Those Baffling Subscripts Arthur W. Friedel a n d David P. Maloney Indiana University Purdue University, Forf Wayne, Fort Wayne, IN 46805 How many oxygen atoms a r e present i n a container with 288 g of 03? (molar mass of 0 3 i s 48.0 g)
cultpartofthebeginningcourse."Severalresearchstudies i n chemistryhave reported t h a t students rely o n t h e u s e of algorithms tosolve problems because they donot have t h e conceptual understanding on which t h e chemistry problems a r e based (3-5).
The items you are about to work are called ranking tasks. The following instructions and H o w m a n y of u s g i v e example show how to work these items. questions like this on exa m s and quizzes to test t h e Each ranking task will have a number of situations, or variations of a situation, that have varying ability of o u r students to values for two or three variables. Your task is to rank these variations using a specified basis. t h i n k ? We h a v e lectured After ranking the items you will he asked to explain how you determined your nddng sequence them on t h e particulate naand the reasoning behind the way you used the values of the variables to reach your answer. t u r e of matter: atoms and Shown below is an example of how to work the ranking tasks. molecules a r e t h e things of which matter is composed. EXAMPLE We have told t h e m about atomic mass, molar mass, Shown below are eight situations where a cmt, which is initially moving, has a force applied to and Avogadro's number. We it such that the force will cause the cart to come to a stop. All of the carts have the same initial h a v e given t h e s t u d e n t s speed, but the masses of the carts vary, as do the forces acting on them. sample problems i n class, and t h e students have been Rank these situations, from greatest to least, on the basis of how long it will take each cart to a s s i g n e d p r o b l e m s for stop. homework. None of them have been exactly like t h e one given above. Few students i n my experience get the above problem correct. Do we feel good t h a t this problem discriminates between those who have developed t h e concepts w e h a v e to t e a c h a n d t h o s e who h a v e not? Are you happy or sad when few students answer this problem correctly? Is i t useful i n separating t h e A students from t h e B students? Does the ability to solve such a problem indicate t h a t t h e C students know t h e differGreatest 1 B 2 *F 3 4 4 5 D,s 6 7 8 Least ence between a t o m s a n d molecules, the meaning of All eight of these carts take the same time to stop. the subscript and the relaPlease carefully explain your reasoning. Z Zlrink f i e t i ~ e d p o b d s m &e tionship among the atomic mass, t h e molar mass, t h e w e l e r ~ ~ h o .qo n . Z d ;v;de d 4mc bv mPrses mass of t h e substance, and and used t h e r n l b n a nurnbe~s. Avogadro's number? Bent ( 1 ) says: "the mole concept i s as easy as rolli n g off a log . . . F o r t h e Notice in this example that in one case two of the situations, and in another case three of the mole concept i s merely situations, produced the same value of the ratio used to determine the ranking, and that the letters common sense applied to for the ones that tied were written in the same answer blank showing they were ranked equally. a n atomic model." NoneIn the same way it is possible that ALL of the arrangements will give the same result for a theless, Dierks, Weninger, particular basis. If that occurs, and only if that occurs, the option of all equal, or all the same, a n d H e r r o n (2)c o n t e n d should be chosen. t h a t "Historically teache r s of beginningchemistry have named t h e mole concept as t h e most diffi- Figure 1. Ranking Task ExplanationIExample Page.
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Yarroch (6)found that a large percentage of the students, who were doing well in a high school chemistry course, had a weak understanding of the meaning of the subscript in a chemical formula. Lythcott (7)investigated the reasoning and procedures used by students who had correctly solved some stoichiometry problems. She found that many students could get the rieht answers without ;nderstaGding the chemistry. Nakhleh ( 8 ) reviewed a varietv of misconceptions relating to number of students'difficulties. One of her recommendations was that "educators should help students begin to understand the differences between atoms, molecules, andions."
Given Information Unknown #a, Formula
#Za
Mass, MM, Formula
#a
#a. Formula, MM Mass, gaw, Formula #a, Formula, gaw
Single-Element Ranking Tasks
Real Element M-C
Imaginary Two-Element Element M-C Ranking Task
NA
NA
NA
Atoms of Z
Atom Total
03
M3
Z Atom Total
P4
ld
#a
Masses of Substances Number of Atoms
Ss
Rs
Mass in Containers Number of Z Atoms
Mass
Actual Mass
Ss
D8
Mass
I
NA
(fa: Total number of atoms. MM: molar mass, gaw: gram atomic weight, mass: actual mass present
Figure 2. Nature and relationships among tasks in study
a
Name
Atom total The eight figures below depict containers that hold various amounts of eight pure substances. The molar masses of these substances are given in the box below:
Nature of the Investigation I n a n e a r l i e r study (9) we D6- 120g L6 - 75g R, - ZOg T, - 90g found that students had consistent and stable alternate In each figure we are told the actual mass of the substance in the container. strategies for certain prohlems. So in this study we set out to Rank these containers, fromgreatest to least, based on the TOTAL number of atoms present. In other words, put first the container which has the largest total number of atoms in it, and put last determine what our students the container that has the fewest total number of atoms in it. are able to do with the information presented in class about moles, molecules, atoms, gram atomic weight, Avogadro's number, and molar mass. Although many researchers advocate indepth interviews in situations like this, we wanted to develop tasks which large numbers of students could do and which would provide teachers with the information needed to improve t h e teachingllearning situation in their classes. We wanted to know what strategies our students were using so we could help them change to more appropriate strategies. The students in this ongoing study were in a variety of college general chemistry classes t a u g h t by t h e same teacher. The classes included CHM 100, Greatest1 2 3 4 5 6 7 8 Least ( l a t e r changed to CHM 111) Preparation for General ChemAll the containers have the same number of atoms. istry, (Prep CHM), for students Please carefully explain your reasoning. who have never had chemistry before, s t u d e n t s who had i t more t h a n four years ago, or students who were reluctant to take general chemistry or counseled out of it for a varietv of reasons; CHM 101, ~ e c t u r i sin Chemical Science for Engineers (Eng CHM) for which there is a Figure 3. Atom Total Ranking Task
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1 How many oxygen atoms are present in a conta ner w th 288 g of 0 3 7 (molar mass of 0, s 48.0 g)
2. There are 1.8 x losatoms in a sample of P4. What is the mass of this sample? (Molar mass of P4 is 124 g.) a) 9.3 x 10-l8 g b) 3.1 x 10-l7 e) 5.6 x lo6 g d) 1.5 x 10-l6 g
3. How many atoms of sulfur are in a sample of 963 g of Ss? (gram atomic weight of S is 32.1 g)
4. There are 2.41 x loz4atoms in a sample of 58. What is the mass of this sample? (gram atomic weight of S is 32.1 g) a) 16.1 g b) 7.74 x loz5.
Figure 4. Real element multiple-choiceRems We designed a series of paper and pencil tasks that involved systematic variations on a basic situation. The tasks presented students with information about three of the following featuredvariables: unit masses such as molar masses or gram atomic weights, masses of the substances that might be used in actual chemical reactions, information about the number of atoms in each molecule, and the total number of atoms present. They then had to respond to the situations on the basis of the fourth featurefvariable. The tasks varied in whether the total mass or the number of atoms was unknown, and in whether the unit mass provided was the molar mass or the gram atomic weight. Two types of tasks were used, the familiar multiplechoice format and a new format called ranking tasks. In ranking tasks students are shown a series of situations that they must rank from greatest to least on some specified basis. These ranking tasks are supply-type items (ll), so the students have to generate an answer rather than choose one. The situations are chosen so that different strategies for ranking them produce different letter sequences as the answers. The ranking tasks are novel formats for most students so some examples and practice are provided before the students work the items of interest. Figure 1 shows the instruction page that accompanies the first ranking tasks assigned to the students. 'Ibis introduction to ranking tasks emphasizes the possible occurrence of ties in the rankings, and explicitly instructs the students about how to indicate such ties. These ties are an important aspect of the ranking tasks because they usually are critical in distinguishingranking sequences generated by alternate strategies. In these items, because of the constraints of the task format which needs several of the situations to have the same numerical values, we used imaginary elements. This was necessary because no two real elements form multiple atom, single-element substances that have the same molar mass. This use of imaginary elements clearly raises the question of whether students would respond differently if the
task involved real elements instead. To investigate this possibility we created multiple-choice items, directly parallel to each of the ranking tasks, which involved real elements. These multiple-choice items had alternate incorrect answer possibilities that were derived by applying the common alternate strategies found in the raukine .. tasks. We also crwited multiplt!-choice items that had imaginary elcmonts that wcre oxc~ct~arallels for each of thc real multiple-choice items. There were 16 tasks used in this study. These are shown schematically in Figure 2. The first four ranking tasks involved just one type of "element" in each compound. The two sets of multiple-choice tasks were analogous to the ranking tasks. These two sets differed in that one set involved real elements while imaginary elements were used in the other set. Finally, the last set of ranking tasks had substances that each had two "elements" in them. We will briefly describe each task. The Atom Total ranking task (See Fig. 3.) provided information about the mass of material present, the formula for the substance, and the molar mass. Students were to rank the variations on the basis of the numbers of atoms oresent -~~~ in thc samples. The Masses of Substances task is the "flipside" of the Atom Tot111task in that the civen ---inform:ltim ----- -.-. is the total number of atoms present, theformula, and the molar mass, and the goal is to rank on the basis of the amount of substance actually in each container. The Number of Atoms task provides information about substance mass, gram atomic weight, and number of atoms in each molecule. The goal is to rank on the basis of the number of atoms present in each container. The difference between Atom Total and Number of Atoms is that the former has molar masses while the latter has gram atomic weights. The "flip-side" task for Number of Atoms, the Actual Mass ranking task, had the total number of atoms, the chemical formula, and the gram atomic weight as the given information. Students were to rank the containers on the basis of the mass of each substance. Please keep in mind that for all the ranking tasks there were two indicators, the ranking sequence and the explanation, of the strategy the student used. The "real" multiple-choice items are shown in Figure 4. The 0 3 item is parallel to the Atom 'Ibtal ranking task. The P4 item is analogous to the Masses of Substances ranking task. The Ss item is parallel to the Number of Atoms ranking task, and the Sg item is parallel to the Actual Mass ranking task. There were also four multiple-choice tasks that were identical to the four just described except that they had imaginary elements instead of real elements. This was done because the ranking tasks involved imaginary elements, and we thought it was important to check on the effect of real versus imaginary elements in the multide-choice format. W: then designed four ranking task3 involving substances that hcrl two diflercnt "elcmmts" in them. lhese werc dew4 oped to determine whether the subscriptwas more meaning11 in a context where the molecules contained two, or more, differentkinds of atoms. The Atoms of Z ranking task had the total number of atoms in the sample and the chemical formula as the given information. Students were to rank the variations on the basis of the number of Z atoms in the sample. This task, which had no analog in the original four ranking tasks, was designed to determine how students handled the subscript in this most basic two-element case. If students cannot determine what fraction of the total number of atoms are Z atoms, then they are not likely to be able to solve more complex problems. The Z Atom lbtal task, which is analogous tothe Atom Total ranking task, had the molar mass, the actual mass of the sample and the chemical formula as the given information.
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Table 1. Percentages of Students Using Specified Strategies
Table 2. Percentages Chooslng Specified Answers on Real Element Multiple-Choice Items
Atom Total
0 3 ltem
Prep CHM Eng CHM Gen CHM
Correct
Alternate
Misc
n
30% 60% 39%
36% 33% 51%
3456
53 57 123
7% 1 0%
PrepCHM Eng CHM GenCHM
54% 49% 48%
B
Ca
D
n
13% 6% 5%
1756 41% 44%
6% 3% 4%
254 39 110
Masses of Substances Correct
fi
(MM)(#a)
P4 item Misc
n
41% 18% 19%
54 60 133
MM
PrepCHM Eng CHM GenCHM
26% 52% 37%
9% 5% 11%
26% 25% 33%
PrepCHM Eng CHM Gen CHM
A~
B~
C
D
n
13% 24% 25%
62% 63% 66%
20% 10%
5% 3% 2%
250 39 110
7%
Ss ltem Number of Atoms
Mass
Correct
MM
Prep CHM Eng CHM GenCHM
27% 37% 43%
26% 2756 41%
Mass (sub) gaw 26% 22% 9%
Misc 21% 14% 8%
n 66 63 129
Prep CHM Eng CHM GenCHM
34%
48% 31%
B
C
Da
n
11% 8% 6%
3%
51% 43% 58%
230 39 112
2%
4%
Ss ltem Actual Mass Correct (#a)(gaw)(sub) (#a)(gaw) Misc sub Prep CHM 17%
28%
14%
42%
n 214
MM: molar mass, gaw: gram atomic weight, #a: number of atoms, sub: subscript.
These were to be ranked on the basis of the number of Z atoms in the sample. The Mass in Containers task, which is analogous to the Masses of Substances task, had the total number of atoms in the container, the molar mass and the chemical formula as the given information. The actual mass in the container was the basis for ranking these. Finally, the Number of Z Atoms ranking task, which is analogous to the Number of Atoms task, had the gram atomic weights, the actual mass of the sample and the chemical formula as the given information. These were to be ranked on the basis of the number of Z atoms in the sample. The tasks listed above were the major focus of this study, but we also had two other ranking tasks that we used to check on the possibility that the difficulty was associated solely with students' comprehension of chemical formulas. These two tasks were identical to Atom Total and Masses of Substances except that they had diagrams of the molecules instead of the chemical formulas. If the students'difficulty is just with the chemical formulas then this change in representation to diagrams should produce significantly better performance. Findings and Results We report the results on the four single-element ranking tasks, then the multiple-choice tasks, and finally the two element molecule ranking tasks. We need to explain how we identified the strategy each student used. We chose a conservative criterion, no more than one deviation from 902
Journal of Chemical Education
PrepCHM GenCHM
A
B
C'
D
E~
n
9% 3%
19% 7%
32% 41%
9% 0%
30% 48%
233 29
'Correct. 'common alternate.
the ranking sequence associated with a strategy Any sequence that differed by more than one deviation was assigned to the miscellaneous category. It is likely, based on students' written explanations, that more students were attempting to use the correct or alternate strategies but miscalculated or failed to show ties in their sequences and, consequently, were assigned to the miscellaneous category. So the values reported below underestimate the percentages using either the correct or alternate strategies. The results for the four single-element ranking tasks are shown in Table 1. The Atom Total ranking task requires the individual to divide the substance mass by the molar mass and then multiply by the number of atoms in each molecule. (Avogadro's number can be ignored because it is in all of the calculations. Interestingly students generally did not ignore Avogadro's number until late in the course and after working several ranking tasks. Even then only a minority left Avogadro's number out of their calculations.)As the results in the table show between a third and one half of the students, simply stopped when they had calculated the number of molecules, or equivalently the number of moles for those who ignored Avogadro's number. There are two common alternative strategies for the Masses of Substances ranking task. One multiplies the molar mass by the number of atoms given, between one fourth and one third of the students tested used this strategy The second alternative strategy divides the number of atoms present by the molar mass, about 10% of the students used this strategy Notice that this latter strategy is
Table 3. Percentages Choosing Specified Strategies on Two-Element Ranking Tasks
Atoms of Z
Prep CHM Eng CHM Gen CHM
Correct
Alternate
Misc
n
38% 66% 50%
17%
45%
7%
25% 30%
24 59 56
20%
Z Atom Total Correct Alternate Alternate PrepCHM Eng CHM Gen CHM
38% 40% 30%
1
2
17% 2% 26%
4% 19% 7%
Misc
n
42% 31% 37%
24 53 61
Misc
n
68% 40% 42%
19 57 54
Misc
n
15% 22% 20%
20 53 55
Number of Z Atoms Correct Alternate Alternate PrepCHM Eng CHM Gen CHM
16% 44% 28%
1
2
5% 12% 17%
11% 4% 13%
Mass in Containers Correct Alternate Alternate Prep CHM Eng CHM Gen CHM
20% 40% 22%
1
2
65%
0% 6% 9%
32%
49%
a dimensional disaster, but students using the strategy gave no evidence that they were aware of the problem. These two problems involve two quantities, number of atoms and molar mass, which are only indirectly related. The particles associated with molar mass are molecules, while the unit mass associated with atoms is the gram atomic weight. So to go between molar mass and number of atoms it is necessary to know the number of atoms in each molecule; i.e., the information provided by the subscript. In the two tasks above most of the students failed to use this information because they ignored the subscript. So if they simply ignore the subscript because they don't think it provides any useful information, they should be able to solve problems where the subscript is ignorable. In the Number of Atoms ranking task the student only needs to divide the substance mass by the gram atomic weight. For this task there were two common alternative strategies that were used by noticeable percentages of the students. One alternate strategy, which was used by a fourth to a third of the students tested, divides the actual mass of the substance by the molecular (molar) mass of the substance. The second strategy divides the actual mass of substance present by the gram atomic weight and then multiplies by the number of atoms in each molecule. This second alternate was used by about a fourth of the students tested. For the Actual Mass ranking task all the students needed to do was multiply the total number of atoms pre-
sent by the gram atomic weight. The most common alternate strategy multiplied the number of atoms present by the subscript and then multiplied by the gram atomic weight. The other commonly used alternate strategy divided the product of the number of atoms and the gram atomic weight by the subscript. In other words both of these strategies used the subscript, instead of ignoring it, and they used it in an entirely inappropriate way. So the results on these two tasks show that the students were not simply ignoring the subscript. A significant percentage of the students was ignoring the subscript where it was needed and using the subscript where it could be ignored. But maybe the students were using these strategies on these ranking tasks because the "elements" involved in the tasks were imaginary. To check whether this was the major source of the difficulty we constructed parallel multiple-choice tasks that used real elements. The results on these multiple-choice tasks are shown in Table 2. Results on the real element multiple-choice items paralleled the results on the ranking tasks in the sense that the dominant strategies were the same alternates and the correct answer. The performance was better on the two items, Ss and Ss, where the subscript could be ignored than on the other two items. For the Ss item, which is parallel to Number of Atoms, about half of the students worked the item correctly. Answer choice A on the Ss item corresponded to the most common alternate strategy on Number of Atoms, around one third of the students chose that answer. Very similar results were found on the Sg item, where about one third chose the correct answer while another third chose the most common alternate. Performance on the other two multiple-choice items was different. On the 0 3 item, which is parallel to Atom Total; about half chose the answer corresponding to the most common alternate strategy on the ranking task. The correct response was chosen by about one third. On the P4 item, which is parallel to Masses of Substances, better than 60%chose the answer corresponding to the most common alternate strategy on the ranking task. The correct response was chosen by about 20%of the students. The similarityin terms of percentages choosing correct and alternate answers and the same alternate procedure being employed between the results for the multiple-choice tasks and the ranking tasks strongly supports the contention that the use of imaginary elements on the ranking tasks was not the primary source of the difficulty with these exercises nor wasthe uifandiar ranking task format the major source of the audenLs'~lifficulties.Addtional cvidtmce that the use of imaginary elements was not a major source of difficulty comes from performance on the parallel multiple choice items with imaginary elements. The results on the multiple-choice items with imaginary elements were extremely similar to those for the multiple-choiceitems with real elements in both uercentazes correct and ~ercentaeeschoosine common alterLate stratepies. The results discussed so far show that these tasks. in whichever format presented, clearly were problematic for the students as demonstrated bv the relativelv small Dercentages getting them correct. More student; get the Ss and Ss items, where the subscript can be ignored, correct than either of the other two, but even for these items there are more than a third, counting miscellaneous strategies, of the students who use the subscript. Unfortunately, even having the larger percentages of students work the S6 and Ss items correctly may not be all that reassuring that these students really understand why their solution is appropriate. There is evidence that a number of those who ignore the subscript are doing so not because they realize that is the correct way to work the task, but because they have an algorithm that they can apply
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here and which in this case gives the correct answer. But the algorithms the students are using on these tasks; i.e., dividing the substance mass by the unit mass and multiplying the unit mass by the number of particles, are the same ones they used on the other two tasks where those algorithms were inappropriate. But maybe the problem is due to the fact that the snbstances in these tasks are single-element substances. Maybe if the substances had two different elements the subscript would take on more meaning for them. (Actually it is more likelv that students will have more trouble with situations involFmg two trlwnmts htwnw the situation is more comokx. but it is worth rherkinc.. lb test this wssibilitv we crekted four ranking tasks that had two-element cokpounds. Three of the four tasks were analogous to single-element multiple-choice items, the fourth was a more basic task. Results on the four two-element molecule-ranking tasks are shown in Table 3. As shown in the table the majority, 55%overall, worked the basic Atoms of Z task successfully. Of the unsuccessful students 14%divided the total number of atoms by the fraction of Z atoms in a molecule, and the rest, 32% of the total, used individual strategies. This result is somewhat encouraging because the majority did use the subscript information properly. Unfortunately on the Z Atom Total ranking task, which is analogous to the Atom Total ranking task, when unit masses are added to the problem the percentage of students who solve the problem correctly drops noticeably. Only 35%of the students tested worked this task correctly Only 15%simply calculated the number of molecules and used that as their basis fir ranking the variations mlternnte 1 ; I l ' r calculated the number of molecules and then multiplied this value bv the ratio of the number of Z atoms in a molecule to the nGber of atoms in a molecule (alternate 2); and finally, 39%used individual strategies. Notice that the more common alternative strategy here is the same as that for Atom Total; i.e., to calculate the number of molecules and quit, but fewer students are usina that stratew with this rankine task. We infer from these reskts that more of the students &e attempting to use the subscrint information here. but thev are not doine so correctly. The results on the Number of Z Atoms rankinz task sunport the same inferences. Only 35%of the students tested worked this task correctly, 14% calculated the number of molecules and then multiplied that by the ratio of the number of Z atoms in each molecule to the total number of atoms in each molecule (alternate I), 9% calculated the number of molecules and quit (alternate 21, and 41%used individual strategies. Again most of the students are attempting to use the subscript information, but they are not doing so correctly. Finally, the Mass in Container ranking task elicited a pattern very similar to that of the Masses of Substances ranking task. On this task only 29% of the 128 students tested used the correct strategy, 45%multiplied the total number of atoms by the molar mass (alternate 1). and most of the remaining 26% used individual strategies. The strategy of multiplying the molar mass by the number of atoms is exactly the same here a s on the Masses of Substances task, and clearly that is a very appealing strategy for many students. So a t this point there is strong evidence that these students have little real understanding of what information the subscript conveys. These students frequently ignore the subscript information when they need it, use it when they could ignore it, and/or use it incorrectly in both types of situations. Clearly there is a problem with these students'understanding of this aspect of chemical symbolism, hut perhaps the diff~cultyis only with the formula representation. To check this possibility we constructed ranking
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Journal of Chemical Education
tasks parallel to the earlier single-element ones, but instead of having the chemical formula, diagrams of the molecules were oresented. The students performed better on these ranking tasks, but the performance was still far from perfect, and the major alternative strategies were still strongly in evidence. For the Atom Total ranking task 31% of the 226 students tested used the correct strategy, 41%calculated the number of moles and quit, and 29%used individual strateeies. On the Masses of Substances task only 21%of the ~ 1 0 ~ s t u dents tested used the correct stratem, 30%multiplied the number uf toms by the molar mass, i 0 ~d~\.ldrd i ;he number of atoms by thc mnlar mass, and 39'i used mdnmiunl strategies. So even changing the representation from the chemical formula to diagrams of the molecules does not remove these students' difficulties with the distinction between atoms and molecules in these tasks. Herron (12)has oointed out that ~ r o b l e mcom~lexitvhas a major effect on &dent The pro~lems"used in these investigations were two-step prohlems where one of the two steps was a n algorithm that was very familiar to the students. Our results indicate that manv of the students applied the algorithm, but then failed-to carry out the second step. However, we do not believe the difflcultv found in these-investigations is due primarily to the two". step nature of the tasks. Evidence to support our belief comes from results on single-element ranking tasks where the subscript information is presented in words. The vast majority of students readily worked these verbal versions correctly even though they were two-step tasks also. The results reported above involve some situations where s t u d e n t s were tested shortly a f t e r studying stoichiometni and some where thev were tested a t the end of the semester. This was done beiause there are a couple of ~ o s s i b l esources of student confusion with the earlier teskng. When tested shortly after studying stoichiometq, the students are new to the ranking task format, consequently, they may have to spend so much time trying to determine how to deal with the task that they loose track of the chemical aspects of the task. Another potential source of confusion for the students early in the course is the possibility that they are thinking of the subscript as a descriptorlidentifier, as it usually is in mathematical notation, rather than a s useful information about the structure of the particles of the substance. Strong efforts were made to assure that these sources of confusion were not viable for the testing a t the end of the semester. The students worked a number of ranking tasks by the time they worked the ones on the final exam, so any unfamiliarity with the task format was long past. The instructor returned the first-ranking tasks the students worked and went over them with the class answering any questions about why and how to work the tasks. With these measures there was some imorovement when the students worked the ranking tasks i n the final. Unfortunately, the improvement was usuallv rather small. and it did n i t extend to the parallel multipie-choice items. Implications and Instructional Suggestions One rather clear implication of this investigation is that many of the students in the study don't realize that the subscript in a chemical formula contains meaningful and important information. While i t is likely that most students who are asked directlv what the subscriot means will respond with an appropriate statmwnt, theiesults of this inwstig!ation imply that if the subicrir)t is a wwt of'a problem task the infbrhation it provides &ay b'missed. There are a t least two possibilities for why this could be happening. One is that students really don't understand the difference between atoms and molecules and the rela-
tionships between them. The second possibility is that students have an algorithm for the kinds of problems used in this investigation, and they simply are applying that algorithm in a rote manner. They are -plugginn -- - and chugging -- - to get an answer to the task. These two oossibilities for whv students struggle -- with the subscript information are not necessarily independent. It is possihle that many students employ algorithms hecause they do not have a solid understanding of the situations thev are confronted with. Pickerine (13)has com". . . the ahility to solve a problem, while mented desirable in itself, does not seem to imply much real understanding of microscopic reality, and it is this understandine that is a t the heart of chemical science." Students appear to have significant difficulty connecting the macmsconic uhenomena thev observe with the microsco~icmodels chemists use to explain the phenomena. Without such a connection, the microscopic models may well he a set of abstract terms to he memorized and used in a rote manner. So it may be that getting students to work prohlems similar to those used here with understanding will require instruction that enables the students to make th;macrosco~ic-microsco~ic connection. IF we treat the two possible sources of this difficulty independently, what are the instructional implications? If the students are applying algorithms to the prohlems then we need to eet them to think of the problem tasks in different ways. There is evidence from problem-solving research (14.15) that students. because thev lack knowledge - in the domain, often employ a general prohlem-solving strategy called means-ends analvsis. In usine this eeneral stratem. or heuristic, the solverUidentifiestge probiem goal, which is a numerical value, for the tasks considered here, then s h e h e works to reduce the difference between where helshe is and the answer. If the goal is a numerical value the most reasonable way to get it is to employ mathematical relations. Applying this heuristic means that the answer they are to find and the equations they can use to find it are the primary focus of their attention while solving the problem. That does not leave much of their cognitive resources to devote to dealing with the situation from a conceptual perspective. If our goal is for students to learn to think of the situation in a qualitative, conceptual way we may well need to change to tasks that have such thinking as the only, or at least the ~rimarv.way to reach the problem goal. For example, chinginithe nature of the task from finding a numerical value to determining whether a calculation is correct, or to identifying why a specific value is incorrect, reauires the student to analvze how a ~rohlemwas solved in order to specify where the process went wrong, or to verify that the process was legitimate. One possibility suggested by the results of this investigation is to give the students one of the tasks with the common alternative answer, ask them to find out what is wrong with the answer, why it is erroneous, and how the task should he done. Another more challenging problem is the following:
teat
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Astudent says that 7 moles of R5combine with 5 moles of L4 oraduce L-R.. She savs because there are 55 (x Avoeadro's number) of atoms in the prnduut that means 11 moles of product is produced. Is rhis studrnt correct? Explain. to
A
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This task is more challenging for two reasons. One is that the student's answer could he right or wrong, and the second is that the students have had little or no experience
with prohlems where the premise is erroneous. Other such tasks that require the students to process the information in different ways can he designed to break students dependence on algorithms. If the prohlem is with the students'understanding of atoms and molecules then more effortneeds to be devoted to building a solid knowledge base on this issue. It is to the teacher's advantage to remember that she is dealing with very abstract concepts (atoms, molecules, and moles) and continually remind the students of the relationship hetween the microscopic and macroscopic worlds. Do the students think of matter as continuous or discrete? Can the students explain why helium leaks faster from a latex halloon than from an aluminized mylar one with respect to holes in the balloon and the tiny helium atoms? Analogies can be useful, hut it is important to identify what features of the analogous system can be transferred and what features cannot. It also is important to emphasize repeatedly the difference between the macroscopic phenomena and the microscopic models used to explain that phenomena. Using multiple representations such as words, formulas, and diagrams on tasks, and emphasizing skill and ahility in going back and forth between these representations is another way to strengthen students' understanding of the ideas of atoms and molecules. Conclusions The results of this study show that the students in this study had significant diff~cultymaking effective use of the information about the number of atoms in a molecule provided by the subscript in a chemical formula. This difflculty appeared in problems involving imaginary elements and real elements hoth and nccurred in three different prohlem situations on two different task fnrmats. The errors were systematic in the sense that the students had specific alternative strategies that they used rather than responding a t random. Two hypotheses for the nature of the difficulty were considered. One was that students were employing algorithms in a rote manner, and the other was that they were not distinmishine atoms from molecules. This latter ~ossihility ties into Nakhleh's recommendation about instructors h e l ~ i n estudents distineuish atoms from molecules that wecitgd in the introduciion. We believe the results of our investieations reinforce the importance of fostering .. this distinction in studrnta hccauw doing so should cnablc students to make cffc!ctive usc of tho r;ubscripl informnt~onin chemical formulas and lead to more meaningful problem solving.
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Literature Cited 1. Bent.H.A. J. Chsm. Edue. 1989.62.58. 2. Dierks, W.; Weninger, J.; Hermn, J. D. J Chm. Educ 1995,62,34%350. 3. Nurrenbern, S. C. Roblem~solvingbehaviors of concrete and formal operational high =tudentswhen solringchemistry requiring formal reasoningskilb; DiasenafioriAbsiraeisInfernorionoi 1980,40,4966A: University
Micmfilms No. DEM 60~05926,167. 4. Gabel, D . L.;Shemood, R. 0.;Enochs. L. J.Res. Sci. lkach. 1984.21,221-233. 5. Gsbel, D. L.;Shenvood. R. D. J. Re*.Sci. nub. 1984,21,84&851. 6. Yarroch,W L. J. Res. Sei. lkach. 1985,22.449459. 7. Lythmtt. J. J Chem. Edue. 1990.67,24G252. 6. Nekhleh, M.B. J. Chm. Educ. 1992,69,191-196. 9. Friedel, A. W; Maloney,D. P Sei. Ed. 199'2, 76,6%78. 10. Ms1oney.D. P J. Coll. S c i Zkoeh. 1987.16 510-514. 11. Marshall, J. C.; Hales. L. W Classmom Tpsf Consfrucfion;Addison-WwwIey:Reading, MA, 1971. 12. Herron, J. P In h w o r d a Scirntiiic Prodiii ofSei4nn Education: Gardner, M.; Greeno, J. G.: Reif, F.; Schoenfeld, A H.; Dissesa. A; Stage, E., Eds.; Lawrence Erlbaum Assaiates: Hillsdale, NJ, 1990; Chapter 2. 13. Pieketing,M.J Chem. Educ. 1990.3.256255. 14. Larkin, J. H.; McDemottJ.; Simon, D. P.; Simon, H. A. Cogn Sci. 1980,4,317345. 15. Sweller, J.Cogn. Sci. 1988, 12,251-285.
Volume 72 Number 10 October 1995
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