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[ Published as: Harrison, A., & De Jong, O. (2005). Exploring the use of multiple analogical models when

teaching and learning chemical equilibrium. Journal of Research in Science Teaching, 42, 1135-1159.]

Exploring the use of multiple analogical models when teaching and learning chemical equilibrium

Allan G. Harrison1, Onno De Jong 2

1 Central Queensland University, North Rockhampton, Queensland 4702, Australia

2 Utrecht University, Utrecht 3584CC, The Netherlands.

(1) A.harrison@cqu.edu.au

(2) O.dejong@phys.uu.nl

Abstract

This study describes the multiple analogical models used to introduce chemical equilibrium, examines the teacher’s reasons for using models, explains the model development during the lessons and analyses the student understandings that were derived from the models. A case study approach was used and the data were drawn from the observation of three consecutive Grade-12 lessons on chemical equilibrium, pre- and post-lesson interviews, and delayed student interviews. The prominent analogical models used in teaching were: the ‘school dance’, the ‘sugar in a teacup’, the ‘pot of curry’, and the ‘busy highway’. The lesson and interview data were subject to multiple, independent analyses and yielded the following outcomes: The teacher planned to use the students’ prior knowledge wherever possible and he quickly responded to student questions with stories and extended and enriched analogies. He planned to discuss where each analogy broke down but did not. The students enjoyed the teaching but built variable mental models of equilibrium and some of their analogical mappings were unreliable. Female students disliked masculine analogies, the students tended to see the elements of the multiple model set in isolation, and some did not recognize all the analogical processes that were the focus of the teaching. Most students learned that equilibrium reactions are dynamic, occur in closed systems and the forward and reverse reactions are balanced. We recommend the use of multiple analogies like these and insist that teachers always show where the analogy breaks down and carefully negotiate the learning outcomes of multiple models.

Introduction

The past 20 years has seen a growing interest in the teaching and learning of science using models (e.g., Bent, 1984; J. Gilbert, 1993; S. Gilbert, 1991; Harrison & Treagust, 2000a; Mayer, 1992). In line with this interest, the present paper focuses on the use of models in teaching and learning chemistry. There is good reason for this interest because most chemistry explanations use analogical models that are agreed upon by scientists, teachers or textbook writers (Harrison, 2001). For instance, formulae and equations symbolically represent reactions, ball-and-stick molecular models help students visualize shapes, and the kinetic theory models matter and helps explain reaction dynamics. Models like these change little from lesson to lesson, year to year and textbook to textbook because these “pictures and models of molecules are important [learning] aids” (Pimental, 1963, p. 31). Another reason for using models is the belief that modeling is the essence of science and that students should

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create, modify and refine models to demonstrate their growth in understanding (Grosslight, Unger, Jay & Smith, 1991). Pimental points out that “a model grows as it is tested in an ever-widening range of experience” (1963, p.34) and often its complexity increases with its usefulness. In a physics example, Cosgrove (1995) described how Grade-9 students, who were given time and opportunity, evolved an effective model of electric current. In school chemistry, however, modeling is mostly limited to students exploring bonding and molecular shapes using proprietary model sets (e.g., Harrison, 1997) with little creative modeling taking place. The most adventurous examples of modeling in school chemistry are the multiple models that teachers use to explain kinetic theory, chemical reactions and equilibrium. Consequently, this paper examines the use of multiple analogical models by a teacher as he explained chemical equilibrium to his Grade-12 students.

Background

Models enhance understanding because some part(s) of an everyday object or process resembles some part(s) of a scientific object or process: for example, atoms in a flask are like “super-rubber balls confined in a box” (Parry, Deitz, Tellefsen & Steiner, 1970, p.21). When an analogical model is used to transfer or construct information, the analog (or source) refers to structures, phenomena or processes in the everyday object or event that credibly map onto structures, phenomena or processes in the target scientific concept (Duit, 1991; Gentner, 1983; Glynn, 1991). In the above example, the shared attributes or positive analogy between the analog and target comprise: moving balls in the box are like moving atoms in the flask; elastic collisions between balls are like perfectly elastic inter-atom collisions; and balls hitting the box wall apply a force like the pressure atoms produce when they bounce off the flask’s wall. If students think the air between balls is like the space between atoms, this may reinforce the alternative conception that there is a substance between atoms and molecules (Lee, Eichinger, Anderson, Berkheimer & Blakeslee (1993). Every analogy breaks down somewhere and unshared attributes like this one explain why Glynn (1989) and Duit (1991) call analogies, “two-edged swords”. Authors like Duit, Gentner and Glynn also point out that the analog is a simplified representation of a target and focuses attention on analog--target similarities in a way that facilitates scientific inquiry.

In some models the analog’s structure, phenomenon or process is exaggerated to attract students’ attention to the analog attributes that teach about the target. This often is the case in chemistry when everyday objects like billiard balls model sub-microscopic particles. Focusing on specific aspects of a target not only implies certain similarities between model and target, but also implies that there are certain differences between model and target. Thus, models can facilitate scientific inquiry when they describe, explain, and predict relationships and functions. A rich overview of literature on the nature and significance of models in science education is given in Gilbert and Boulter (2000).

Categorizing Models

Models can be categorized by looking at their mode of representation, the way they are classified and the conceptual demands different models place on their users. Under mode, external representations can be described in terms of appearance, for instance, concrete models (such as a 3D plastic heart), visual models (such as a drawing of the heart), and mixed mode representations (Gilbert and Boulter, 2000). Internal representations can be described in terms of personal cognitive interpretations of a target, which are dynamic in character and often are called mental models (Greca & Moreira, 2000). In their study of textbook analogies and models, Curtis and Reigeluth (1984) classified analogies under three headings based on each analogy’s degree of elaboration. The most common type was simple analogy where the writer said something like “a cell is like a box” or “activation energy is like a hill” and left the student to interpret how a cell is like a box. The second type, enriched analogy, includes the grounds for the likeness. For example ‘activation energy is like a hill because you have to add energy to reacting substances to start the reaction’. Thus, the difference between a simple structural analogy and an enriched functional analogy is the addition of some form of causation and the enriched analogy is more explanatory. Curtis and Reigeluth’s third type, extended analogy, contains multiple simple and/or multiple enriched analogies. The “eye is like a camera” analogy is an extended analogy (Glynn, 1989) because the grounds on which an “eye is like a camera” are stated in each case and there are multiple shared attributes in the analogy (and some limitations or unshared attributes).

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Harrison and Treagust (2000b) proposed a typology of analogical models that accounts for the increasing complexity and conceptual demands of models. This classification arranges model types in a continuum ranging from simple to complex, concrete to abstract as summarized below: concrete scale models (atoms are like balls) pedagogical analogical models (ball-and-stick molecules) iconic symbolic models (formulas and equations) mathematical models (PV = nRT), theoretical models (kinetic theory) concept process models (e.g., the four redox models). The predictive power of chemistry models is especially evident in the periodic table and systematic models like reaction mechanisms. In school chemistry, models are embedded in the language and conceptual structure of chemistry and contrast with the more everyday analogies that are used to model biology and physics concepts. Everyday analogies are the system of water flowing in pipes (Glynn, 1991) and the continuous train for electric circuits (Dupin & Johsua, 1989), and multiple pumps for the function of the heart (Venville & Treagust, 1993). In contrast, the embedded nature of chemical models can desensitize teachers and students to the models’ analogical and metaphoric nature. We note that Duit (1991) comprehensively reviews the literature on the nature and use of analogies and models in science education.

The Use of Analogical Models in School Science

Studies of analogy use in the classroom are not often compared with studies on the use of other kinds of models. Indeed, some teachers say that they are wary of analogies because their students misinterpret them but are happy to use models (Harrison, 2001). One group of studies examined the use of analogies in learning to solve quantitative problems (e.g. Friedel, Gabel, & Samuels, 1990; Klauer, 1989), while another group focused on concept learning promoted by analogy use. In the latter group, most studies focused on how teachers used analogies in their regular teaching (e.g. Dagher, 1995; Jarman, 1996; Treagust, Duit, Joslin & Lindauer, 1992). A minority of studies also examined student learning in classrooms when analogical models were used (e.g., Dupin & Joshua, 1989; Harrison & Treagust, 1993) but most of these studies were concerned with physics and biology topics.

Thiele and Treagust (1994) observed four chemistry teachers’ teaching and found that they used planned analogies to explain abstract concepts to the whole class and inserted spontaneous analogies when groups of students demonstrated a lack of understanding. The teachers drew on their own experiences as a source of analogs and half of the teachers provided clear statements of analogical limitations. Later still, Harrison and Treagust (2000) described conceptual change learning when multiple models were used to teach atoms and molecules in Grade-11 chemistry. Multiple models are more popular in physics (e.g., “bridging analogies”; Clement, 1993), therefore, this study is interested in the effectiveness of multiple models in explaining chemistry concepts like chemical reaction rates and equilibrium.

The effectiveness of analogical models is related to the teacher’s explanatory ability, the students’ familiarity with the analog, the ability of both to map the analogy and interpret its conceptual similarities, and to intrinsic factors in the model itself. Gentner (1983) therefore recommends analogies whose surface similarities provide easy student access to the analogy and also develops conceptual relationships. No analogy covers all features of the target under consideration but multiple analogies can address most aspects of a conceptual target (Gentner & Gentner, 1983). For this reason, multiple analogies are more effective that single analogies, especially when the target concept is complex and ill-defined (Thagard, 1992).

The ways teachers present analogies and models also is important. Glynn (1989) proposed a six-step Teaching-With-Analogies (TWA) model comprising: 1) introduce the target concept, 2) check student familiarity with the analogical context, 3) establish the analogy, 4) map the shared attributes, 5) summarize the analog-target outcomes and 6) show where the analogy broke down. Harrison and Treagust (1993) evaluated the TWA model and showed that a better order was 4) map the shared attributes; 5) negotiate when and how the analog broke down; and 6) summarize the learning outcomes. The modified model still had too many steps for teachers to remember in class and they often left out steps 5) or 6). Researching the ways teachers plan, teach and reflect on their lessons led Treagust, Harrison and Venville (1998) to construct and evaluate the Focus-Action-Reflection (FAR) guide. Here, the in-class Action involved only steps 4-6 of the TWA model (likes, unlikes and conclusion) and teachers found these steps easier to remember and implement.

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Studying Equilibrium Teaching

With these ideas in mind, the present study reports an expert chemistry teacher’s use of multiple analogical models in his chemistry classroom when teaching chemical equilibrium. Chemical equilibrium is considered to be complex, counterintuitive and its learning is limited by common alternative conceptions (Hackling & Garnett, 1985; Niaz, 1998, Bucat & Fensham, 1995). It is one of the central organizing topics in chemical education, and includes several important sub topics, such as reversible reaction, reaction rate, energy effects, and dynamic equilibrium. Many students experience difficulty understanding chemical equilibrium and believe, for instance, that the forward reaction goes to completion before the reverse reaction commences, or hold the idea that at equilibrium no reaction is taking place, so, ‘nothing happens’. van Driel and Gräber (2002) provide a detailed overview of students’ conceptual difficulties with equilibrium.

As far as we know, there is just one study of analogy use in lessons on chemical equilibrium (Thiele & Treagust, 1994). That study used the student ball/dance analogy (increasing student activity leads to an increase of the number of body collisions and is like increased particle kinetic energy increasing reaction rate) and people moving into and out of a shop to represent the rates of forward and reverse reactions. Thiele and Treagust (1994) reported the teachers’ use of the analogies but did not explore their student’s understanding. The present study attempts to fill this gap by studying both the teacher’s presentation and the students’ understanding of multiple equilibrium analogies. Specifically, this study focuses on the teaching and learning of chemical equilibrium as a dynamic event in a closed system where the forward and reverse reactions run simultaneously and with the same rate. Because the study is naturalistic, it is presented as a case study, involving one teacher and his students. The study focuses on qualitative aspects such as the ways analogies are used in the classroom and the meanings that the students derived from the teacher’s analogies.

Method

The study is part of a large 3-year funded project studying teacher explanations. The study of chemical equilibrium fitted an invitation by a cooperating teacher, whom we call Neil Scott, to study him and his chemistry class. Studying analogical models is a central element of the project and both authors have extensive experience in studying and analyzing analogy use. Two research questions guided our study of Neil’s analogical models and the learning experiences of his Grade-12 students:

1. Which multiple analogical models are used to develop the conditions for chemical equilibrium, how are the analogies developed, and for what are the teacher’s intentions?

2. What sense do students make of the multiple analogical models used by their teacher to help them to understand conditions of chemical equilibrium?

Participants

Neil is an experienced chemistry teacher who has taught chemistry for 18 years at four senior high schools. His present high school is a large metropolitan fee-paying college (Grades 8-12) in Queensland, Australia. Neil, is the science curriculum leader, a role he has held for eight years at this and his previous school. He is a supporter of the school’s view on teaching, namely. Teaching-For-Understanding (Perkins, 1992), and he also has developed a series of understanding goals from the current chemistry syllabus.

The class involved in the study consisted of 11 students (three female, eight males) in Grade-12 (average age: 17 years). This is their second year of senior chemistry and Neil has taught this class for most of this year. Neil estimates that this class is slightly lower achieving than his previous Grade-12 classes.

Chemistry Classroom Context

The lessons that were relevant to the study occurred over two days. During the first day, a double lesson (80 minutes) was given, starting with a recapitulation of the particulate nature of chemical reactions, and factors that influence reaction rate, followed by the introduction of activation energy, reaction profile diagrams, and the conditions for chemical equilibrium. The next day, a single lesson (40 minutes) further elaborated the concept of dynamic equilibrium. Teacher demonstrations or practical work for students were not included in these lessons.

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The topics were presented and discussed in an interactive way with Neil and his students asking many questions. Despite the presence of observers (both authors) in the classroom, the interactions seemed normal. Neil and the first author are familiar with each other because of preceding common research projects. The students were not familiar with the observers, but welcomed us and, after a short time, appeared to take little notice of us. The lessons proceeded in an open and natural way with minimal interference from the observers.

Data collection and analysis

Research data were collected at specific moments, as indicated in the overview below.

- Pre-lesson interview (lesson 1 and 2), focused on teaching intentions

- Chemistry lessons 1 and 2 (double lesson) on aspects of chemical equilibrium

- Post-lesson interview (lesson 1 and 2), focused on teaching reflections

- Pre-lesson interview (lesson 3), focused on teaching intentions

- Chemistry lesson 3 (single lesson) on features of dynamic equilibrium

- Post-lesson interview (lesson 3), focused on teaching reflections

- Delayed interviews with students, focused on the analogies that have been taught

The teacher interviews were conducted by both authors and audio-taped. The lessons were also audio taped, with both authors observing and making detailed notes, including copies of all Neil’s whiteboard notes, diagrams, and drawings. Seven student interviews were conducted about 10 weeks after the lessons to gain an insight into the students’ medium-term conceptual understanding of chemical equilibrium (Treagust, Harrison, Venville, & Dagher, 1996). The student interviews were conducted by the first author, focused on everyday equilibrium examples, and were audio-taped. All interviews and lessons were transcribed verbatim.

The data were analyzed from an interpretative phenomenological perspective (Smith, 1995) and aimed at producing interpretations that are credible, transferable, and dependable (Guba & Lincoln, 1989). To enhance rigor, all of the transcripts were independently read and interpreted by both authors. When necessary, the observation notes were used. Where possible, each interpretation was considered from a confirming and a disconfirming perspective. Comparing and discussing the individual analyses (investigator triangulation, Janesick, 1994) yielded what we believe are coherent conclusions. The interpretations are reported in as transparent a way as possible so that the reader can interrogate the findings and recommendations. We recognize, however, that our preference for analogical model-based explanations may color our findings. For this reason, we have presented rich data to enable the reader to critically share the analogical models that Neil used to explain equilibrium.

Data, Interpretations and Findings

In the course of the three lessons, Neil discussed 10 analogical models (see Table 1). One of the models was proposed by a student (Table 1, item 9). Five of the models are reported here and are discussed in detail (Table 1, items 1, 7, 8, 9, 10) because they were given much more attention in the classroom than the remaining models. The data, interpretations and findings are presented in the order in which they appeared in the lessons.

Table 1

Analogical models, in order of appearance in the lessons on chemical equilibrium

Analog (familiar situation) Target (science concept)

1. School dance (restricted version) Chemical reaction, reaction rate depends on rate of boys colliding with girls

2. Up- and down skier Activation energy, energy in before energy out

3. Air flight including route details Reaction mechanism, many steps produce the overall effect

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4. Assembling a model aircraft or car Reaction mechanism, many steps, some parallel like assembling two identical wings

5. Balancing on a see-saw Physical equilibrium; force x distance is balanced on each side

6. Being normal and insane Physical equilibrium is like being mentally stable

7. School dance (elaborated version) Conditions for chemical equilibrium; couples committing and breaking up is continuous, rate committing = rate breaking up, commitment room smaller than the hall (incomplete reaction) and the hall is sealed

8. Excess sugar in a teacup (covered) Dynamic nature of equilibrium; cup sealed, rate dissolving = rate precipitating; process continuous, temperature dependent

9. Pot of curry (lid in place) Dynamic nature of equilibrium; amount of water evaporating = amount condensing; continuous while simmering, sealed pot = closed system

10. Busy highway Dynamic nature of equilibrium; rate cars entering = rate of cars leaving, collision rate is important

Pre-lesson interview (lesson 1 and 2).

Neil announced that he was concluding reaction rates and that he would use this as a foundation for chemical equilibrium. We asked him about the problems he expected to encounter and he spoke at length, and in rather general terms, about the conceptual and practical difficulties involved in teaching abstract concepts. From these discussions, we derived an image of Neil as a ‘thinking teacher’ who reflects on ‘what went well last time, what went wrong, and how to do better next time’.

The discussion gravitated to his planning for the next lesson on equilibrium and he stated his intention to use an analogical model. He wanted to impress on the students that an analogy is always artificial, and he wanted them to find the flaws, to find where the analogies break down because, in his opinion, all analogies have limitations. The planned analogy for bridging reaction rates and equilibrium was the ‘school dance’ scenario where everyone is blind-folded. Boys have stubble and girls have their hair in pony-tails. Feeling the head of a potential partner is the only way boys and girls can pair off. Boy-girl meetings represent compatible atom collisions, and when the couple goes to a side room (the “commitment room”) and commits to each other, this represents a chemical reaction and bonding occurs. After linking the effects of concentration, temperature and surface area on reaction rate, he intended to raise a problem: In the analogy there are 500 boys and 500 girls in the hall to start with, but the commitment room can only hold 250 couples. Only when one of the 250 couples in the commitment room splits, can a new couple enter and commit. This introduces the idea of balanced and dynamic forward and reverse reactions. According to Neil, teenagers who make and break relationships easily understand the dynamics of this analogy. When asked about the functions of the school dance’ model, Neil responded by saying that he tries to present something that the students can visualize, that can help them understand what is happening in the microscopic world and entertains and maintains their interest.

Neil then talked at length about the dynamics of teaching and learning analogies, especially in the case of low achievers, whom he would help by asking them to retell the analogy and relate it to the target in their own words. He concluded that he intended using the analogy as a lead in to equilibrium, because he felt the students have not previously encountered reversible reactions. He also told how he likes to introduce analogies without telling the students where he is going. He felt that if he started by announcing the concept he wanted to explain (e.g., activation energy) “they will tune out”. He enjoyed keeping them wondering.

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Finally, Neil presented a Teaching-For-Understanding concept and criteria sheet (performances of understanding) for reaction rates and equilibrium. The criteria closely match the processes and concepts that he intended to develop using his equilibrium story.

Chemistry Lessons 1 and 2 (double lesson)

(a) The ‘school dance’ analogy (condensed version). Neil started with a recapitulation of the “school dance” analogy and its links to chemical reaction and reaction rates.

Neil Let’s go back to the dance in the gym … what do you remember about the dance in the gym?

Sts. Filled with people … Half boys, half girls.Neil … you remember that … What happened?Sts. Everyone had to walk around till they found someone they could partner up with, then

they had to go to … the commitment room.Neil … where they made life long commitments with one another … and once you took the

blindfolds off you knew what you were stuck with … you remember that far … What did the commitment represent? The fact that the two things got together, what was that representing?

Sts. Bonding … a reaction.Neil Right, the bonding, the two things coming together which is in general terms, what? [St.

A reaction] A reaction. So, we were looking at a chemical reaction … What were some things that we changed?

St. The number of people in the room.Neil Right, we changed the number of people. What does that actually represent? [St.

Quantity] OK, we’ll push the word concentration … So, we’re looking at the number of people in a room, so that’s changing concentration, the number of people. Right, did anything else vary?

Sts. The way they could find someone of the opposite sex … If they were running instead of walking, they found someone faster.

Neil OK, what would that represent if they were running rather then [St. Heat] … we change the temperature and they now become faster, so that will represent … [St. Rate] A faster rate. So … we can say if you increase the concentration of the reaction of the reactants what happens to the rate? [St. It increases]. And you increase the temperature? [St. Increases] Increases. Now … when it came to surface area … that was where I felt it broke down. Did we come up with something? I know someone came up with something.

St. Bigger basketball court.Neil If we made a bigger room, is that changing the surface area of the actual reactants or is that

something else? St. … any part instead of just being the head region, any body part.Neil OK, so, that’s heading into dangerous territory … instead of just above the neck, the whole

body becomes an area, so if you actually touch feet.... So that is a greater chance of … two combine and go off to the commitment room. Actually I’m very pleased that you remember so much.

This episode shows that the ‘school dance’ analogy was not only used to highlight chemical reaction itself, but also several factors that influence reaction rate, namely, concentration, temperature and surface area (see Table 2). When talking about the surface factor, Neil indicated that the analogy had broken down, but he did not discuss this issue further, despite his clearly stated intentions in the pre-lesson interview. Neil recapitulated the analogy in a very interactive way, checking that the students shared his version of the story before developing it further, although he did not ask any low achieving student to tell the whole story in his or her own words, despite his clearly stated intentions in the pre-lesson interview.

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Table 2

The “school dance” analogy for chemical reaction, reaction rate, and equilibrium conditions

Analog (familiar situation) Target (science concept)

Chemical reaction

Dancing students in the gym (reactants) Moving and colliding particles

Commitment between students Chemical bond (products)

Reaction rate

Number of people in the gym Concentration effect on reaction rate

Speed of running students Temperature effect on reaction rate

Extent of body contact Surface area effect on reaction rate

Equilibrium conditions

Couples going in and out the commitment room Simultaneous forward and reverse reaction

Couples going in and out at the same time Rate of forward = rate of reverse reaction

Gym doors are sealed Reaction system is closed

(b) Intermezzo with more analogical models. After the “school dance”, Neil explained two related concepts using five more analogies. His activation energy explanation included the up- and down-hill skier; then he explained reaction mechanisms in terms of routes using the detailed itinerary of his trip from the Gold Coast to New York; and concluded with an analogy of how to assemble a model aircraft in stages. He also talked about everyday equilibrium using the analogies of a see-saw and losing your equilibrium and becoming insane.

(c) The ‘school dance’ analogy (elaborated version). In the third lesson, Neil introduced conditions for chemical equilibrium by asking his students the following question (can a forward and reverse reaction run at the same time).

Neil A plus B gives AB, but at exactly the same time, AB is breaking up by giving A plus B together … Can that happen?

St. … It’s a cycleNeil A cycle?St. Does it really?Neil … Can A and B be combining to form AB, at the same time another AB is breaking up to

form A plus B?St. But only in theory, right? It would not happen in real life…

Neil repeated his question and subsequently returned to the ‘school dance’ analogy. He used the difference between the dance floor and the commitment room in the following way (see also Figure 1).

Neil I need a little bit of energy to combine A and B to form AB and I need more energy to break up AB into A and B.… Now believe it or not, … a reaction doesn’t necessarily take all of A and all of B to form AB and go until every A and every B is used up.

St. What are you trying to say?Neil I’m trying to say this is important … a reversible reaction leads us into the chemistry of

equilibrium ... St. You said that while A and B react to create AB, AB creates A and B at the same time…

Would it keep going?Neil Let’s go to the commitment room. OK, AB, A and B. Now we have a slight problem. We

have 1000 students in the gym. If eventually each one of the 500 girls find the 500 boys and they go to the commitment room, I’m going to have 1000 students in the commitment room, which is a good deal smaller than the gymnasium. So, I may have to stand at the door … and count. I know the commitment room can take 500 people. 500 is 250 couples.... the

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250th couple walks in and I tick them off, the 251 st couple comes to the door, and I say two things. “Sorry, you can’t come in, the commitment room is full”, or I make a [public] announcement, “If anyone in the commitment room has found someone by chance that they don’t particularly want to spend the rest of their life with … one couple may go out into the gym and search again. Put your blindfolds on and off into the gymnasium.”

St. One couple?Neil One couple, AB. The minute they come out they are sent into the world. Now, that allows

the other couple … they come into the commitment room because I can allow 250 couples. So someone else comes to the door, and I say, “anyone else picked unwisely, fine you can leave. You’re off the hook, someone else can come in”. Now I think you can see how this can happen simultaneously … together. Now for this to work, what are my conditions?

St. You can only have so many in the room Neil Yeah, I can’t have people coming in and out of the gym, the gym is sealed…. That would

ruin my concept if I didn’t bolt the doors of the gym. So I’ve got to have my system closed from the world, that’s one of the conditions of equilibrium…. So A and B can form AB, AB forms A and B, so that can happen simultaneously … In summary, we’ve talked about reaction rates…. For equilibrium to occur, we discovered … What is the rate of people per minute committing as opposed to people uncommitting?

St. It’s the sameNeil … Another thing that defines equilibrium is not only, it’s a closed system, but the rate of

the forward reaction equals the rate of the reverse reaction, another thing that defines equilibrium.

This lesson episode shows that the ‘school dance’ analogy was elaborated to highlight three chemical equilibrium conditions: (a) forward and reverse reaction run simultaneously, (b) with the same rate, (c) in a closed system (see also Table 2).

Our perception is that separating the reactants (in the gymnasium) from the products (in the commitment room) creates too distant an image (from the students’ cognitive point of view) of forward and reverse reactions. In a reaction, the reactants and products are in contact and it is the balanced interaction of reactants and products that produces the equilibrium state. We suspect that the students would benefit from a recapitulation in the form of ‘this is how the “school dance” is like a reversible reaction’ and ‘this is how it is unlike a chemical reaction’. Analogical stories are seductive: they interest students and provoke questions and discussion that lead the teacher into a false sense of success concerning his or her explanation. Only a final comparative analysis of the analogy’s shared and unshared attributes can provide this confidence. Finally, we would remark that Neil taught in an interactive way, but, again, low achieving students were not asked to show their understanding by rephrasing (parts of) the analogy in their own words. This may have been intentionally designed to not embarrass low achieving students.

There is an unshared attribute in this analogy that has been ignored. The “closed system” concept is easy to understand with respect to the “commitment room” but is not a logical conclusion for the gymnasium. Indeed. Any number of students could be in the gymnasium so long there are more than 250 couples. In chemical equilibrium, the relative reaction concentrations in both the gymnasium and the “commitment room” are important.

Figure 1: Neil’s diagram on the whiteboard of the dance room and the commitment room.

Post-lesson Interview (Lesson 1 and 2)

“school dance” in the gymnasium

1000 students

500 boys

500 girls TEACHER

“commitment room”

250 couples

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When asked to comment on what he thought went well and satisfied his expectations, Neil was pleased with the students’ remembrance of the ‘school dance’ in the restricted version. On probing specific aspects of the lesson, Neil stated that he was also happy with the students’ recognition that reversible reactions can occur simultaneously with the same rate. However, he added spontaneously that he has been too quick in dealing with the dynamic aspect and, for that reason, he felt he had not explicated this feature sufficiently. As a result, he planned to next lesson spend more time of talking about dynamic aspects, probably by firstly focusing on physics equilibrium, for instance, equilibrium in a solution or in gases. Neil insisted on dealing with physical equilibrium before he attempted chemical equilibrium. This agrees with the beliefs of many teachers (e.g., Tyson, Treagust & Bucat, 2000).

Neil stated that he thinks his students can understand a physical process better than a chemical one because they are much more familiar with them. He said that he is very keen to access his students’ prior knowledge, including their daily life experiences. He also indicated that some students were not able to understand the chemical nature of reversible reaction in the first instance, but as soon as they thought about the commitment room setting, the concept of reversible reactions made sense to them. For that reason, analogical models are very important to him. Neil chose the dance room analogy because he thinks that it is an example of a “working analogy”. This comment is intriguing: Neil sees the use of his analogical models as a process that connects his and his students’ working memories. The analogy is an instance where he and the students can interact by mapping their everyday knowledge onto the concepts he knows that they need to understand. ‘Working analogy’ is an important cognitive and social constructivist idea for teachers who engage as closely with their students as does Neil.

Pre-lesson Interview (Lesson 3)

From the reflective interview on lesson 1 and 2, Neil has already suggested introducing the features of physical equilibrium. Looking forward to lesson 3, he wanted to start with a recapitulation of the three conditions for chemical equilibrium, and apply them to physical equilibrium. He stated his intention to examine the situation of excess sugar in a cup of tea.

Neil also intended to use the ‘busy highway’ analogy. He said he frequently uses this analogy when teaching the kinetic theory version of evaporation or teaching equilibrium between evaporation and condensation. He enjoyed this kind of analogy, because the students are just earning their driver’s licenses. He felt that his students were familiar with the difficulties of merging into and out of traffic. He once used a similar analogy, namely, the ‘peak hour train’ analogy, but found that this analogy did not work because his students did not often use the public transport system. The analogy was relevant to him, but not to his students.

Chemistry Lesson 3 (single lesson)

(a) The ‘sugar in a teacup’ analogy. Neil began the lesson with a verbal quiz on the students’ recall of the conditions for chemical equilibrium. In order, the students indicated that the system must be closed, the rate of the forward reaction is the same as the reverse reaction, and the processes are dynamic. At this point, a student asked Neil to elaborate what was meant by ‘dynamic’. Neil immediately launched the intended story of sugar in a cup of tea in the following way.

Neil In England at 4 o’clock … everybody stops, and they offer you one lump or two, … and we’ve put in 5 lumps, stirred very, very nicely … of course, that tea is hot, but what happens is that the tea becomes cold. What happens to the sugar? You’ve all experienced this if you’ve over sugared your tea.

Sts. Lumpy at the bottom … sweet.Neil Sweet and, syrupy, isn’t it? That’s all the sugar at the bottom of your tea…. we put sugar

in our tea, the tea cools and sugar falls out, so what does it say about the solubility of sugar in that tea. What sort of solution is that? [St. Saturated] … meaning we can’t add any more sugar without it dissolving … Let’s say that you dissolved … a million sugar molecules, right? If I add a million and one molecules, is that one molecule going to dissolve? [St. No] … what does this have to do with equilibrium? Let’s have a look at

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the characteristics of equilibrium. Is that cup of tea in terms of sugar a closed system if I put the sugar basin away, they can’t get at it any more.

St. We can assume so.Neil It’s not quite [closed] at the moment but what could I do to make it one? St. Put a lid on it.Neil … put a lid on it, or seal it up .... So we can’t get any more sugar in, the tea cannot

evaporate … Now rate of forward process versus rate of reverse process. What is the process we’re talking about here? [St. Sugar dissolving] Sugar dissolving … if one molecule out of the million that I have there comes to rest, how many particles will now be here? A million minus one, which is 999999, does that mean that one molecule be able to dissolve? [St. Yep] … So what we’ve got is a situation where for every one molecule that actually comes out of solution and forms a solid, another molecule can dissolve … what can we say about the rates of those processes?

St. They’re the same.Neil They’re the same, looks … that’s looking … like an equilibrium … Jon asked before, the

point about why is this dynamic, and I think this is the hardest point to get across to you … imagine you can see the sugar on the bottom. You can see through the cup … Now, I can sit and look at that, and after about two minutes nothing appears to be happening. But you as a chemist are well aware that for every molecule that solidifies, another one dissolves, and another molecule from up here may find it’s way down and another one may find it’s way up and so on … a kid will tell you nothing’s going on there, but the process is dynamic, in terms of you’ve got solidifying occurring, you’ve got dissolving occurring, all at the same time, so it’s dynamic on a molecular level.

Neil used the ‘sugar in a teacup’ analogy to highlight the meaning of ‘dynamic’ but in his analogy, the dynamic process that is meant (particles solidifying, while others are dissolving simultaneously) cannot be observed, as he admitted to his students.

(b) The ‘pot of curry’ analogy. Immediately after the sugar in the tea episode, a student called Mal offered a variation on the analogy by asking the following question:

Mal Is that happening when you’ve got like food in a pot and you’ve got a lid on, and when some evaporates at the same time some is condensing and dropping down at the same time?

Neil Ok, very good … Now for all intents and purposes is that a closed system if I’ve got the lid on pretty tight? [St. Yeah] not completely closed, but it will do…. Now, you do a recipe and they tell you add this and that, simmer for 20 minutes with lid on. Why are they telling you to do that? Why leave the lid on?

St. Liquid stays in the pot. Neil And the liquid’s got to stay in the pot, why?St. Cause otherwise it’ll all evaporate and everything will like go dry.Neil Exactly, end up with some sort of curry that’s just bits of dried up chicken with bits of

dried out vegies, and little lumps of curry sticking to it … I put it on the stove. What’s happening to the water? [St. Evaporates] Evaporates, what happens when it hits the lid? [St. Condenses] It cools and condenses … it drops back into the solution, so you pretty well know you’re going to get the consistency you want for the curry. … [It] is an equilibrium if we’ve got a closed system … and the level remains constant … the rate of evaporation equals the rate of [St. Condensation].

This episode shows that the ‘pot of curry’ analogy, proposed by a student, is more easily related to daily experiences because there are more observable cues than in Neil’s ‘sugar in a teacup’.

(c) The ‘busy highway’ analogy. After these two analogies, discussion about the dynamic feature of equilibrium continued. During the discussion, Neil introduced a new analogy in the following way:

Neil Who’s got their licenses?Sts. We’ve got our Ls [learner driving license]. Neil … If you live on a major road … and your street runs into a major road, right, there it is

[Neil is drawing Figure 2), three lanes in one direction and three lanes in another

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direction. Whoever taught you to drive, probably taught you to drive in a quiet time. So here you are, at the STOP sign lined up ready to do a right hand turn into traffic. Now, if this is 2am in the morning … streets are free there is no car to block you … At 5, there’s a couple of early birds getting out to start jobs so … let’s draw a couple of cars in here, a couple there, one there. At 8 o’clock in the morning, and this is the major road into the city? [Sts. Packed]. Packed, bumper to bumper, cars everywhere heading into the city, and a few going the opposite way. Now, let’s now go to the really bizarre situation … you don’t obey the STOP sign, you don’t look for traffic, you simply go out. Right in peak hour what would happen to you? You’d hit a car, … [St. Roll it]. Yeah, roll it whatever, and, you would be spun back into there or you’d cause the other car to be spun off the road. You’d cause complete traffic chaos, right? Now, I bring up that example, because it demonstrates to some degree, why vapor pressure happens. Why we get condensation happening, after evaporation. Now you’re in your driveway or at the end of your road at 4am in the morning and you just go straight out. There’s nothing to stop you … But you get to a point where … what happens to the number of cars on the road? [St. It increases]. Right, so when suddenly a molecule evaporates, it hits another molecule. Now that molecule may even go back [into liquid], that can happen. Or it causes this molecule to hit that molecule to hit that one, to hit that one to knock it back. The very dynamic situation comes to a point where there’s just too much traffic on the road, for another car to get in [8am]. So for every car that goes on the road, another car has to come off. I suppose you could always think of a car park. If this was a car park I can come in but there comes a point where the car park is totally full of cars.

St. Well what happens when like you get water condensing, a big drop, drops down.Neil … if you want to take my analogy, there might be little car parks on the side of the road

with some cars parking for a while until it gets really full until a truck comes, loads them on and takes them away to be dropped.

St. But it wouldn’t go back on the road.Neil No, it ‘d have to go …St. Down the service stationNeil I should have actually drawn it out here, taken off back to where they came, and just

remember: analogies have their limitations.

At the final point, the ‘busy highway’ analogy is breaking down, as Neil indicated. However, he did not discuss why the analogy broke down, just as he did not discuss the problems in linking the ‘school dance’ analogy (body contact issue) to the surface area effect on reaction rate. The remainder of the lesson was devoted to explanations of dissolved oxygen and carbon dioxide in open and closed water bodies, without any use of analogies.

Post-lesson Interview (lesson 3)

When asked about the analogies used in the lesson, Neil did not comment on the ‘sugar in the teacup’ story, probably because that went to plan (see pre-lesson interview). However, he noted that halfway through the ‘busy highway’ analogy, he actually thought that a car-park analogy may be even a little bit better. He said that by describing the highway situation at 4 am, he would be able to get into the car-park situation. He thought it may get the message across. When asked, “explain why you were thinking about it”, he replied by saying that he became aware that his students can see when a car-park gets full, they cannot get in there, so they have to wait for a car to come out before another can go in. He thought it was more structured. Later he remarked that he will not change direction in the middle of an explanation. When asked if he used ad hoc analogies, he said he did, but sometimes he had “fallen flat on my face”. Neil’s opinion reinforces Treagust et al.’s (1998) recommendation that teachers should avoid ad hoc analogies and models in favor of carefully prepared or tested analogies.

Delayed interviews with students

About 10 weeks after the lessons about chemical equilibrium, seven students of Neil’s class were interviewed about the analogies that had been taught. They were interviewed in couples (Jon and Mark, Sue and Jim) or individually (Peter, Mal, and Andy). The students conducted two simple experiments and the one that prompted the responses is reported below. Students were asked to “place 40-50mL of water in a beaker, add 2-3 teaspoons of table salt, stir, and explain what happens”. If,

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during the ensuing discussion, they did not mention one of Neil’s analogies, they were asked, “can you recall a story that Mr Scott used to help you understand these ideas?”

(a) Jon and Mark’s Interview. After the students had conducted the ‘salt in water’ experiment, they concluded that not all salt dissolved. One of the students explained it this way:

Mark Dissolving solid NaCl is split up into chloride ions and sodium ions … till it reaches the point where no more Cl ions or Na ions can fit the solution anymore, so that’s saturation. And then … more sodium and chloride ions are trying to dissolve and that’s forcing other ions to form a solid again.

Int. It doesn’t stop?Mark No, it’s still happening. You just won’t notice anything happening because you’ve still got

the same amount … you can see … the same amount … of salt left in the bottom of the beaker, but on a more atomic level, things are actually changing shape. You’ve got individual atoms and that dissolving, you know what I mean?

Int. … it’s not the same atoms?Mark It’s not the same atoms you’re looking at, they’re always changing throughout the solution.

Several pieces of Mark’s explanation resemble parts of Neil’s analogical model of the excess sugar in the cup of tea. When cued about Neil’s analogies, Jon mentioned the sugar in the teacup (analogous to the salt in water instance) and both students applied the ‘busy highway’ analogy in the following way.

Jon Sort of like a freeway that’s packed and maybe one person would drive onto the freeway and another one would be bumped off … You can only fit another car in if one comes off.

Mark … it represents how the chemical comes to be in equilibrium. For example the analogy to the beaker, you’ve just put your salt in so that’s like at 2am in the morning when you can just go straight onto the freeway, don’t have to wait. So the salt can just dissolve straight in the water. At 10am it’s like really busy and you can only come onto the freeway if someone else gets off and so that sort of goes back to the beaker. Only one salt crystal can dissolve if one salt crystal is formed.

Int. Does that give you a mental image … of what’s happening? Mark Yeah.Jon It works for some people. I don’t usually go on analogies. Int. … why don’t you go for analogy? Jon … if you think too deeply into the car thing, you can say what if people had died? It can

get out of hand … there are sections where the analogy falls apart. Analogies do that.

Again, several pieces of the student’s explanation resemble parts of Neil’s analogical model. For instance, Mark’s statement “only one salt crystal can dissolve if one salt crystal is formed” is like Neil’s statement at the ‘sugar in the teacup’ analogy “for every one molecule that comes out of solution and forms a solid, another molecule can dissolve”. The mechanism for Mark’s equilibrium process seems to have its origin in Neil’s explanation and this leads us to conclude that Neil’s analogical stories were effective for these students. Jon’s comment on the meaning of analogies is salient because few students are sensitive to the weaknesses of analogy. Jon’s reasons highlight some difficulties students encounter with analogies that are teacher favorites.

(b) Sue and Jim’s Interview. After the ‘salt in water’ experiment, Sue and Jim gave the following explanation.

Sue Particles that are in the water are in equilibrium with the particles in the solid. So that when the particle in a solid dissolves, it rebounds off the others, and knocks another particle that was in the liquid back to a solid, so that the solid will change shape but only slightly, because of that.

Int. … so you say that some will dissolve …Sue The same number of them will …Jim Come out of solution.Int. … Mr. Scott used an analogy … Sue Tea cupNeil … then he talked about cars …

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Jim Cars on the freeway … a flow of cars was going that way … it’s very difficult to hop on, but if there was a very low amount of cars going through, then it’s easy to hop on. And I think what he meant by that was the amount of salt that can dissolve in a solution is easier when there’s a lower amount of salt already dissolved. So that means like a low amount of cars. Whereas if there was a high amount of salt that was in the solution that would mean that there’s like a high number of cars, so it would be more difficult for one little salt particle to get onto the freeway.

For part of the time, Jim reasoned from the chemical process to the traffic situation, whereas Neil used the traffic density to explain what happens in the teacup. This suggests that the analogy has either served its purpose or was not (now) necessary to direct Jim’s thinking. Sue’s commented “it rebounds off the others, and knocks another particle that was in the liquid back to a solid” and this statement closely resembles Neil’s “busy highway” analogy. He said that if a car entered the dense traffic at 8am it was likely to be hit and bounce off the road or knock off a car already on the road. This attribute of Neil’s model appears to have been assimilated by Sue into her equilibrium explanation.

When Sue and Jay’s opinion was sought about the effectiveness of Neil’s analogies, they responded thus:

Sue … really good because you remember them more than you remember a text-book explanation.Jay You’d remember his analogies better … but I think the tea-cup was a better analogy than the

cars because it’s more …Sue To the point … you say cars, and a lot of the girls just switch off …Jay Boys remember cars, that’s how I remember.

The students’ views showed the importance of model relevance, especially differences between boys’ and girls’ interests, but this does not show in Neil’s stories. Sue told how the girls switch off when he starts talking about cars and airplanes (in the ‘assembling an aircraft’ analogy). Jay raised another key idea: he preferred the teacup explanation because it is closer to the point. This suggests that for him the ‘sugar in a teacup’ analogy is nearer to the science event that he needs to explain.

It seems that the difference in settings between the excess sugar in tea (afternoon tea) and excess sodium chloride in water (the chemistry class) is sufficiently everyday versus science to make one an analogy for the other. This difference may be strong enough to require the mapping of analogical attributes between the “sugar in a teacup” and “solid sodium chloride in a saturated solution”. We claim that students comparing both instances felt that they needed to make decisions as to what does match (positive analogy) and what does not (negative analogy). As Jon indicated in the former interviews observation, “there are sections where the analogy falls apart”.

(c) Peter’s Interview. After he had conducted the ‘salt in water’ experiment, Peter commented on this experiment as follows.

The salt down the bottom, you’ll see the same amount of salt but it’s not necessarily the same salt that’s there. What will happen is the salt down the bottom will keep mixing and as it mixes with the water, other particles will sedimentate in the bottom…. at equilibrium, it’ll continually stay at the same mixture except not the same particles of salt will stay at the bottom.

Peter’s comments raise questions about his understanding of the interaction between the solid sodium chloride and the solution. He calls the solution “the water”, talks about “mixing” and says that salt “will sedimentate”. These statements suggest that he may have a weak understanding of scientific dissolution and precipitation. His statements show that Neil’s choice for analogical models does not assure us that all students will adequately map the analogy between everyday and scientific concepts.

Peter remembered the ‘school dance’ analogy, but in his version, this analogy involved only equal numbers of blindfolded boys and girls pairing up and going into a small room (he did not give any reason) from which, when full, the boys come and go for no apparent reason. His recall of the analogy is confused indicating that he did not, at this time, transfer meaning from the model to the equilibrium event. This raises questions about the interdependence of the target on the analog and the analog on the target for meaning making in analogical thinking. Despite his confused and ineffective mapping, Peter claimed that Neil’s story helped him make sense of the salt solution equilibrium. Indeed, he

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concluded that multiple analogies help you “get your head around” different concepts. In summary, his responses suggest that he did not see the process ideas inherent in the analogy and therefore failed to understand the dynamic balance that is called chemical equilibrium.

(d) Mal’s Interview. Mal explained the ‘salt in water’ phenomenon by saying that there is a limit to how much salt can be dissolved in the water. As he observed the salt on the bottom, he stated that some particles will dissolve and others will “un-dissolve”, or keep going like a dynamic equilibrium. He could not recall Neil’s ‘sugar in the teacup’ analogy, but he remembered the ‘busy highway’ analogy as follows.

Mal I remember him using that analogy … it sounds a difficult analogy for an equilibrium situation because just cars rushing past, and it doesn’t have a constant amount, oh, actually it does make a lot of sense now because it’s reached it’s saturation point because it’s got as many cars as the road’s going to be able to hold at one time or safely without people tailgating, and yeah there’s no room to come on unless someone comes off, which is an equilibrium situation of one particle dissolving so another particle can un-dissolve.

Int. An equilibrium.Mal Yeah. … I picture what he’s talking about in his analogy and I apply that, and I learn it, but

it’s not a thing I use to remember. I don’t you know, sit in a test and go what was the story with the cars, because I’ve used the story to learn it… [emphasis added]

Mal thinks that he does not need the analogy, but the transcript shows otherwise. Until he started recalling the analogy (he needed a cue) he had no process ideas on which to base his thinking. When he started to ‘argue’ with the analogy, he discovered that it did apply, that it was a cognitive “hook” on which he could hang his explanation. His statements at the end of the interview, beginning with “I picture … I apply … I learn …” suggest that as his mental model of the analogy emerged, his understanding grew. This is in line with functions of analogies as aids-to memory and scaffolds for the construction of conceptual understanding (Treagust et al., 1996). When used well, analogies and models do play a crucial role in memory, learning and explaining.

(e) Andy’s interview. When asked to explain his observation of the ‘salt in water’ experiment, Andy gave the following explanation.

Some of the salt molecules have dissolved into the water but some stayed at the bottom because the water’s saturated. It’s like reached its maximum, can’t fit any more salt molecules in, can’t dissolve any more salt molecules, so that’s why they’re still sitting at the bottom …. I think it’s sort of like the equilibrium too … some are dissolving while others are un-dissolving, like it’s become saturated.

Andy could not explain the dynamic nature of equilibrium, so he was cued with Neil’s analogy of the school dance. Andy remembered much of the story but did not map the ‘school dance’ process attributes onto dynamic solution equilibrium. Nor did he recall the ‘sugar in teacup’ episode. He seemed unable to convert his equilibrium descriptions into causal explanations of dynamic forward and reverse processes. He claimed he liked Neil’s stories and analogies because they were, as he said, pretty hilarious and easy to understand, but he was unable to use their embedded processes to extend his chemistry knowledge.

The other analogy that Andy remembered was the ‘busy high way’ analogy that he described as follows.

A car trying to get onto the highway … or a car park. When one car comes out, it means another one comes in because it means an extra sort of space, and you can’t really cram any more cars in there if there’s no extra space, so you just can’t sort of go in. Only when one car goes out another one can come in…. That’s at equilibrium …

Subsequently, Andy described a ‘full car-park’ analogy. This corresponded with Neil’s original intention to use the same analogy, although Neil did not develop it in the class. This evidence suggests that the ‘busy highway’ analogy did lead to another analogy and to some useful understandings of equilibrium for Andy.

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Discussion and Conclusions

The transcripts show that discussion was the norm for Neil’s lessons and that multiple analogical models played a prominent role in his teaching. For this reason, his lessons were an excellent case for examining the efficacy of multiple models.

(a) Teaching perspective on multiple analogical models. From a teaching perspective, and based on the reported findings, we would like to make the following comments.

Neil recognized the difficulties his students faced in understanding the three conditions for equilibrium; thus, he followed common practice by starting with a familiar example like the ‘school dance’ which he then transformed using an analogy-example of excess sugar in a cup of tea. The ‘school dance’ was an interesting stratagem because it was used as an advance organizer for what was to follow. He did not tell the students where he was going and he insisted that this was an effective strategy for him. He has a pedagogical reason for ‘keeping them wondering’: he wants students to see that his ideas in science are sensible, he wants them to like science and he hopes they will find chemistry useful.

Neil’s stories indicate a high level of reflection and attention to student interests. But Sue shows that Neil is not as well connected to student interests as he thinks! Still, his stories were carefully rehearsed and he made a conscious effort to marry difficult concepts to everyday stories that were familiar to most of his students. The conceptual thread of his analogical models reflects this care: he attended to reaction rates with the ‘school dance’, took an intermission to dealt with reaction profiles, then back to the ‘school dance’ to develop incomplete and reversible reactions. When he saw the need to link reversibility and activation energy, he added the ‘downhill skier’. Once the question of mechanism appeared, his assembly of a model aircraft analogy was needed. Neil’s working analogy metaphor is evidence of his conscious need to hold student interest; and, as the pre- and post-interviews show, he had a clear vision of the three equilibrium conditions that the students needed to understand.

But he’s not finished yet: the next lesson he immediately grasps the opportunity to develop the ‘sugar in tea’ analogy when asked, “is the reaction balance dynamic”? The ‘simmering stew’ question receives a similar response. Everything seems so natural but Neil’s planning shows that his marriage of content and pedagogy was planned and honed over many years. We claim that this is an instance of pedagogical content knowledge (PCK) in action; or, as Cochran, deRuiter and King (1993) better call it, pedagogical content knowing. He knew what he wanted to explain and had the pedagogical tools to explain the concepts. A feature of expert PCK also is the ability to explain difficult concepts to a diverse audience and this brings us back to Sue’s comment. She stated very clearly that girls and boys do not have the same interest when talking about analogies (boys like car analogies more than girls). Neil, although he is be aware of differences in gender and interest, does not always show this his choice of his stories. It is important to remember that interactive explaining and answering student questions is a complex action and teachers cannot always be expected to remember everything!

Neil’s reflection that, “I think kids can see when a car-park gets full …”, and his conclusion, “I think [the car-park analogy is] more structured” is indicative of his continuous search for the ‘most effective story’ (emphasis added). Neil was confident of his student’s ability to understand; nevertheless, he is always seeking a better way to explain; and remember, he has done this for 18 years. Like equilibrium, Neil is mentally moving backwards and forwards between analogies, working his analogical models to find the best sense.

If working analogies are to promote relational understandings, the shared and unshared attributes must be carefully mapped with the hearers (Duit, 1991, Harrison & Treagust, 1993; Treagust et al., 1998). But even more is needed: after mapping the positive and negative analogies, the analogical model needs to be summarized as a causal explanation that links the analog processes to the target concepts to show how and why the analogy works and where it breaks down. The interview data suggest that the students lacked precise understandings of this kind. In the pre-lesson interview, Neil declared his awareness that all analogies and models break down somewhere, yet just once did he mention a limitation of his analogies to his students. Disparities between teacher’s intended and actual behaviors are not uncommon (Treagust et al., 1998) and are likely related to the pressure of unexpected questions and management situations that arise in class.

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Teachers can close a ‘good story’ once it has been well received and they think that the students understand it. We cannot always make these assumptions. Effective teaching formatively assesses the development of all key ideas. For some analogies, Neil did do this. He took valuable time to recapitulate and elaborate the ‘school dance’. The relational benefit of his analogies could, though, be further enhanced if he had given 2-3 minutes at the end of analogies like the ‘school dance’, the ‘sugar in the teacup’ and the ‘busy highway’ to summarize each model’s outcomes with respect to equilibrium.

(b) Learning perspective on multiple analogical models. From a learning perspective, and based on the reported findings, we would like to make the following comments.

The case of Mal and his argument with the busy highway’ analogy is very interesting. As he tries to discredit the model, he realizes that the model does make a lot of sense. We hear him mapping dense traffic (“following safely without tailgating”) onto concentration and saturation. He sees traffic entering (dissolving) and exiting (precipitation, condensation) and thinks that equilibrium is a cycle and then he makes his eye-opening discovery that the analogy enabled him to picture, apply and learn. And he thinks that the story was non-essential!

Drawing on Gentner’s (1983, 1988) ideas, we here see how the surface similarities permitted him to access the analogy and, as he argued with it, he finds the deep systematic ideas that Neil had planted for him to find. We like to analogize Neil’s quest for more structured stories with Gentner’s ‘structure mapping’ (the comparison may not be as strong as we suggest) and Mark’s story also is reminiscent of Gick and Holyoak’s (1983) analogical story, called ‘The general’.

Sue talked about particles “rebounding off others” and colliding with other particles. The only analogy-example invoked by Sue was the ‘sugar in the tea’ which she did not elaborate. Given the preponderance of classroom stories and the similarities between the ‘sugar in the tea’ and the ‘salt in water’ event, both Sue and Mark may have mapped the positive analogies between the instance and the event without needing to call on the “school dance” or the “busy highway”. This is possible when we recall that most of the classroom learning was couched in analogical stories. They had many individual analogies to choose from or they could derive their equilibrium knowledge from the sum of all or some of the stories. The level of uncertainty inherent in this answer indicates that further study of Neil’s class (and other classes of other teachers) is needed.

We tentatively claim, however, that this case bolsters our confidence in the efficacy of multiple analogies when they are connected and presented in a systematic way. Remember, Neil eschews ad hoc analogies and loathes ‘changing horses in midstream’. These are qualities worth emulating. The one enduring criticism, from the students’ need point-of-view, is his unfulfilled intention of discussing where his analogies break down. This study reinforces a previous finding: unless teachers are vigilant, they often fail to negotiate the negative analogy. We therefore reiterate Treagust et al’s (1998) recommendation that the Focus—Action—Reflection (FAR) approach be the preferred format for planning, presenting and reflecting on analogical teaching.

(c) Final comments on multiple analogical models. Notwithstanding the study’s commitment to rigor and detail, the compilation of ‘thick descriptions’ of Neil’s intentions and teaching, and the school’s willingness for us to interview students; there are evident limits to our ability to access teachers’ and students’ thinking during teaching and learning in science. Norman (1983) warned that there will be significant differences between what teachers and students say they think and intend to do, and their actions and conclusions. This is apparent when Neil says he will present the ‘school dance’ analogy – he does – but he presents four others as well. He says he will map the unshared attributes of the analogy, thinks that he has done so, but the evidence suggests otherwise. Students say they do not need a particular analogy, yet show a dependence on it when explaining equilibrium. Another interesting question whose answer is prejudiced by our inability to ‘see inside their thinking’ is why some students relied on the “busy highway”, others the ‘school dance’ and some even used the ‘car park’ analogy that Neil hesitated to introduce.

Perhaps these data provide some answers to the question of what happens when multiple models are used to teach science? One simple answer is that students choose the representation that is most meaningful for them. Of course, this does not answer the deeper question of which combination of multiple models is most efficacious; still, Neil’s multiple models appear to possess a coherency that

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enabled his students to choose ‘the best for them’ – a choice based on their own epistemological preferences. But this raises another question: what mix of modeling epistemology and ontology is effective for different students? And so, the quest goes on. We research these questions to refine our conceptual framework of how modeling benefits both learners and teachers. Cases like this are excellent ‘thought provocateurs’ as we try to make sense of scientific modeling.

Science education researchers are self-critical of their ability to answer questions about how to better understand what teachers and students are thinking and doing when learning science. At present, naturalistic phenomenography (of which this study is an example) is the best available method, and we believe that we are making progress. We claim that we have provided evidence on which decisions designed to improve teaching can be confidently made. Classrooms are dynamic social environments where many different minds from many different backgrounds with many different preferences, meet and learn. It is important that we maintain our participant-researcher partnerships because, to date, they provide useful windows into teaching and learning science, with and without the use of analogical models.

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