Integrating Social and Representational Aspects of Problem Solving across the Highschool Curriculum Proposal to the Cognitive Studies for Educational Practice (csep) Program Mcdonnell Foundation 2.1 Employment Demands for Reasoning and Communica- Tion Skills

ions a ects the complexity of reasoning with those representations (see e.g. Levesque 1988, Stenning & Oberlander 1995). Put simply, the most e ective system of representation is generally the one which is just expressive enough for the abstractions the task requires, but no more. If it were more expressive, it would be less e ective, thanks to its o ering large but unnecessary inference spaces. Over the past several years, we have been concerned to develop expressiveness theory by investigating how people perform with representations of di ering expressiveness. In particular, we have been testing hypotheses concerning the way people learn various reasoning skills, and the ways in which the representations they acquire depend both upon their prior abilities, and upon the representations used in teaching them. Our research on this topic has demonstrated that expressiveness theory can provide a principled basis for the empirical study of the e ectiveness of teaching methods in real classrooms in formal subjects at freshman college level. Hyperproof is an example of new media in teaching (mentioned above). It is a computer environment for teaching rst order logic in which the machine's interactive redrawing capacities enable quite novel uses of graphics. An HCRC study of Hyperproof's use in a Stanford freshman class was carried out, with a balanced control course using conventional sentential methods. Expressiveness theory allowed us to make predictions|based on the semantics of the graphical presentations|about what would determine students' learning experiences with the system. Analysis of the data con rmed these predictions (eg Stenning, Cox & Oberlander, 1995; Cox, Stenning & Oberlander, 1995; Oberlander, Cox, Monaghan, Stenning & Tobin, 1996). Although this study was focussed on di erential e ects of media in teaching, it almost incidentally produced one of the very few extant demonstrations that logic teaching can substantially improve general reasoning and problem solving skills. Although there is strong belief in the literature that such generalisation will not take place, this belief is not based on studies of the impact of real logic teaching (as opposed to brief experimental interventions). We do not doubt that logic teaching could fail: but this study is an existence proof of its possible success. The data revealed strong aptitude treatment interactions (ATIs). Hyperproof is extremely good at teaching logic to some students. Subjects classi ed as diagrammatic reasoners responded to Hyperproof's graphical approach better than less diagrammatic students. However, diagrammatic students responded far less well to a syntactically (i.e. traditionally) taught course. Expressiveness theory allows detailed analyses of the mechanisms underlying di erences in thinking style which underpin the ATIs. Again, the importance of exibly addressing di erences in learning styles has recently been highlighted by the US Secretary's Commission on Achieving Necessary Skills. The crucial point about learning a logic is that it involves adding a new representational system to an existing repertoire. Some ways of teaching it seem to key into the richness (or otherwise) of that repertoire. That is, using multiple representations to teach logic works well for those who are adept at moving 8 between multiple representations. Students who bene t most from HP are those who are adept at translating between alternative modes of representation|a model for this whole project! These analyses provide the basis for re nement of the teaching method. But they also throw up the question of whether the styles are malleable and should be modi ed by curriculum revision, or whether they are more xed, and should be ameliorated by tting teaching methods to di erent styles. Thus, further classroom intervention would allow us to determine empirically the extent to which translational-aptitude can be improved, by extending students' representational repertoires. 3.3 Bringing social and representational issues together Students need both a better grasp of argument, and better general knowledge of representations. The teaching problems presented by these two domains are intimately connected. When we have come to put detail onto this program we nd that teaching derivation involves students' learning to ignore structures in expositional discourse which signal asymmetries of authority for information. Furthermore, these linguistic structures are eliminated in graphical representations. This observation invites the hypothesis that graphical representations have a particular function in teaching relations between exposition and derivation. For example, in exposition (say in a story) the two sentences Some of the boys are running and Some of those running are boys carry quite distinct implications about what is already known to the hearer, and what is new information. The subject/predicate structure determines di erent information packaging (see Vallduv 1992). But the student learning derivation has to learn that these di erences are logically immaterial and the two sentences are derivationally equivalent. Our data on quanti er interpretation reveals that this is a real problem for a sizable group of students. They have trouble disengaging from the expositional model of communication. Graphical systems of elementary logic such as Euler's Circles (see Stenning & Oberlander 1995 for a full semantic reconstruction of this method) actually remove information packaging structures entirely. The diagram for the two example sentences is wholly symmetrical. It is a quite general truth about the semantics of graphical systems that they have no systematic information packaging. We conjecture that graphical representations are often used in science and math for removing expositional structures which are di cult to eradicate from linguistic presentations. In Vygotskyan terms, learning the autonomy of representations requires abstracting them further and further from their embedding in social procedure. We are currently engaged in a teaching study which explores the relations between the kind of interpretation problems students exhibit, the method of teaching elementary logic, and the outcome in terms of subsequent reasoning behaviour. We conjecture that the students for whom relinquishing information packaging interpretations of quanti ers is a real problem will particularly bene t 9 from graphical presentations of the logic which remove information packaging structures. This is merely an example of the interaction of social understandings of communication and representational choices for problems. We believe these observations and analyses can be repeated for other areas such as the teaching of independence in reasoning with uncertainty, and in many other areas of problem solving. Summary: Communication consists of both exposition and derivation| both the building up of shared assumptions and the derivation of new ways of representing what participants share. Current divisions of the curriculum tend to separate analysis of the social relations determined by knowledge in communication (taught in the humanities) from the properties of representations of knowledge. And the teaching of di erent representations has grown scattered across the science and mathematics curriculum. We propose to bring these two aspects of communication and reasoning back together by making explicit, both for student and for teacher, relations between the skills of argument analysis and representation selection. Our purpose is to improve the generalisation of formal knowledge about reasoning in the solution of real-world problems. 4 Description of the proposed instructional or educational intervention. Our aim is to develop teaching interventions which can bring out the links between social and representational aspects of reasoning and communication, and thereby have an impact on students ability to combine and generalise their skills. We have already observed the tension between developing self-contained modules as opposed to coordinating teaching across subject areas. We adopt a strategy of developing a module rst with the express intention of embedding teaching interventions across the curriculum, and of taking the material into teacher training courses (see Section 4.3 for details of target student groups). In order to make our approach vivid we present here some examples from the research literature. They are not intended to be realistic classroommaterial, but rather to illustrate how theory relates to practice. We follow the examples with a more generalised description of a terrain for the teaching of communication and reasoning which is at present based on our undergraduate teaching experience. 4.1 Illustrative Examples 4.1.1 Example 1: Social Embedding Here is an example of a communicative misunderstanding: [Teacher]: All the bears in Omsk are white. One day Ivan went to Omsk and saw a bear. What colour was it? [Student]: I don't know. I wasn't there. You'll have to ask Ivan! 10 What has gone wrong in this conversation? Taken from Luria's famous ethnographic material, this dialogue illustrates misunderstanding easily analysed in terms of exposition and derivation. The teacher intends the syllogism's premisses to be taken as an exposition of material hitherto unknown to the student to be taken on trust in de ning a hypothetical world. The student is then supposed to derive a conclusion from it. Instead, the student assumes that exposition should continue, and pleads ignorance (the legitimate ignorance of a story's audience). Presenting the misunderstandings of others is one technique for making vivid the examples from which we seek to teach the underlying principles. Using video-clip examples of misunderstandings motivates students (there is a wealth of material available from comedy such as Monty Python's Argument Clinic), but also helps with application of learned principles to real situations. A good teaching method is to alternate the students' perspective from diagnostician (as the above example is presented) to reasoner (now you solve this problem, or persuade someone to `buy this argument'). 4.1.2 Example 2: Alternative forms of representation There are many di erent ways of representing the same information. For example, we can write a paragraph as a representation of the distances between three cities: C is ve miles from B. B is two miles from A. A is four miles from C. This text is reasonably well written but we could improve it. On the other hand an obscurely written text could contain exactly the same information but make things really hard: The northernmost city is ve miles from the city which is neither most northern nor most southern. The town which is neither most eastern nor most western is two miles from the town which is on average furthest from the other two . . . Alternatively, we could decide to organise our statements in a table of inter-city distances. Tables can also be organised in di erent ways, and there is a close relation between tables and maps. With the right organisation of the rows and columns, the zeros in the table become the city positions on the map: Alphabetical Long./Lat. Map Table Table A B C B A C A 0 2 4 C 5 4 0 C B 2 0 5 B 0 2 5 B C 4 5 0 A 2 0 4 A 11 An o ce manager must assign o ces to six sta members. The available o ces are numbered 1{6 and are arranged in a row, separated by six foot high dividers. Therefore sounds and smoke readily pass from one to others on either side. Ms Braun's work requires her to speak on the phone throughout the day. Mr White and Mr Black often talk to one another in their work and prefer to be adjacent. Ms Green, the senior employee, is entitled to O ce 5, which has the largest window. Mr Parker needs silence in the adjacent o ces. Mr Allen, Mr White, and Mr Parker all smoke. Ms Green is allergic to tobacco smoke and must have non-smokers adjacent. All employees maintain silence in their o ces unless stated otherwise. 1. The best o ce for Mr White is in 1, 2, 3, 4, or 6? 2. The best employee to occupy the furthest o ce from Mr Black would be Allen, Braun, Green, Parker or White? 3. The three smokers should be placed in o ces 1, 2, & 3, or 1, 2 & 4, or 1, 2 & 6, or 2, 3, & 4, or 2, 3 & 6? ... Figure 1: Room allocation problem What we choose will depend on what has to be done with the information and who it is for. Looking up cities is easier if they are alphabetical. But which form of representation we can use is also determined by what information we have. Maps depend on having complete information. Tables also need entries to be classi ed in both coordinates. Texts on the other hand can cope with any combination of missing data. We can see these generalisations played out in all examples of tables from other parts of the curriculum. 4.1.3 Example 3: Combining social and representational issues Figure 1 presents a simple well de ned problem which is easy to transpose into any number of real-world guises. It is also possible to add a dimension of negotiation to the problem by giving insoluble constraints and setting the problem of how to get people to agree to relax them to a possible solution. Such negotiation is often taught by role play methods. But even the simple problem brings out a wide range of representational strategies from students, some e cient and some obstructive. Students use tables, plans, and textual rearrangement among other strategies. Even understanding the language of these problems requires overcoming various implicatures which would operate if this were fully cooperative exposition. 4.1.4 Example 4: Expressiveness and representation selection The `poets' problem in Figure 2 is interesting for a number of reasons. Answering the questions in fact requires solving a syllogism for each. Most of the di culty is identifying the premisses relevant to each answer. This is made more di cult by the fact that the premisses appear to be relational, whereas their relevant form is categorical. They are also not in a helpful order. Each may be solved by rules or by diagrammatic methods. What the problem will not permit 12 Professor Kittredge's literature seminar includes students with varied tastes in poetry. All those in the seminar who enjoy the poetry of Browning also enjoy the poetry of Eliot. Those who enjoy the poetry of Eliot despise the poetry of Coleridge. Some of those who enjoy the poetry of Eliot also enjoy the poetry of Auden. All those who enjoy the poetry of Coleridge also enjoy the poetry of Donne. Some of those who enjoy the poetry of Donne also enjoy the poetry of Eliot. Some of those who enjoy the poetry of Auden despise the poetry of Coleridge. All those who enjoy the poetry of Donne also enjoy the poetry of Frost. 1. Miss Gar eld enjoys the poetry of Donne. Which of the following must be true? She may or may not enjoy the poetry of Coleridge. She does not enjoy the poetry of Browning. She enjoys the poetry of Auden. She does not enjoy the poetry of Eliot. She enjoys the poetry of Coleridge. 2. Mr Huxtable enjoys the poetry of Browning. He may also enjoy any of the following Poets, except: Auden Coleridge Donne Eliot Frost ... Figure 2: The poets problem is construction of a single comprehensive diagram, though many students persist in trying to do this. We have even seen this impossible strategy recommended in a test-crammer! In fact diagrams are insu ciently expressive to capture the abstractions of the whole problem. Each of these features is is an entry point for teaching about representational strategies. 4.2 Curriculum Our approach is to make as much use as possible of materials that have already been developed for teaching about the social and representational issues that we are targeting. We see our main contribution as providing an overarching theory which can make the interelations between these materials clear at di erent levels to both teacher and student. We will start by collecting material from across the existing highschool curriculum, and from the research literature, to see how best to t the material into our framework. What follows draws heavily on material developed for an interdisciplinary freshman undergraduate course called Human Communication which we have developed over the last two years. This is taught to a wide range of freshman students from arts and sciences. In moving the material down the curriculum (and ability range) to highschool we are aware that a more `how to' approach will be necessary. On the other hand, experience of teaching freshman students has underlined their need for explicit teaching on these topics. Here we illustrate by taking some topics which we know to be relevant and 13 Social embedding Representation selection analytical reasoning negotiation, argument analysis expressiveness, transformation, redescription reasoning in uncertainty stereotypes, propaganda, persuasion, base-rates sample and population, variability, sequences vs. kinds of sequences Table 1: Example topics classi ed by areas of problem solving and by social and representational aspects Social embedding Representation selection english argument analysis concept mapping oral skills debate, presentations graphics for data presentation history role playing timelines social science stereotypes, persuasion graphical exploratory data-analysis science problem formulation and negotiation matrix graphics Table 2: Emphasis on components of problem solving by areas of the curriculum. Bold denotes current emphasis. placing them in a matrix organised by the social vs. representational dimension. In teaching freshman undergraduates we have found the division between analytical reasoning and reasoning in uncertainty a useful division of the subject matter of problem solving. At a theoretical level, logic underpins the rst, whilst statistical reasoning the second. Within the psychological literature on reasoning, work such as that of Wason on conditional reasoning, and Gentner on analogical reasoning relates to the rst area. Work by Tversky and his colleagues dominates the second. Classifying illustrative topics by these two divisions of problem solving, and by our distinction between social and representation aspects we derive Table1. While we teach these kinds of reasoning and their underlying theories explicitly at university level, our strategy here would be to nd them implicitly in existing highschool subjects. These topics are already found in the curriculum, but the balance of social and representational aspects correlates broadly with the humanities/science dimension in the curriculum, as Table 2 illustrates. 4.2.1 Analytical reasoning By analytical reasoning problems we mean the kind of problems found in the Analytical Reasoning scale of the Graduate Record Exam (e.g., Examples 2{4). We choose them because they are designed to require limited domain knowledge, and to correlate with general reasoning skills needed at University. Absolute level of di culty can be adjusted for the student level. 14 Social Embedding In raw form these problems tend to be rather well de ned and rather arti cial, but by embedding them in more realistic social circumstances they can be used as a vehicle for teaching the negotiation of wellde ned problems from ill-de ned ones, and for teaching the role of derivation in argumentation. Here we would expect to call on materials from the the Argument Clinic, from our own materials, and from critical thinking courses. Representational SkillsHere we already know from our own work that rerepresentation skills are highly signi cant, and are improved by logic teaching. We already have some principles derived from expressiveness theory which condition the kinds of representation best used for di erent problems|editing of text, networks, tables, set-diagrams, layout diagrams etc. (in approximate order of increasing expressiveness). We have recordings of students solving problems (and also failing) which can be used to teach other students how to approach the problems. Another advantage of this domain is that it is possible to relate the representations to their uses elsewhere in the curriculum, and to work through enough examples to aid student generalisation about representations. 4.2.2 Reasoning under uncertainty Drawing on Tversky's work on the heuristics which people use for reasoning in uncertainty, we can bridge from topics in English, such as anecdotalism; in Social Studies such as propaganda and prejudice; right across to statistical and mathematical issues about variability. Social Embedding Anecdotalism is the practice of tempting a reader to make generalisations on the basis of stereotypical interpretations of single-case anecdotes. Tversky shows how this rhetorical device is based on peoples' heuristics for reasoning in uncertainty. Anecdotalism can be taught as a positively useful persuasive communicative device (often necessary, for example when teaching), as well as a sinister propaganda tool. What the student needs is a principled understanding of the invited inferences, and systematic methods of questioning what assumptions are smuggled in. Much of what is required can be thought of as an understanding of the relevant base-rates. Representational SkillsGraphical devices are widely used for teaching the basis of statistical reasoning both in industry (e.g. statistical process control for machine operators is taught graphically) and in highschool and college maths and social science. Most of these graphics are representations of distributions and the concept of population and variation. There are also graphical methods for teaching the core statistical concept of independence based on a metrical interpretation of Venn Diagrams (see e.g. Cheng, 1995). More sophisticated graphical methods can reach up to the derivation of Bayes-theorem (see Shimojo and Ichikawa 1989). 4.3 Student Groups We have therefore identi ed two niches in the current Scottish Highers curriculum which will receive support for our teaching interventions from schools. The 15 rst is 6th year (16{17 yr olds) highers students who have completed a rst year round of highers assessments and will be going to college the subsequent year. These students are less timetable-constrained than in their previous year, and represent the minimal educational step from the freshman undergraduates we have studied hitherto. The second niche is 5th year (15{16 yr old) students enrolled in GSVQ (General Scottish Vocational Quali cation) modules for Problem Solving (modules 1{4). These are academically less able students than the previous group, making it perhaps more di cult, but certainly more important, to develop a successful program. Because these GSVQs are already accredited 40 contact hour modules, but are quite generally speci ed in terms of teaching outcomes rather than methods, this latter group give us an immediate opportunity to demonstrate to schools that teaching reasoning and communication can have an impact on students' performance across the curriculum. 5 Experimental design In previous work we have always stressed the importance of having a control group representative of current practice|the Hyperproof study used a conventionally taught `sentential' logic course with established text, with students drawn from the same population. The proposed project presents the extra difculty that there is little established practice in integrating curriculum around core skills of reasoning and communication. There is no established practice for teaching this topic. However, we have the advantage that although there is no accepted teaching practice, there are evaluation benchmarks externally set against which schools and students' performance are to be measured. Problem solving may not be seen as a subject, nor taught by an accepted method, but it is systematically a separate component of the way all work is graded. The GSVQ modules have associated assessments which distinguish factual knowledge from its deployment in problem solving. The Scottish O ce is about to publish its evaluation methods for `core skills'. There will therefore be large bodies of students taking these tests which have not experienced our teaching interventions. In some cases these evaluation criteria are themselves of recent origin, and our research may reasonably expect to produce evidence for the improvement of evaluation, as well as re ecting on our experimental teaching interventions. In previous work, we have found individual di erences between students' thinking styles, and aptitude treatment interactions to be powerful analytical tools in understanding the processes underlying observed performances. Thus the Hyperproof study yielded some gross information about relative performance of teaching methods, the information gained from analyses within the Hyperproof course across di erent pre-course aptitude groups yielded far more insightful information into what was good and what was bad about the teaching methods, and for whom. Accordingly, we will use similar designs in the proposed study. Pre-course 16 and post-course aptitude tests will be used to assess progress and to identifystyles of thinking and how they change after teaching. These tests will berun on control groups receiving no systematic intervention, but being taughtsimilar curricula outside our interventions. As much student problem workingas possible will be recorded (both process and product) for analysis. This will bedone using video tape. In some cases use of software environments is possible andappropriate for this function. Our existing theoretical frameworks are su cientto indicate some of the important individual di erences in response we can useto base process analyses on.Gross analyses will be based on standard regression techniques for apti-tude treatment interactions. We have developed a wide range of special pur-pose computational linguistic analyses and visualisations of the complex teach-ing/learning discourses that arise from this sort of study (see references forexamples). We believe more insight is gained from tailoring analyses to themental processes which turn out to be important in determining learning out-comes than to the application of standard statistical techniques applied to thegross data of student grades. A combination of theoretical predictions aboutwhat semantic properties of representations will be critical in their use, and pre-course measurements of thinking styles, take the approach beyond psychometricmethods to the study of the underlying mechanisms.Our theoretical goal is to understand the relationship between the socialand representational aspects of reasoning and communication, and how to teachthem. To this end we need to understand the relations between the individualdi erences we have observed in students' models of communication, and theirrelations to the individual di erences observed in students' responses to teachingwith di erent kinds of representation. This will involve modelling the mentalprocesses underlying the performances. We can only do this by experimentingin realistic teaching situations where we use (and re ne) the `ruling' metrics ofstudent performance across the curriculum.If the theoretical goals are met and the project develops a successful in-tervention and can show how this can be integrated into teaching across thecurriculum within the constraints of schools' operations, then it will be possi-ble to make a speedy introduction into the teacher training curriculum throughMoray House's training function.5.1 TimelineYear 1: Develop curriculum module with highschool teachers seconded tothe project by adapting our existing communication and reasoning cur-riculum, working with small groups of students.Year 2: Evaluate e ects in `normal class teaching' with `designer' teachersin assessed modules.Year 3: Repeat teach/evaluate cycle. 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