Chemical Education Today
Reports from Other Journals
Chemistry Problem-Solving: Symbol, Macro, Micro, and Process Aspects by William R. Robinson
We know many of our students have difficulty with problems that combine the fundamental concepts of chemical stoichiometry: the particle description of atoms and molecules, the mole concept, and/or chemical change. It has been shown that many students develop algorithmic techniques to solve such problems yet never develop an understanding of the scientific concepts behind those techniques (1–4). Teachers in introductory courses often provide algorithmic formulas for solving problems rather than requiring students to use reasoning combined with an understanding of the concepts underlying the problems. For example, Thompson (5) reports the following rhyme for converting percent composition to an empirical formula: “Percent to mass/Mass to mole/Divide by small/Multiply ‘til whole”. Some texts use the equation M1V1 = M2V2 (where M represents molar concentration and V represents solution volume) to handle calculations in titrations. Such algorithms, which may seem helpful at first glance, actually promote an approach that hinders meaningful learning and true understanding. Students use these crutches without understanding what they mean and their limits. The empirical formula algorithm, for example, cannot handle problems in which the quantity of one component is given in grams and the quantity of the second component as number of atoms. The equation M1V1 = M2V2 cannot be used with a titration that involves polyprotic acids, but many students do not recognize this and cannot understand why it does not work. Rote, algorithmic-type teaching and learning hinders chemistry students’ development of conceptual understanding and higher-level thinking skills. Because they lack understanding of concepts in chemistry and the required mathematical skills, many students are unable to draw the proper relations among the various concepts and the quantitative representations of these concepts (6). This prevents them from coming up with a well-reasoned solution to the quantitative problem at hand (2, 7–10). To obtain the “correct solution” these students memorize a variety of algorithmic techniques rather than attacking the problem using the basic concepts (1–4). A lack of understanding of introductory concepts hinders understanding of subsequent topics and leads to even more reliance on memorized techniques. Students who are new to chemistry face problems connecting symbols that describe chemical processes with the quantitative information that a formula provides; many do not recognize the reasons for the stoichiometric relationship between atoms and molecules, reactants and products, and so forth (11). Johnstone (12) suggests that students must master three levels of chemistry concepts in order to avoid these problems. The first is the macroscopic level, which deals with visible phenomena (such as how much solid salt dissolves in water). The second is the submicroscopic level, which deals 978
with particles (disruption of [The benefit] is most the ionic lattice by water molecules). The third level is the noticeable in students symbolic level, which represents matter in terms of studying at the low chemical formulas and equa+ tions [NaCl(s) → Na (aq) + and intermediate Cl–(aq)]. Johnstone believes that emphasis of the three mathematics levels. levels and the associations among them will facilitate transformations students need to perform. In particular, students must first thoroughly understand how to convert a symbol into the meaningful information it represents. Only then can they cope with the quantitative computation. Multidimensional Analysis The study by Dori and Hameiri (13), “Multidimensional Analysis System for Quantitative Chemistry Problems: Symbol, Macro, Micro, and Process Aspects”, describes an approach to classifying and analyzing introductory stoichiometry problems. [Editor’s Note: A related article by Dori and others begins on p 1084.] The Multidimensional Analysis System (MAS) provides a format that students can use to classify introductory stoichiometry problems based on three transformation levels (symbol ↔ macro, symbol ↔ micro, and symbol ↔ process) and the complexity of the transformations required within each level. In this column we do not have time to fully describe the MAS process except to point out that students who use the process spend part of their problem-solving time analyzing the specific transformations and their complexity in the problems they are solving. These activities are intended to move students from an algorithmic learning mode toward a conceptual mode in which they can evaluate and analyze the various concepts they have studied as they apply them to the problems at hand. Dori and Hameiri summarize the MAS technique as follows: MAS is a method of systematically focusing student and teacher attention on important concepts concerning the mole. Unlike rote learning that applies only algorithms as to how to plug the right number in the right equation without deep understanding of the underlying concepts, MAS does not provide a specific solution. Conversely, it fosters conceptual thinking while reflecting on the four levels of chemistry understanding to classify the problem, which then leads to one or more ways to solve the problem at hand. One cannot correctly classify the problem without the deep understanding of the symbol, macro, micro, and process levels and the transformations among them (13).
Journal of Chemical Education • Vol. 80 No. 9 September 2003 • JChemEd.chem.wisc.edu
Chemical Education Today
Reports from Other Journals Table 1. Comparison of Week 9 Post-Test Mean Scores by Mathematics Level Control
Experiment
Mathematics Level
N
Mean
N
Mean
t Value
p
Mathematics II (3 units)
21
25.90
20
63.35
-7.2237
< .0001
Mathematics I (4 units)
18
37.03
36
65.14
-4.7293
< .0001
Honors (5 units)
21
62.98
20
77.79
-2.3853
< .0500
MAS combines transformation with complexity by assigning a number of complexity levels to each of the three transformations students commonly encounter in introductory stoichiometry problems. Transformations
Macro ↔ Symbol Transformations Transformations are required between the macroscopic (mass) to the symbol (for example, chemical formula). Converting among different measurement units—from grams to milligrams or kilograms—can add additional complexity. The level of complexity increases in the order: no mass involved, mass in grams, mass in units other than grams, and mass comparisons as in empirical formula determinations. Micro ↔ Symbol Transformations Transformations are between the microscopic (particulate) level of a substance and the symbolic level for the substance. The complexity within this level increases in the order: no particles involved, particles in moles, both particles and Avogadro’s number involved, and comparison of particles. Process ↔ Symbol Transformations These transformations are between chemical processes and the set of symbols in a chemical equation that specify a process. The order of increasing complexity is: no process involved, process involved with chemical equation provided, process involved with no equation provided, process and limiting reagent involved. The Study The research population involved in the study of MAS consisted of 241 students in 10 classes from six high schools in Jewish and Arab sectors from both rural and urban areas in Israel. This population was divided into an experimental group and a control group based on the teachers’ willingness to teach according to the MAS. Both experimental and control classes used the same syllabus. Quantitative chemistry was taught for about 20 hours over a period of 9 weeks. All the students took the same set of examinations, administered every three weeks. The difference between the experimental and control groups was the teaching method. The six classes of the experimental group learned quantitative chemistry using the Mole Environment studyware (14) and analyzed prob980
lems from the studyware using the MAS framework. The control group students studied in the traditional method without any intervention. The students in the control group were administered the same battery of tests as the experimental group students. In the control setting teachers usually focused on demonstrating specific examples of solutions to problems of various types. Rarely did they take enough time, if any at all, to discuss the concepts involved in the mole-related problems. Discussion The results indicate that, in general, the achievements of both groups increased over time. However, the gap between the experimental group and the control group widened as the course progressed. After nine weeks of instruction with MAS, an analysis of covariance using the pre-test as a covariant showed the experimental group benefited significantly more from the MAS intervention compared to the control group. The effect is most noticeable in students studying at the low and intermediate mathematics levels. Students at the lower mathematical levels showed the greatest difference in scores on the last test in the series of three. Table 1 reports these differences. A complicating factor in interpreting this study is the use of specially designed Mole Environment studyware (14) that assisted the experimental group in working with the Multidimensional Analysis System. The studyware included problems that required various combinations of transformation and complexity levels. Students progressed to higher transformation levels at their own pace. For example, a student who completed the symbol ↔ macro level at Complexity Level 2 was able to solve problems that contained the concepts of mole, molar mass, and measurement units, but was not presented with problems containing Avogadro’s number, particles, and chemical processes. At a more advanced stage, a student who completed the symbol ↔ process level at Complexity Level 3 was able to solve problems that involved a limiting reactant in a process for which no equation was provided. Each incorrect response provided the student with feedback explaining why the answer was wrong. Although the use of studyware is an intervening factor, the interaction with the student is relatively simple and provides immediate feedback and gradual exposure of the problems according to the individual student’s pace. In the control groups, teachers spent little time, if any, discussing the concepts involved in mole-related problems.
Journal of Chemical Education • Vol. 80 No. 9 September 2003 • JChemEd.chem.wisc.edu
Chemical Education Today
Reports from Other Journals Instead they usually demonstrated specific examples of solutions to problems of various types. Essentially, these students were exposed to rote learning that applies algorithms that involve plugging the right number into the right equation without deep understanding of the underlying concepts. On the other hand, multidimensional analysis systematically focuses students’ attention on important concepts concerning the mole and encourages conceptual thinking about the four levels of chemistry understanding as students classify their problems. One set of examples of two students in the study solving the following multiple-choice question illustrates the difference in the way students in the two groups analyze problems. Question How many atoms are there in 0.75 mol of water? (a) 2.25 (b) 4.5 ⫻ 1023 (c) 13.5 ⫻ 1023 (the correct answer) (d) 18 ⫻ 1023 Explain your answer.
Response, Control Student To transfer from moles to atoms we must multiply by Avogadro’s number, so 6 ⫻ 1023 times 0.75 is 4.5 ⫻ 1023, so (b) is the answer. Response, Experimental Student I read the problem and realized that I have no process at all and I don’t need to deal with molar masses. I soon realized that I need to deal with particles; so I asked myself what particles do I need to consider. The problem asks about atoms but we are given water, which consists of molecules. This molecule is H2O, so it has 3
982
atoms: 2 H and 1 O. Having realized this, all I had to do is multiply 3 atoms by Avogadro’s number to get the number of atoms in one mole, then by 0.75 to get the final answer, (c).
Literature Cited 1. Beall, H.; Prescott, S. J. Chem. Educ. 1994, 71, 111–112. 2. Gabel, D. L. In International Handbook of Science Education; Fraser, B. J.; Tobin, K. J., Eds.; Kluwer Academic: Dordrecht, The Netherlands, 1998; pp 233–248. 3. Lythcott, J. J. Chem. Educ. 1990, 67, 248–252. 4. Wandersee, H.; Mintzes, J. J.; Novak, J. D. In Handbook of Research on Science Teaching and Learning, Gabel, D. L., Ed.; Macmillan: New York, 1994; pp 179–198. 5. Thompson, J. S. J. Chem. Educ. 1988, 65, 704–705. 6. Gabel, D. L.; Bunce, D. M. In Handbook of Research on Science Teaching and Learning; Gabel, D. L., Ed.; Macmillan: New York, 1994; pp 301–326. 7. Gabel, D. L.; Briner, D.; Haines, D. The Science Teacher 1992, 59, 58–63. 8. Gabel, D. L. J. Chem. Educ. 1993, 70, 193–194. 9. Garnett, P. J.; Garnett, P. J.; Hackling, M. E. Studies in Science Education 1995, 25, 69–95. 10. Noh, T. L.; Scharmann, L. C. J. Res. Sci. Teach. 1997, 34, 199– 217. 11. Koch, H. The Science Teacher 1995, 62, 36–39. 12. Johnstone, A. H. J. Comp. Assisted Learning 1991, 7, 75–83. 13. Dori, Y. J.; Hameiri M. J. Res. Sci. Teach. 2003, 40, 278–302. 14. Dori, Y. J.; Hameiri, M. Internat. J. Sci. Educ. 1998, 20, 317– 333.
William R. Robinson is in the Depar tment of Chemistry, Purdue University, West Lafayette, IN 47907;
[email protected].
Journal of Chemical Education • Vol. 80 No. 9 September 2003 • JChemEd.chem.wisc.edu
