using think-aloud protocols to investigate secondary school teachers’ misconceptions about...

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Using think-aloud protocols to investigate secondary school chemistry teachers’ misconceptions about chemical equilibrium Derek Cheung

Received 30th July 2008, Accepted 11th December 2008 DOI: 10.1039/b908247f

Secondary school chemistry teachers’ understanding of chemical equilibrium was investigated through interviews using the think-aloud technique. The interviews were conducted with twelve volunteer chemistry teachers in Hong Kong. Their teaching experience ranged from 3 to 18 years. They were asked to predict what would happen to the equilibrium system N2(g) + 3H2(g) ⇌ 2NH3(g) if a small amount of nitrogen gas is added to the system at constant pressure and temperature. Analyses of the transcribed protocols indicated that regardless of the teachers’ years of teaching experience, none of the twelve teachers were able to solve the chemical equilibrium problem. An over-emphasis of Le Châtelier’s principle was found to be a major contributing factor to teachers’ problem-solving failures. They held the misconception that one can always cause an equilibrium system to shift to the right by increasing the concentration of a reactant. The interviewed teachers were classified into two problem-solving categories based on their cognitive sources of failures. Excerpts from the transcripts discuss the differences between the two categories as well as the hints given to teachers to help them solve the chemical equilibrium problem.

Keywords: chemical equilibrium, Le Châtelier’s principle, subject matter knowledge, think-aloud method

Introduction According to Abell (2007), three types of knowledge interact with pedagogical content knowledge for teaching of a particular topic in science: subject matter knowledge, pedagogical knowledge, and knowledge of context. Subject matter knowledge is indispensable in teaching (Hashweh, 1987; Grossman et al., 1989; Carlsen, 1991). Gess-Newsome (1999) reviewed the literature on teachers’ subject matter knowledge and discussed its effects on student learning.

“For instance, from this review, teachers having low levels of subject matter knowledge often teach for factual knowledge, involve students in lesson primarily through low level questions, are bound to content and course structures found in textbooks, have difficulty identifying student misconceptions, and decrease student opportunities to freely explore the content either through manipulatives or active discussion.” (pp. 82-83)

Although chemical equilibrium is a very important topic in the secondary school or college chemistry curriculum, both practising and prospective teachers are generally weak in their knowledge of this subject matter (Banerjee, 1991; Quílez-Pardo and Solaz-Portolés, 1995; Quílez, 2004; Ganaras et al., 2008; Özmen, 2008). An important subtopic of chemical equilibrium is concerned with factors affecting the position of equilibrium. Most secondary school teachers teach students how to use Le Châtelier’s principle to predict the direction in which a chemical equilibrium system will shift when it is

disturbed. But application of Le Châtelier’s principle can lead to incorrect predictions. Actually, chemists and chemistry educators have known of the inadequacies of Le Châtelier’s principle for about 100 years (see, for example, Raveau, 1909; Epstein, 1937; de Heer, 1957, 1958; Wright, 1969; Allsop and George, 1984; Gold and Gold, 1984, 1985; Levine, 2002; Cheung, 2004; Lacy, 2005; Torres, 2007). They have shown that there are circumstances where apparently reasonable applications of Le Châtelier’s principle can result in incorrect predictions about the effects of changes in concentration, volume, pressure, or temperature on chemical systems at equilibrium. Because Le Châtelier’s principle has no value for chemistry teachers and students other than historical interest, some researchers have criticized the over-emphasis of Le Châtelier’s principle in the school chemistry curriculum (de Heer, 1957; Allsop and George, 1984; Gold and Gold, 1985; Kemp, 1987; Solaz-Portolés and Quílez-Pardo, 1995; Cheung, 2004). They recommended the use of the equilibrium law, reaction quotient, and the van’t Hoff equation to predict the direction in which a chemical equilibrium system will shift when it is disturbed. Yet many textbooks of secondary school chemistry or college chemistry published in countries such as the USA, the UK, Australia, Canada and China still present Le Châtelier’s principle as infallible. In the USA, for example, a textbook presents the principle as follows:

Le Châtelier’s principle states that the shift will be in the direction that minimizes or reduces that effect of the change. Therefore, if a chemical system is at equilibrium and we add a substance (either a reactant or a product), the reaction will shift so as to reestablish equilibrium by consuming part of the added substance. (Brown et al.,

Department of Curriculum and Instruction, The Chinese University of Hong Kong, Shatin, Hong Kong. E-mail: [email protected]

This journal is © The Royal Society of Chemistry 2009 Chem. Educ. Res. Pract., 2009, 10, 97–108 | 97

2006, p. 650; emphasis in original) It is important to note that predictions based on Brown et al.’s (2006) statement of Le Châtelier’s principle may conflict with experimental facts. If the amount of products formed by a reversible chemical reaction is not equal to the amount of reactants, adding more reactant at constant pressure and temperature may shift a gaseous chemical equilibrium to form even more reactant (Lacy, 2005). Le Châtelier’s principle fails to predict such a shift in equilibrium. Unfortunately, research has found that school teachers are generally unaware of the inadequacies of Le Châtelier’s principle (Cheung, in press; Cheung et al., in press). Furthermore, several researchers have attempted to replace Le Châtelier’s principle by a small number of specific rules with a limited applicability range. For example, van Driel and Gräber (2002) suggested the following specific rule for homogeneous systems:

One of such specific rules may be formulated as: The addition of some amount of one of the substances of a chemical system at equilibrium leads to a subsequent decrease of the concentration [or partial pressure] of this substance, and vice versa. (p. 285, emphasis in original)

Van Driel and Gräber (2002) pointed out that the above rule can easily be applied to homogeneous equilibrium systems, but it can still give incorrect predictions. Although the rule focuses on concentration rather than amount of a substance, Posthumus (1933) showed that there are circumstances where addition of a substance to an equilibrium mixture can further increase the concentration of that substance. It is a matter of serious concern that if secondary school teachers’ knowledge of factors affecting the position of equilibrium is poor, their students are bound to gain an even poorer understanding about chemical equilibrium. To my knowledge, only seven previous studies (Banerjee, 1991; Quílez-Pardo and Solaz-Portolés, 1995; Quílez, 2004; Piquette and Heikkinen, 2005; Özmen, 2008; Cheung, in press; Cheung et al., in press) assessed how Le Châtelier’s principle affects teachers’ understanding of the effects of addition of more reactants or products on chemical equilibrium. Although these seven studies are useful, they used paper-and-pencil tests and focused on problem-solving outcomes, not on the underlying problem-solving processes used by teachers when they tackled chemical equilibrium problems. Often, in the assessment of problem solving skills, the solution to a problem reveals little about the way in which the solution was reached. Until the processes are understood, chemistry teacher educators will have great difficulties formulating effective interventions to help teachers understand chemical equilibrium. It is not known how Le Châtelier’s principle actually affects the thinking processes of teachers. What needs to be documented in detail is the effect of Le Châtelier’s principle on teachers’ cognitive processes while solving chemical equilibrium problems. Herron (1983) also pointed out: “What the experienced chemist writes on paper and what the novice writes on paper may appear to be similar, but examination of problem solving protocols suggests that the thought processes may be very different” (p. 8).

Research has indicated that the think-aloud method is an effective way to collect information about the cognitive processes that a person follows during problem solving (Ericsson and Simon, 1993; Van Someren et al., 1994). The use of think-aloud method to investigate problem solving processes in science education is not new. In the area of chemistry education, work has been done by researchers to investigate problem solving in topics such as the mole concept (Staver and Lumpe, 1995), organic synthesis (Bowen and Bodner, 1991), bonding (Mason et al., 1997), volumetric analysis (Anamuah-Mensah, 1986), chemical equilibrium and thermodynamics (Thomas and Schwenz, 1998), and molarity (Gabel et al., 1984). The study by Camacho and Good (1989) is probably the only published work on the use of the think-aloud method to investigate the problem-solving processes used by teachers when solving chemical equilibrium problems. However, their study did not focus on the inadequacies of Le Châtelier’s principle. Specifically, this qualitative exploratory study used the think-aloud method to seek to answer the following two research questions: 1. How does Le Châtelier’s principle affect secondary school

chemistry teachers’ cognitive processes while solving a gaseous equilibrium problem involving addition of more reactant at constant pressure and temperature?

2. What hints do secondary school chemistry teachers need to find out the solution to the gaseous equilibrium problem?

Design of the study The context and participants

The Hong Kong educational system at secondary school level comprises three years of middle school (Secondary 1-3) and four years of high school (Secondary 4-7). Science students study chemistry as a separate discipline in Secondary 4-7. Chemical equilibrium is taught in Secondary 6. In 2008, I interviewed a convenience sample of twelve experienced secondary school chemistry teachers in Hong Kong. I contacted them by email. They were former students of my university and selected on the basis of their willingness to participate in the study. All the twelve teachers, seven males and five females, majored in chemistry. They had received formal preparation in teaching and obtained a postgraduate diploma in education (see Table 1). Their chemistry teaching experience ranged from 3 to 18 years (mean = 8.8 years).

Procedures for conducting the interviews

The 12 interviews were conducted by me between March and May 2008. Ten interviews took place in my office and two interviews were held in the teachers’ schools. Before the interview was started, the teachers were informed of the purpose of the interview and assured that their participation was voluntary and confidential. Then the teachers were handed a piece of paper with the following chemical equilibrium problem. They were asked to think aloud about how they would solve it.

98 | Chem. Educ. Res. Pract., 2009, 10, 97–108 This journal is © The Royal Society of Chemistry 2009

Table 1 Background information about the teachers

Tea Qua eaexp

T1 Mal B.S 3

cher Sex lification Y rs of teaching erience

e c., Dip.Ed. T2 Male B.S

M.E15

T3 Fem B.S 8 T4 Male B.S 4 T5 Fem B.S 5 T6 Mal B.S

M.E8

T7 Male B.SM.P

4

T8 Male B.SM.E

15

T9 Fem B.S 18 T10 Fem B.S

M.P11

T11 Mal B.S 6 T12 Fem B.S 9

c., Dip.Ed., d.

ale c., Dip.Ed. c., Dip.Ed.

ale c., Dip.Ed. e c., Dip.Ed.,

d. c., Dip.Ed., hil. c., Dip.Ed., d.

ale c., Dip.Ed. ale c., Dip.Ed.,

hil, PhD e c., Dip.Ed.

ale c., Dip. d. E

The reaction N2(g) + 3H2(g) ⇌ 2NH3(g) is at equilibrium in a reactor fitted with a movable piston. The amounts of N2, H2, and NH3 in the equilibrium mixture are 0.510 mol, 0.197 mol, and 0.204 mol, respectively. The total volume of the gaseous mixture is 1.00 L. Predict the direction of the shift in the equilibrium position if 0.140 mol of N2 gas is suddenly added to the equilibrium system at constant temperature and pressure. Clearly show your calculations.

The above problem was tailor-made to serve as a diagnostic assessment tool. Le Châtelier’s principle was not explicitly mentioned in the problem, in order to avoid response bias. Apparently, the problem is an algorithmic problem-solving question dealing with chemical equilibrium. Actually, a good understanding of the principles of chemical equilibrium rather than just an algorithm is required to solve it. Each interview was conducted in Chinese and progressed through two phases. The first phase aimed to collect data to answer the first research question. The teachers were asked to solve the above chemical equilibrium problem while thinking aloud so as to assess their cognitive processes and diagnose where difficulties arose. According to Ericsson and Simon (1993), the think-aloud method may slow down but does not change the course of cognitive processing. However, specific hints or requests to supply reasons can alter cognitive processing. Thus, to avoid diverting their natural problem-solving processes, I did not provide teachers with any hints or specific feedback during the first phase of the interview. I just motivated them to keep talking by saying, “I see,” “What are you thinking now?” or “Please tell me what you are writing.” I gave the following instruction to think aloud, which was adapted from Ericsson and Simon (1993):

The purpose of my study is to find out how chemistry teachers think when they solve chemical equilibrium problems. Try to think aloud as you work on the problem given. Tell me everything that passes through your head during your work searching for the answer to the chemical equilibrium problem. It is most important that you keep talking. You can write down anything you wish as well as use a calculator.

The second phase of the interview aimed to collect data to

answer the second research question, and began when the teachers were unable to solve the chemical equilibrium problem and asked for hints, or when the teachers said that they had finished solving the problem, but were unaware of having gone wrong. Because the second phase of the interview aimed to help the twelve teachers to reflect on their heuristics and enhance their understanding of chemical equilibrium, I gave them some hints or reinforcement. However, I did not provide the solution directly. The degree of intervention varied with teachers, depending upon the nature of difficulties encountered by the individuals.

Data analysis

All interviews were audio-taped and fully transcribed according to guidelines given by Bogdan and Biklen (2007). The length of the interviews ranged from 30 to 55 minutes, with the average being about 38 minutes. The resulting protocols and the accompanying written calculations on the answer sheets served as the source of data for this study. According to Clement (2000), there are two major purposes of clinical interview studies: generative and convergent. Analysis of interview data in a generative study aims to produce new observation categories or new elements of a theoretical model of mental processes. Intensive interpretative analysis of transcripts is involved and relatively large sections of transcripts are often presented alongside the researcher’s interpretations. In contrast, convergent studies are confirmatory rather than exploratory in nature. Analysis of interview data in a convergent study usually involves coding of transcripts to provide reliable information about the frequencies of certain observation patterns. Clement (2000) pointed out that both types of study are important. In an area where cognitive processes are unexplored, researchers may start with generative studies, which can provide a good foundation for the design of convergent studies. The purpose of this study was to explore the nature of difficulties that can be observed when the twelve chemistry teachers solve the chemical equilibrium problem. Thus, this research project was a generative rather than convergent study (Clement, 2000). The data were qualitatively analyzed to answer the two research questions. A research assistant and I separately analyzed and interpreted the contents of teachers’ protocols and answer sheets to elucidate the course of cognitive processes as each of the teachers solved the problem. First, we independently identified and categorized the reasoning paths used by the teachers. The protocols were read repeatedly so as to identify the segments of protocols that could show where individual teachers made errors. Then, the research assistant and I compared and discussed the interpretations of protocols until a consensus was reached. The initial set of categories of reasoning paths were refined by collapsing categories of similar nature into one category. We also agreed to drop a few categories that were irrelevant to the two research questions. Finally, we went back to the data and independently classified the teachers according to the consensus category scheme. Several excerpts of the think-aloud protocols were selected to illustrate the characteristics of each category.


Results When analyzing the think-aloud protocols and answer sheets, two categories of cognitive processes emerged. However, none of the teachers gave a completely acceptable answer to the chemical equilibrium problem. The categories were not related to teachers’ years of teaching experience. For confidentiality, the 12 teachers are referred to in this paper as T1-T12.

Category I: Teachers applied Le Châtelier’s principle and ended up having an unsolvable mathematical equation

This category consisted of seven teachers (T1, T2, T4, T6, T7, T9, and T12). Problem solving is a non-linear process, but the major steps followed by six of these teachers (T1, T2, T4, T7, T9, and T12) are summarized in Table 2. The first tactic utilized by these teachers was to read and interpret the chemical equilibrium problem. They underlined or circled key words to aid their problem solving. Then, these six teachers calculated the value of the equilibrium constant based on their understanding of the equilibrium law. They set up the Kc expression below and then performed the calculation of Kc correctly.

However, these six teachers applied the ‘change-then-minimize’ logic of Le Châtelier’s principle to predict the equilibrium shift without fully utilizing the information provided by the problem statement. Owing to their reliance on Le Châtelier’s principle, they ended up having an unsolvable mathematical equation. The following excerpt from the transcript of T1 is representative. The symbol ‘I’ stands for interviewer. Dots ‘…’ indicates a short pause in speech.

T1: The Kc is equal to 10.67. I: What is your next step? T1: According to Le Châtelier’s principle, the position of equilibrium should shift to the right to minimize the increase in the amount of nitrogen. More ammonia will be formed…Now, let’s say the new equilibrium is established and x moles of nitrogen have been used to form the extra ammonia. If V is the total gas volume in the new equilibrium state, then the number of moles of nitrogen in the new equilibrium mixture will be 0.510 + 0.140 – x, and the new equilibrium concentration of nitrogen is equal to 0.510 + 0.140 – x divided by V. For hydrogen, the new equilibrium concentration is 0.197 – 3x divided by V. And for ammonia, the new equilibrium concentration is 0.204 + 2x divided by V. I: I see. T1: Because Kc equals 10.67, all I need to do is to substitute these new equilibrium concentrations into the Kc expression. Let me write it down here… What! (with strong emphasis) There are two unknowns, x and V. But I have only one mathematical equation. The two unknowns cannot be found with a single mathematical equation…I want to find out the value of x because it tells me the amount of nitrogen reacted to form ammonia when the position of

Table 2 Reasoning path IA

Step Problem-solving behaviour 1 Read and interpret the problem. Underline or circle key words. 2 Write the balanced chemical equation and list the number of

moles of each gas in the equilibrium mixture. 3 Set up the Kc expression and calculate the value of Kc. 4 Apply Le Châtelier’s principle to predict the shift in equilibrium

position. Let x be the number of moles of nitrogen used to form the extra ammonia molecules and V be the total gas volume in the new equilibrium state.

5 Express the concentrations of nitrogen, hydrogen, and ammonia present in the new equilibrium mixture in terms of x and V.

6 Set up a new Kc expression and attempt to solve for V and x using the Kc expression.

equilibrium has shifted to the right so as to counteract the change. But I need to know the value of V in order to calculate x.

astysstavucabfk

2

67.10)197.0)(510.0(

)204.0(][][

][3

2

322

23 ===

eqeq

eqc HN

NHK

100 | Chem. Educ. Res. Pract., 2009, 10, 97–108

T1 was not able to get rid of one of the two unknowns. He sked, “Does the problem provide enough information?” I aid, “Yes, the data are enough to determine the direction of he equilibrium shift. Rethink your strategy. Let me know if ou really don’t know what to do next.” T1 checked all his teps again. He switched back and forth between the problem tatement and the Kc expression so as to find out whether here were any steps omitted. He abandoned the Kc expression nd wrote a new one, exploring in what ways the two ariables could be solved. After about four minutes, T1 gave p reluctantly. Unsure how to get rid of the mathematically omplex equation, all the six teachers (T1, T2, T4, T7, T9, nd T12) asked for hints. The following excerpt shows how I egan the second phase of the interview by serving as a acilitator to guide T1 to develop his subject matter nowledge about chemical equilibrium:

67.103197.0140.0510.0

2204.0

][][][

3322

23 =

⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −+

⎟⎠⎞

⎜⎝⎛ +

==

Vx

Vx

Vx

HNNH

Keqeq

eqc

T1: Can you give me a hint? I: OK, I don’t think you can determine the exact values of V and x with a single mathematical equation. My hint is…avoid considering the total gas volume in the new equilibrium state. Assume that after nitrogen is added to the system, there is physical rather than chemical change. Can you find out the total gas volume of the mixture? T1: Physical change? I: Yes, assume that there is just physical change after nitrogen is added to the system. Can you find out V? I mean the new total gas volume. T1: The ideal gas equation can be used to calculate the new volume…PV = nRT…Since temperature and pressure are kept constant, V is proportional to n. I: Yes, V is directly proportional to n. Please continue. T1: So, 1.00 litre divided by 0.510 mol is equal to V divided by 0.140 mol…V is 0.27 and the unit is litre. I: The initial gas volume is 1.00 litre. How come the new total volume is 0.27 litre?...smaller than 1 litre! Are you

This journal is © The Royal Society of Chemistry 2009

sure the new total gas volume is equal to 0.27 litre? T1: Oh yes, there must be a mistake…Ahh, the V should be the increase in gas volume, not the total volume…Yes, the new total gas volume is 1.00 + 0.27; that is 1.27 litres. I: I see. But I suggest you check the calculation again. T1: Let me see…Ah, my calculation of the initial amount of gases was incorrect. It should be 0.510 + 0.197 + 0.204 …the total amount of molecules is 0.911 mol. I: Yes, please find out the value of V again. T1: V is proportional to n. So, 1.00 litre divided by 0.911 mol is equal to V divided by 0.140 mol…V equals 0.15 litre. I: Correct! T1: The total gas volume is 1.00 + 0.15; that is 1.15 litres. I: Correct. What is your next step?

With the aid of hints, T1 managed to calculate the new total gas volume, assuming that the addition of nitrogen merely caused a physical change. However, T1 was not sure what the next step should be and thus asked for additional hints.

T1: Umm…..I don’t know how to make use of the new gas volume…1.15 litres…Any hints? I: You have found that Kc is equal to 10.67…Do you think the Kc value is useful? T1: The Kc equals 10.67 and this is true even when the new equilibrium is established because the temperature is kept constant. I: Yes, this is a very important chemical concept. T1: But… I don’t know the real total gas volume in the new equilibrium state. …I don’t know what to do with 10.67? I: We just assumed that after nitrogen is added to the system, it does not cause any chemical change. Here you have found out the new total gas volume …1.15 litres. If we calculate the Kc again using the new data, do you think the answer will equal 10.67? T1: Umm…Not necessarily, maybe larger or smaller, because the system is not at equilibrium…ahh, yes, we can calculate the new Kc and then check whether the new Kc value is larger or smaller than 10.67. If the new Kc value is larger than 10.67, the system must adjust to reduce it to 10.67…and this can be achieved by shifting the equilibrium to the left. If the new Kc value is smaller than 10.67, the equilibrium must shift to the right. I: Yes, we can make good use of the equilibrium law to determine the direction of equilibrium shift. Because the system is not at equilibrium, let’s call the new value Qc. Now, please calculate its value.

The above excerpt indicates that T1 did not understand how the properties of Kc or the concept of reaction quotient, Qc, can be applied to predict the effect of an increase in concentration of a reactant on chemical equilibrium. Below is the Qc expression written by T1:

T1 was very surprised when his calculations indicated that adding more nitrogen to the equilibrium mixture will produce

even more nitrogen molecules when equilibrium is re-established. He also tried to rationalize why Le Châtelier’s principle failed to make a correct prediction as follows:

T1: Qc is equal to 11.08. Therefore, Qc is larger than Kc, the equilibrium shifts to the right…Oh no, I made a mistake… the equilibrium should shift to the left. More nitrogen will be formed. If nitrogen is added to the system, the equilibrium will shift to form even more nitrogen molecules…This is very strange. Le Châtelier’s principle is wrong! (with strong emphasis) I: Yes, your prediction is correct. The equilibrium will shift to the left. T1: Le Châtelier’s principle cannot make a correct prediction in this case. Why?...(long pause). Oh yes, the piston is movable, so the problem involves more than one variable. The concentration of nitrogen is changed, and the total gas volume is also changed. Ah, I know, Le Châtelier’s principle should be applied to predict shift in equilibrium caused by only one change…I mean…one variable. I: In many textbooks, Le Châtelier’s principle is applied to make predictions about the effects of changes in concentration, volume, pressure, or temperature on chemical systems at equilibrium. Are you saying that all those examples of application of Le Châtelier’s principle involve changes in only one variable? T1: Yes. Last month, I taught about Le Châtelier’s principle in my class and used the Haber reaction as an example to show how addition of more nitrogen gas disturbs the equilibrium. But the volume of the reactor was kept constant. I: Let’s consider the example you just mentioned. Imagine that we have a reactor here. The piston is not movable. So, the total gas volume cannot be changed. If I suddenly inject more nitrogen into the reactor, are you sure only one variable will be changed? T1: The concentration of nitrogen in the reactor must increase…ahh…the total gas pressure must also increase. I: Therefore, when a chemical equilibrium system is disturbed, more than one variable are usually affected.

In addition to T1, four other teachers (T2, T7, T9, and T12) also erroneously believed that Le Châtelier’s principle fails to give a correct answer to the equilibrium problem due to involvement of more than one change and thus they thought that it is a complex rather than a simple equilibrium problem. For example, T9 argued, “This is a complex problem because there are two types of change. So we should not misuse Le Châtelier’s principle. I was foolish to have used Le Châtelier’s principle to solve this kind of complex problem.” After I helped T9 to understand the relationship between the so-called two types of change, she concluded, “The change in amount of nitrogen gas and the volume change are not independent of each other. The increase in gas volume is affected by the amount of nitrogen added. So the problem is not really complex.” At the end of interview, T12 raised this query, “Le Châtelier’s principle is misleading. If a system is in equilibrium and a change is made, the reaction need not shift in the direction that opposes the change. Why is Le

08.11)197.0)(650.0()15.1()204.0(

))(()()(

)/)(/()/(

]][[][

3

22

3

22

3

2

322

23

22

3

22

3 =====HN

NH

HN

NHc nn

VnVnVn

VnHN

NHQ


Table 3 Reasoning path IB

Step Problem-solving behaviour 1 Read and interpret the problem. Underline key words. 2 Write the balanced chemical equation and set up the Kp expression. 3 Calculate Kp. (But the calculation is incorrect.) 4 Recognize that a mistake was made in calculating the value of Kp. 5 Calculate the total number of gaseous molecules in the equilibrium

mixture. List the mole fraction of each gas. Let PT be the total gas pressure. Calculate Kp in terms of PT.

6 Calculate the total number of moles of molecules in the mixture when more nitrogen is added. List the mole fraction of each gas. Calculate Kp in terms of PT. (But he did not understand why the value is not equal to 8.9/(PT)2.)

7 Switch to consider the shift in the position of equilibrium. Predict that the equilibrium will shift to the right on the basis of Le Châtelier’s principle. Let x be the number of moles of nitrogen used to form the extra ammonia molecules. Express the number of moles of nitrogen, hydrogen, and ammonia present in the new equilibrium mixture in terms of x.

8 List the mole fraction of each gas in the new equilibrium state. 9 Set up a new Kp expression. Attempt to solve for x using the Kp

expression but find that there is an x4 term in the mathematical equation.

Châtelier’s principle still included in textbooks published in Hong Kong?” Only one teacher, T6, applied Kp rather than Kc expression to solve the numerical problem (see Table 3). This finding is significant for an exploratory study because it can provide a foundation for the design of future convergent studies (Clement, 2000). The use of Kp expression did not require T6 to consider the change in total volume of the chemical equilibrium system. T6 failed to solve the equilibrium problem because he just plugged in numbers into the Kp expression without really understanding the chemical concepts involved. T6 began the problem-solving process by calculating Kp. But he calculated the value of Kp incorrectly because the total gas pressure was taken as 1 atm and the number of moles of each gas was treated as its mole fraction. The following is his erroneous equation:

T6 got the Kp wrong but luckily discovered the mistake immediately. Below are the Kp expression written by T6 and his value of Kp in terms of the total gas pressure, PT.

Then T6 went on to consider the disturbance. Very quickly, he calculated the total number of moles of molecules in the mixture after more nitrogen is added and listed the mole fraction of each gas. He proceeded to compute Kp in terms of PT as follows:

T6 was shocked to find that the value of Kp is not equal to 8.9/(PT)2. He checked his calculations over and over again. He did not understand why the equilibrium constant had been changed, indicating that he did not recognize that he was calculating the reaction quotient, Qp, rather than Kp. The following excerpt shows where Le Châtelier’s principle further hindered the problem solving of T6.

23

2

3

2

)(3.9

051.1197.0

051.1650.0

051.1204.0

))(()(

22

3

TTT

T

HN

NHp P

PP

P

PPP

K =

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

==

T6: Kp is equal to 9.3/(PT)2. It’s larger than 8.9/(PT)2. Why? The temperature is kept constant…Perhaps the value of Kp is affected by factors other than temperature…(long pause) Ahh, I know…The amount of ammonia in the new equilibrium state should not equal 0.204 mol because more nitrogen has been added. According to Le Châtelier’s principle, more reactants give more products. I: I see. Please continue. T6: Well, let x be the number of moles of nitrogen used to form the extra ammonia molecules. So, I can express the number of moles of each gas present in the new equilibrium mixture in terms of x. Let me write down the answers here…The amount of nitrogen is (0.650 – x) mol, hydrogen is (0.197 – 3x) mol, and ammonia is (0.204 + 2x) mol. So, the total amount of molecules is (1.051 – 2x) mol… Let me list the mole fraction of each gas here…The Kp expression can be written by plugging in the partial pressures…The total gas pressure, PT, appears on both sides of the Kp expression. So, I can cross it out. The expression can be simplified like this…(long pause)

I

)2051.1()2204.0()( 222−+ xxP

Tinsp

3

2

3

2

)197.0)(510.0()204.0(

))(()(

22

3 ==HN

NHp PP

PK

Tproteacthe to eHe comaboKon

23

2

3

2

)(9.8

911.0197.0

911.0510.0

911.0204.0

))(()(

22

3

TTT

T

HN

NHp P

PP

P

PPP

K =

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

==

TfITIw

102 | Chem. Educ. Res. Pract., 2009, 10, 97–108

: What are you thinking now?

9.8)3197.0)(650.0())(( 33

22

3 =−−

==xxPP

KHN

NHp

6: It isn’t a quadratic equation. An x4 term will appear n the numerator and denominator of the fraction. There is o formula to solve this kind of equation. I must have done omething wrong while solving the chemical equilibrium roblem. 6 told me that he had never attempted this kind of

blem in chemical equilibrium before. Like the other six hers in Category I, T6 switched back and forth between problem statement and the equilibrium constant expression xplore a way to make the mathematical equation solvable. checked his calculations repeatedly in hopes that some putational errors would be found. He also expressed doubt

ut the appropriateness of the numerical problem for Hong g secondary school students. 6: Is this chemical equilibrium problem really suitable

or Secondary 6 students in Hong Kong? : Yes, certainly. 6: But how can they tackle the x4 term?…Any hints?

: The chemical equilibrium problem can be solved ithout tackling the x4 term.

This journal is © The Royal Society of Chemistry 2009

T6: Really?...When more nitrogen is added to the mixture, the equilibrium must shift to the right in order to remove some of the excess nitrogen molecules. How can we confirm the right-shift without calculating the exact value of x? I don’t understand. I: Try to avoid concentrating on the confirmation of the right-shift. Think about a way to deduce the direction of shift in equilibrium based on the data given. T6: Umm…How? I: Let’s go back to the Kp expression that you wrote several minutes ago. You found that after more nitrogen is added to the mixture, the Kp does not equal 8.9/(PT)2. This is an important finding. The new value, 9.3/(PT)2, is useful. Do you know why? T6: The new Kp value is useful? …But the temperature is kept constant. The value of Kp should not change. It should equal 8.9/(PT)2. I: That’s correct. The new value, 9.3/(PT)2, refers to the situation immediately after nitrogen is added to the mixture. The system is not at equilibrium yet. How can the system re-establish equilibrium? T6: Umm…Because the temperature is kept constant, the molecules must undergo chemical changes so as to reduce the value of Kp from 9.3/(PT)2 to 8.9/(PT)2. How can this be achieved?...(long pause) I: Take a look at the nominator and denominator of your Kp expression. T6: Ok, the value of Kp can be reduced from 9.3/(PT)2 to 8.9/(PT)2 by shifting the equilibrium to the left. Adding more nitrogen to the ammonia synthesis reaction will shift the equilibrium to produce even more nitrogen molecules. I: Have you ever heard about this kind of change in the position of equilibrium? T6: No, I’ve never heard about this–I mean–adding more reactant to an equilibrium system can produce even more reactant. So, Le Châtelier’s principle is wrong. I didn’t know that Le Châtelier’s principle can make incorrect predictions. Now I understand why you said that Secondary 6 students have the ability to solve this chemical equilibrium problem. All they need to do is to apply the equilibrium law.

Category II: Teachers applied Le Châtelier’s principle and ignored the volume change

Table 4 summarizes the main steps in the reasoning path used by four teachers (T3, T5, T10, and T11). They computed the value of Kc successfully but failed to use the properties of Kc further to predict the direction of equilibrium shift after nitrogen is added to the system. Even worse, they did not recognize the volume change. T5, for example, calculated Kc correctly, but she applied Le Châtelier’s principle at an early stage and ignored the volume change when solving the problem. The following excerpt illustrates how the principle adversely affected her problem solving:

T5: Le Châtelier’s principle tells us that adding more reactants gives more products. The equilibrium will shift to the right hand side. Therefore, more ammonia will be

Table 4 Reasoning path IIA

Step Problem-solving behaviour 1 Read and interpret the problem. Underline or circle key words.2 Write the balanced chemical equation and list the number of

moles of each gas in the equilibrium mixture. 3 Set up the Kc expression and calculate the value of Kc. 4 Apply Le Châtelier’s principle to predict the shift in

equilibrium position. Let x be the number of moles of ammonia newly formed when the equilibrium is re-established.

5 Express the number of moles of nitrogen, hydrogen, and ammonia present in the new equilibrium mixture in terms of x.

6 Set up a new Kc expression. Attempt to solve for x but find that there is an x4 term in the ma hematical equation. t

formed…Let x be the number of moles of ammonia newly formed when the equilibrium is re-established. So, the equilibrium concentration of ammonia in the new equilibrium state is 0.204 + x. The new equilibrium concentration of nitrogen is 0.510 + 0.140 – x/2. And the new equilibrium concentration of nitrogen should be 0.197 – 3x/2. I: What are you going to do next? T5: In order to determine the value of x, I need to use the Kc expression again. Let me write it down here…(long pause) Oh, my God!

I: What is the problem?

67.10

23197.0

2140.0510.0

)204.0(][][

][3

2

322

23 =

⎟⎠⎞

⎜⎝⎛ −⎟⎠⎞

⎜⎝⎛ −+

+==

xxx

HNNH

Keqeq

eqc

T5: The Kc expression is very complicated. Because the coefficient of hydrogen in the balanced chemical equation is 3, the concentration of hydrogen in the Kc expression is cubed. There isn’t a simple way to solve for x…If it is a quadratic equation, then the solution can be found by using a formula….In the denominator of the Kc expression, the new equilibrium concentration of nitrogen also has an x. Concentration of nitrogen times the concentration of hydrogen must result in an x raised to power 4. It’s difficult to solve…(long pause). Secondary 6 chemistry students are not required to solve such a complicated mathematical equation…I must have made some mistakes.

It is important to note that although both T5 and T6 experienced similar difficulties in solving an x4 term, the cognitive sources were different. T5 applied Kc and ignored the change in volume when calculating the concentrations of gases. T6 applied Kp rather than Kc and the use of Kp did not require him to take the volume change into account when calculating the partial pressures of gases. During the second phase of the interview, I provided T5 with the following hints and guided her to develop a deeper understanding of chemical equilibrium:

I: OK, let me give you some hints. You avoid considering the concentrations of gases in the new equilibrium state. Assume that after nitrogen is added to the chemical system, there is only physical change. No chemical change. If you substitute the new concentrations into the Kc expression


and do the calculation again, do you think the answer will equal 10.67? T5: Well…the answer will not equal 10.67 because the chemical system is disturbed. 10.67 is the ratio of concentrations when the system is at equilibrium. I: Good, the answer will not equal Kc. Let’s call it Qc. Do you think Qc is useful? T5: Qc is useful?...But Qc is not identical with Kc, I mean the equilibrium constant….(long pause) I: Think about these questions. If Qc is smaller than 10.67, what will happen to the mixture? If Qc is larger than 10.67, what will happen to the mixture? T5: Uh-hmm, at equilibrium, Kc equals 10.67…(long pause). Ahh, if Qc is smaller than 10.67 after a disturbance, the mixture will bring Qc back to the value of Kc…The chemical system will shift to return to equilibrium. To increase Qc, the concentration of ammonia should increase or the concentration of hydrogen and the concentration of nitrogen should decrease…Well, the position of equilibrium must shift to the right hand side…If Qc is larger than 10.67, then …the position of equilibrium must shift to the left hand side to decrease the value of Qc. I: Correct. Now please calculate Qc. T5: Ok, let me write down the Qc expression here.

Although T5 understood the usefulness of Qc, she still did not recognize the importance of volume change and thus calculated the value of Qc incorrectly. The following excerpt shows what additional hints were given to T5:

T5: Qc is equal to 8.37. It’s smaller than Kc. To increase Qc from 8.37 to 10.67, the position of equilibrium must shift to the right hand side. More ammonia molecules will be produced. That’s it. I: I suggest you check your Qc expression again. Something is missing here…Also, take a look at the wording of the problem statement again. T5: 0.140 mol of nitrogen gas is added to the equilibrium mixture…I already considered this. 0.510 + 0.140 equals 0.650…No errors (laughs)…The calculation is correct. Qc is really equal to 8.37…(long pause) I: Try to consider the total gas volume. T5: Volume? …But the volume is equal to 1 litre. I: Really? The total volume is still equal to 1 litre after nitrogen gas is added? T5: Uh-hmm…Oh, yes, the volume will increase because the piston is movable.

Using the ideal gas equation, PV = nRT, T5 found out the new total gas volume correctly. Then, she calculated Qc correctly and was very surprised to learn that the position of equilibrium will shift to form even more nitrogen molecules. T3, T10 and T11 also applied Le Châtelier’s principle when they set up their Kc expressions. For example, T10 uttered confidently, “This is the famous Haber process. An increase in the concentration of nitrogen must cause the reaction to shift to the right in accordance with Le Châtelier’s principle.” She never checked the volume change during the first phase of

Table 5 Reasoning path IIB

Step Problem-solving behaviour 1 Read and interpret the problem. Circle and underline key words.

2 Apply Le Châtelier’s principle to predict that the equilibrium should shift to the right. Claim that the prediction can be verified quantitatively by comparing the values of Kc and Qc.

3 Write the balanced chemical equation for the Haber process and list the concentration of each gas in the initial equilibrium system.

4 Set up the Kc expression and calculate the value of Kc.

5

Find out the new concentration of nitrogen when the system is disturbed. Set up the Qc expression and calculate Qc. (But the calculation was incorrect because the total gas volume was taken as 1 litre.)

6 Conclude that Qc is smaller than Kc and thus the equilibrium should shift to the right. Point out that the result is consistent with that predicted by Le Châtelier’s pr ciple. in

the interview. T11 held the same misconception and made similar errors. He was not sure how the pressure of the system can be kept constant and thus queried, “How can the pressure remain constant? Adding more nitrogen gas must increase the total pressure. How come the total gas pressure can be kept constant? Impossible!” T3 did not realize that the piston is movable. With the help of my hints, she could pool all the essential ideas together at the end of the interview. She went back to problem statement and said, “I should have read the problem more carefully. The problem statement has specified the conditions clearly. The piston is movable and the disturbance is done at constant pressure. These two pieces of information is important because they imply that the volume of the reactor is not fixed.”

37.8)197.0)(650.0(

)204.0(]][[

][3

2

322

23 ===HN

NHQc

The think-aloud session with T8 revealed that he was the only teacher who could apply the concept of reaction quotient (see Table 5). But he just wanted to use reaction quotient to verify the prediction based on Le Châtelier’s principle.

T8: Kc is equal to 10.67. If more nitrogen is added to the mixture, Le Châtelier’s principle predicts that the equilibrium should shift to the right to counteract the increase in amount of nitrogen molecules. This prediction can be verified quantitatively by calculating the reaction quotient. I: I see. T8: First, I’ll calculate the value of Kc. Then, I’ll calculate Qc and compare Kc with Qc.

However, T8 finally failed to answer the problem correctly because he did not recognize the volume change. He thought that the concentrations of hydrogen and ammonia remained unchanged when calculating the Qc.

T8: The new concentration of nitrogen equals 0.510 + 0.140 mol divided by 1 litre. The concentration of hydrogen equals 0.197 mol divided by 1 litre. And the concentration of ammonia equals 0.204 mol divided by 1 litre. Let me substitute these values into the Qc expression…The value of Qc is equal to 8.37 and the units should be dm6 mol-2. Let me check my calculations again…Yes, 8.37 dm6 mol-2. So, Qc is smaller than Kc. The equilibrium must shift to make Qc equal Kc. To increase the value of Qc, the concentration of ammonia may increase or the concentration of nitrogen


may decrease. Well, this indicates that the position of equilibrium must shift to the right. This is the solution. So, the prediction based on detailed computations of Kc and Qc is consistent with the prediction made by application of Le Châtelier’s principle. I: I’ve interviewed several chemistry teachers. So far, you’re the only one who can utilize the concept of reaction quotient. But the value of Qc is incorrect. I suggest you check the problem statement and your Qc expression again.

T8 calculated the value of Qc again but still found that Qc is equal to 8.37. I asked T8 to read the problem statement carefully. Below are his responses:

T8: Ahh…the piston is movable. The volume will expand…larger than 1 litre. I: That’s correct. Can you determine the total gas volume immediately after nitrogen is added to the equilibrium mixture? T8: The ideal gas equation, PV = nRT, can be applied to calculate the new volume. In the original equilibrium mixture, the total amount of molecules is equal to …0.911 mol. After 0.140 mol of nitrogen is added, the total amount of molecules will be equal to 1.051 mol. So, the new volume is…1.15 litres. I: Correct! T8: Let me set up the Qc expression again. The new concentration of nitrogen equals 0.510 + 0.140 mol divided by 1.15 litres. The concentration of hydrogen equals 0.197 mol divided by 1.15 litres. And the concentration of ammonia equals 0.204 mol divided by 1.15 litres.…The value of Qc is equal to 11.08, which is larger than Kc…But according to Le Châtelier’s principle, Qc should be smaller than Kc. Otherwise, the equilibrium will not shift to the right. Let me calculate Qc again…(long pause) The value of Qc is really larger than Kc.

T8 was very surprised when he found that the equilibrium should shift to the left to form more nitrogen molecules. At the end of the interview, T8 understood that the conditions under which the Haber reaction is disturbed are very important because they can result in opposite shifts of the equilibrium. He also understood why Le Châtelier’s principle cannot make a correct prediction in this case; that is, when Le Châtelier’s principle is applied, it is assumed that only the concentration of nitrogen is changed while the concentrations of hydrogen and ammonia remain unchanged. Actually, the concentrations of all the three chemical species are changed because the pressure is kept constant by fitting the reactor with a movable piston.

Discussion The interviews found that the twelve experienced secondary school chemistry teachers were good at calculating the equilibrium constant. One possible reason is that calculating equilibrium constants is a key part of the school chemistry curriculum in Hong Kong. Teachers usually provide students with a lot of practice in classroom. Textbooks also have this kind of calculation exercises. However, the interview data revealed that these experienced teachers still had serious difficulties in solving a simple chemical equilibrium problem,

indicating that the ability to calculate Kc or Kp does not ensure a deep understanding of the properties of equilibrium constants. Analysis of the data found that their difficulties had resulted from three major sources: too much reliance on the logic of Le Châtelier’s principle; inability to consider the properties of equilibrium constant and reaction quotient; and inability to recognize volume change. All twelve teachers held the misconception that one can always cause a reversible reaction to shift to the right by increasing the concentration of a reactant. Clearly, the ‘change-then-minimize’ logic of Le Châtelier’s principle adversely influenced their understanding of chemical equilibrium. Analysis of the contents of the answer sheets revealed that they showed the mathematical equations but not the details of reasoning that led to them. Only five teachers put down Le Châtelier’s principle explicitly. Thus, without the think-aloud method, it would have been difficult to identify Le Châtelier’s principle as the source of misconception about chemical equilibrium for all the teachers. Seven teachers were in Category I. Their ability to solve the problem was chiefly constrained by the lack of conceptual knowledge of equilibrium constant and reaction quotient. This finding is not unexpected, because many textbooks do not utilize the equilibrium law and reaction quotient to predict the effects of changing conditions on chemical equilibrium. An inspection of textbooks of high school and college chemistry found that many textbook writers do introduce reaction quotient (e.g. Van Kessel et al., 2003; Irwin et al., 2006; Ash and Hill 2008), but they do not link it to analysis of factors affecting chemical equilibrium. For example, the textbook written by Irwin et al. (2006) includes both Le Châtelier’s principle and reaction quotient. Yet this textbook gives a 4-page introduction to the use of Le Châtelier’s principle to analyze how concentration, volume, and pressure affect the position of equilibrium, without mentioning reaction quotient. This wastes the opportunities to demonstrate the usefulness of reaction quotient. The fundamental concept of chemical equilibrium is thus neglected in favour of Le Châtelier’s principle that has no theoretical basis. Similarly, although Ash and Hill (2008, pp. 453-464) introduce reaction quotient, they do not discuss how the effects of changing conditions on chemical equilibrium can be directly predicted on the basis of the properties of the reaction quotient. Five teachers were in Category II. They did not pay attention to the conditions under which nitrogen is added to the system. They just focused their attention on the variable whose change is most evident. This is a way of thinking found in many students who have misconceptions in science (Talanquer, 2002). The interview data showed that, like their students, these five chemistry teachers also possessed this kind of narrow thinking. In addition to a change in the concentration of nitrogen, they should have noted that the concentrations of hydrogen and ammonia are also changed when the equilibrium system is disturbed at constant pressure and temperature. Never during their problem solving processes did they point out the change in total gas volume. One of the possible reasons for this is that many textbook writers (e.g. Wong and Wong, 2005; Irwin et al., 2006; Ash


and Hill, 2008) do not specify the conditions under which an equilibrium system is disturbed. For example, Ash and Hill (2008, p. 469) ask students to predict the changes in the equilibrium concentrations in 2NH3(g) + energy ⇌ 3H2(g) + N2(g) if nitrogen is removed from the system, without specifying whether the volume and temperature are kept constant. It is worth noting that the chemical equilibrium problem used in this study is not anomalous and fits well in the Hong Kong school chemistry curriculum; it is indicative of a category of problems that cannot be solved by Le Châtelier’s principle (Lacy, 2005). Unfortunately, few chemistry textbooks discuss this kind of constant-pressure problem. Also, it is important to note that this kind of constant-pressure problem is not particularly complex. As de Heer (1958) pointed out, when chemical equilibrium systems are disturbed, changes are seldom restricted to one variable, and multiple changes constitute a complex process only if they are independent to each other. Adding more nitrogen gas to the N2(g) + 3H2(g) ⇌ 2NH3(g) system at constant pressure and temperature is not a complex process because the increase in volume is affected by the amount of nitrogen gas added. For example, if we add nitrogen gas and modify the temperature of the system simultaneously, then these two changes will constitute a complex process. Few past studies provided excerpts of teachers’ cognitive processes to document their misconceptions about chemical equilibrium. A unique feature of this research design is that the think-aloud technique was not only used to gain access to the teachers’ minds but also to promote further learning by providing hints to address their misconceptions. However, a disadvantage of this technique was that it was very time consuming to collect and analyze the think-aloud protocols.

Conclusions and implications Chemistry educators worldwide have long been asking the question: how do we best help secondary school students understand chemical equilibrium? Students are always considered to be the source of problems whenever they have misconceptions about chemical equilibrium after teacher instruction. However, the results of this study revealed that acquiring a degree in chemistry and gaining chemistry teaching experience do not ensure that secondary school teachers have a deep understanding of chemical equilibrium. The think-aloud protocols provided evidence for two categories of difficulty experienced by teachers while solving the chemical equilibrium problem. The teachers failed to solve the problem not because they made purely computational errors, but rather they held a misconception; that is, one can always cause a reversible reaction to shift to the right by increasing the concentration of a reactant. This misconception is due to an over-emphasis of the ‘change-then-minimize’ logic of Le Châtelier’s principle in our school chemistry curriculum and textbooks. Understanding secondary school chemistry teachers’ difficulties is critically important because teachers cannot help students understand what they themselves do not understand (Hashweh, 1987; Carlsen, 1991). I believe that

there are three implications from this exploratory study. First, this paper presents empirical evidence about how application of Le Châtelier’s principle will create barriers to understanding the properties of equilibrium constant. Therefore, if secondary school teachers continue to rely on Le Châtelier’s principle to teach factors affecting chemical equilibrium, there is little hope for their students to develop a deep understanding of the properties of equilibrium constant. Chemistry teacher educators must share the responsibility for the transmission of subject matter knowledge to prospective teachers (Grossman et al., 1989). They should teach the inadequacies of Le Châtelier’s principle explicitly in their chemistry teaching-methods courses in order to enhance preservice and inservice teachers’ knowledge for teaching chemical equilibrium. It is important to bear in mind, when analyzing factors affecting chemical equilibrium, that changes are rarely restricted to only one variable. The problem used in the present study may be treated as a numerical exercise. Second, professors of physical chemistry often use a thermodynamic approach to teaching chemical equilibrium at university. They discuss the quantitative relationship between the free energy change for a reaction and reaction quotient. They also show that when a system is at equilibrium, the free energy change is equal to zero and the reaction quotient is equal to the equilibrium constant. The teachers who participated in the present study were all chemistry majors. However, prior to being interviewed, eleven of them were not clear about what the concept of reaction quotient is. This indicates a gap in chemistry learning at the university level. Professors of physical chemistry may need to think about why chemistry majors have not developed the ability to reason beyond the ‘change-then-minimize’ logic of Le Châtelier’s principle. The curriculum contents of some undergraduate physical chemistry courses may also need to be revised to address this gap in chemistry learning. Third, the chemistry curricula or syllabuses prepared by examination boards worldwide should be reviewed, because research has indicated that the required school curriculum is the most powerful determinant of teacher content knowledge (Arzi and White, 2008). It is unfortunate that the school chemistry curricula developed by examination boards or governments in many countries recommend teachers to teach only Le Châtelier’s principle. For example, in Australia, the Board of Studies (2002) has included only Le Châtelier’s principle in the chemistry syllabus. Thus, educating chemistry educators worldwide on the inadequacies of Le Châtelier’s principle should be a high priority. The findings from this qualitative exploratory study allowed chemistry educators to begin to delineating the categories of cognitive processes that account for secondary school chemistry teachers’ difficulties in solving chemical equilibrium problems. But the study has limitations and I wish to suggest two directions for future research efforts. The findings are based on a small convenience sample of chemistry teachers. It is likely that there are other categories of cognitive processes used by chemistry teachers. More research should be undertaken to identify other reasoning patterns. Another limitation of this study is that I investigated


teachers’ thought processes as they worked on one particular type of chemical equilibrium problem. Research (Cheung, in press) has confirmed that secondary school chemistry teachers also have difficulties in solving other types of equilibrium problems (e.g., the effect of addition of an inert gas on chemical equilibrium, the effect of a decrease in the volume of the reaction container). Further research is needed to investigate how school chemistry teachers’ subject matter knowledge can be enhanced so that they can transfer their learning to solve new problems in chemical equilibrium.

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