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Estimation – An Empowering Skill for Students


Lynn S. Penn, Department of Chemistry, Drexel University

10/26/17 to 11/01/17

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



As a rule, today’s students have a poor sense of the expected size of a physical quantity, and they cannot tell if their computed value is realistic or preposterous.   Students place great faith in the correctness of calculators, but they fail to acknowledge that they can make mistakes in typing numerical values or in entering mathematical operations.  These factors together not only contribute to poorer grades on exams and problem sets in their coursework but also lead to embarrassing mistakes on group projects and in the professional workplace.

Wrong answers can be greatly reduced if the students 1) adopt the practice of presenting the solution to a problem in several reasonable steps and 2) learn to rapidly and reliably estimate the numerical value of each step until the final result is reached.  While the first step may be assumed to be routinely practiced by most students, today’s students tend to carry out the complete computation process on the calculator before writing down the final result (which is often wrong).   Even when instructed to “show their work,” they write down only one intermediate step, a practice that fails to allow them to check their work and that offers no insight to the instructor as to where the student went wrong.    Even those students who have previously been taught to write out multiple explicit steps still may generate the wrong answer, because they are not able to estimate, and therefore are not able to check,  the numerical values associated with each step.   

Approach for the Teaching of Estimation

The present paper provides the details to an approach that has been observed to improve the students’ ability to make computations correctly and also to enhance their confidence and speed in working problems.  In learning this approach, the students are forced to observe more closely the sizes of the intermediate steps and of the final answers in various types of problems.    While the approach presented here is based on personal observations, it was developed through trials with many students over several years of teaching in the physical sciences.  Typically one class time (one hour) early in a course is devoted to this approach, which is composed of training in estimation skills and in step-by-step presentation of the computation.  The number of steps required for adequate display of a computation is determined by the limitation that only a few (no more than 2 or 3) algebraic operations are allowed for each step.   Below, we describe the approach explicitly in terms of five successive exercises.  All of these exercises must be done without calculators.   Calculators may be used, at the instructor’s discretion, after each exercise for comparison of calculator values with estimated values.

The first exercise is for the purpose of overcoming the students’ assumption of invincibility in making calculations.    A good way to make them realize their vulnerability is to give them a nontrivial equation in class and to ask them all take out their calculators and compute the answer.   A good example is the equation for enthalpy change for the process of heating a mole of copper at constant pressure from 500 K to 1000 K.  The heat capacity coefficients and the formula are supplied:

Typically 15 - 20 % of the students get answers that differ from the correct answer by significant amounts.  The students have no trouble doing the integration; where they stumble is on the arithmetic operations required after plugging in the temperature limits.   They are taken aback when they state their answers orally and there are so many different answers, some of them orders of magnitude different from others.   This is enough to convince them that there is indeed a problem.  After this, the students must work without calculators.

The second exercise addresses the trouble the students have in computing quotients and products accurately when these quotients and products are not round numbers presented in an instantly comprehensible form.   Therefore, the first exercise they are asked to do is to express a set of provided quotients as a larger number divided by a smaller number, where both numerator and denominator are multiplied by the appropriate powers of ten.   This does not mean conversion to scientific notation  --  rather, numerator and denominator must be converted to numbers that are readily divided mentally without mistake.  It is important to recognize that students find it easy to do mental division when the numerator is larger than the denominator.   Completely worked examples are shown in Figure 1 below.


Figure 1.  Examples of conversion of quotients to final values.

The instructor can show the students how to reduce one or two initial quotients to a final number, and then can give the students a longer list of initial expressions to work themselves.  The extra time required to change the numerator and denominator to numbers multiplied by powers of ten is offset by the rapidity and confidence with which the students can carry out the division.   After doing a half a page of this exercise, the students are proficient.

The third exercise is to have the students round the numerators and denominators of realistic quotients to simple numbers, use powers of ten to convert the numerator to a larger number than the denominator, and mentally compute the final answer.  Examples are shown in Figure 2.


Figure 2.  Examples of conversion of quotients to mentally operable expressions.

After the students have been shown once how to do this, they can tackle a half a page of initial expressions to do themselves.   For the estimate to be as close as possible to the actual answer, an increase (or decrease) in the numerator should be paired with an increase (or decrease) in the denominator.    Although the final answers to these quotients are estimates, typically they are well within 10% of the answer one would obtain with a calculator.  After half a page of this type of exercise, the students are proficient.

The fourth exercise is more complex but is still not very time consuming.  Students are given more complicated initial expressions and are asked to show their work in multiple steps.  They are instructed to do no more than two arithmetic operations per step, where a step is understood to be the material between equal signs.   Figure 3 below shows three examples, completely worked out in steps.  The instructor might demonstrate by working out one initial expression to the final answer, but then the students should be given several different initial expressions and should develop the final answers themselves.  


Figure 3.  Examples of more complicated expressions to be converted to a single numerical value.

The last expression in Figure 3 is an example of one that frequently gives the students trouble; they find it difficult to subtract two quotients correctly.   Conversion of these quotients to those having larger digits in the numerator and smaller digits in the denominator improves the students’ ability to correctly make the subtraction.   Note that the powers of ten are the same for both terms in parentheses, i.e., both terms involved in the subtraction.   Clearly, there is room for individual differences in rounding the numbers as well as in choosing which powers of ten to use.  However, the use of multiple steps allows student, instructor, and colleagues to understand the process in an instant.   The calculator values for the first, second, and third examples are 0.999 x 10-23, 1.35 x 10-10, and -0.966, respectively.  The estimated values, shown in Figure 3, are all within 11% of the calculator values.   Although the exercise shown above seems lengthy, it can be executed fairly quickly.  In addition, the closeness of the estimate to correct calculator values boosts the students’ confidence.  It should be remembered that the simplification and estimation involved in obtaining a final answer obviate any emphasis on significant figures.   After doing one page of this type of exercise, the students are proficient. 

The fifth and final exercise is to estimate the final values for some typical physical science equations, given the constants and the values of the variables.   This exercise, like the previous ones, must be done without a calculator.   Three examples are given below.

  1. Compute P, the pressure of one mole of methane gas confined to a 250-mL volume. 1

The calculator yields 73.3 bar for this example, and the estimated value of 80 bar is within 10 % of the calculator value, an agreement that again enhances the students’ confidence.   

  1. Use the generic empirical formula for heat capacity of copper2 to compute ΔH for one mole of copper at constant (atmospheric) pressure for a temperature increase from 500 to 1000 K.2

The final value is positive, as it should be for the enthalpy change of a system to which heat is added.  The last line allows the students to see that the leading term in the equation is the dominant one and that the second and third terms are relatively small and serve as refinements on the value of the dominant term.  Had they not written out the solution step-by-step, they would not be able to gain this insight.  The estimated answer of 10.9 kJ is within 1 % of the calculator answer of 10.8 kJ.

  1. Use the generic empirical formula for heat capacity of calcium titanate2 (CaTiO3) to compute ΔH for one mole of CaCO3 at constant (atmospheric) pressure for a temperature increase from 298 to 1000 K.


This estimated value is less than 5 % different from the calculated value of 85 kJ.   When the students get to the point where they estimate, without calculators, the final values for realistic physical science equations, they are ready to do problems on an exam without calculators.   For such an exam, this instructor counts their answers correct if the final values are within 15% of the value correctly obtained by computer or calculator.   Students taking this type of exam typically do well and do not require extra time.  

The approach described in the present paper was introduced a few years ago by the author because of the many errors made by students in their computations, on exams, on in-class problems, and on student projects.   The described approach has greatly improved the performance of the students in these tasks.  In addition to better performance on exams and better technical communication, undergraduate and graduate students who have been taught to practice estimation and step-by-step display of calculations save time and avoid mistakes by actively using this approach to check computed values needed in their research activities.  An anecdote that illustrates an unexpected benefit of estimation and step-by-step display was related by a student who forgot to take her calculator to an exam in a course where estimation and step-by-step display was not the practice.  She used the approach presented in the present paper and was given credit for correct answers by an impressed instructor. 


The approach described in this paper leads to achievement of good estimating skills in college students in a minimum of time.  They employ these skills to check their calculator work on exams, in problem sets and in class projects.  In addition, they become committed to the practice of writing the solutions to problems in multiple steps, each of which can be checked readily by estimation, whether during an exam, during classroom projects, or by the instructor during grading.  Students in classes in which estimation has been practiced typically get better grades on exam problems than students in classes where estimation was not introduced. 


1.  Donald A. McQuarrie and John D. Simon, Physical Chemistry: A Molecular Approach, University Science Books, Herndon, VA, 1997.

2.  David R. Gaskell and David E. Laughlin, Introduction to the Thermodynamics of Materials, 3rdh edition, Taylor & Francis, Washington, DC, 1984.


Dr. Penn:
This is the part of the conference where we are asked to ask questions rather than make comments -- but I am going to sneak in that I cheered repeatedly as I read your paper. In my experience both personally and as an instructor, when the math of chemistry becomes understood and makes sense, calculations become quite enjoyable.
Two questions if I may:
1. In your experience, what’s the ratio of students who in the end feel good about their understanding of their ability to “pencil and paper” estimate as a check on calculator use versus those don’t appreciate that skill? How much pushback might be expected at various levels of courses.
2. Supposing a chemistry department were considering a policy that would require chemistry majors to either pass a one-semester course in “math for calculations in the sciences” with a focus on “paper and pencil” estimates in calculations, or meet this requirement by passing a test on “calculation math without a calculator” offered prior to the start of each semester. Based on your experience, what advice pro and con would you offer to department members?
-- Eric (rick) Nelson

About 90% of the students feel empowered.  There is not much pushback at levels from freshmen to seniors college level.  The problem about new course, however short (even a weeek-end course) is that today's universities see eveything in terms of budget and in terms of allowed numbers of credits, which means that it is hard to get something remedial introduced.  If there is a day off, I often give the lesson on estimation, and a surprisingly large number of students come.  

Cary Kilner's picture


I see you are addressing mathematics issues that typically arise within a physical chemistry course, where students are differentiating and integrating. But you cite problems students have negotiating calculations with complex fractions, and doing estimations using powers of ten within improper fractions.

Since your students are in upper-level chemistry courses, how much of a problem do you see for your students in dealing with fractions? My study of NCTM (National Council of Teachers of Mathematics) materials has shown that problems with an understanding of fractions can persist even into the upper grades, but we assume mastery.

What other issues with more elementary mathematics skills have you also encountered? In his paper, Rick will be discussing the “lack of automaticity and fluency with basic arithmetic skills.” I note that Diana Mason, in the second paper from this week, cites exactly the same idea.

It seems like your estimation exercises are precisely the place where students might fall down from such a lack of basic memorized skills. Do you agree that students need to be able to access basic operations, as well as multiplications tables from long-term memory, to be successful with this kind of stepwise estimation that you are advocating?

Thank you!



I do agree that students need to be able to do basic operations, including multiplications table from long-term memory, to be successful with stepwise estimation.  I have found that the students' mastery of fractions is poor.  However, when the fractions are converted to ratios of very round numbers, with the numerator larger than the denominator (or the reverse if the numbers are 5's and 10's) , the students can "see" the result of the division very quickly.  Being able to do operations quickly makes it much more fun for the students and motivates them.  Lynn P

ChemEd XChange in August had a post about fractions that I found to be helpful when talking to students about some of the math in their chemistry classes.

Basically, the link suggests changing the way we talk about math with students helps them to understand how we are applying it to chemistry. The change "only requires trading one Latin preposition you already use – “per” – for its two-word English meaning: “for every.”"

SDWoodgate's picture

I think that it is useful to point out to students that per implies a division.  For example grams per litre can be calculated if the mass in grams is divided by the volume in litres.

I can't take any credit for this and I don't remember where I read it (hopefully someone her might), but another option for "per" is to use "for every".  The instructor that discussed this talked about the changes in her students understanding by making this simple change.  

Hi Dr. Penn,

As someone teaching physical chemistry, I can really relate to your paper! I think the math heavy reputation of pchem intimidates students even before they start. The example questions you showed are not "easy" but doable, which I thought was great to boost students' confidence in their math skills.  I have a couple related questions:

1. You mentioned that the student performance after the math practice improved. How did you assess their improvements? Was there a specific type of algebra question you used as a marker? Do you think it's the repeated practice upfront that got them to do better or the boost in their confidence?

2. Did you notice any improvements to more complicated algebra exercises, such as exponential and natual log calculation? From the 2nd law on, not able to estimate exponential or natual log becomes more and more of an issue in assessing the quality of their answers. 

3. I think being able to plot or visualize physical processes is a bigger problem for physical chemistry students. Beyond calculation, did you try any other methods of plotting functions or drawing diagrams, etc. to help students getting a real sense of what they calculated?

Thank you. 

Isabel Green

1. I assessed by exam.  I could see where they used the back or side of the page to estimate and check the calculator answers.  Also, on some exams I require no calculators, and they do very well, whereas without some practice in estimating they are an order of magnitude wrong. 

2. I have required estimation only for simple exponents.  I have not used it for natural logs.  On some exams I have said, get your answer without a calulator and leave the logarithmic expression in logarithmic form. 

3. We do plot, just as part of making the students see how useful; it can be.  We have plot questions on exams. But often, they can use calculators for coputing points to plot.   

Drs. Penn and Green,

Being a conference coordinator, I was allowed a peek at the upcoming papers.

On the topic of log calculations, Paper #4 by Dr. Leopold includes a persuasive argument that both base 10 and natural log numeric values can be quickly estimated with good precision without a calculator by memorizing just a few base 10 logs of whole numbers. The method uses simple, stepwise, paper and pencil arithmetic that students need to keep fresh. It fits well into Dr. Penn’s estimation procedures, and is especially applicable to the many types of log calculations encountered in 2nd semester gen chem and beyond.

Do take a look when the paper is posted next week. I think you’ll like it!

-- rick nelson

That is great.  Thanks, I will read that paper.  I would love to add logs to my estimation exercises.

Students lacking math skills is a very common problem in teaching general chemistry. I would love the idea of a statewide conference for general chemistry teachers to improve math skills in chemistry students. 

I use online homework system. I try to assign math reviews along with the chapter homework. To get a serious feedback from students I usually assign certain points from them. . You can also ask maths department to have basic algebra seminars and encourage your students to attend to those seminars. If we can have a prerequisite of having basic algebra before taking general chemistry would be helpful as well.


I really think that this comment was meant for Paper 2's suggestion of a Texas statewide meeting of the general chemistry minds. I know that at UNT the prerequisite for enrollment in general chemistry is completion of college algebra. However, enforcement of the "rule" has been overlooked for many years.  Next fall (2018) we have an e-system in place that will prevent students from enrolling in a course of which the published prerequisites have not been met. Will it make a difference in the success rate (grades of A,B,C)? That is the big question. The experienced professors at UNT have openly stated that it will not make a difference.  We will see. 

Another demographic that we are looking into this semester is whether or not a student is co-enrolled in a mathematics class (any!).  There is something to be said about having completed your mathematics requirement for your degree and being "done" with math.  That part of the brain is no longer being used and knowledge decay is rapidly progressing.  What effect will this have on success in chemistry?  Any insights?



Diana - We use Math ACT (23 and up) as a pre-req for Gen Chem I and it seems to have helped.  The alternative path of completion of college algebra does not seem as effective maybe because they don't have the long term exposure to the math skills that are demonstrated through the ACT scores.  Given the strong correlation that we see and that others have seen between Math ACT and Gen Chem grades, I can't imagine how it wouldn't make a difference in success rates.  

Several years ago, my university eliminated differential equations from the required curriculum in chemistry.  In my opinion, this did precisely what you mentioned -- allowed the chemistry majors to dispense with math requirements early in their college careers and not use that part of their brains anymore.  This is just an opinion.  


I enjoyed reading the detailed explanations of how you teach your students to estimate numerical answers, and I think it is a valuable skill. I have questions and comments about two points you make.

1) In your conclusion, you state "Students in classes in which estimation has been practiced typically get better grades on exam problems than students in classes where estimation was not introduced." What are the two groups of students you are comparing? Did you teach all of them yourself? Are you comparing cohorts before you introduced estimation with cohorts after you introduced it? How big is the difference in their grades? Is the data sufficiently robust that you could present it, or is it more anecdotal?

2) I agree with the first sentence of your introduction, "As a rule, today’s students have a poor sense of the expected size of a physical quantity, and they cannot tell if their computed value is realistic or preposterous.". You addressed the second part of the statement in your paper. I'm wondering how you help your students with the first part, i.e. the poor sense of expected magnitudes of quantities. Are your students expected to memorize some anchoring values of physical quantities, e.g. that the volume of a mole of gas under ambient conditions is about 25 liters, or that it takes about 25 kJ to raise the temperature of a metal by 1000 K? These would help students to put the answers to the final three problems you show into context. For the gas example, a mole of methanol at atmospheric pressure takes up about 25 liters, but here you confine it to 0.25 liters, so the pressure will be much higher than one bar, which the estimation shows it is. For the two heat capacity examples, the result for heating up copper falls nicely near the ballpark figure we expect (about half of 25 kJ because the change in temperature was 500 K instead of 1000 K). On the other hand, the result for the ionic compound is higher than what would be expected for a metal, which is intriguing and might prompt a discussion about how molar heat capacities of metals, ionic compounds and molecular compounds differ.

Again, congratulations on a thought-provoking paper!



The lack of ability to estimate begins early. One of the questions on the MUST (see Paper 2), is to convert 1/20 to a decimal number (without using a calculator.) When you get a response of "5" for example, are they just not paying attention to decimal placement, or if you get an answer of ".20" is there a question of what is happening inside their brains.  Just changing the given fraction to x out of 100, should provide the student with some insights and get them better prepared when it comes to p-chem type courses.  This first paper has some excellent insights on how (even at the gen chem level) we could place more emphasis on estimation. 


REply to question 1:  My answer is based on my own experience, as "chemical education" is not my area of expertise.  The people I am comparing are those in the same physical chemistry class in successive years.  This , of course , is not a controlled study, but they students who have learned to estimate and who are asked to work without calulators get more of the same tpye of question correct than those who use calulators and have not been taught how to estimate.  In addition (and this is from several eyars of teaching the same course) the students who are allowed to use calcultors and who have been tauigght to estimate, do check their answers (by working the problem on the side or below their calculator work on the exam paper) and make corrections to their calculator work if needed. Many students have commented to me that their confidence level goes up when they know how to estimate.   


Reply to question 2:  I continually try to impart to students a notion of relative sizes.  For example, a buollion cube of solid will increase by 10% (not much but still an increase) when melted.  Upon vaporization, the liquid becomes the size of a cardboard box.  I always try to connect amounts to things they are familiar with, but, alas, they are familiar with less and less these days.  I used to compare thermal coefficients of expansion of ceramics and metals by asking them to recall using hot water to loosen the screwtop lid of a glass jar.  They no longer relate to this at all, as they no longer open glass jars ar any jars.  No one today has pulled a nail rapidly out of a piece of wood and gotten burned from the hot nail (mechanical work into heat).  At present , it is very challenging to come up with examples of physical principles (or even chemical principles) that they themselves have experienced.  Lynn P 

Thank you for the entree into an important discussion on heat capacity.  Lynn P

Hello Lynn,

Excellent paper. I love this idea of teaching students to estimate in Physical Chemistry courses to evaluate the validity of their answer. However, I am wondering if you have used estimation lessons in General Chemistry courses? If so, what were the outcomes? Was the pushback comparable or worse than you experienced with physical chemistry students?


Hi Lynn:

Nice work! Thanks for sharing your work on estimates. Estimating and having a feel for numbers (related to physical quantities) is so important in science. I am curious about Gen Chem classes as well.

Students are allowed to use calculators in my Gen Chem courses. Estimates are great way to compare numbers using calculators, especially in the context of numerators and denominators. So many topics related to exponents and numerators/denominators in Gen Chem courses, unit conversions, kinetics, equilibrium, etc. and the list goes on.  It is so very important to use parenthesis correctly in calculators to obtain correct + meaningful results. 


I have not tried the estimation techniques in general chemistry. Lynn P

Oh, erroneous use of parentheses might be one reason why the calculator answers can be so wrong.  I will mention this henceforth, in order to convince them that the estimation exercises they are about to do are worthwhile.  Lynn P 

Rich Messeder's picture

Back in the 1970s, I was working with my lab partner in physics. Came time to write up the lab, he was always faster than I at getting results on his calulator. I was pretty hot stuff on my algebraic (TI), and wondered how he could possibly be faster. So I asked...and for my trouble, I got a bunch of RPN and the stack nonsense each time I asked. I scoffed. Finally I asked him to show me. At the end of his demo, I decided to sell the TI and buy an HP. I've never gone back. Years later. as I was working with 2 physics PhDs at a nuclear facility, one of them asked my opinion about calculators. I launched into a discussion of the benefits of RPN and the stack. The other physicist scoffed, and said that algebraic was a more natural system. We were getting ready to do a reactivity calc, which was complicated. We had it written in "solved" form, which involved a lot of numerators and denominators and trig and roots, etc. I proposed a test. As I began the calculation, I noted the scoffer was busy writing...I asked what he was doing...he said that he was rewriting the expression so that he could keep track of ((())). I finished way before. Grass is natural. Neither algebraic entry nor RPN are. But in terms of performance and minimizing mistakes, RPN beats the pants off algebraic. In several decades of real engineering, I have seen but a few calculations that needed more than a 4-level stack, so I opine that undergrads should be fine with that. I have more stories if the same vein. There is no keystroke benefit to RPN...nearly all calculations in algebraic and RPN use the same number. The performance boost is all in not keeping track of ((())) and being able to focus on the expression. If you think it is hard to learn, try this. My wife was dead set against learning RPN...for years. One day, as she was deep into the household budget and bills, her calculator died. She came to me to borrow one of mine. She said that she was not interested in hearing a lecture about RPN, she simply wanted to finish her work. As she went along, she asked questions every now and again. The next day she came to me and said that I ought to buy that new HP calculator that I wanted (tongue in cheek, you see...I had no such plans, and she knew it), because she "needed" the one she was using, and had no intention of returning it! Children can be taught RPN as easily as algebraic, and I opine being relieved ot tracking ((())) justifies it.

This paper lists multiple arguments for requiring estimation practice. Permit me offer a few more.

I try to keep up with research on the scientific study of how the brain solves problems. Since 2010, cognitive scientists have been in broad agreement that “working memory” (where the brain reasons) can easily process information that has previously been well-memorized, but has severe limitations when processing less familiar knowledge.

As one implication, cognitive experts say that facts and procedures used often in a field must be well-memorized to avoid creating bottlenecks during reasoning. If a student must use a calculator for what should be recallable or a quick mental calculation, working memory tends to clog, and confusion results.

Among the fundamentals used often in science are arithmetic and algebra. Recall of these facts and algorithms, say the science, needs to be “automated.” But even after initial memorization, recall declines unless it is practiced with some frequency.
The rounding, estimation, and step-wise problem solving advocated in this paper require students keep sharp what needs to stay sharp for reliable success in calculations.

Students need practice using simple numbers to apply algebraic rules sequentially. If they can’t solve with simple rounded numbers, when the numbers and calculations become complex, how will they be able to remember what calculator buttons to press when?

Requiring estimation with some frequency would seem to be doing science and engineering majors a favor.

-- rick nelson

I have noticed that the amount of material committed to memory by undergraduate (and graduate) cjeistry and chemicacl engineering majors has declined over the years.  I was recently working through the estimation exercises with a chemical engineering class, and no one could remember any of the multiples of 12.  I recommended that, while in their cars or on public transportation, they simply practice counting by 3's, by 4's by 6's,etc. and then backwards.  However, this is hard for them to do becasue they are so busy with their cell phones (which I do not allow in class, incidentally).   Lynn P 

I frequently refer to the ULM (Shell et al., 2010) when talking with others on what we can do in the classroom to improve learning.  The bottom line is that if the students are not engaged, they will not learn, AND if they are not motivated to be engaged, they will not learn.  How do we motivate and how do we engage students in any course at any level? Love to hear any insights?  

I want to emphasize that memorization is necessary.  We all memorized words in order to learn to speak.  Sometimess I tell my students that htey know they are able to memorize, because they learned to speak that way.  (Grammar, which comes later,words to use and put together.)   I also note to my students that they used to memorize phone numbers and sports stats, but I fear that memorization of those things is going away with the increased use of hand-held devices.  Lynn P

A two page handout or quick online reading assignment on when memorization is important in learning chemistry, how it relates to conceptual understanding, and how to memorize efficiently and effectively, is posted for free student and instructor use at


I was a co-author, it is copyrighted, but the publisher/owner has given permission to distribute it free.

-- rick nelson


This brings up a question in my mind. Are memorized math facts and other facts like memorized names of polyatomic ions stored in a similar fashion in the WM?

Diana –

The recently verified scientific rules that govern how the non-expert (student) brain solves problems can be stated in several ways, including

• During reasoning, the “working memory” where the brain solves problems can recall, hold, and reason with essentially unlimited amounts of well-memorized information, but very little information that has not previously been well memorized.
• Your problem-solving skill in a field depends on how much you know about the field: How much knowledge you have memorized and linked to other memorized knowledge.

In 2008, five of the leading US experts in cognitive science (David Geary, Valerie Reyna, Robert Siegler, Susan Embretson, and Wade Boykin) wrote:

“[T]here are several ways to improve the functional capacity of working memory. The most central of these is the achievement of automaticity, that is, the fast, implicit, and automatic retrieval of a fact or a procedure from long-term memory…. [I]n support of complex problem solving, arithmetic facts and fundamental algorithms should be thoroughly mastered…, rather than merely learned to a moderate degree of proficiency (Geary et al. NMAP, 2008).

Those rules also apply in the physical sciences. University of Virginia cognitive scientist Daniel Willingham writes,

“In each field, certain procedures are used again and again. Those procedures must be learned to the point of automaticity so that they no longer consume working memory space. Only then will the student be able to bypass the bottleneck imposed by working memory and move on to higher levels of competence.” (Willingham, American Educator, 2004)

In first-year chemistry, what fundamentals “are used again and again” that need “automatic retrieval from memory?” I would suggest the list would include the fundamentals of arithmetic and algebra, plus the names, symbols, and periodic table location for the about 40 elements encountered most often in first-year problems, and the names and formulas for the most frequently encountered monatomic and polyatomic ions. The elements of knowledge of math and science are stored and connected in a similar manner in the neurons of long-term memory.

Most chem students and instructors figured out the value of memorizing fundamentals long before cognitive science discovered why it was necessary. But in some chemistry education philosophies, it was popular for quite some time to deride “rote memorization.”

Scientists who study how the brain works are telling us that thorough memorization of fundamental vocabulary and relationships is a key first step in learning.

For additional detail, see “Cognitive Science for Chemists” at www.ChemReview.Net/CogSciForChemists.pdf

-- rick nelson

The President at UNT told us about "disruptive technology", and as an example he shared, it was like the investion of the cell phone camera that replaced the need for having to buy a regular camera.  Maybe we need some disruptive education (in Texas) because what we are doing is not working.  Memorization is probably a most important first step.  


On this the last day of conference discussion of Dr. Penn’s paper, here’s what I personally think I have found.

Using estimation is valuable if not essential as a way to require students to keep their fundamental math facts and procedures “automated,” Cognitive science says those skills need to be kept sharp if students are to keep their “train of thought” when solving complex problems.

If students do not know how to solve a calculation that has simple rounded whole numbers without a calculator,

• they will have a very difficult time remembering what sequence to use on a calculator, and
• more important, we can’t certify they know chemistry. Being able to do simple arithmetic and algebra is a central component of a quantitative science.

Based on our experience, we know that without estimating the answer, student calculator answer in complex calculation is often wrong, and the student will not know it.

People deserve a chance to shoot at Dr. Leopold’s paper next week, but I think the final judgment on Dr. Penn’s approach combined with Dr. Leopold’s will be a conclusion that close to all of the calculations of general chemistry and physical chemistry can have answers estimated well within 30% using paper and pencil mental math, without a calculator, and gotten to proper significant figures with only a “4 function” ( + - x / ) calculator.

Then I think the question will be: Because calculator answers are easily checkable, if we as instructors do not require students in gen chem to estimate them, when most of those students are headed into medicine, nursing, and engineering jobs where lives are riding on correct answers, and unchecked calculator answers are often wrong, are we as instructors being socially responsible?

I was not asking myself this question before reading this and some following papers in this conference, but I am doing so now.

Is it a fair question to ask?

-- rick nelson

Could the link to the MUST test be given?  I can't find it in the original paper.

Thanks.  is the link in the paragraph just before the Introduction. 

It's also posted here:

Bob Belford's picture

Hi Patricia and everyone,

Please let me know if this link does not work,

But it should be a public annotation that should work in any browser, and if I did it right, it takes you to the part of the article where the link is :-)




Milind Khadilkar's picture

I am aware that since the discussion deadline for Dr. Lynn's paper has expired, I ought not to post this.... (and hope the system will ensure that the posting does not succeed!), but this is my first visit here since the conference started... so if it does get posted, please excuse my intrusion.

Two quick queries:

1. Since the problem is there for all branches (not unique to Chemistry), should it not be addressed at a different level?

2. Surmise: "If this problem has been there for long, it must be present in a section of the junior faculty too... to some extent.". Is my surmise unfounded? Trust so.