In a recent contribution to ChemEd X "Stoichiometry is Easy", the author states that he has "vacillated over the years between using an algorithmic method, and an inquirybased approach to teaching stoichiometry." I would like to suggest that there is another approach to mastering stoichiometry and that it should precede the algorithmic one: it is the conceptual approach based on a particle model to represent the species involved in chemical reactions.
Most teachers, and I include among them those responsible for writing the items for the highstakes tests the author describes, have a tendency to equate quantitative fluency with a genuine understanding of the underlying processes. After all, if students can correctly determine the mass of a product to be expected or which is the limiting reactant, then certainly they must know what is going on during the reaction, right? Unfortunately, as researchers such as Craig Bowen and Diane Bunce^{1} have shown, typical quantitative test items don't probe whether students have persistent naïve conceptions about chemical reactions and processes.
The 5step algorithm "Stoichiometry is Easy" to the tune of "Hark the Herald Angels Sing" described by the author is catchy and appears to be effective. But l believe that learning any algorithm is most effective when it follows an introduction stressing conceptual understanding. As Dudley Herron^{2} wrote about algorithms:
For similar reasons, I teach efficient algorithms for such routine tasks as balancing chemical equations (after I am convinced that the student knows what a balanced equation is and why we want one) and encourage students to use them. I emphasize the point that the algorithm should be sensible (i.e., we know what the product of the procedure means) but should not require them to think any more than necessary. Indeed, the purpose of an algorithm is to reduce the load on working memory and save time. [My emphasis]
In the approach advocated by Modeling Instruction in High School Chemistry^{3}, students use particle diagrams depicting the reaction mixture before and after a reaction has occurred to make the point that the balanced chemical equation relates numbers of particles, not mass or volume, the quantities we typically use to measure how much stuff is involved. The use of a BCA (beforechangeafter) table, similar to the ICE table used in a quantitative treatment of equilibrium mixtures, helps students connect the particle diagrams to a more convenient way of representing the ratio of reacting species. Consider the example below:
2 H_{2}S  +  3 O_{2}  →  2 SO_{2}  +  2 H_{2}O  
Before  4  xs  0  0  
Change  4  6  +4  +4  


After  0  xs  4  4 
Note: xs is shorthand for "excess"
The first examples involve calculations that can be done in one's head and easily related to particle diagrams. By this time students have learned that moles are simply weighable amounts of given species, so they readily accept that the ratios of coefficients relate to numbers of particles. What happens if the information about the situation is given in terms of mass? One has to apply techniques learned in an earlier unit to convert the givens to moles, chemists' counting unit. If the desired quantity is mass (or volume), then that calculation is done on the side using the molar mass or molar volume as the required conversion factor.
The particle diagrams are especially useful when dealing with limiting reactant problems. Consider the reaction in which water is produced when hydrogen and oxygen gas react.
2 H_{2} + O_{2} → 2 H_{2}O
The reactant mixture might look like the box at left. Students are encouraged to cross out reacting species and draw in product species until the reaction can no longer proceed. They should end up drawing a product mixture as shown in the "after" box.
Before  After 

The corresponding BCA table appears below:
2 H_{2}  +  O_{2}  →  2 H_{2}O  
Before  4  4  0  
Change  4  2  +4  


After  0  2  4 
Once students recognize the connection between the numbers in the table and the particle diagrams for these "obvious" examples, it's a straightforward step to examining cases where one might have to guess which reactant is consumed first. Consider the reaction between aluminum and iodine to produce aluminum iodide.
2 Al  +  3 I_{2}  →  AlI_{3}  
B  0.50  0.72  0  
C  0.50  0.75??  +0.50  


A  0  ??? 
Here, the typical student guess  that the reactant with the fewest number of moles available is limiting  leads to an obvious problem when one multiplies the 0.50 moles of Al by the 3/2 ratio given in the balanced equation. Students reassess and assume that all of the iodine must be consumed in the reaction and complete the table correctly as follows.
2 Al  +  3 I_{2}  →  AlI_{3}  
B  0.50  0.72  0  
C  0.48  0.72  +0.48  


A  0.02  0  0.48 
This approach not only shows the correct number of moles of aluminum iodide produced, but also how many moles of the excess reactant remain.
One might argue, "Well, isn't using the table just a different algorithm?" This might be true if it weren't for the fact that the instructor explicitly connects the values that populate the table with particle diagrams when the students first encounter its use. When a serious effort is made to make the "steps" in a procedure sensible to students, they are more likely to understand what they are doing as they are doing it.
References
 Bowen, C. and Bunce, D., "Testing for Conceptual Understanding in General Chemistry" The Chemical Educator, Vol. 2, No 2, 1997
 Herron, J. Dudley. The Chemistry Classroom, The American Chemical Society, 1996
 American Modeling Teachers Association, http://modelinginstruction.org
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Comments 2
Exactly the introduction I would encourage
Larry,
Thanks so much for your comment and for the excellent explanation of using modeling to introduce the topics of what a balanced equation means and how stoichiometry relates to the equation. In my article, I was focusing on ONLY the stoichiometry lessons, but I had stated that in the preparation lessons to that topic, we spend quite a bit of time with the meaning of a chemical equaiton:
Balancing Equations – we spend time describing the meaning of a chemical equation, modeling the particulate nature of what is going on, relate that to lab experiences. . .
Because my article focused on the stoichiometry and math portions, I did not include an exhaustive explanation or set of worksheets, labs, and classroom exercises that I use when we discuss what a balanced equation means, and how to interpret that both on a molecular level and on a macroscopic level. Your blog post does a very nice job of showing exactly the type of thing that my classes would do during that portion of the class. Getting students to think in terms of particles is essential for real understanding in chemistry  if you ask my students, they will tell you there is hardly a day we do not stop whatever else we are discussing and take time out to model (sometimes getting out of our seats and pretending we are the molecules) or draw "what this looks like to a molecule."
The first session on stoichiometry includes in lecture and Socratc discussion the same type of "ICE"like table (and pictures) that you illustrated. But again, I did not give all of the exact lessons, but only the principal worksheets, labs, and activities. I am glad that you presented this to give others an idea of how modeling works and how to incorporate it into this topic.
Please rest assured that I know the importance of the difference between simple mathematical fluency and real conceptualization of chemical phenomena. Drawing and describing the molecular basis of almost every topic in chemistry was a regular part of my assessments. I completely agree that if students can ONLY do the math, that is peachy, but unimportant  I expect them to do the chemistry. I cannot say how many times I repeated that idea, so much that many students could probably even quote me: Let's understand what we are doing, let's model the system. First we will do the chemistry, THEN we will work on the math. Do not get ahead of yourselves. I know you can all work a calculator, but if you do not have the chemistry work accompllished, then you haven't done the work for my class. You might want to go down the hall and see Mrs. Cadillli or Mr. Burke, if all you want to do is the math. You can ask them to grade your papers and tests. But if you want a grade in this class, doing the math is not the point . . . let's do the chemisry first!
I would have loved to have the modeling workshops available in Southern California at a time I could participate while I was teaching  then again, I don't know if my students would have put up with any more modeling. I cannot think of anything that makes chemistry more accessible than taking the time to "look" at what is happening on a molecular level. I really appreciate the extra input. May I add a comment to the main article referencing your blog post?
D.
BCE vs Stoichiometry
When using the BCE model do you turn to the traditional stoichiometry model to finish up the unit? I am planning on adopting the BCE model, and am interested in others' experiences with this.