Are Our Students Conceptual Thinkers or Algorithmic Problem Solvers

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Are Our Students Conceptual Thinkers or Algorithmic Problem Solvers? Identifying Conceptual Students in General Chemistry Mary 6. Nakhleh Purdue University, West Lafayette, IN 47907-1393

Bright students who perceive the world somewhat differently from the traditional math-oriented sciencelengineering students sit in every large general chemistry lecture. These students desire to explore the why of chemistry more than the how of chemistry. That is, they are more interested in the concepts than in algorithmic problem solving. Second-Tier Students in General Chemistry Tobias ( I )has studied these students and their reactions to general chemistry courses. She calls these students 'second-tier students" and urges that some effort be made to recruit them into the study of chemistry. I became interested in the problem of identifying these students who have the ability to study chemistry yet are not attracted to the discipline. This student population could be viewed as a potentially rich source of recruits into the scientific disciplines, which have experienced steady declines in majors for several years (2).

Testing for Conceptual Thinking and Problem-Solving Skills However, no way has been devised of identifying, and thus remitine. these students in the twicallv large lec-

1I 1

ture sections of general chemistry wurses. This study created a short, simple test that might help identify these students in general chemistry by investigating differential performance on conceptual and prohlem-solving questions. The test consists of five matched pairs of questions. Each oair deals with a soecific area of chemistrv. One auestion of each pair is phrased as a n algorithmic problem-solving ouestion. while the other ouestion is uhrased as a concevwhose solution requires understanding bf La1 the principles of the topic rather than a n algorithm. Constructing Pairs of Questions I identitied five areas of chemistry that were taught in each of the four general chemistry courses offered in the fall semester at a large midwestern university I then constructed five pairs of questions so that each pair dealt with one specific area of general chemistry (see Figures 14). pair 1:gas laws pair 2: equations pair 3: limiting reagents pair 4: empirical formulas pair 5: density

Within each pair, one question required the student to manipulate a formula or work 1 0 100 mole of hydrogen gas ocxples 600 mL at 25 'C an0 4 08 atm If !he vol~me through a n algorithm to find a numerical sos nela conslant, wnat wll be tne pressue of !he samp e of gas at -5 'C? lution to a problem. The second question of the pair required students to use their conA. 4.54 atm 8.' 3.67 atm C.6.00 atm ceptual knowledge of the topic to select an D. 2.98 atm E.4.08 atm answer. In pairs 1, 3, and 5 the conceptual questions required the students to interpret 2. The followingdiagram represents a cross-sectionalarea of a rigid sealed steel tank drawings. Conceptual questions in pairs 2 filled with hvdroaen , - aas - at 20 'C and 3 atm oressure. The dots reDresent the distribution and 4 required students to interpret text of all the hydrogen molecules in the tank. ' only. The test items were randomly inwrporated into the final exams of the four first-semester freshman chemistry courses: remedial, sciencdengineering major, chemistry major, and honors. The bulk of the students were in the course for science and engineering majors. A p proximately 1,000 students were involved in the study Which of the following diagrams illustrate one probable distribution of molecules of hydrogen gas in the sealed steel tank if the temperature is lowered t o 4 'C? The boiling Hypotheses point of hydrogen is -252.8 'C. There were three initial hypotheses.

I

I

Figure 1. Question pair forgas laws.

52

Journal of Chemical Education

That the remedial course would contain the highest concentration of conceptual thinkers, rather than algorithmic problem solvers. That the honors students would demonstrate hoth conceptual thinking and algorithmic problem solving. That the sciencelengineering majors course would contain some conceptual thinkers but mostly algorithmic problem solvers.

I.Potassium, vanadium, and iron crystallize in a body-centered xbic unit cell. Given the lengths of the unit cell edges and the 3tomic weights listed below, which of these elements has the high?stdensity (is the most dense) ?

1. Calculate the maximum weight of NH3 that could be producedfrom 1.9 mol of hydrogen and excess nitrogen according to the following reaction.

>otassium:a = 5.250 A Vanadium: a = 3.024 A lron: a = 2.861 A AW = 39.098 AW = 50.942 AW = 55.847 A. Potassium 6. Vanadium C.' lron D. They all have the same density. E. Not enough information isgiven.

2. Any quantity of Cu in excess of one mole will always react )~ with two moles of AgN03 to produce one mole of C U ( N O ~and two molesotAg. Therefore we knowthat 1.5 molesot Cu will react with two moles of AgN03 to produce 215.74 grams of Ag. Which of the following concepts is the only concept NOT associated with these statements ?

2. The drawings below aredrawn to scale and illustratethecrysal structure of mbidium, niobium, and molybdenum. The atomic p eights of these elements are roughly equivalent. Which of these ?lementshas the lowest density (is the least dense)?

A. Chemical reactions involve the rearrangement of atoms about one another. 6. In an ordinary chemical reaction mass is not created or destroyed. C. Identical compounds are always composed of the same elements in the same proportion by mass. D: NQ-is easily reduced. E. The number ot moles of products formed in this case are determined by the number of grams of AgNOl available. Figure 2. Question pair for equations LbMwhun

1. Which is the limiting reagent when 2.0 mol of CO2 reacts with 2.0 mole of S2 to form COS and 0 2 ? A.'CO,

0. S2

C. COS

W u m

Mldlm

A. Niobium 6.' Rubidium C. MolyWenum D. They all have the same density.

D. 0 2

E. Not enough informationis given.

E. There is no limiting reagent. Figure 5. Question pair tor density.

2. Atoms of three different elements are represented by 0.0, and e. Which is the limiting reagent when two 00 molecules and two EX20 molecules react to form 00. and Wa. A. 00 B.'Wa* C. OIZI* E. There is no limiting reagent.

Figure 6 illustrates the possible categories.

Data Analvsis

D.rn

I

Figure 3. Question pair tor limiting reagent.

C. FeSi3 D. Fe2Si E.' Fe3Si

2. Two moles of Hz gas are known to combine with one mole of O2 gas to form two moles of a substance called water, which we write as H20. Which of the following concepts is NOT associated with understanding this statement ?

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AlCO: algorithmic question correct; conceptual question wrong AOCI: conceptual question correct; algorithmic question wrong AOCO: both questions wrong AlC1: both questions correct

1. What is the empirical formula of a compound if a sample of the compound contains 1.0 x loz3 Si atoms and 0.50 mol ot Fe atoms? A. FeSi 6. FeSi,

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T h e resuonses were cateeorized a n d t h e n freauencies were tabulated. Acorrect answer o n a conceptual question was coded as C1: a correct answer o n an aleorithmic auest i o n was coded as A l . Thus, four possible combinations of responses for each p a i r were possible.

I

Conceptual Thinking High High

Meaningful problem solving; good understanding

Many successful chemistry students

Algorithmic

A. Chemical reactions involve the breakingand rearranging of chemical bonds. 0. Chemical formulas show the ratios of atoms in a molecule. C. The moles of Hz, 02,and H20 are proportionally related to each other. D.' Chemical formulas show the spatial arrangement of atoms in a molecule. E. The number of moles of water formed are determined by the number of moles of H, and 02. Figure 4. Question pair for empirical formulas.

Problem Solving

LOW

Second-Tier Students who are more interested i n why than how

Many unsuccessful chemistry students

Figure 6. Possible categories of students in general chemistry classes.

Volume 70 Number 1 January 1993

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Table 1. The Frequencies of Response Categories by Question Pair

students Als

Cla

Table 2. Significance Levels of Differential Performance on Algorithmic and Conceptual Questions for All Courses

AlCl AlCO AOCl AOCO

Gas Laws

Remedial

SciIEng

Majors

Honors

Remedial

131

32

29

102

Gas Laws

0.000Ia

0.0001'

0.0001a

0.0391a

SciIEng

715

450

397

318

Equations

0.6406

0.000Ia

0.0005a

0.0005'

Majors

47

23

19

28

0.4545

0.O04Za

Honors

35

28

27

8

0.0018a

0.4531

0.7905

1.00

Equations Remedial SciIEng

104

100

66

38

672

437

372

300

Majors

53

41

40

13

Honors

36

22

21

15

Limiting Reagent Remedial

86

82

37

49

Majors

30

26

20

10

Honors

33

21

19

14

SciIEng

Empirical Formula Remedial SciIEng Majors

42

54

41

1

Honors

31

34

29

2

Density Remedial SciIEng Majors Honors

37

124

88

67

57

634

632

507

127

47

46

39

8

34

34

31

3

T h e N s indicate algorithmic questions, and the C's indicate wnceptual questions.

Table 1gives the frequencies in each response category Significant Differences By inspection, these data indicate that many students can answer an algorithmic question about a chemical idea but cannot answer a conceptual question dealing with the same topic. A chi square analysis could not be used to test these differencesfor significancebecause the same student answered both questions; the questions were not independent of each other. Therefore, McNemar's test was used to test the significance of the differences between performance on the muceptual questions and their mathematical counterparts. The analysis tested the correlation between the number of students who answered a mathematical question of a pair correctly and those who answered a conceptual question of the same pair correctly. The probability level was set at 0.05. Therefore, a probability less than 0.05 means that the results are statistically significant and probably not caused by chance. As can be seen from Table 2, eleven of the comparisons are statistically significant. This indicates that there was indeed a difference in performance between questions 1 and 2 in each pair. 54

Journal of Chemical Education

Limning

Reagent Empirical Formula Density

~0.0001~

0.0001a

0.8503

'Significant at p s 0.05.

Questions Concerning the Gas Laws

The questions on the ideal gas law were adapted from studies by Nurrenbern and Pickering ( 3 , 4 ) and Sam-ey (5) that involved the general chemistry classes at three other large universities. These questions were specifically included in the present study for two reasons. First, they are questions that probe an area of chemistry found in every type of introductory course. Second, the frequencies and significance levels they reported for their question pair were similar to the frequencies and significance levels for the gas law questions generated by the present study. This comparison provided a useful check on the validity of the data in the present study. Nurrenbern and Pickering reported significant differences (p < 0.05) between conceptual and algorithmic gas law questions for all groups in their study (N = 205). In their study, 65% of the students in all categories could work the algorithmic problem, but only 35%could correctly respond to the conceptual question. This corresponds very well to the data in this study. From Table 1, a simple calculation indicates that 85% of the students (N = 1,090) could successfully answer the algorithmic gas law question (Al), but only 49% could correctly answer its wnceptual counter~art(Cl). In answering the conceptual and algmithmic questions, students in this studv fared sliehtlv better than theircounterparts in ~ u r r e n i e r nand-pickering9s study, but the basic pattern is the same. This comparison provides some evidence that the differencesin performance found in this study are indeed valid and that they follow a trend noted in other large schools with general chemistry programs. Results

Three interesting fmdings are noted in Table 2. First, performance on the questions for equations are not significantly different in the remedial course, although the initial hypothesis stated that differences were expected. This might be explained by the fact that the professor in the course tries to incorporate conceptual ideas in both the lectures and the exams. Second. the fact that chemistrv maiors had no sienificant on cmiting reage& is indifferen& in their terestine and lareelv unex~lained.Because even student in the c&se hadd&laredcbemistry as a major, perhaps these students have been more willing to construct their knowledge in terms of atoms and molecules. Third, Table 2 shows no significant differences in performance on density questions except in the remedial course. These questions probed algorithmic and wnceptual understanding of density. A reasonable speculation is that the concept of density has been presented to students many times in their high school career and that many students

Also, only a few responses (10%)were categorized as low algorithmidlow conceptual (AOCO). However, the fact that a full 31% of the responses were located in the low conceptualhigh algorithmic quadrant of many able stuthe table is a cause for concern. Apparently .. dents were leaving their first semester of chemi~trywith good algorithmic problem-solving skills hut weak understanding of conce&s and principl& Reviewing the Hypotheses The honors students tended to have greater success in answering both the algorithmic and conceptual questions, which supported the initial hypothesis. However, the remedial students did not have greater success with the conceptual questions as opposed to the algorithmic questions. This fmding does not support the original hypothesis. Finally, the sciencelengineering majors course did appear to contain a mix of algorithmic and conceptual students, which again supports the original hypothesis. InvestigatingStudent Preferences This studv has indicated directions for future research. Students' preferences for either conceptual thinking or al..eorithmic nroblem solvine were not investieated. This might be a critical issue because Tobias (11indicates that her second-tier students often could work the problems but found them boring. Her students wanted learn more about the theories and principles of chemistry and less about the strategies used in problem solving. The best wav to investieate this issue would be to conduct i n d i v i d d interview; but some information could be gleaned from an exam question asking students to indicate their preferences. These answers could then be correlated with their performance on the algorithmic vs. conceptual questions. Also, follow-up interviews could be conducted with students from each quadrant in the truth table to determine if the students had been indeed correctly identified by the test.

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Figure 7. Algorithmic and conceptual means for all courses on gas laws, equations, and density. had gained both algorithmic and conceptual understanding of the topic. comparing across Courses The pairs dealing with gas laws, equations, and density were chosen to investigate differences across courses because onlv these three nairs were included on everv exam. significant performa&e differences were obtainei across (u). and densitv courses for gas laws (D = 0.016).. eauations @ = 0.006).Figure 4 depids the mean score on algorithmic and conceptual questions for these three pairs. Because there were three pairs, the top score in each category was three. Figure 7 clearly shows that there were significant differences in performance within courses and across courses. These data support the initial hypothesis that there would be differences in performance across courses. Assessment by Quadrant This strategy of randomly placing paired questions on examinations provides a reasonablv auick wav to assess the understan&ng of general chemishj. students a t the algorithmic and conceptual levels. Students did seem to dih d e into four When all responses were counted, 49% fell in the high algorithmichigh conceptual (AlC1) quadrant of the table, and 31% were in the high algorithmidlow conceptual (AlCO) quadrant. Fewer responses (10%)fell into the low algorithmichigh conceptual (AOC1) quadrant, where it was originally thought second-tier students would reside.

Conclusion Chemistry apparently appears less and less interesting or inviting to large segments of our student population. The current emphasis on algorithmic problem solving in our general chemistrv courses mav have somethine to do witgthat trend. By integrating b h h the concept& and alenrithmic understandine of a tonic in chemistrv and then teking that integration, &tructok might make"chernistry more attractive to second-tier students. Acknowledgment This study was supported by funds provided by Sheila Tobias. The author gratefully acknowledges the assistance of graduate student Phemie Dandashli in analyzing the data. The author also thanks George Bodner and Bill Robinson of Purdue and Maurice Schwartz of Notre Dame for suggestions for questions. Literature Cited 1.Tobias, S. Theyk Nof Dumb. Thsy'rr Diffemd: SfcUing fha S m n d ?IReaeareh *?: Corporation:l\rcson, AZ, IssO. 2.Heylh. M.Chem. and Engn. News May 20,1981,W 0 . 3.Numenhem, S.; Piekering,M.J Cham. Edue. 1987.64. 5508-10. 4.Rekering.M. J.Chem.Edur 1880.67.234-255. 5.Sawrey, 9.J Cham. Educ IssO,67,25&254.

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