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PQtutor, a quasi-intelligent tutoring system for quantitative problems in General Chemistry


Karsten Theis, Department of Chemical and Physical Chemistry, Westfield State University, Westfield, MA.


The best tutors are humans who give a student the appropriate amount of guidance necessary for learning while helping the student stay confident, motivated and focused. So-called intelligent tutoring systems, trying to replicate the discipline-specific and the psychological dimensions of expert human tutoring, require enormous investments and are not accessible to the larger student population. In a six-month sabbatical effort, I created PQtutor, a free online tutor designed to help students work out homework problems closely related to worked examples they already studied. The software is built on top of an online calculator for science learners (J. Chem. Educ., 2015, 92, 1953–1955) and uses problems from an open textbook, making PQtutor accessible in terms of both technology and cost. PQtutor works by comparing student input to a model answer in order to generate prompts for finding a path to the solution and for correcting mistakes. The feedback is in the form of questions from a virtual study group suggesting problem-solving moves such as accessing content knowledge, reviewing the worked examples, or thinking about the meaning of the question and the answer. In cases where these moves have been exhausted but the problem remains unsolved, the tutoring system suggests seeking intelligent human help.

PQtutor, a quasi-intelligent tutoring system for quantitative problems in General Chemistry

Karsten Theis, Ph.D., Dept. Chemical and Physical Chemistry, Westfield State University



The best tutors are humans who give a student the appropriate amount of guidance necessary for learning while helping the student stay confident, motivated and focused. So-called intelligent tutoring systems, trying to replicate the discipline-specific and the psychological dimensions of expert human tutoring, require enormous investments and are not accessible to the larger student population. In a six-month sabbatical effort, I created PQtutor, a free online tutor designed to help students work out homework problems closely related to worked examples they already studied. The software is built on top of an online calculator for science learners (J. Chem. Educ., 2015, 92, 1953–1955) and uses problems from an open textbook, making PQtutor accessible in terms of both technology and cost. PQtutor works by comparing student input to a model answer in order to generate prompts for finding a path to the solution and for correcting mistakes. The feedback is in the form of questions from a virtual study group suggesting problem-solving moves such as accessing content knowledge, reviewing the worked examples, or thinking about the meaning of the question and the answer. In cases where these moves have been exhausted but the problem remains unsolved, the tutoring system suggests seeking intelligent human help.



Learning to solve quantitative problems in college chemistry courses is a challenge for many students. Students have to integrate general problem-solving skills, chemical insight, unit algebra and error propagation into their solution, and learn how to tackle multi-step problems on paper. Students who get frustrated or stuck while doing homework seek various forms of help (Fig. 1). While there is a growing market for paid online supplements such as intelligent tutoring software or online human tutoring, not all students can afford these. There is also a growing amount of free help available on the internet's question-and-answer sites, but these contain material that can be wrong or misleading. In contrast, high-quality open textbooks are becoming available at no cost to students (and paid for by donations and support from foundations and not-for-profit organizations), for example those published by OpenStax [external link]. Striving to provide a free but valuable tutoring system supplementing a free textbook to students of General Chemistry, I spent a sabbatical writing PQtutor, an online tool to help students practice solving quantitative problems.

Figure 1.  Different ways of getting help with homework


Purpose of PQtutor

One goal of PQtutor is to allow students to practice quantitative problem-solving without losing sight of the chemistry context. Students give names to all quantities in the calculation, and they can easily include chemical or mathematical notation, figures or links to chemical data in comments. Moreover, when students submit their solution, they are presented with a multiple-choice follow-up question designed to elicit reflection on the meaning of the answer. These "What does the answer mean?" questions are inspired by the final step in the COAST problem-solving recipe used in W.W. Norton's chemistry textbooks by Gilbert et al. [external link]. Finally, chemical data is available within PQtutor (e.g. molar masses, table of quantities and units) and via links to outside sources (e.g. periodic table, thermodynamic data). Integrating the scientific context into the calculator helps bridge the divide between algorithmic and conceptual aspects of learning chemistry.

A second goal of PQtutor is to remind students of sound problem-solving strategies while they are working alone outside of the classroom. At different stages of solving a problem, students can request help from software agents that come in the form of quasi-intelligent peers in a slightly odd study group. Asking for help triggers reminders of general problem-solving strategies first, followed by more specific hints that do not give away the answer. Hopefully, the way hints are delivered will keep frustration and boredom at bay while encouraging students to think on their own.

PQtutor is limited in that it can not give help with higher level cognitive tasks (i.e. it is not intelligent). Thus, this homework system helps students to develop low-level and mid-level skills in quantitative problem-solving, making time to work on high-level skill development (such as identifying problem type or solving novel problems) in the classroom and in the laboratory.

Solving problems with PQtutor – use as a calculator

PQtutor works through a browser, and input is plain text entered via the keyboard (with some typing shortcuts provided), while mathematical and chemical notation is typeset in the output (the details of using the calculator are described in an online manual). Each command in PQtutor starts with a name, describing the quantity one is defining or calculating, followed by a value or mathematical operations on quantities already defined. For example, to calculate the density of a sample with a volume of 54.3 mL and a mass of 81.4 g, you would type the following into the user interface:

   V_sample = 54.3 mL

   m_sample = 81.4 g

   ρ_sample = m_sample / V_sample

Typing "return" or pressing “go” results in the output shown in Fig. 2. Variable names and math are typeset in the results for better readability. For example, fractions are written with a fraction bar, and subscripts are used for chemical formulae and for variable names. The calculator takes care of the lower level mechanics such as number arithmetic, unit algebra and error propagation, and catches errors such as trying to add a mass to a volume, or taking the logarithm of a quantity that is not dimensionless. (While these are low-level errors, it does not mean textbooks are free of them, see supplemental information on "dimensional rigor" below). Thus, the calculator ensures that work done by the students is mathematically sound.

Figure 2: Output from a simple calculation in PQtutor


Homework with PQtutor

PQtutor lets students work on homework problems that are very similar to problems already discussed in the class or in the textbook. The problems currently available are all adapted from an open educational resource, the Chemistry textbook published by OpenStax [external link]. In the textbook, worked examples are paired with similar problems to which the numerical answer, but not the solution path is given. The latter problems are used as homework problems. Thus, students can refer back to the worked examples to learn how to solve this type of problem (they can access these directly in PQtutor, e.g. example 17 of chapter 3 [view in PQtutor]). The instructor would post a link to the homework problem, in this case the follow-up to example 17. When students click on the link given by the instructor, the calculator displays the question, and students can proceed to enter commands (Fig. 3 and [view in PQtutor]). If students are unsure of how to proceed with a problem, or they just want feedback on what they already entered, they can request help from a virtual study group by just pressing go without any input. Hints come in two flavors, feedback or pumps; feedback points out problems with the work so far, while pumps are leading questions suggesting what the student might do next. These two types of hints are discussed in detail below. Once a student submits an answer, PQtutor shows a “What does the answer mean?” question, allowing for reflection on the calculation and its meaning. A short video illustrates the mechanics of solving a problem, submitting it to PQtutor and sending a report to the instructor.


Figure 3: PQtutor presenting a homework problem.  The question is shown in maroon on the top. Commands to the calculator are typed into the empty box.


The virtual study group

There are four characters in the virtual study group named Q, B, C, and L (Fig. 4). Each one of them represents a different set of skills, strategies or tactics to tackle quantitative problems. Q has general quantitative problem-solving skills such as considering the relationship between knowns and unknowns, dimensional analysis, picking up on common patterns (e.g. two-state problems), algebra to solve for an unknown and some general troubleshooting strategies. B has a firm grasp of the big picture tools chemistry has to offer, such as the periodic table, symbolic notation (chemical formulae and equations), conventions for naming different types of quantities, and data tables available in the textbooks. C is the expert on the specific chapter, the concepts that are introduced, the common pitfalls associated with the chapter, and the chapter-specific tools information the textbook offers. L, finally, likes to look up how others have worked out similar problems, primarily by referring to the worked example (but will probably start consulting the primary literature before she graduates). The following two sections discuss how the study group responds to different student input, illustrated by live demonstrations that show these characters in action.

Figure 4: Members of the study group and their respective special power


Feedback from the study group

PQtutor checks student input by comparing it to a model solution (the model solution for example 3.17 is shown in the supplemental information below under the heading "authoring new questions"). Given quantities are checked for dimensions [demo 1], value [demo 2], number of significant figures [demo 3] and sensible name [demo 4]. Results of calculations are matched with those in the model solution. If there is a no good match, the study group will comment [demo 5]. If there is some unexpected input, there will be feedback as well [demo 6]. Feedback is given in general terms, but there will be some more specific information (often about which line of input might be incorrect) when hovering over the character's picture. Feedback never directly states that something is incorrect – it is up to the student to critically examine their work and correct it if necessary. The reason for this is twofold: On the one hand, the software’s assessment of the students’ work is not flawless (see supplemental information "alternative solutions" below), on the other hand, the students should develop a critical mindset and be responsible for their own conclusions.

Pumps from the study group

The best human tutors do not give away the answer to homework, but give just the right amount of scaffolding so that students can figure out solutions by themselves. This requires substantial knowledge about the psychology of learning in general and about the discipline and the student specifically, none of which PQtutor has. Instead, PQtutor turns the model solution into a graph of dependencies (Fig. 5), matches the student answer to the solution graph, and identifies reachable subgoals. Then, it proceeds to ask guiding questions about missing data from the question [demo 7], missing data from other sources [demo 8], relationships between knowns and unknowns, and subgoals and goals [demo 9]. Like for the corrective feedback described in the previous section, the pumps are ordered from general to specific. Students who take pride in figuring out a solution on their own will not receive more help than they need (i.e. as soon as one of the prompts gives them an idea on how to proceed, they can try on their own). Students who are trying to game the system and max out the hints will receive plenty of hints, but the solution is never revealed directly. Because there might be students who are unable to find a solution even with the help of PQtutor, it is good practice to discuss the problem or at least show the solution at the beginning of the next class.


Figure 5: Solution graph for the example shown in Fig. 3


Chemistry resources in PQtutor

Sometimes, students are not aware of resources in their textbook that might help with quantitative problem-solving. PQtutor integrates these tools into the hints given by the virtual study group (there is also a study tools page [PQtutor page]). The B character shows links to the periodic table, the appendices, and information about formulae introduced up to a given chapter. The Q character shows links to a table of quantities and units along with their definitions up to a given chapter. As mentioned above, the L character always seeks information from the matched worked example. Finally, the C character shows links to the chapter summaries and the explanation of keywords in the given chapter. If students adopt the strategies modeled by the virtual peer group, they will learn to make better use of these resources in the textbook (and discipline-specific resources like databases and primary literature later on).

“What does the answer mean?” questions

When students are ready to submit their solution to a homework question, they have to answer one final multiple-choice question before moving on. These questions are called "What does the answer mean?" because they have the purpose of connecting the numerical answer to the chemistry the student is learning in a given chapter. The questions come in three flavors. Some ask whether the magnitude (or sign) of the result could have been anticipated from an estimate, for example:

  How does the density compare to that of water (water has a density of approx. 1 g/mL)?

   a) It is lower because it sank to the bottom

   b) It is higher because it sank to the bottom

   c) It is lower because it floats at the surface

   d) It is higher because it floats at the surface

Others ask for an interpretation of the calculated result, for example:

  Based on the pressure you just calculated, the airbag ...

   a) will collapse because the pressure is much lower than atmospheric pressure

   b*) will feel soft like a birthday balloon because the pressure is similar to atmospheric pressure

   c) will feel semi-hard like a basketball, which is inflated to 1.5 times atmospheric pressure

   d) will feel hard like a bicycle tire, which is inflated to 7 times atmospheric pressure

. Finally, some go after common misconceptions, for example:

  What happened?

   a) The volume increased because the distance between the particles increased

   b) The volume increased because the size of the particles increased

   c*) The volume increased because the number of particles increased

   d) The volume increased because the balloon got stretchier over time

In all cases, the questions ask the student to make a connection between the result of their calculation and concepts that they are familiar with or have just learned. The idea is to nudge students to think about what their result means, and to check whether they might want to re-examine their calculation to troubleshoot it.


Field testing and further development

Teaching General Chemistry I and II this year, I will give students homework through PQtutor to find out how the tutoring system works in practice. For each homework submitted, PQtutor reports the student answer as well as the hints, if any, given by the virtual study group. In aggregate, these data will show how much students rely on the hints, at which stage in problem-solving they request hints, and how many hints they view before proceeding with the calculation. The submitted homework will include any comments students entered, giving those students who do not succeed at solving a problem an opportunity to formulate where they got stuck, and what additional information they might have needed to continue. By talking to my students about their experiences using PQtutor, I can also gather anecdotal data about their academic emotions, in particular whether they experienced boredom or frustration while solving interacting with the online tool. Based on this data and feedback, I will be able to fix any problems with individual homework questions, fine-tune how the virtual study group interacts with the students (clarity and tone of the hints), and address any annoyances with the mechanics of entering commands. I will also be able to decide which of the many ideas for additional features would be worthwhile to implement (e.g. suggestions for more challenging questions after successful problem-solving or for foundational problems after failed problem-solving attempts, interpreting chemical equations, algebra or free-form responses entered by students, measuring student response time or asking them about how they are doing to decide on what type of hints to offer, giving students detailed feedback after they have studied a model answer upon submitting their work). Finally, this first field test with students will provide important pointers on how one might rigorously test the impact of PQtutor on the development of quantitative problem-solving skills in students of general chemistry.



I would like to acknowledge Westfield State University and my home department Chemical and Physical Sciences for granting the sabbatical that allowed me to work on this project. I would also like to thank the Biochemistry department of the Max-Planck-Institute of Chemical Ecology in Jena for their gracious support in form of office space, and Estonia-based artists Vladimir and Maksim Loginov of for allowing me to use their characters for the virtual study group. Finally, I would like to thank past and present students for their patience in testing and using the PQcalc and PQtutor software.


Supplemental information


Support for chemical notation and algebra

While PQtutor does not know any algebra (and will not help solving a mathematical formula for an unknown), it does support students in doing algebra by allowing them to write arbitrary math in comments. While PQtutors knowledge of chemistry is very limited (it does have a database of the fifty or so compounds most commonly mentioned in the textbook), it allows students to write chemical formulae and chemical equations in comments. In a live demonstration (i.e. a link to PQtutor with pre-filled input), you can experience how to more fully document a solution using comments integrated into the calculations [demo 10]. You can also click on different parts of the chemical equations to see how you can easily paste stoichiometric coefficients into the actual calculation.


Alternate solutions

Even though PQtutor is an example tracer working off a single model answer, it is able to adapt to many variations in student answers gracefully. The order of steps is flexible as long steps are independent of one another. Combining two steps into a single one is tolerated, as is using an intermediate variable to break a single calculation into two. Naming variables is flexible, as long as the root of the name matches the dimensions, and its value is correct or close to correct. However, some homework problems have two correct solution pathways that are sufficiently different to get PQtutor into trouble. For example, consider calculating the pOH from a given hydronium ion concentration (in aqueous solution at room temperature). The model answer might first calculate the pH, and then subtract it from 14 to get the answer. The student answer, in contrast, first calculates the hydroxide concentration via Kw and then takes the negative logarithm to get the answer. In this case, the relationship between the two solutions is not clear to PQtutors algorithm (one solution depends on knowing pKw while the other depends on knowing Kw). This could be solved in two ways. Either the question is changed to explicitly give either Kw or pKw, and asks to use this quantity in the calculation, or (which would require a modification of the software), two model solutions are allowed and the one that matches the student work better is used.


Dimensional rigor

There are at least three instances where typical general chemistry textbooks are less rigorous in their treatment of dimensions, scales, and units than PQtutor requires. The first concerns the dimensions of concentration in the context of equilibrium thermodynamics (and the logarithm function). In relationships such as pH = -log([H+]) or ΔG = -RT ln(K), the argument of the logarithm has to be dimensionless. However, introductory textbooks usually lack separate notations for concentration (units mol/L) and concentration divided by the standard state (unitless, equal to activity in the limit of infinite dilution). In PQcalc, the convention is that c_solute refers to a concentration, and [solute] to the dimensionless quantity. To convert between the two, multiply or divide by the standard state (1 mol/L for solutes). The second concerns dimensions of thermodynamic quantities ΔH, ΔG, and ΔS. Depending on the textbook and the chapter, these are either extensive quantities (e.g. units of J or J/K) or molar quantities (e.g. units of J/mol and J/(K mol)), but often the same symbols are used (upper-level textbooks sometimes use the convention that bold-faced quantities are molar to distinguish the two possibilities). In some cases, the formula used imply molar quantities (e.g. ΔG = RT ln K) or extensive quantities (ΔS = k ln (W1/W2), ΔS = q_rev / T). In PQtutor, the convention is that thermodynamic quantities H, G, and S are molar. If necessary, homework questions are changed to reflect that (for example, instead of calculating the enthalpy of burning a sample of a substance, the question will ask for the molar enthalpy of the reaction as well as the heat produced given how much reacts). The third concerns temperature scales, often treated less than rigorously in introductory textbooks. Internally, PQcalc always uses the Kelvin scale for temperature. To be able to enter and display quantities using the Celsius scale, PQcalc allows separate units for absolute temperature and temperature differences using the Celsius scale. Absolute temperatures are entered with units °aC, temperature differences with units °ΔC. This is relevant for the conversion into the Kelvin scale. For example, 25 °aC converts into 298 K, while 25 °ΔC converts to 25 K. If you subtract 0 °aC from 25 °aC, the answer is 25 °ΔC, if you try to add 0 °aC to 25 °aC, you get an error message. While ugly, this seems to work reasonably well.


Authoring new questions

It is straightforward to author new questions, and it requires minimal work. First, you need a description of the problem in plain text. It will be formatted just like user input in PQtutor. Second, you need a solution that PQtutor understands, i.e. input for a correct solution. Third, you need to write a multiple choice follow-up question, with the correct answer marked by an asterisk. For example, here is the information for the example 3.17 used throughout this paper:

    <h3>Example 3.17b: Determining the Mass of Solute in a Given Volume of Solution</h3>

    How many grams of CaCl2 are contained in 250.0 mL of a 0.200 M solution of calcium chloride?



    V_solution = 250.0 mL

    c[CaCl2] = 0.200 M

    m[CaCl2] = ?

    n[CaCl2]  = c[CaCl2] V_solution

    m[CaCl2] =n[CaCl2] M[CaCl2]


    Think about it: If this is an aqueous solution, we can estimate the density as about 1 g/mL.

    What is the mass of the solution, and what is its main component?

    a) 250g, CaCl2

    b*) 250g, water

    c) 4 kg, CaCl2

    d) 4 kg, water

PQtutor has no other knowledge about this question, and generates the feedback and prompts from the model solution and from the logic of the underlying mathematics. It does have some general knowledge about dimensions, units, and commonly used symbols for certain quantities covering the concepts covered in General Chemistry I and II.

04/26/18 to 04/28/18


Cary Kilner's picture

This is certainly a heroic effort, and another valuable contribution to the online tutor regimens for struggling gen-chem students. 1) I would suggest that some students will find it useful and perhaps interesting, and be willing to negotiate the perhaps steep learning curve for its use. Others may bail at the formidable work that will be required to master this early version. 2) I am still a strong proponent of ALEKS as an "intelligent tutor." Its attribute is its ability to keep the student learning and practicing at the periphery of his/her knowledge and understanding. However, in my discussion with the company, they acknowledge that it was not designed to teach higher-order problem-solving, and that this is the role of the live instructor. If PQtutor works out well, it may supplement this valuable function. 3) Either way, I deign to my colleague from our recent ConfChem, Eric Nelson, that students must master their fundamental skills and understanding through practice to put them into long-term memory, before being expected to be able to process problem-solving at a more sophisticated level. Hopefully, PQtutor will do this before dropping the student into the abyss. 

Thanks for your encouragement. I wanted to comment on the learning curve. It is true that PQcalc's user interface is quite different from what students might have worked with in the past. However, they have some immediate benefit (help with units and significant figures) and learn some skills that might be helpful in the long run (organizing their work, working with a text-based system). Also, because the calculator is an open educational resource, they can use it in any context at any time once they figured out how to use it efficiently.

I think the movement toward open access educational materials is very interesting.  This is a great example of supporting that effort.  

I admit up front that I know little about coding, but since you mention the proliferation of ‘intelligent tutoring software’, I was curious if that was a direction you think PQtutor might move into?  That is, if you assign a set of questions or a topic, the student performs exceptionally or poorly and the software adjusts for the student’s performance level.

Another, more basic, question- would PQtutor be able to generate random values within parameters, so each question could be unique to a student?  For example, in the CaCl2 example you give, couldPQtutor generate a random volume between 125.0 - 275.0 mL for each student?

I'll start with the last question, parametrisation of questions. Yes, that would be fairly easy. I would want to avoid questions that involve impossible values like a 7.5 M NaCl solution, though. Also, the OpenStax textbook already has hundreds of end-of-chapter problems, so it would be possible to assign those. Because the arithmetic portion of the problems is done by the computer, changing the numbers does not increase the difficulty - you would just use the same recipe again. Also, to clarify, the numerical answers to all of the questions currently offered in PQtutor are given in the textbook. What is not given is the path to solving the problem.

Concerning the question about making PQtutor intelligent - I do not have the resources to do that, and I am not sure if it would make sense. We human instructors are intelligent, and we should use that intelligence where it matters most. I have taught GenChem in a flipped model in the past, where students work on the hard problems in class. I think I prefer teaching my students the high-level skills myself, and leave the low- and mid-level skills to an electronic system if I have to, not the other way around.

Karsten --

Two things you write about that I think are especially important. The first is the end question in the COAST algorithm: “Think about what the answer means,” a variation on “Is the answer reasonable?” Dr. Penn’s paper in the fall Math ConfChem spoke to the question of teaching students to recognize when an answer is “preposterous.” To start solving with, “What is a likely reasonable answer?” and end on “is the answer reasonable?” would seem to be good steps in solving any problem.

Second, I appreciated your attention to units. Dimensions are a powerful tool to guide problem solving during initial learning, but chemistry’s reliance on equations that involve concentration rather than unitless activity require that this “shortcut” be addressed to be consistent with dimensional homogeneity so the science and math make sense together.

Here’s my question: Would it be wise to steer students toward a specific but widely applicable algorithmic approach? A specific example: Should we encourage students at the first step of the question analysis to write the unit wanted? To help focus on where they are going?

Those in science who study how the brain solves problems say that experts can solve problems in their field by general reasoning strategies because they have so much knowledge of the field in long term memory, but undergraduates do not, and for them, solving using recalled algorithms is the only thing that works. Trying to apply general problem solving strategies would be great if it worked for undergrads, but doing so has been well documented to quickly overwhelm the limits of working memory. and confusion tends to result.

So, can the software be adapted to encourage the use of widely applicable algorithms for chem calculations? Especially algorithms that rely on dimensional analysis? Do you think our field can agree on what algorithmic strategies are best?

-- rick nelson

Thanks for your support concerning the meaning of an answer, and paying attention to dimensions!

I am ambivalent about algorithmic strategies. I think they have their place for calculations that are common like limiting reactant and yield - an ICE or RICE table is great for that. On the other hand, I am weary of recipes that allow students to tackle specific problems but are not generalizable. There is this triangle strategy to solve for any of the three variables in an equation of the form d = m/V. That's fine, but what if the equation is of the form p V = n R T? So in that case, I think giving the strategy is a short-term fix, and students could have spent some time learning algebra with d = m/V that they then can apply when it comes to p V = n R T. Also, they could have spent time looking at the dimensions.

In the same way, dimensional analysis will not work for all problems, so it can't be the only strategy. It sort of falls apart in a dilution problem (and labeling your concentration and volumes as "stock" or "diluted" comes in handy), and really stops working when you have topics with lots of dimensionless variables (when switching from concentrations to activities, for example, or when discussing pH, [H+] and friends).

Cary Kilner's picture

Again aligned with Eric Nelson, I suggest that (and this is what I do in my own instruction within the Chem-Math Project) students will first memorize the elementary and fundamental mathematical procedures; algebra, prealgebra AND arithmetic. They will then, by various means as structured by a pedagogical content expert, be brought to understand how these procedures are to be used in a specific chemistry context, i.e. their conceptual basis. At that point students will evolve, or can be shown, algorithms that will simplify the procedures so they can be quickly used in a laboratory situation (which is, of course, the purpose of chemical calculations). So algorithms, to my mind, come last, not first.

Another point I shall make from the Chem-Math Project materials is that in our science we are mostly dealing with situations involving ratio-proportional reasoning. So unit-analysis is simply straight algebra used in the service of the inherent ratios present in our calculations and conversions. As I have pointed out on other occasions, I often found that my honors students would recognize this, and spontaneously evolve the "equivalents" approach to stoichiometry that us old-timers were taught before unit-analysis became the "universal elixer" to address beginning chem-students' mathematics difficulties. When I asked them to use UA for stoichiometry exercises they would complain that they better understood, and would prefer to use, the equivalents approach (multiplying the formula-weights by their respective equation coefficients and setting up proportions). So I see a lot of the argument regarding issues students have in using UA is not addressed by teaching algorithms, but by helping them learn their basic algebra. (As a caveat, I would suggest this is even more important in later gen-chem topics like thermo where logarithms enter in and the proportional relationships are more obscured. But do note that most of equilibrium consists of ratio-proportions.)

Dear discussants, if I'm being too dogmatic, feel free to rein me in.

What I like to research is: What do the experts who study how the brain solves problems say about facts, algorithms, and conceptual understanding in quantitative sciences?

Two short and very readable article on this by experts, speaking for consensus science, are
by Willingham (2009):
and Clark et al. (2012):

In part what they say is:
--Facts must indeed be memorized first. When the brain solves problems, it must rely almost entirely on facts and procedures recallable from LT memory
-- Students must solve problems by applying algorithms. Avoiding the measured bottlenecks in working memory is what algorithms do.
-- Students need to be taught to apply “standard algorithms.” Many algorithms may work, but teachmore than one and students tend to mis-remember which steps to follow. If different books in a nation use different algorithms, kids get confused. The US teaches different arithmetic algorithms than much of Europe, but all US textbooks for most of our history (thankfully) have taught the same ones. Which algorithms are standard in chemistry?
-- Students gain conceptual understanding as they solve problems successfully using algorithms. “Neurons that fire together wire together” during problem solving. That wiring is the substance of conceptual understanding.
-- The conceptual understanding that students use during problem solving is IMPLICIT not explicit. They need an intuitive (implicit) sense of what to do.
Knowing why they do what they do (explicit understanding) does not help during problem solving. That’s why chemists can solve algebra calculations intuitively without being able to state math theory. Experts in a field need explicit understanding. Chemists are not experts in the why of math. Over 90% of kids in gen chem are aiming to be science majors, but not chem majors. They need implicit understanding at this stage of their learning curve.

So I think science would say that what matters on understanding depends in large part on implicit vs. explicit.

This is all newly discovered science, verified just in the last 10 years. But the science of how the brain works is not a minor concern for those of us who teach.
Many popular approaches in chem ed still ask students to solve problems by reasoning like a scientist, which cognitive science says they cannot do. See Clark above. Ah, but paradigm shifts are not easy, and there is indeed much new science to learn about learning.

- rick

After skimming the Clark paper, I can comment on how PQtutor fits into the model of explicit instruction. First, here is a quote from the paper on this type of instruction:

"In a math class, for example, when teaching students how to solve a new type of problem, the teacher may begin by showing students how to solve the problem and fully explaining the how and why of the mathematics involved. Often, in following problems, step-by-step explanations may gradually be faded or withdrawn until, through practice and feedback, the students can solve the problem themselves. In this way, before trying to solve the problem on their own, students would already have been walked through both the procedure and the concepts behind the procedure."

The problems from OpenStax Chemistry are paired, with one worked example discussed in the textbook and also available as a full solution in PQtutor (so that students can see the mechanics of using the tool as well). When the homework is presented, students see a blank slate in the answer box, but they can refer back to the worked example. If students need more scaffolding, they can invoke the virtual study group. As I mention in the presentation, there should also be some feedback in class after the problem was assigned as homework.

So the intention is that homework occurs after a full explanation of the procedure and the concepts, at the stage where "step-by-step explanations may gradually be faded or withdrawn".

SDWoodgate's picture

I agree totally with what you say about algorithms.  I used to use the RICE table a lot, but no longer do so because I think that it is better to foster approaches which are also applicable to more complicated problems.  BUT, having said that, there are momre fundamental strategies that can be applied generally, particularly to multistep problems that are so common in Chemistry, like identifying the unknown quantity and giving it a symbol.  In chemistry it is appropriate to also include with the symbol a chemical formula.  Likewise make a list of all of the given data in symbolic form.  Create a pathway which links the known and the unknown. Identify additional quantities which are required and mathematical relationships between quantities to be calculated.  It is difficult to force students to think in this general way if left to their own devices. It is easier to force this way of thinking with an on-line system.  We need to embed the habit of analyzing before punching numbers into a calculator.  Your system is indeed partly putting in place more basic strategies like this one.

Yes, I do try to get students to name quantities before they start calculating with them. I made it as easy as I could to enter meaningful names in PQtutor. So the concentration of magnesium chloride is typed in as c[MgCl2], and is formated as 

If you want to show that this is your unknown, you just type c[MgCl2] = ?, and the online tutor will consider that as your goal. On the other hand, if you entered all the knowns and then ask for help, there will be a prompt asking you to think about what you are trying to determine in your calculation.

In the COAST problem strategy I reference in the paper, this is the second step called "Analyze", after "Collect and Organize", which is naming the knowns. Sometimes, I give students an entire page on an exam for a multi-step problem, and explicitly ask them to write down the knowns and unknowns in the first part, write down the relevant formula in the second part and solve it for the unknown, do the arithmetic in the third part and answer a question about the meaning of the result in the last part.

The argument I make in the JChemEd paper about the PQcalc tool is that when students work with a pocket calculator, the chemistry context is lost. Because quantities are given names in PQcalc, I'm hoping the context is preserved, and students see the patterns (e.g. they learned the relationship c1 V1 = c2 V2 for dilutions, and they just named the knowns c_diluted, V_diluted and c_stock, and decided that the unknown is V_stock).

rpendarvis's picture

General chemistry has become extremely algorithm heavy.  Students tend to learn patterns of solving problems with some difficulty.  They get so bogged down in these methods that I am concerned about whether or not they truly understand the material.  For example, my nursing chemistry students have trouble understanding why elemental calcium is not a component of bones.  I think we need to do more with understanding the concepts rather than focusing so intently on working through algebraic representations.  This would probably bring about more genuine interest in the subject.  How many of us went into chemistry because we thought gas laws were fun?   Thanks - Richard

You are right, it seems that the pendulum swung a bit too far to the quantitative side after a long time where the first chemistry course was walking element by element through the periodic table (the latter was my experience in the 1980s in Germany). I teach a research experience course for chemistry majors where I ask students to improve a GenChem experiment or design one of their own. Many students took one of the experiments and added a twist they cared about (for example, using the gas law to build an airbag that inflates but does not pop). However, one student just picked one of the only experiments in our lab manual that was not quantitative (which we had never done). The experiment was called "Some non-metals and their compounds" and included the preparation of CO2, SO2, NO, NO2, H2S, NH3, and elemental oxygen, nitrogen, chlorine and bromine. In his end-of-the-semester presentation, he said something like "I became interested in chemistry because of the smells and the colors of different substances, not because I wanted to measure something." He also mentioned every chemistry graduate should know the smells of some of the most common gases (some toxic) that might be produced in a chemical reaction.

Excuse me if I go a bit off-topic. Karsten Theis wrote, "You are right, it seems that the pendulum swung a bit too far to the quantitative side...".

I've been pondering this issue for years. The main motivation for me to write a textbook was related to this, but I decided that the problem for beginning chemistry students wasn't that there was too much math, but that it came too soon. The chemistry-first version of my text postpones most of the math until mid-text, giving the students time to learn the basic concepts of chemistry early. I think if we start our beginning courses with unit conversions and moles instead of descriptions of chemical structure and reactions, we run the risk of giving students the impression that chemistry is more about math manipulations than chemical changes. You can read more about this at


I teach a nursing chemistry course, and found a textbook that does something similar, in this case get to the organic chemistry three weeks into the semester, after an introduction to atoms and bonding at the particular and the macroscopic level. The topics rich in math (acids, bases, buffers; equilibrium reactions) come later in the semester, and we wrap up with enzymes and metabolism, again a topic that is quite relevant for their further studies but sometimes gets short shrift.

I will look more into your textbook. I get the feeling that sometimes the lab componenent of a general chemistry course drives the order of topics. If we had more meaningful experiments that bring home the particular nature of matter, or aspects of chemical reactivity that are qualitative, then maybe it would be easier to do atoms first or "chemistry-first". I put quotes around the latter - shouldn't a chemistry course be "chemistry-always"?

Circling back to the topic at hand, PQtutor could play a role in a revamped order of things because it would allow for some intensive practice of quantitative aspects inside and outside of the class room. On the other hand, there are some parts of PQtutor - not yet developed fully - that allow working on the symbolic representations of chemistry, such as writing chemical formulas and chemical equations.

Cary Kilner's picture

Now you’re preaching to the choir! We seem to have totally lost sight of the importance of descriptive chemistry in college as well as in high school, given the rise in computer simulations and the fear of litigation.

Early in a course I provide my students the definition of chemistry that I wrote: the study of the composition and properties of matter and the changes it undergoes. In lab and class in wet-chemistry activities and demonstrations, we explore phenomena. When we get to the tough quantitative stuff, students are willing to work on it, having been provided a context for it. Allow me to share a couple of anecdotes.

In fourth-grade, when I had begun to work in my basement chemistry lab, I had a teacher, Mrs. Stevens, who was an expert on butterflies. We would go out in the field behind the school at recess with nets and collect specimens. Then we would bring them into the classroom to put into her NaCN killing-jar before pinning them up on a board to key and label. (Would you ever find this today?) Intrigued by “cyanide” of course, I precociously asked her if I could smell it and, of course, she let me waft it. What an interesting odor! But now I knew it and could thereafter recognize it.

In 1994 I was taking an advanced inorganic lab-course for my MST degree. We were using a tube-furnace to synthesize a chromium compound, and one of the products was phosgene that was vented up the hood. I had often read that its odor was that of “newly-mown hay,” and I’ve always wanted to investigate it, so I stood near the end of the tube and tried to waft some. The TA saw this and flipped out at me, also reporting it to the course professor who severely admonished me. I had checked out the toxic-limit concentration in Hawley’s chem-dictionary and found it was not too much lower than that of HCN, H2S, CO, CS2, and CCl4.

In my high-school course I, too, had students generate gases; H2 to determine molar-volume in one experiment and its flammability in another, O2 to investigate combustion, and NH3 to make a fountain. Students were, of course, then much more willing to work on the quantitative properties of gases.

I had meant to respond to Tanya Gupta’s article but read it too late to post these remarks, which are appropriate here. Her simulations with the stoichiometry of combustion of hydrocarbons present a nice bridge from gen-chem to the organic course for which she developed them. I suggest an extension to the simulation that could be incorporated for an additional challenge. Have students determine which hydrocarbons provide a “greener” source of energy by producing a greater ratio of water to carbon dioxide—the longer carbon-chains or the shorter chains, e.g., compare the ratio of combustion products in methane (LNG) to that of propane (LPG).

In lab or class, also perform the “whoosh-bottle” demonstration with methanol, then ethanol, then propanol, to see the water (and perhaps excess alcohol) pour out after the reaction. One can even then use butanol and observe that it will not “whoosh,” due to too low a vapor-pressure to produce an explosive mixture. Such a demonstration may provide greater interest in working on the simulation.

As a caveat, there have been some fears about using this demonstration, due mainly to reports of folks using too large a container or using glass containers. I use soft-plastic gallon-milk containers that have proved completely safe, and also allow students to watch the wave-front move down through the gas mixture.

Long-live demonstrations and descriptive chemistry! "Silver chloride is a pale green gas," as Derek Davenport famously reported in JCE in 1970, having heard a student report this as a fact.


In the last 6 or 7 years, I've become the chemical weapons guy at the Middlebury Institute of International Studies at Monterey. In the class I teach, I talk about the use of phosgene as a chemical weapon in WWI and the likely use of cyanide by Iraq against the Kurds in the 80s (and of course, the use of cyanide in the Nazi gas chambers). Phosgene crosslinks proteins much like its cousin formaldehyde, which has a similar structure, and therefore disrupts a variety of functions in the body, and cyanide disrupts cellular respiration. Cary...although your story about these two toxic chemicals is interesting, I vote no on going back to the days where sniffing them was considered OK. 

By the way, I appreciate your use of unit analysis in the place of dimensional analysis. It's a much more intelligible phrase for students.

My personal view is that the new discoveries on how the brain works are in part bad news, but overall very good news.

The bad news is that initially, students need flashcards, etc. to move new facts and procedures into memory, which is rarely fun, and then they need to use them to solve problems in different contexts.

The good news is: Thanks to progress in understanding how to structure written text to help students with comprehension, and what new software such as PQTutor can effectively do, much of what is now covered in lecture can be covered in self-study, and this frees up more time in class with students to solve problems, do demos, talk about social implications, smell interesting liquids, etc.

The best news is: Applying the sequence called for by cognitive science of: First thoroughly memorize new facts and procedures, then apply them, and then have some fun with them, students become very good and very engaged problem solvers. Applying the science gets good results.

- rick