CHAPTER 4 METHOD DEVELOPMENT AND RATIONALE
4.6 DERIVATION OF PROPOSITIONAL KNOWLDGE STATEMENTS 74
From the literature I identified several potential problems with propositional knowledge.
Firstly, according to Mintez and Novak (2000) propositional knowledge within a field of study should form a coherent whole, showing consistency by reconciling internal contradictions.
These authors also contend that propositional knowledge should be agreed within the academic community of a discipline. However, Eybe and Schmidt (2001) caution that putting forward a
“system of knowledge statements” (p 220) may infer that there is a single view of acceptable chemistry, rather than the system simply giving the frame of reference against which the researchers will compare the students’ ideas. As already discussed (see Sections 2.3.1 and 3.2.1) the nature of science means that scientific conceptions are not fixed; they are human constructs and modern meaning for terms may differ from the original (Hall, 1930; Schmidt, 1991; Taber, 2002). Consequently, I anticipated not one set of propositional knowledge but several, each within the context of a particular historical model. Furthermore, each model should have transparency in that the propositions could be justified within the conceptual framework of the appropriate scientific paradigm.
A further problem was raised by Stains and Talanquer (2007) concerning the relationship between ‘accepted’ understanding and that which practitioners actually use. In particular, their interviews with university lecturers revealed a strong association and corresponding lack of differentiation between some pairs of concepts. For example, some staff associated the label compound with O2, or N2 because they were both molecular species, although scientists generally accept oxygen and nitrogen as elements. Furthermore, Bowen (2005) differentiates between “ready-made-science” or school science presented as unproblematic and “science-in- the-making” as practised by scientists, which is messy but needs to be defensible.
Consequently, lack of agreement among scientists about particular propositional knowledge may present further complications. Nonetheless, there should be some consensus so that meanings are resonant with or shared by experts (Mintez & Novak, 2000), rather than being my own understanding.
A third possible problem is that expert chemists may reject conceptions which are deemed acceptable among school pupils, considering them incomplete or even incorrect. For instance, Taber (2002) notes that abbreviated definitions are often introduced to novices because they may only use a concept in limited contexts, but this could leave students unaware that their conception is not generally applicable. Moreover, Hawkes (1994) maintains that students tend to retain what they learn first, so an introductory qualitative description should lead correctly
into the quantitative models that students will encounter later. However, as Nelson (2003) and Bucat (2004) argue, it makes no sense to plunge a novice chemist into a formal definition as agreed by the International Union of Pure and Applied Chemists (IUPAC). Nelson then makes practical suggestions of “pragmatic definitions” which are as simple yet precise as he could make them. In a similar manner, de Vos and Verdonk (1996) prepared a summary of the particle nature of matter in the form of propositional statements. They made no attempt to win the approval of expert scientists for this summary, because it was used to evaluate introductory school textbooks. Instead, the summary was agreed to be valid by science education researchers. This was the researchers’ community of practitioners. Accordingly, I would seek something suitable for students, without being wrong in the view of expert chemists.
Mintez and Novak (2000) also emphasise parsimony which is evident when an individual understands a topic, so propositional knowledge should not include superfluous information such as extraneous explanations or unnecessary propositions in their conceptual structure.
However, a fourth problem lies in Shulman’s (1986, p 11) warning: “the representation of knowledge in the form of propositions has both a distinct advantage and a significant liability.”
Because propositions strip away the superfluous, they are economical but at the same time decontextualized. Furthermore, being discrete statements, they are hard to remember, especially as lists. Propositional statements are the ‘bare bones’, which teachers and textbook authors need to transform into learning experiences. Accordingly some way of integrating the statements would be needed.
The challenge in this study was therefore to outline acid-base models in sets of discrete propositional statements against which student conceptions might be compared, which could still be integrated into a whole. The statements should represent the different historical models authentically, yet be understandable and appropriate in the school context. They should certainly be acceptable to a community of practitioners, in this case chemistry education expert opinion. Notwithstanding de Vos and Verdonk’s reservations mentioned above, ideally they should also be acceptable to expert chemists. Could such a coherent set of statements of acceptable knowledge reflecting the contexts of different historical models be compiled? This led to the fifth and final research question: Does the set of propositional knowledge statements derived through analysis of student difficulties reflect appropriate knowledge for teaching and learning acid-base models? In order to answer this question, two sub-questions were formulated:
5a. How well do the propositional statements reflect curriculum models for acid-base chemistry?
5b. What are the implications of the propositional knowledge for teaching and learning acid- base chemistry?
Accordingly, based on arguments given above, criteria given in Table 4.5 were developed for propositional statements relating to each acid-base curriculum model (see Section 2.1.3).
Table 4.5 Criteria for acceptable propositional statements
Propositional knowledge derived from mapping student difficulties would be evaluated against criteria representing five aspects of acceptable student understanding. By this means I would determine how well they met the ideal as shown in the final column, rather than give a dichotomous acceptance or rejection. The first criterion of pragmatism was introduced to ensure that statements are appropriate for students rather than experts. However, there was a problem because the age groups in this analysis were not restricted (see Section 3.5) – which age students should I consider? A solution, allowing me to accommodate many ages of students, would be to let the difficulties themselves guide the particular propositional knowledge statements, rather than starting the research by specifying propositional knowledge, as recommended by Treagust (1988, 1995). For example difficulties with an operational model would indicate statements at an operational level of macroscopic observations, appropriate for younger students. Conversely, difficulties with calculating pH of an extremely dilute solution, which would probably be encountered at tertiary level, would be addressed by propositional statements pertinent at that level. However, to address Research sub- question 5a, it was still necessary to evaluate whether these propositional statements represented the whole or only limited aspects of the acid-base topic as taught in high schools. Accordingly the set of
Aspect Propositional statements should... How it will be evaluated.
Propositional statements will be....
1 Pragmatism Be age appropriate Determined by the difficulties concerned Compared with curriculum statements 2 Parsimony Avoid superfluous propositions and
examples.
Phrased in terms of general principles, with specific applications given as examples to indicate prototypes and boundaries of concepts.
3 Consistency Be coherent within each model. Integrated as concept maps for each model.
4 Transparency Maintain the integrity of the hard-core for historical models.
Able to define the context and limitations of each model.
5 Consensus Be acceptable to chemists. Checked against publications in chemistry education and chemistry and evaluated by expert chemists.
propositional statements would be compared with three typical curricula (Ross & Munby, 1991;
Nakhleh & Krajcik, 1994; Independent Examination Board, 1997). The first two publications give extensive propositional knowledge to represent acid-base topics as taught in high schools, the first as a concept map and the second as a set of propositional statements. The third publication outlines a South African curriculum that has been superseded, but it was retained in this analysis because the current South African high-school outcomes based curriculum does not feature acid-base chemistry as a distinct topic (Department of Education, 2003).
Next, according to Mintzes and Novak (2000), parsimony requires that the propositional knowledge focuses on the core principles of acid-base chemistry and that each application or example is there for a reason. For example Herron (1996) recommends that concept analysis requires specific examples and non-examples to indicate the extent and limitations of a concept as shown by Criterion 2. Criterion 3 involves internal consistency – the propositional knowledge defining a field of study should form a coherent whole. A concept map comprises a number of propositions, each linking at least two concepts. It is a useful way to ensure propositional knowledge is integrated without contradictions (Novak, 1996). Furthermore, concepts represented as nodes with attendant propositions will provide a context for the propositions. Transparency means that the propositions for a given model can be defended within the scientific paradigm concerned. This paradigm needs to be defined and its limitations made clear as in Criterion 4. Finally, to satisfy Criterion 5, Consensus or agreement within the community of chemists can be established through first checking propositional statements against original chemistry and chemistry education publications which distinguish the models concerned, then expert chemists can evaluate the propositional statements.
To ensure that propositional knowledge would meet the criteria given in Table 4.5, certain checks were instituted which ran concurrently with developing the list of propositional statements. Figure 4.4 below shows the processes of selecting, comparing and synthesising in diagrammatic form, with more detail than was possible in Figure 4.1; the next sections explain it further.
Figure 4.3 Flow Diagram to show derivation of propositional statements
4.6.2 Selecting the propositional knowledge statements
As mentioned in the previous section I allowed the difficulties to determine suitable propositional knowledge rather than starting with a fixed idea of what to include. At the outset, mapping between difficulties and propositional statements identified some of the required propositional knowledge. This process was found to be reciprocal and iterative. In many instances, mapping a student difficulty back and forth to propositional knowledge led to both clarifying and increasing the set of acceptable propositional knowledge statements. A starting point was any propositional knowledge given as being scientifically acceptable in the publications from which the data on student difficulties were extracted. For example Nakhleh and Krajcik (1994) gave their synopsis of textbook presentations concerning the general principles of the Brønsted acid-base model, separated according to four representational systems, which they term: macroscopic, microscopic, symbolic and algebraic. As already stated, not all authors were as clear as this, and some gave no orientating framework of acceptable knowledge. Where possible this information was extracted in one or more of the following five forms:
CATEGORISING Propositional statements
Initial List
Description of Student difficulties from Published Research
Chemistry Literature Historical and other journal articles COMPARING
Concept Maps for each Model SELECTING
Propositional statements published with conceptions research Chemistry education literature
Teacher experience
Modify List rewording differentiate
models Modify List
expand rearrange clarify
Modify List Corrections Revise
Propositional Statements for each Model
Propositional statements Final List SYNTHESISING
INTERPRETING RESOLVING
MAPPING &
HONING
• Lists of separate propositions (only Nakhleh & Krajcik, 1994)
• Paragraph(s) describing general acid-base chemistry principles (e.g. Schmidt, 1995)
• Acceptable answers to specific open-ended probes (e.g. Ouertatani et al., 2007)
• Acceptable answers to multiple-choice items.
• Acceptable principles given as part of the text in the discussion section.
To ensure consistency within each of the Arrhenius and Brønsted models, I also made lists of propositional knowledge from the following sources for comparison:
• Outlines of Arrhenius and Brønsted Models by Oversby (2000a) and de Vos & Pilot (2001).
• Original historical papers, in the original or English translations (for example Arrhenius, 1903; 1912; Brønsted, 1926 and Lowry, 1923)
• Historical studies such as Bell (1969) and those in the Journal of Chemical Education such as Kolb (1978).
• IUPAC definitions for modern expert knowledge (McNaught & Wilkinson, 1997).
This propositional knowledge was typed verbatim into a Microsoft Word document, as separate statements from reports, along with the corresponding reference to its source. Since many authors reported students as having much less conceptual understanding of bases than acids, I kept statements about bases separate from those for acids so they would have equal prominence.
As a result, definitions of acids and bases are not given simultaneously as in Nakhleh and Krajcik (1994). In order to categorise the statements, each item of propositional knowledge was also prefixed with key words such as ‘Base, Arrhenius’ or ‘strength, Brønsted’ which were based on the 13 broad topic headings that had been found workable (see Section 4.5.2). In this way statements could be easily sorted into categories. The consistency of propositional statements within a topic could then be evaluated, and those which suggested consensus were adopted.
Textbooks were not consulted at this stage for the reasons outlined in the literature review, namely, that content analyses around the world have shown a preponderance of mixed models in the acid-base section (see Section 2.5.3). By the same token, schoolteachers were also not consulted when deriving propositional statements because Justi and Gilbert (1999) had found that much of their content knowledge was derived from school textbooks. At this stage, I also did not consult chemistry experts because, as Furió-Más et al. (2005) found, due to their tacit knowledge experts may flip-flop between models without making the change overt.
The outcome of the selecting and categorising phases, was a fairly comprehensive list of possibly overlapping statements (indicating consensus), each with a reference, and sorted into broad categories according to topics. These were ready to be mapped to the student difficulties in the comparison and synthesis phases as described in the next section.
4.6.3 Using student difficulties to make missing propositional knowledge overt With both data segments on student difficulties (Section 4.5.2) and propositional statements sorted into the same broad categories, at least one propositional statement could now be allocated to each data segment. The propositional statements chosen were given decimal numerical codes, indicating some sort of hierarchy of concepts. These were the codes used for the ‘fine-sort’ described in Section 4.5.3). The outcome of that sorting process was groups of data on student difficulties corresponding to particular themes of student conceptions.
On examining the propositional statements alongside similar student conceptions, I experienced a key moment in the development of the research process. I found that I could barely restrain myself from rewriting propositional statements. Only the demands of accurate reporting of research prevented me from altering my original list. I examined the intense emotions within myself and I realised I had moved into ‘teacher mode’; imagining what the student needed to know in order to address or pre-empt such a difficulty. In their original from, the propositional statements could not sufficiently address the nature of the difficulty – perhaps further examples were needed, or I should clarify or extend the statement. This is exactly like the cyclical process I had adopted as a teacher, when each year I made notes, based on difficulties I had identified, of how I should modify the curriculum material for next year’s students. Likewise I was sure that I could not allow these statements to remain as they were; they had to be changed.
So I retained the originals, but immediately added in alterations alongside them. I realised that, in this way, I was using my PCK to make overt my more expert knowledge to address student difficulties. This is the tacit knowledge that had been missing and which needed to be engaged when deriving propositional statements.
The literature records similar processes. A first example is from a series of articles on teaching the nature of a chemical reaction. In one of these, de Vos and Verdonk (1987a) show how a definition is modified as student responses are studied, making underlying terminology clearer with each iteration. For example: “We could counter these objections by defining identical as
‘not differing from each other in any way except position and motion.’” And then later: “We now declared objects to be identical if they did not show any difference, except in position or in
motion” (p 694). These subtle changes were made in response to student feedback. As second example of this intuitive process is from Nussbaum (1998). When discussing how to identify proper instruction strategies, he advocates a first step of cognitive analysis of content. Such a cognitive analysis goes beyond a mere content analysis, or hierarchy of concepts as advocated by Herron, 1996. Rather, by using intuition and psychodidactic knowledge, which may include input from research on student misconceptions, a cognitive analysis relies on good understanding of subject matter and a natural tendency to delve into the subject matter so as to expose the deeper basic assumptions and their conceptual implications. Nussbaum thus sees a link between student conceptual difficulties and intuitively exposing the content which should be included in instruction. These two examples gave my intuitive responses to student difficulties some validity.
The new propositional statements which I had added (synthesis) were also coded, and the decimal system I had adopted proved its worth in that subdivisions could be made and new ideas incorporated into the hierarchy. An example follows to illustrate the process. I started with the statement: “Acids and bases affect the colour of indicator dyes differently” (Nakhleh &
Krajcik, 1994). This was made into distinct statements for acids and bases:
• Indicators have characteristic colours in acidic solutions (code 2.1.1.2)
• Indicators have characteristic colours in basic solutions (code 3.1.1.2)
Data for difficulty P9 (Section 7.3.1.2) suggested students thought the colour was inherent in the acid so a new explicit statement was introduced:
• Indicators are substances added to solutions of acids and bases (code 6.1.1)
Then difficulty P20.1 (Section 7.4.3.3) indicated that students believed an indicator assisted with neutralization. This necessitated another statement:
• Indicators are substances that change colour at certain pH values. (code 6.1.2)
Through this process one statement has been expanded to four; such resolution of statements into finer detail was frequently warranted. A recommendation to keep propositions in the form of subject – predicate thereby linking only two concepts (Finley & Stewart, 1982; Liu, 2001) was attempted but proved to make the list pedantic. For example, propositional statement 2.1.1.7 links at least four concepts as shown by the / divisions: Acids / and some metals / react chemically / to produce hydrogen. Moreover these subdivisions made the propositional knowledge unwieldy and did not facilitate clarity for educational practitioners. Finally, the original list of propositional statements that had been derived from literature sources in the selecting phase (see Section 4.6.2) was used to verify the new expanded statements. In a few cases, it was necessary to resort to textbooks when verifying these. This verification sometimes
showed discrepancies between my wording and that of experts and my statements, requiring closer examination of chemical principles, with propositional statements being subsequent reworded accordingly. At the end of this stage there was a composite list of propositional