Evaluating Students' Conceptual Understanding of Balanced

Research: Science and Education edited by

Chemical Education Research

Diane M. Bunce The Catholic University of America Washington, D.C. 20064

Evaluating Students’ Conceptual Understanding of Balanced Equations and Stoichiometric Ratios Using a Particulate Drawing

Amy J. Phelps Middle Tennessee State University Murfreesboro, TN 37132

Michael J. Sanger Department of Chemistry, Middle Tennessee State University, Murfreesboro, TN 37132

In 1987, Nurrenbern and Pickering (1) demonstrated that chemistry students who can solve mathematical problems often have more difficulty answering particulate-level conceptual problems covering the same topics. Since that time, several researchers have corroborated these results using the particulate drawings used by Nurrenbern and Pickering (2–5), while others have identified additional areas of student difficulty using other particulate pictures (6– 15). One of the particulate pictures used by Nurrenbern and Pickering, which appears in Figure 1, measures students’ conceptual understanding of balanced chemical equations. I have used this particulate picture, and pictures like it, in my introductory general chemistry courses when teaching students to balance chemical equations. These pictures are quite useful in teaching students the difference between subscripts and coefficients, identifying limiting and excess reagents, and in general are helpful in teaching the chemical concepts behind balanced equations. Most students eventually recognize that (c) is the best description of the chemical reaction shown in the picture, but over the years, several students have tried to convince me that (d) is also a correct description for this reaction. My response to them is that although (d) is a correct description of the system shown in the picture, (c) is the best description of the chemical reaction since we do not include unreacted species in the bal-

The reaction of element X ( ) with element Y represented in the following diagram.

( )

is

3X + 8Y → X3Y8

(b)

3X + 6Y → X3Y6

(c)

X + 2Y → XY2

(d)

3X + 8Y → 3XY2 + 2Y

(e)

X + 4Y → XY2

This research study asked students to write a balanced equation for a particulate picture and to perform two stoichiometric calculations that require the use of their balanced equation (Figure 2). The particulate picture is identical to the one used by Nurrenbern and Pickering (1) with the exception that instead of using X and Y as unidentified elements, I used carbon (C) and sulfur (S). This makes it possible to perform the stoichiometric calculations since students will need the atomic weights of the starting materials. The first stoichiometric calculation was chosen because the mass of product should be the same whether students used the balanced equation similar to the correct answer (c) [a 1:1 ratio of C to CS2] or distractor (d) [a 3:3 ratio of C to CS2] from Figure 1. This question was used to assess the students’ abilities

These questions concern the reaction of carbon and sulfur ( ) represented by this diagram.

( )

1. Write the balanced equation for the chemical reaction shown above. 2. How many grams of the carbon–sulfur compound can be produced from 75.0 g of carbon? 3. How many grams of sulfur are needed to react with 33.0 g of carbon?

Figure 1. Multiple-choice question used by Nurrenbern and Pickering (1).

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Research Question and Methodology

Additional Information: MW(C) = 12.0 g/mol; MW(S) = 32.1 g/mol

Which equation describes this reaction? (a)

anced chemical equation. I also point out to them that (d) suggests that the combining ratios of X:Y is 3:8 instead of 1:2. Many of these students responded that even though they left the excess Y atoms in the equation, they recognized that the stoichiometric ratio between X and Y is 1:2. This research study will determine whether students leaving unreacted chemical species in the balanced equation recognized the proper reacting ratio of the starting materials.

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Figure 2. Stoichiometry questions from the test instrument used in this experiment.

Vol. 82 No. 1 January 2005

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Journal of Chemical Education

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Research: Science and Education

to properly perform stoichiometric calculations regardless of whether they provided a correctly balanced equation. Unfortunately, this question also yielded the correct answer if the student forgot to consider the mole:mole ratio of the two chemicals altogether. The second stoichiometric calculation, on the other hand, should have yielded different masses depending on which balanced equation students used (i.e., a 1:2 or a 3:8 ratio of C to S). This study measured and evaluated two student skills: (1) the ability to convert particulate pictures into the symbolic representation of a balanced chemical equation, and (2) the ability to use the symbolic balanced equation to solve mathematical problems associated with stoichiometry. All of the students were formally taught to use algorithms (conversion factors) to solve simple stoichiometry problems like the ones in this study. Students enrolled in eight sections of a first-semester introductory general chemistry course at Middle Tennessee State University (N = 156) were asked to answer the questions appearing in Figure 2. Student responses were categorized based on their balanced equations, and the stoichiometric calculations performed by these groups of students were then analyzed for inaccuracies and inconsistencies. Results for the Balanced Equation Question Table 1 contains a summary of the balanced chemical equations provided by these students, along with the number of students providing this equation. The most common response, reported by 59 of the 156 students (38%), was 3C + 8S → 3CS2 + 2S, followed by the correct response: C + 2S → CS2, reported by 24 students (15%). The other responses (with the exception of C + 3S → CS2 + S, exhibited by 5 students or 3%) all showed some confusion between the use of subscripts and coefficients (16, 17). For example, some students classified the three unconnected squares in the

initial container as “C3” instead of “3C” and the eight unconnected circles as “S8” instead of “8S”. Similarly, some students classified all of the starting materials as one chemical object: “C3S8”. This confusion also appeared in the students’ representations of the final container, labeling the two unconnected circles as “S2” instead of “2S” and the products as “(CS2)3” instead of “3CS2”. It is interesting to note that although many students had trouble with the subscripts and coefficients for C and S as starting materials (37 students or 23%) and S as a “product” (34 students or 22%), very few incorrectly labeled the carbon–sulfur product as anything other than “3CS2” (6 students or 4%). In general, 68 of the 156 students (44%) showed at least some confusion between the use of subscripts and coefficients in converting the particulate picture into a balanced chemical equation. Yarroch (17) reported that 9 of the 14 highachievement high school students (64%) interviewed in his study showed difficulty differentiating between subscripts and coefficients, and were likely to draw “3H2” as six connected circles instead of as three pairs of connected circles. Yarroch concluded that the unsuccessful students’ conceptions of “subscript” and “coefficient” were nonexistent. These students were likely to view both of these ideas as being the same, defining them both as “the number of symbols present” and “just a number”. While 59 students from this study (38%) wrote a balanced equation consistent with distractor (d) and 24 students (15%) wrote a balanced equation consistent with the correct response (c) from Figure 1, none of the 156 students wrote balanced equations consistent with distractors (a), (b), or (e). This is consistent with data Sawrey (2) reported involving freshmen chemistry students and by Phelps (4) involving senior-level chemistry teaching majors, in which less than 1% of the total population of their students chose distractors (a), (b), or (e). It appears that these three distractors are not attractive alternatives to students. The results of this research

Table 1. Data on Student Responses to Exam Question 2 Regarding Balanced Equations Example Number

Student-Generated Balanced Equations

Number Using This Equation

1

3C + 8S → 3CS2 + 2S

59

Incorrect: additional “spectator” species

42

2

C + 2S → CS2

24

Correct response

21

3

C3 + S8 → 3CS2 + S2

21

Incorrect: subscript and coefficient confusion

7

4

C3 + S8 → 3CS2 + 2S

10


4

5

C3 + S8 → (CS2)3 + S2

6


2

6

C + 3S → CS2 + S

5

Incorrect: additional “spectator” species

4

7

3C + 8S → 3CS2 + S2

4


0

8

C3S8 → 3CS2 + 2

3


0

9

C3S8 → 3CS2 + S2

3


1

other unique responses

21

Incorrect responses containing additional “spectator” species and/or demonstrating confusion between subscripts and coefficients

5

132


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Characteristics of Students' Setup To Solve Question 2


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Number of Correct Responses to Question 2

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suggest that the following equations may be more effective distractors for identifying student difficulties in differentiating between subscripts and coefficients: • X3 + Y8 → 3XY2 + Y2 where the coefficients for the X and Y particles are listed as subscripts (exhibited by 21 students or 13%) • X3 + Y8 → (XY2)3 + Y2 where all of the coefficients are listed as subscripts (exhibited by 6 students or 4%)

In Sawrey’s study (2), 12% of the students chose the correct response (c) and 87% chose distractor (d). A comparable percentage of students in this study (15%) wrote the correct balanced equation. This suggests that students in Sawrey’s study who may have preferred balanced chemical equations other than the five choices listed in the multiple-choice question were most likely to choose distractor (d) as the next best alternative. Phelps (4) reported that 32% of the senior-level chemistry teaching majors chose the correct response (c) and 68% chose distractor (d). Although the proportion of students choosing the correct response is still less than those choosing the incorrect response, it does appear that they have a better understanding of the concepts associated with balancing chemical equations than freshman-level chemistry students. Results for the First Stoichiometric Calculation Table 1 lists the number of students providing a proper setup for the first stoichiometry problem, organized by the balanced chemical equations provided by these students. Student responses were evaluated not only for the correct answer, but also for the underlying stoichiometry concepts demonstrated by the students. Many student responses that were not completely correct were still categorized as having a proper stoichiometric setup. The most common of these errors was calculating the molecular weight of CS2 incorrectly, followed by simple mathematics errors (many students had completely correct algorithms but reported incorrect numerical values). Students correctly assigning subscripts and coefficients performed much better on the first stoichiometric calculation than students who had difficulty distinguishing between these two concepts. While 67 of the 88 students using equations (1), (2), and (6) provided a proper stoichiometric setup for this problem (76%); only 14 of the 47 students using the other six equations provided a proper stoichiometric setup (30%). This difference in proportions is statistically significant (z = +5.24, p < .0001), and suggests that students who understand the concepts of subscripts and coefficients are also better at performing simple mathematical calculations in chemistry. It is interesting to note that of the 13 students who were able to provide a proper stoichiometric setup for this question using equations 3, 4, and 5, eight (62%) used the term “C3” and the molecular weight of 36.0 g/mol in their setup. The other five used the term “C” and the atomic weight of 12.0 g/mol even though this was inconsistent with their balanced equation. Other common errors included the assumption that 75.0 g of carbon would yield 75.0 g of carbon–sulfur compound because their ratios are 1:1—that is, misinterpreting a mole:mole ratio as a mass:mass ratio (18, 19); using www.JCE.DivCHED.org

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the molecular weight of “3CS2” instead of “CS2” in the algorithm; and using a molecular weight of 12.0 g/mol for “C3”. Results for the Second Stoichiometric Calculation One of the goals of this study was to determine whether students who left unreacted species in their balanced equations would recognize the correct reacting ratio of the two starting materials when performing stoichiometric calculations. This section focuses on the work of the two sets of students who provided balanced equations containing unreacted sulfur atoms [equations (1) and (6)]. I also limited my analysis to the subset of students in each group who were able to provide a proper setup for the first stoichiometric calculation. What follows is the analysis of the second stoichiometric calculation for 42 students using the first equation in Table 1 (3C + 8S → 3CS2 + 2S) and 4 students using the sixth equation in Table 1 (C + 3S → CS2 + S).

Equation 1: 3C + 8S → 3CS2 + 2S Of the 42 students using equation 1 in Table 1, eight (19%) performed calculations recognizing the 1:2 reacting ratio of C and S. Five of these students used the traditional algorithmic ratios (conversion factors) to convert mC to nC to nS to mS that were used in class (three used a 2:1 ratio and two used a 6:3 ratio). The other three students used the percent composition of CS2 (15.75%C and 84.25%S) to determine the ratio of mC to mS. The first student used the following ratio: 33.0 g/15.75 = x/84.25 to solve for the mass of sulfur. The second student used a similar ratio to determine the total mass: 33.0 g/x = 15.75/100, then subtracted 33.0 g from the final answer to get the mass of sulfur. The third student determined the amount of carbon present (using 33.0 gC and 12.0 gC/molC), and multiplied this amount by the molecular weight of two sulfur atoms (64.1 g S2/mol S2) to get the mass of sulfur. Twenty-two of the 42 students (52%) using equation (1) used the 3:8 ratio from their incorrectly balanced equation in their algorithmic calculations. Surprisingly, 7 of the 42 students (17%) used a 3:2 ratio from their incorrectly balanced equation in their calculations. These students not only had difficulty recognizing the proper reacting ratio, but also confused the concepts of reactants and products by calculating the amount of sulfur left over as a product (right side of the equation) instead of the amount of sulfur used as a reactant (left side of the equation). Equation 6: C + 3S → CS2 + S Of the four students using equation 6 in Table 1, three (75%) performed calculations recognizing the 1:2 reacting ratio of C and S; all of them used traditional algorithmic ratios to convert mC to nC to nS to mS. The other student (25%) used traditional algorithmic ratios to convert mC to nC to nS to mS, but used the 1:3 ratio of carbon to sulfur from the incorrectly balanced equation. Other common errors demonstrated by the 156 students in this study included the assumption that 33.0 g of carbon would require 66.0 g of sulfur since it is a 1:2 ratio—that is, misinterpreting a mole:mole ratio as a mass:mass ratio (18, 19) and ignoring the 2:1 mole:mole ratio by assuming that moles of carbon equal the moles of sulfur (18, 20).


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Conclusions A total of 156 students were asked to provide free-response balanced chemical equations for a classic multiplechoice particulate-drawing question (Figure 1) first used by Nurrenbern and Pickering (1). The most common balanced equation was 3C + 8S → 3CS2 + 2S, where students left the unreacted (excess) sulfur atoms in the equation, followed by the correct equation C + 2S → CS2, exhibited by 15% of the students. The most common student errors demonstrated confusion between the concepts of subscripts and coefficients (i.e., writing “C3” instead of “3C” for three independent carbon atoms, etc.) (16, 17) or included unreacted chemical species in the equation. Balanced chemical equations consistent with two of the five choices used in Nurrenbern and Pickering’s question appeared in this study, and they were the two most popular balanced chemical equations provided. However, none of the 156 students in this study provided balanced equations consistent with the other three distractors. The results of this study suggest that equations such as X3 + Y8 → 3XY2 + Y2 and X3 + Y8 → (XY2)3 + Y2, where some or all of the coefficients are listed as subscripts, would be more attractive choices to students having difficulty distinguishing between subscripts and coefficients and would ultimately provide instructors with more useful information regarding their students’ conceptual understanding. The students were also asked to perform two stoichiometric calculations using their student-generated balanced equation. The responses to the first question were evaluated to determine whether the students had a grasp of simple stoichiometric calculations and the responses to the second question were used to evaluate the students’ concept of the proper reacting ratios of the two starting materials. Fewer than half of the students who demonstrated confusion between subscripts and coefficients provided a proper setup for the first stoichiometric calculation while more than half of the students showing no confusion between these concepts provided a proper stoichiometric setup. This implies that students who better understand the concepts of subscripts and coefficients are also better able to perform simple stoichiometric calculations. Of the students writing balanced equations containing unreacted chemical species, only 23% recognized that the proper reacting ratio was not the same as the ratios in their balanced equation. On the other hand, 64% of these stu-

dents used the stoichiometric ratio directly from their incorrectly balanced equation, demonstrating that even though they can use the balanced equation in a stoichiometric algorithm correctly, they do not understand the chemistry concepts that a balanced equation is trying to convey (1). Literature Cited 1. Nurrenbern, S. C.; Pickering, M. J. Chem. Educ. 1987, 64, 508–510. 2. Sawrey, B. A. J. Chem. Educ. 1990, 67, 253–254. 3. Pickering, M. J. Chem. Educ. 1990, 67, 254–255. 4. Phelps, A. J. What They Don’t Know and Why: Improving the Teaching of Chemistry through Misconceptions. Presented at the 225th National Meeting of the American Chemical Society, March 2003, CHED#1282. 5. Deming, J. C.; Ehlert, B. E.; Cracolice, M. S. Algorithmic and Conceptual Understanding Differences in General Chemistry: A Link to Reasoning Ability. Presented at the 226th National Meeting of the American Chemical Society, September 2003, CHED#299. 6. Gabel, D. L.; Samuel, K. V.; Hunn, D. J. Chem. Educ. 1987, 64, 695–697. 7. Nakhleh, M. B. J. Chem. Educ. 1993, 70, 52–55. 8. Nakhleh, M. B.; Mitchell, R. C. J. Chem. Educ. 1993, 70, 190–192. 9. Williamson, V. M.; Abraham, M. R. J. Res. Sci. Teach. 1995, 32, 521–534. 10. Zoller, U.; Lubezky, A.; Nakhleh, M. B.; Tessier, B.; Dori, Y. J. J. Chem. Educ. 1995, 72, 987–989. 11. Smith, K. J.; Metz, P. A. J. Chem. Educ. 1996, 73, 233– 235. 12. Lee, K.-W. L. J. Chem. Educ. 1999, 76, 1008–1012. 13. Sanger, M. J. J. Chem. Educ. 2000, 77, 762–766. 14. Raviolo, A. J. Chem. Educ. 2001, 78, 629–631. 15. Mulford, D. R.; Robinson, W. R. J. Chem. Educ. 2002, 79, 739–744. 16. Al-Kunifed, A.; Good, R.; Wandersee, J. Investigation of High School Chemistry Students’ Concepts of Chemical Symbol, Formula, and Equation: Students’ Prescientific Conceptions. ERIC Document ED376020. 17. Yarroch, W. L. J. Res. Sci. Teach. 1985, 22, 449–459. 18. Schmidt, H. -J. Int. J. Sci. Educ. 1990, 12, 457–471. 19. Olmsted, J., III. J. Chem. Educ. 1999, 76, 52–54. 20. Oliver-Hoyo, M. J. Chem. Educ. 2001, 78, 1425–1428.

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Evaluating Students' Conceptual Understanding of Balanced

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