as an experienced professor of general, analytical, inorganic and physical chemistry, could wish my students to understand and to be able to apply for their chemical purposes.

1) I do not disagree that Maple can be used as a good tool to for mathematics and data analysis in chemistry. I have used it as well as Mathamatica and SageMath for exactly those purposes myself in research and teaching. Mostly I was asking if I was understanding your arguments correctly? Might SageMath actually be a better option based on your criteria than Maple or R because SageMath costs less than Maple and SageMath has the full set of symbolic mathematics, statistics, etc. tools available within it, some of which are lacking in R?

2) I only make my students use data manipulation centric plotting packages for manipulation and presentation of experimental data for formal reports and presentations. Maybe I shouldn't be trying to get my students to produce nearly publication ready plots, but if I have them use a package like Origin, IgorPro or SciDavis (open source) by default I get things turned in that are nearly publication quality except for mistakes in labeling, units or outright errors in data analysis. That puts the students a step ahead of where they are if they are using the default plotting in a math package or Excel. It also makes it much easier for me to evaluate the plots. I do have my students use the plotting in math packages for exploration and visualization of mathematical representations (eg. wavefunctions, potential surfaces, trajectories, kinetic modelling...).

From your response, I believe you think that I am on the wrong track teaching separate tools for data analysis and symbolic math. This comes back to my musings on cognitive load. I'm not sure where the balance is. If the students are learning a math package is the cognitive load lower with them also learning the ins and outs of the commands/menus necessary to import experimental data, generate plots of the data with proper legends, point markers and such inside one of these packages or using a package that defaults to more-or-less professional quality plots of experimental data? Do you have any relevant comparisons? I've found it difficult to handle most of my own data plotting needs in math packages, so have stuck primarily with the data manipulation centric packages for my teaching in laboratory classes.


Of course, completely free software such as SageMath, or Maxima or Reduce or ... , is preferable for students to software that requires an expenditure, even if only half the cost of a common printed textbook, but the experience that Erick Castellon and I have reported in the paper under discussion is based on the use of the interactive electronic textbook Mathematics for Chemistry with Symbolic Computation, which is a sequence of Maple worksheets.  Much effort in Maplesoft company has been, and still is, being devoted to extend and to expand Maple as a vehicle for teaching and learning mathematics, such as in the classrooms of precalculus and calculus in particular, in the form of 'clickable calculus' as I already mentioned, and in the form of  'Student' packages for Basics, Calculus, Linear Algebra, MultivariateCalculus, Precalculus, Statistics, VectorCalculus etc.  As far as I am aware, Mathematica lacks such capabilities to anything like that extent, and Maxima within SageMath has practically nothing specifically for pedagogical purposes.  Again all that effort at Maplesoft is for the general teaching of mathematics, whereas my book is designed to be applicable to students of chemistry who undertake various mathematical operations as part of their chemistry courses.

About SageMath having the "full set of symbolic mathematics statistics etc. tools", I think that you will find that, for instance, the capabilities of SageMath for the solution of ordinary-differential equations, for instance, fall far short of what Maple has, and for partial-differential equations SageMath has practically nothing.  As an example of some relevance to chemistry, one can solve directly with Maple the Schroedinger (partial-differential) equation for the hydrogen atom in all four coordinate systems, but with no other software.  Each product of mathematical software has its strengths and weaknesses, but Maple excels in most cases.

The experience that Erick Castellon and I report in our paper is based on the use of the specified textbook, which operates only with Maple.  If you wish to translate that book into Maxima, please feel free to proceed with my encouragement.  One should bear in mind that, because Maxima has been under development for half a century involving hundreds of programmers, the syntax is less regular than that of Maple, so that the learning curve is steeper, and less 'Help' is available.  The syntax of MuPad was, in the wake of Maple and Mathematica, even better than that of Maple, but that project ended prematurely for lack of support of its developers.  Furthermore, there is, to my knowledge, no equivalent "Mathematics for Chemistry" based on Maxima or Mathematica.

The fact is that, if you wish to use my book Mathematics for Chemistry, and so to experience the success of your students comparable with what Erick and I have reported from our students (whose first language is not even the English language of the textbook), you must run Maple underneath it. 

Thanks that clarifies the situation for me. As my school seems to have given up on our access to Maple recently, I will have to explore translating some of your exercises to alternative systems.


What this discussion has made clear is that mathematical (statistical) software for instruction does not exist in the context of a single course or a department, or even a college in a university.  For best effect, indeed almost FOR any effect, there has to be an almost universal agreement to use a selected platform.  To abuse a simile, you do not teach general chemistry in the language of your choice.   

The benefits of this across the curriculum and the student's academic career at an institution are obvious, but something that needs to be seriously discussed.  Especially in the case where the software is not cost free, universal local agreement on the software to use would be vital. 

The place to start would be the undergraduate education committee of each department which would then consult with other departments.  

The point of the preceding comment is unclear to me.  "You do not teach general chemistry in the language of your choice", but clearly general chemistry is taught with a textbook or a language of one's choice; in Costa Rica general chemistry is taught in Spanish.  I am aware that in some departments of chemistry there are even multiple textbooks of general chemistry in use, which one depending on the instructor of a particular section among several that are commonly required to accommodate students of the large number who enrol in this course in its variaties.

Another aspect to clarify is that the approach presented by Erick Castellon and myself in our paper under discussion is directed to students not just in general chemistry but throughout the undergraduate chemical career.  The interactive electronic textbook Mathematics for Chemistry with Symbolic Computation would be likely to be of little or no value for remedial purposes of students entering general chemistry with deficient understanding and capability for simple arithmetical or mathematical operations.

The most important aspect of our approach to the teaching of mathematics to students of chemistry is the systematic manner in which software for symbolic (and much other) computation is introduced and maintained throughout the teaching and the learning of that mathematics, from arithmetic to statistics and including everything between. In courses delivered in a department of mathematics, a particular instructor might use software to a limited extent for Calculus I, and then another instructor might not use that software for Calculus Ii, or, worse, use some other software, and so on in the progression of courses required for chemistry students.  

The "power" that our students have acquired, as documented in our paper, arises from the systematic use of one particular software for the teaching, learning and doing of mathematics (in chemistry courses, for instance) at all stages, and even after their graduation.  To use the specified textbook requires underlying Maple; if an instructor wishes, for one reason or another, to use another program for computer algebra, he or she can, with a systematic approach, enable the students to reap comparable benefits, but much effort would be required to duplicate or to translate the material in that textbook.

Our course, our textbook is not magic, not unique, but it represents a medium that has been developed and used over two decades; the proof of the effectiveness lies in the cited references.

Apparently we agree.

Teaching with one mathematical application across the curriculum is much more powerful.  Each instuctor using a different application confuses students. 

Consider, for example a student taking differential equations and physical chemistry in the same term if their instructors used Maple on the one hand and Mathematica on the other.  This would especially be the case if they had had no previous experience with either.  On the other hand, if the "University" math app of choice had been introduced in the freshman year in the context of typical freshman courses such as general chemistry, calculus or general physics, they would be much better off.

My view of this is that we can do very little on the basis of a single course but have to think of the problem in the context of the multi-year student experience, not just in chemistry but across the curriculum.  

Getting agreement in a department, let alone a school or college will be a long and hard process but worthwhile. 

Bob Belford's picture

John and Erick,

Do either of you have a youtube style screen-capture video that could allow those of us who do not have Maple the opportunity to see some of the features you are discussing? You know, watch you navigate and show some of the features in action?  If you made one, we could also embed it in your paper.





The short answer is no, we -- or at least I -- have made no "youtube style" video according to what you have suggested.  In the article by Michael Monagan and myself in Journal of Chemical Education, cited in the list of references of our manuscript, there are some examples that, even though on a printed page, might indicate the flavour of the presentation of knowledge.

Perhaps videos available at will fulfill your request.  In our courses from which our experience is reported, essentially we teach mathematics to our chemistry students in roughly the same way that mathematics is taught with Maple in schools and universities around the world, but we inject any applicable chemical application or illustration at appropriate points, and we use graphics of high quality and animations wherever practical.  I recall in particular how the lower jaws of the watching students dropped as I presented an animation of concentrations of reactants, intermediates and products with time in an oscillating chemical reaction, but that occasion was not within the teaching of pure mathematics, rather using Maple in teaching physical chemistry.

If I consider all of the newsletter contributions, one thing that strikes me is that most of the featured tools are text-based rather than graphical user interfaces. This is a way of interacting with a computer that many students are not familiar with when they enter college. On the other hand, text-based computing (formerly called programming or scripting, now often called coding) is a relevant skill.

There are some common features (paying attention to syntax, assigning values to variables, writing understandable code, re-using contributions from others) that students would learn in a computer science course, but could also pick up in these discipline-specific or discipine-adapted tools discussed here. Once students are familiar with one of these tools, they will probably find it easier to pick up the next, whether this is for another course or after graduation.

Much emphasis in the development of Maple in recent years has been devoted to 'clickable calculus' and similar tools.  I have not incorporated this approach in Mathematics for Chemistry, but it would be entirely practical for an instructor to combine this approach with selected material from that book, seeking to derive the maximum advantage and benefit of both methods.

It does seem that it would be much more effective use of student effort and time to have some consistency. Unfortunately, I think you are correct that it will be very difficult to get agreement across a whole college. Thus choosing the software will probably be a slow process. This runs afoul an additional issue I see with software: the abilities and useability of any single piece of software changes quite rapidly, plus newer software often comes out that does things better and sometimes worse. This compounds the problem because we may need to change what we are using to keep up with the technology. In this case, I do not mean 'keeping up' in the negative sense of being trendy, but in the positive sense of preparing our students for the current and near future situation.

So, what ideas are there for working with these constraints or circumventing them? Here are a couple of my random thoughts:

1. Develop more software agnostic teaching methods. A tongue firmly in cheek idea: revert to paper textbooks and paper and pencil? A little more seriously, maybe we need a collaboration between people comfortable in each of the available tools generating a text with instructions for working in any of the tools? The coding text _Numerical_Recipes_, that I mentioned before, has example code available in multiple programming languages.

2. Treat the issue more the way we treat instrumentation in chemistry labs. Make sure students understand the general capabilities of the instrument (software) and then give them specific instructions for doing what we need them to do with it.

I think this is a quite important issue. Many students are not adept at mathematical manipulations and even those that are cannot do it as fast by hand as they can with computer assistance. Thus to keep up in the future, they will need to be aware of and probably use tools like Maple, Mathematica, R, SageMath...

Please add your ideas to my list.

Thanks for engaging.  There are somethings that could make the process easier

First, if the institution supported one of the applications it becomes the place to start.

Second, to the extent that the undergraduate education committee or the Chair can institute adoptation in general chemistry or general chemistry lab or before, that becomes a point of leverage.  Since most departments have their own gchem lab manuals, the software app can be integrated into the lab classes.

Third, another department might have already made the choice, for example mathematics, or physics.  

Fourth, in institutions where there are compactified science department (eg Natural or Physical Sciences) choosing a uniform symbolic algebra application might be easier.

But let's be honest.  It is never going to be easy.

So here I will insert a blurb for the LibreTexts project I am involved in.  Since it is easy to modify the LibreTexts for your own classes, it is, perhaps, not a bad place to start to begin integrating the math apps into your textbooks.

Erick Castellon's picture

About using mathematical software, the key point is to remark the students to learn some, even by their own means. They could choose, of course, but here comes the importance of having the capability of symbolic computation as offered by Maple. For graphics there are plenty of options to use freely (one can write a function equation in the Google browser and receive as answer the graph), but the power of handle symbolic equations, to differentiate or integrate, to solve differential equations, that is no so spread.

All that vague generality is useless for the purpose of improving the present capability of students of chemistry to perform mathematical operations, which those students apply in chemistry (and other) courses.  If you seek to achieve that capability some time in the future, all that rigmarole and obfuscated organisation might work. If you seek to achieve that capability now, this semester or next, deliver a course (or better three courses, to replace the courses that might be taught in the mathematics department) in mathematics based on whatever symbolic software you might select, with the appropriate teaching materials -- i.e. a textbook based on that software.

How practical is that idea?  Its implementation requires an instructor of chemistry to learn both the mathematics and how to use the software, to design the course or courses, and to compose and to generate a textbook.  That action might require two or three years, as I know from my personal experience.

Alternatively, obtain a licence for Maple, install it and use the (free) interactive electronic textbook Mathematics for Chemistry with Symbolic Computation; all that action might require an entire week before the classes might begin!  Maple is a commercial product but is not costly: for instance I am aware that a particular university a few years ago paid $180,000. per annum for a licence for Microsoft Office but only $5000. per annum for Maple.  ($5000 might be equal to the tuition fees of one student for one year!)  Other software such as Maxima or SageMath is free, but for the purposes of teaching mathematics for students of science, technology, engineering and mathematics it falls far short of what Maple can achieve.

We can teach mathematics to our students with the formidable success that is documented in our paper under discussion, but we can not make the blind see.