The Chem-Math Project

THE PROBLEM:

Many of our students equate chemistry with the mathematics they fear. What can we do to improve their academic experience? For years we have been struggling with this issue. In September we begin high-school college-preparatory chemistry and many students cannot do simple unit-conversion and do not understand scientific notation. We think they should have learned these skills in their previous physical/general science and mathematics classes. We begin college general chemistry and our freshmen cannot use unit-analysis and don’t perform scientific notation correctly. If they are matriculating STEM students, shouldn’t they have already learned this in high school? We sometimes unfairly judge the students but they were never taught the math skills they need to use in science, e.g. unit-conversion and expressing orders-of-magnitude. They never had a chance to practice them to mastery, and they never understood them conceptually in the first place. Thus, if we want them to know how to use chem-math, we have to teach it to them ourselves.

WHAT IS CHEM-MATH?

In his paper, Eric Nelson will describe a study that looked at math reasoning skills as compared to computation; this distinction is close to my definition. I shall define formal-math as the skills and ability that are measured by the SAT–Math and ACT exams as well as the calculus placement test written by my mathematics colleague at the University of New Hampshire. This material we can describe as pure mathematics, representing the traditional study of numbers and their operations that is found in the mathematics curriculum as abstract concepts and taught in their courses. Chem-math I postulate to be the skills and ability to use arithmetic and simple algebra in expressing the proportional relationships that are found throughout chemistry exercises and problems.

It is important to make the distinction that exercises are used for the practice of specific skills. They involve applying algorithms that may or may not have been well-learned. Eric’s argument is that they must, in fact, memorize these algorithms. My argument is that they must ultimately understand what the algorithm means. A problem is “what you do when you don’t know what to do” (Wheatley via Bodner 1987, 1994). Typically, a problem will use previously “learned” algorithms that have been drilled in exercises, but students are not overtly told which ones to use. Students must also “unpack” the written material of the problem as another distinct skill. However we define problem-solving, they have to be able to select the appropriate algorithms and have memorized their mechanics. Much of the research regarding students’ troubles with mathematics in science refers to Piaget and his idea of formal thought as the mature reasoning stage of intellectual development involving ratio/proportional reasoning, conservation principles, and logico-deductive reasoning (Piaget 1958, Herron 1975, 1978). Note his differing use of the word, “formal,” and the dichotomy between this kind of thinking, formal thought, related to the measurements of tangible substances and materials studied in chemistry, and the kind of thinking related to ideas found in the formal mathematics taught in mathematics classes. At UNH I encountered students strong in formal-math who did poorly in chemistry, and I found students poor in formal-math who then excelled in chemistry, indicating two distinct domains however we choose to label them.

Students’ experience with the mathematics curriculum in the middle and high-school years has been principally with formal-math. When I initially tried to introduce chem-math to the algebra teachers at my high school in an effort to integrate our courses, one teacher told me, “Oh, I could never pass your class! Chemistry is too hard; I couldn’t do it!” Here was an exemplary teacher who had taught many of my chemistry students. She was highly-respected and her students who arrived in my classes had learned their formal-math very well. They were easily able to transition over in chem-math when I introduced it because I connected the formal-math skills they had learned to their applications to chemistry. But when I began teaching chemistry recitations in college, I encountered many students who did not have a strong foundation in formal-math; they viewed its use in chemistry with trepidation and as a stumbling block. I heard some of my recitation students say they were in life-science majors to avoid the mathematics found in chemistry and physics. One resolute student emphatically told me “I hate mathematics but I love science!”

We cannot expect students to have learned chem-math from their mathematics teachers, who typically teach formal-math using no units and no applications in the physical sciences. Therefore, we must provide our students with requisite instruction in the mathematics techniques they will need, show them how and why these are used, and provide sufficient time for them to practice so their confidence can be built up. They will then find they can do this simple mathematics and thus it is possible learn the chemistry. In his paper, Eric will speak of the importance of students memorizing their fundamental mathematics skills. If they cannot quickly rearrange an equation to solve for a denominator, they may make mistakes that drive them to think what one student told me, “It’s impossible to learn chemistry because it’s all math!”

Another student articulated the following as a viable strategy for her: “I think it is okay to first memorize, until you are comfortable with the equations, and then learn where it comes from.” However, this only works for a mature learner who is not unduly intimidated. And it presents the controversy regarding having students drill on practice exercises without understanding them. Should we work to ensure that understanding precedes their introduction into problem-solving, or will understanding follow as a result of reconciling the cognitive dissonance resident in problem-solving? More research is needed since situations are different, students are different, the purposes of courses are different, and the chemistry contexts of various problems differ. Each instructor must do what he has found works best with his students. For some recent research on this issue, see Jo Boaler’s Ted Talk:  https://www.youtube.com/watch?v=3icoSeGqQtY

Our students (or our mathematics colleagues) might ask, “What does solving for a term in the denominator have to do with chemistry?” It occurs frequently in proportional-reasoning situations and chemical calculations are rife with ratios and proportions. Chemistry is the study of the composition and properties of matter and the changes it undergoes (my definition). If students associate the mathematics they will need to use with the entirety of chemistry content, and can’t rearrange an algebraic equation, it’s our responsibility to disentangle the two before they will thoughtfully attend to our subject. We cannot merely tell and show students what we want them to learn about the mathematics they will use; we need to carefully present chem-math to them as they do chemistry, and give them practice in small group-learning situations where they can collaborate with their peers and seek immediate clarification from the instructor when necessary.

In my early collaboration with a high-school algebra teacher, I began to develop a series of Chem-Math Units that spelled out and described the specific mathematics skills that comprise chem-math. As this project evolved, we presented them in several professional-development talks and workshops and elicited feedback from chemistry and physics/physical-science instructors. A Power-Point presentation of this material can be found in the Supplementary Section of the ConfChem.

THE RECITATION FORMAT

For underprepared and anxiety-ridden students, chem-math instruction cannot be done in a large lecture class or online, since many of these students have not yet acquired sufficient independent study-skills. They need the “friendly,” small-class environment with which they are familiar from HS. And I have found that often the students who most need the chem-math help will not reach out for it; they must be identified by an appropriate diagnostic instrument. Much research has focused on “mathematics” itself as a reliable predictor of chemistry success, so institutions typically use a test of formal-math (Sadler & Tai 2007). But since it does not separate out chem-math it potentially misplaces two important segments of the population; those who don’t need the chem-math recitation and those not identified who do need it. Thus, an effective instrument must be written and evaluated. Some institutions have used a combination of HS class-rank with results from the TOLT or GALT test (q.v.) that do measure proportional-reasoning. Such a recitation class needs to be integrated as a required part of the course for those identified students and appropriately-described, so as to not be perceived as onerous but as a way to help them succeed. Hence, my preference has been to describe it as an “intervention” rather than a “remediation.” This accords with the definition of chem-math as a collection of mathematics skills that the students may already have, but have not been shown how to use in the physical-sciences. (Thus we will also describe these students as “underprepared,” rather than “at-risk.”) This class needs to be led by a pedagogical content expert and not a graduate student who may not have any experience or expertise unless he or she is specifically trained, nor even be interested in teaching. A successful model can be found in the University of Colorado Learning Assistant Program:  https://laprogram.colorado.edu/

SEMANTICS

Notice the distinction I made earlier in using the word, formal. I have been forced to use it in two contexts with two different meanings and thus must clarify it to prevent confusion. Formal-math is not proportional-reasoning, whereas the use of “formal” by Piaget focuses on proportional-reasoning. Part of the Chem-Math Project pedagogy consists of anticipating such misunderstandings of important words and symbols that are distinctions found in chemistry. These can act as barriers to understanding in laboratory, study, and homework. For instance, when I use the word, substance, I mean “one kind of matter; an element or a compound.” But students will not understand such specificity unless the instructor overtly makes this distinction and consistently reinforces it, given the widespread colloquial use of the term. I asked students in recitation to share their definitions of density, an important concept they should understand from previous physical-science experience, on their portable white-boards; see Figure 1. Why did the student write this? In elementary and middle-school we teach that “matter is anything that has mass and takes up space.” But it’s a circular definition; what is “mass?” He has used the only part of the definition that he does understand, and thus his understanding of the mathematics of density as a ratio is impugned.

Figure 1 – density definition

I also have students examine and discuss the Greek alphabet before we use such symbols in class, to see similarities and differences with some of the English letters. We may use the symbol “C,” for a constant in a variety of contexts, and must differentiate it from “c,” the speed of light when used as such. In one particular laboratory experiment at UNH involving indigo violet to investigate a rate of reaction, the freshmen were seeing “k” used in three different ways. It was being used as a rate-constant, a Beer’s law constant, and an equilibrium constant but the TAs were not pointing this out. As a result the students were having great difficulty with the calculations.

PHYSICAL-SCIENCE FUNDAMENTALS

To use and understand chem-math, students need to have their physical-science fundamentals in place but if they do not, this is where the chemistry recitation is vital. We must put cow-magnets in their hands to “feel” Coulomb’s law in its force formulation, and not the prevalent textbook energy formulation that they cannot feel. Once this is done, a review of inverse and inverse-square variation can be explored (fig 2). Buoyancy is another important concept involving the mathematics of the gas-laws. When such basics have not been previously studied, an entire separate course may be needed to fill them in, either before or concurrent with general chemistry. In his paper, Eric will describe it as “Computation for the Sciences,” whereas I like to call it a “Q-course,” for a quantitative-reasoning course as I read it enacted at the University of Connecticut: http://geoc.uconn.edu/quantitative-competency/ . In such a course, much measurement will be done and dimensional-analysis practiced (aka unit-analysis, the factor-label method, the fence-pole method). Students will learn how to operate on the unit designation as an algebraic symbol, investigate unitary-rates, come to understand “scaling,” and do many conversions of measurements.

Figure 2 – Coulomb’s law graph

THE LANGUAGE OF MATHEMATICS INSTRUCTION

Another important pedagogical point is if we are to harness what students have learned in their formal mathematics classes, we must use the language that they have heard used. My high-school collaborator in the Project made this very clear to me, which led me to study National Council of Teachers of Mathematics (NCTM) materials. He said it’s not just “solve” as an exhortation, but “combine like terms, distribute terms, and solve,” Miraculously, I found that I had much greater success when I used this verbiage. I could not get students to understand how to approach mixture-problems when I told them we have “two equations with two unknowns,” or “simultaneous equations.” But when I used “a system of equations,” they knew exactly what to do because that was how it had been taught in their high school mathematics classes. As a symbolic example where a clarification is needed, I had always been using y = mx + b, then found newer HS mathematics books were using y = a + bx. One can see the confusion that may result for some students without clarifying that two different formulations exist, and “b” represents completely different components of linear regression. Note that the NCTM now emphasizes the use of “rate,” as in “rate-of-change,” rather than “slope” to better prepare students for the study of calculus. To provide more practice with the slope/intercept formula, I developed a worksheet in which students draw and label thermometers on paper and use these to develop the conversion formulae for Fahrenheit/Celsius, then continue on to do several fairly sophisticated operations (fig 3).

Figure 3a,b,c  – thermometer worksheet

THE AFFECTIVE DOMAIN

Students can confuse almost anything when they are struggling to survive. I call this Educational Darwinism, and we need to be vigilant in detecting this phenomenon. They mis-use technology when they employ calculator “E-notation.” They unnecessarily combine it with standard notation by using “10•E(exponent)” and are off by an order of magnitude. I implore them to translate all calculator display to standard-notation when doing work on paper. Some will write out “E” as “e,” and need to be reminded that it stands for the base of the natural logarithm. I needn’t discuss how they must to be re-taught the law-of-logarithms and do powers-of-ten calculations by hand rather than on an eighty-dollar calculator! Our underprepared students can become completely demoralized by an accumulation of such confusions, and they may subsequently decide that they cannot learn chemistry when in fact they can. Then Educational Darwinism will kick in. Students with this frame of mind have given up any hope of understanding chemistry but are desperate to pass the course in order to not lose the credits or have to repeat the course. They will do whatever they believe it takes to survive the course and earn at least a D^- and these will not be logical or rational actions. As we anticipate and clean up such confusions, we build up their confidence that they can learn chem-math, and they will then be more amenable to accessing the conceptual aspects of chemistry. We need to provide them with alternative approaches to its study rather than the survival modes they may have elected, and move our chem-math instruction into the affective domain. These future STEM students most need this supplemental coursework. To illustrate the concept of ED, here are some exact student quotes:                                      

 “I still hate math and will always be out just for survival.”                                                                        
“I personally picture myself surviving through chemistry, probably not getting much out of it as the workload that is put forth within the class and lab draws too much attention away from other classes” (another important issue). “I just want the grade.”                                                                                       
This next student is “asking” for appropriate chem-math instruction:                                                                         
“– students would be far more comfortable if they could at least complete chemistry calculations then learn what it actually means. Sadly, survival basis will be my way of getting through chemistry.”

We will also find students following the Principle of Least Cognitive Effort (Herron 1996), doing only the minimum they think will contribute to getting the grade they seek, usually a required C^- to remain in their elected majors. They will spend precious energy on other subjects they find easier to confront. They will not seek understanding of the chem-math but only “correct answers.” As we hear physics teachers say, students will “look for an equation that uses the variables found in a problem and then plug-n-chug.” In his paper, Eric will argue that choosing the correct equation to use IS half the battle. However, with thorough chem-math instruction in recitation, they should also achieve a conceptual understanding.

RESEARCH IN MATHEMATICS INSTRUCTION

Why have my students always had trouble with empirical-formula/molecular-formula calculations? I have found they refuse to put the smallest number of moles in the denominator. Why is this? My study of NCTM materials subsequently informed me that a misunderstanding of fractions can persist from the elementary years into upper-level mathematics courses (NCTM 2002). Students simply do not like improper fractions! They want all fractions to have a value of less than one, even after you have consistently stressed to “divide all moles by the lowest number of moles in the compound.” In the second semester, we also find students having trouble with complex-fractions. They have forgotten how to “invert and multiply.” So a variety of small issues such as this sum up into “chemistry is all math! And since I can’t do math, I can’t do chemistry!” as one student despondently stated.

Students can come in to our classes with much trepidation but still want to matriculate with a STEM major. One student told me, “I love science but I hate chemistry!” Does this seem enigmatic? Here are some other quotes from students in recitation that have informed my work:

 “I make stupid mistakes like x-1 instead of 1-x” (in mixture-problems).                                                       
“I just need help applying my math to ‘real life’ chemistry problems and solving.”                      
“It helps when they (chem-teacher) write a lot of verbal notes on the board rather than just mathematical equations –“                                                                                                                     
 “I think a lot of my problems stem from lack of confidence when it comes to mathematics. I also tend to get really frustrated and muddled and then I get upset and more frustrated because I get confused and I stop being able to function mathematically.”                                                             
“The most useful information was when we learned the underlying meaning for the problems we were trying to solve.”                                                                                                                        
 “Although I hate math and thus consequently chemistry due to its high math content it was good seeing the reasoning behind the equations and that helped.”     

Here are some examples of how students actually understand this formal-math/chem-math distinction and will state an ersatz definition of chem-math: 

“I don’t think I’ll ever understand chem. I don’t get the concepts and can’t put math and chem together. It is frustrating because math is one of my strong points but mixed with chem I get lost.”                                                                                                                                                 
“I feel that if math was not taught as an abstract subject from the beginning, I might know how to apply it to chemistry problems.”                                                                                                                
“Math comes much easier to me with science to back it up or apply to.”                                          
“Chemistry is in-depth and the problems are chem-math, so the teacher has to provoke a good understanding of not only the math processes, but why we are doing the math to begin with.”    

THE USE OF PROOFS AND DERIVATIONS                                                                                   

One strategy I have found helpful for building mathematical confidence is to give students proofs and derivations involving chemistry content. However, it is not done independently; it is done in three-person groups, on portable white-boards, and in a small recitation class where the instructor can easily monitor every group to get them through the confusion and frustration that can arise. I have my students verify diatomic gases using Avogadro’s hypothesis and the formation equations for HCl, H₂O, and NH₃. They calculate the universal gas-constant, R, using the combined gas-law and the conditions for an ideal gas at STP. They derive the gas-density formula from the ideal gas-equation and the density formula—but they have to be reminded that “mol” is the unit found on the variable, “n.” They compare the tabulated densities of several gases with their molar-masses to validate the relative constancy of molar-volume. They convert electron-volts per atom to kJ per mole. They derive the Henderson-Hasselbalch equation from the Ka equilibrium expression, removing much of its “mystery.” They see why the acid/base fraction can be written in either inversion, since following the law of exponents the log of that term can be either added or subtracted to the kPa to calculate the pH of the buffer. And I have found they enjoy revisiting the quadratic formula when solving for the pH of weak acids and bases. By providing the time and support for students to work out such material accords them the opportunity to see that they actually can do sophisticated mathematics—not as an abstraction but in service of the chemistry they are studying. Figure 4 shows the exuberance (hence growing confidence) manifest by one mathematically-challenged young lady when her group was the first to convert R in liter-atmosphere units, used in the first semester, to R in energy units used in the second. Such a conversion allowed students to see the “similarity” in their values, 8.31 vs. 0.0821 (or 8.21•10^-2) depending upon the units, and not then confuse them when used in their respective contexts.

Figure 4 – “Bam!”

MULTIPLE REPRESENTAT IONS

The NCTM standards (Curriculum and Evaluation Standards for School Mathematics; 1989) stress the use of multiple representations: numeric, symbolic, graphical, and verbal, to develop better conceptual understanding. Given that students are slaves to their graphing calculators, I have seen how many have forgotten what a graph is supposed to represent. Thus I have developed a variety of hand-graphs for students to draw in recitation. When I assign them, it is always interesting to find that students rarely complain and actually enjoy the tactile aspect of using a straight-edge and a sharpened pencil (and colored ones) to make a visually-attractive product, properly labeled and titled. They do this individually but then chose one to rewrite on their group white-board for display to their classmates. (See Figure 2 for the example of work we did to understand Coulomb’s law and to compare the force and energy formulations side-by-side.)

One exercise I designed involved using two graphs, one that required learning to use a French curve, to create a third. They each graph the vapor-pressure of water vs. temperature from a table they find themselves from paper materials or online, and I provide them the barometric pressure at several altitudes for another graph. From these they create a third graph of boiling-point of water vs. altitude. I give them the altitude of several cities to determine the boiling point at those locations, and they then compare their individual values for accuracy. Thus, they saw why it is necessary for a chef to understand geographically-varying cooking times based upon the not-necessarily single boiling-point of water.

COMMON MECHANICS

Rather than lecturing, my current procedure is to hand out one of a series of “New Basic Data Sheets” I have written for students to read silently in class, which I then verbally reinforce as we work on chem-math. I’m sure most readers instruct their students in procedures they favor, but see the reasoning behind these particular steps. In the worksheets I have studied from our UNH Math Placement Test, I saw the wide variety of shoddy, ineffectual procedures students have brought from high school (fig 5a). I work hard in the beginning to get all students to abandon their various bad habits and move to this one sequence of steps. Some do come with good practices as you can see (fig 5b). Using one consistent method cleans up their thinking and allows me to quickly inspect their work as I pass by their tables and intercede as necessary. I have also included an example of what using this procedure should look like (fig 6). It also provides an example from my previous discussion of using proofs and derivations with gas-law work.

Figure 5a,b  – sloppy and neat work

Figure 6 – good example

New Basic Data Sheet #0:     

This is Chem-Math

The Three Rules:

1. Every number represents a measurement of a physical quantity, and must have a unit written with it.

The only exception is a ratio, where the units have been factored out. Examples are specific gravity and specific heat, where the density and heat capacity of a substance are compared to that of water.

2. Every measurement must be correctly expressed. Don’t drop those zeros; the decimal places convey information about the measuring instrument that was used to obtain it.
3. Every calculation must be correctly rounded; your calculator will spew forth many unnecessary digits.

Follow the rules of significant figures (aka significant digits).

A useful procedure to follow:

Your thought process is more important than your answer. If your problem setup is correct, you can easily correct an error in the arithmetic. You must show all of your work clearly so an examination of your mechanics (your solution process) can occur.

1. The result from your calculator follows the “equals” sign. It should have extra significant figures so that you can use it in a future calculation without introducing rounding error. An arrow then says you are rounding to the correct number of significant figures, with your answer and units boxed in for clarity.
2. Use a dot or parentheses for multiplication. An “x” denotes an unknown variable without units in an algebra problem. From the formula that appertains you will already have a variable symbol, like P, V, T, t, and m (for pressure, volume, temperature, time, and mass respectively). However, you may use “x grams =,” or “x liters =” in stoichiometry problems.
3. Don’t use slanted lines for divisors in your work; break that habit. Use a horizontal divisor line so that you can see exactly what you divide by and what you multiply. (A slanted line is acceptable in print, such as when writing laboratory reports, to avoid having to displace a whole line below it. Your equation editor can do this for you.)
4. For a number less than 1, without a digit to the left of the decimal, always include a zero to clearly demarcate the decimal.
5. Collect all of the mathematical values found in an exercise or problem, with their units, in an inventory list before beginning the mathematics.
6. Rearrange your formula to solve for the unknown variable FIRST, before entering numbers with units. Use your inventory to identify the unknown variable and indicate it with “ = ? ”
7. Group similar variables in your formula together.
8. Then, as you have been previously taught, “distribute terms, combine terms, and solve,” and show all of your work neatly.
9. Use pencil or erasable ink instead of crossing stuff out and making a mess.

Regarding 3) above, it may seem like a minor point, but I find many cases where a student has made an arithmetic error due to a slanted divisor-line that had become almost vertical. See Figure 7 for an example. (The use of the fence-pole method of unit-analysis places a long horizontal divisor first, on which all of the conversion-factors are then sequentially placed, so it’s not possible for the line to slant.)

Figure 7 – slanted divisors

THE USE OF ON-LINE HOMEWORK

At UNH we experimented with nine online homework systems. We employed ALEKS, an intelligent tutor, both as an optional but for-credit HW and in other years it was required. It presents both chem-math and its formal-math prerequisites. https://www.aleks.com/ The more mature students liked it because it is rigorous; they were diligent in using it and saw a positive result. Its principle advantage is that the work each student does is individualized, based upon periodic assessments of readiness to learn specific material. Thus, it has internal quality-control. Students cannot study for what they are not ready, and they must perform to show that they are. They are not credited for time-on-task but for competence. The less mature students were impatient and intolerant of the periodic assessments and did not derive the benefit to accrue. Some students were exasperated with ALEKS at first, but came around when they began to see results in better skill and understanding. Again, this all depends upon the role of the affective domain and students’ ability to persevere in the face of adversity.

Recently I have begun to use Khan Academy for online HW and help. https://www.khanacademy.org/science/chemistry Many of my students like it because they can choose to study what they need when they need it, and they find the human element in its presentation more engaging. The short mathematics, chem-math, and conceptual pieces are helpful for students with short attention spans. But unlike other systems, there is no student interaction; use is essentially passive unless it leads to work on instructor-assigned accompanying HW. Accountability for HW credit is solely obtained by the time spent viewing Sal Khan’s vignettes. Both of these online resources originated as mathematics resources so chem-math is, in fact, embedded in their chemistry tutorials. And I would be remiss if I did not cite the excellent chem-math tutorials of my ConfChem co-moderator, Eric Nelson at http://chemreview.net/

As a last example, I provide four examples of work that show what can be accomplished by underprepared students when their confidence is properly built up in an effective small-class setting. Figure 8 was the result of a mini-experiment they performed in recitation. Each group massed out one reagent and calculated what was needed of the other to freeze the reaction beaker to a block of wood. Groups who finished early then estimated the enthalpy change. For problem-solving practice I tried to provide realistic problems whenever possible. One week after a hike, I passed out power bars to see their labels. I told them I ate one when I climbed a 4000-footer that weekend (although they had to go online to find the actual vertical distance). I gave them my weight and asked them to use Hess’ law to calculate if one bar would provide sufficient energy to elevate my weight that height. We were all surprised to find it to be true. Another week I had heard on NPR that each gallon of gasoline combusted in a car produces twenty pounds of CO₂, so I asked each class to verify this. The first example below shows how one group got off the rails and reverted to previous bad habits I was striving to rectify. The work is correct but it is very hard to follow and we will not allow this (fig 9a). Two more examples demonstrate how successful working groups individualized their approaches. One used good chem-math technique and decided to refute the NPR statement (fig 9b). The third used the fence-pole method and understood that the NPR statement was valid within an order of magnitude (fig 9c). When groups finished early, I often gave them an additional chem-math task so that students at all levels of ability would be challenged. This last group went ahead and calculated the volume of CO₂ their mass would represent. See the humorous remarks embedded in their work, as well as the varying ways students can learn to approach a problem once they understand the mathematics.

Figure 8 – stoi/thermochem experiment

Figure 9a,b,c – CO₂ problem

REFERENCES

Bodner, G. (1987). The Role of Algorithms in Teaching Problem Solving; Symposium on Algorithms and Problem Solving. J. Chem. Educ. 64(6): 513-514.

Bodner, G. (1994). On Solving Chemistry Problems. CHED Newsletter, Spring. ACS.

Herron, J. (1996). The Chemistry Classroom; Formulas for Successful Teaching. American Chemical Society, Washington, D.C.

Herron, J. (1975). Piaget for Chemists. J. Chem. Educ. 52(3): 146-150.

Herron, J. (1978). Piaget in the Classroom. J. Chem. Educ. 55(3): 165-170.

NCTM (2002). Making Sense of Fractions, Ratios, and Proportions. 2002 Yearbook.

Piaget, J. & Inhelder, B. (1958). The Growth of Logical Thinking from Childhood to Adolescence. Basic Books, Harper-Collins, New York, NY.

Sadler, P.M. & Tai, R. (2007). The Two High-School Pillars Supporting College Science. Science 27(317): 457-458.