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Simulating Creative Reasoning in Mathematics Teaching Tomas Bergqvist and Johan Lithner
Research reports, No 2, 2005 in Mathematics Education
Simulating Creative Reasoning in Mathematics Teaching Tomas Bergqvist and Johan Lithner Abstract. The purpose of this paper is to study how regular teaching simulates non-routine problem solving in the sense that the presented reasoning is creative and the conclusions are justified by explicit reference to mathematical properties of the components. In the situations analysed this is sometimes the case, but in rather limited and modest ways. The teaching is mainly focussed on presenting algorithmic methods by solving routine tasks.
Contents 1. Introduction 1.1. Quality in teaching 1.2. Observation of practice 1.3. Practice and change 1.4. Teacher - student interaction 2. Theoretical framework 2.1. Creative reasoning 2.2. Imitative reasoning 2.3. Simulation of creative reasoning 2.4. Research question 3. Method 4. Data presentation and local analysis 4.1. Insufficient explanations 4.2. Solving equations: Presentation of algebraic rules 4.3. An inequality: Non-creative explanations 4.4. Finding Angles: Creative questions 4.5. Proving a theorem: Creative reasoning without reflections 5. Global analysis and results 5.1. Mathematical foundation 5.2. Creativity 1
2 3 4 5 5 6 6 8 10 11 12 12 13 15 16 18 20 22 22 23
5.3. Reflection 5.4. Focus 5.5. Summary 6. Discussion References
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1. Introduction For several years the mathematics education community has been aware of the difficulties in helping students to develop fundamental mathematical abilities like conceptual understanding, problem solving ability, critical thinking, and creative reasoning. Mathematics is often reduced to a large set of isolated incomprehensible facts and procedures to be memorised and recalled at written tests. (Skemp, 1978) distinguishes between ‘instrumental understanding’ and ‘relational understanding’ of mathematical procedures. The former can be apprehended by a person as ‘true’ (relational) understanding, but is only the mastering of a rule or procedure without any insight in the reasons that make it work. Other research has pointed to the relation between weak conceptual understanding and a procedural focus, e.g. Tirosh and Graeber (1990); Tall (1996); White and Mitchelmore (1996). Previous research, Lithner (2000a,b, 2003b) and Bergqvist et al. (2003), on mathematical reasoning indicate that students often rely on non-creative and mathematically superficial strategies when they meet problematic situations. The reasoning is dominated by what is remembered and familiar rather than focusing on intrinsic properties of the mathematical objects. In these studies it seemed like the students didn’t hold the belief (see Schoenfeld (1985) for a discussion about beliefs) that problematic situations could be solved with creative reasoning. Due to the “astonishing complexity of mathematical learning”, as it is formulated by (Niss, 1999) when summarising the results of research in mathematics education, many factors will affect the ways that students learn to reason mathematically. These include influences from school and from the individual’s home and community cultures Brenner (1998). Concerning research in both traditional and alternative classroom practices, (Hiebert, 2003) summarises the baseline conclusion as “students can learn what they have opportunity to learn”. So a crucial question arises: what opportunities do they have to 2
develop different forms of mathematical reasoning? In a series of studies different parts of the learning environment are analysed, using the mathematical reasoning framework presented in Section 2 below. In Lithner (2004) it was shown that a majority of items in calculus textbooks were possible to solve with superficial reasoning, and in Lithner (2003b) that students use such reasoning. In ongoing studies students’ seat-work and course tests are analysed. In this study we focus on the presentations made by teachers, and their reasoning as apprehended by students. If we want to foster reflecting and creative students, what kind of reasoning should they meet? In what ways can the teachers provide this to the students? One possibility would be for the teacher to present genuine problem solving on the black board. The main objection to this is, of course, that a genuine problem to the teacher is normally mathematically to difficult for most of the teacher’s students. One alternative is to show the students a simulation of a solution of a non-routine problem by reasoning as if it was the first time the teacher solved it. The general research question of this paper is: In what ways are simulations of creative reasoning utilised by the teachers as a means to help students to learn mathematical problem solving? In the Sections 1.1 to 1.4 other research on classroom practices will be presented. 1.1. Quality in teaching. Analyses of quality in teaching has been made by some researchers, often connected to implementation of changes. (Artzt and Armour-Thomas, 1999) suggests a framework for examining teachers’ instructional practice. The authors argue that the value of the model lies in its usefulness when it comes to enabling teachers to reflect over their own teaching. In the framework a lesson is divided into phases and dimensions. The phases are the initiation, the development and the closure of an instructional episode. within each of these three phases, instructional dimensions are examined. The dimensions are tasks, learning environment and discourse. They are chosen since the NCTM Standards NCTM (2000) suggests them as areas of instructional practice that might foster learning with understanding, a central conception in this study. Our research would in this framework partly concern the task -dimension (i.e. sequencing and modes of representation), and partly the learning environment-dimension (modes of instruction). A similar framework is suggested by (Kahan et al., 2003). Their starting point is somewhat different in that they look at the importance of 3
teachers’ content knowledge in their teaching. The lessons are here divided into processes and elements. The authors argue that the difference from (Artzt and Armour-Thomas, 1999) is that they identify teaching processes rather than phases. Processes can overlap and be ongoing, while phases are temporal stages. The four processes are preparation, instruction, assessment and reflection. For each of these there are six elements of teaching. The authors propose that by looking into several of the 24 “cells”, it is possible to reach a better founded conclusion concerning the teacher’s mathematical content knowledge, but also concerning the pedagogical content knowledge. In relation to this framework, our research questions would concern the mathematical development and sequencing during instruction. An other kind of structure in order to analyse how teachers develop a concept in the classroom is proposed by (Bromme and Steinbring, 1994). Starting from a model of communicated meaning, the meaning of meaning triangle, object - symbol - concept, they identify the extent to which a concept is described using an object or using a symbol. The quality of the teaching becomes higher if the teachers develop both sides of a concept, both the object and the symbol. The authors mean that even more important for the quality of the teaching is how the teachers treat the relationship between object and symbol.
1.2. Observation of practice. The difference in the structuring of a lesson between expert and novice teachers has been examined by (Leinhardt and Greeno, 1986). The authors argue that teaching “rest on two fundamental systems of knowledge, lesson structure and subject matter ” (p. 75). The study focus almost entirely on the lesson structure. The main difference between expert and novice teachers found in the study was that the experts both more flexible and more consistent in their structure. More flexible in the sense that they quickly could change what routine they would use, and more consistent in that the routines, once decided upon, was well-practised and mutually known to both the teacher and the students. Teachers’ preferences of methods for problem solving was examined by (Leikin, 2003). The results suggests that teachers choose different problem solving strategies depending on the goal, whether it is to solve the problem, to explain the problem, to teach the problem or a solution they like. One interesting finding is that when the goal is to teach the problem, teachers 4
prefer a strategy which is more convincing, and not a strategy which makes the problem easier to solve or easier to explain. Observations of teachers’ practice can also be found in several other studies. In (McDuffie, 2004) teachers’ reflective thinking were investigated in relation to their pedagogical content knowledge (PCK). Teachers’ use of manipulatives in teaching mathematics was looked upon by (Moyer, 2002), and teachers awareness and handling of the fact that students have a tendency to finish algebraic expressions (3x + 2 = 5x) was examined by (Tirosh et al., 1998). 1.3. Practice and change. Analyses of practice connected to changes of some kind is also present in the research literature. (Simon and Tzur, 1999) propose a methodology for studying teachers development in relation to reform, creating accounts of practice, which can help researchers to find the specific pedagogical problems related to teachers development from traditional to reform teaching. Results of specific inservice courses for teachers is also an important issue. Teachers’ change of practices as a result of a problem solving course was analysed by (Chapman, 1999). A comparison of three groups of teachers, participating in different programmes for change was made by (Saxe et al., 2001). Observation as a means for implementing change has also been investigated. (Grant et al., 1998) found that teachers observing lessons performed by reform minded teachers is an effective approach, at least if the observing teacher already has a vision or mental image of a better way to teach. According to (Nilssen et al., 1998) observations in teacher education can be a way to improve the teacher student’s practice, but then it is necessary to let the student discuss the observations with a mentor or supervisor. 1.4. Teacher - student interaction. Communication in the classroom is an interesting research focus. One example of a study in this area is a case study by (Brendefur and Frykholm, 2000) They investigated two novice teachers’ interest in and ability to use communication in the classroom. They found that there was “an interesting contrast between the two teachers and their ability and willingness to implement various forms of communication in the classroom” (p. 144). One of them really tried to give the students opportunities for communication, while the other believed that instructions from the teacher was the best way to enable student learning. 5
Two other examples are (Forster, 1999) and (Forster and Taylor, 2003), where analysis of a teacher’s interaction with a student presented. In these studies the teacher and researcher were the same person. In the study by (Forster and Taylor, 2003), the goal was to identify communicative competencies when the use of graphics calculators.
2. Theoretical framework The framework is a summary of Lithner (2003a) which is a theoretical structuring of the outcomes of a series of empirical studies (Section 1) aiming at analysing characteristics the relation between reasoning types and learning difficulties in mathematics. 2.1. Creative reasoning. ‘Reasoning’ in this paper is the line of thought, the way of thinking, adopted to produce assertions and reach conclusions. It is not necessarily based on formal deductive logic, and may even be incorrect as long as there are some kind of sensible (to the reasoner) arguments that guide the thinking. Argumentation is the substantiation, the part of the reasoning that aims at convincing oneself, or someone else, that the reasoning is appropriate. In particular, in a task solving situation, which is called problematic situation if it is not clear how to proceed, two types of argumentation are central: 1) The strategy choice, where ’choice’ is seen in a wide sense (choose, recall, construct, discover, guess, etc.). This choice can be supported by predictive argumentation: Will the strategy solve the difficulty? 2) The strategy implementation, which can be supported by verificative argumentation: Did the strategy solve the difficulty? In this paper, creative reasoning is seen as what you do when you solve non-routine problems Schoenfeld (1985). According to (Haylock, 1997) there are at least two major ways in which the term is used: i) Thinking that is divergent and overcomes fixation. ii) The thinking behind a product that is perceived as creative by a large group of people, e.g. works of arts. Central here are the creative aspects of ordinary mathematical task solving thinking, thus notion ii) is not very useful here. (Haylock, 1997) sees two types of fixation. Content universe fixation is in terms of range of elements seen as appropriate for application to a given problem: useful knowledge is not seen as useful. Algorithmic fixation is shown in the repeated use of an initially successful algorithm when this becomes inappropriate. (Silver, 1997) argues that “Although creativity is being associated with the notion of ‘genius’ or exceptional ability, it can be productive 6
for mathematics educators to view creativity instead as an orientation or disposition toward mathematical activity that can be fostered broadly in the general school population.” He adds that students hardly experience mathematics as the highly creative intellectual domain it is. Silver sees fluency, flexibility and novelty as the core components of creativity. In school tasks, one of the goals is also to achieve a high degree of certainty, but one crucial distinction from professional tasks is that within the didactic contract Brousseau (1997) of school it is allowed to guess, to take chances, and use ideas and reasoning that are not completely firmly founded. Even in exams it is accepted to have only, for example, 50% of the answers correct, while it is absurd if the mathematician, the engineer, or the economist are correct only in 50% of their conclusions. This implies that it is allowed, and perhaps even encouraged, within school task solving to use forms of mathematical reasoning with considerably reduced requirements on logical rigour. (P´olya, 1954) stresses the important role of reasoning that is less strict than proof: “In strict reasoning the principal thing is to distinguish a proof from a guess, a valid demonstration from an invalid attempt. In plausible reasoning the principal thing is to distinguish a guess from a guess, a more reasonable guess from a less reasonable guess.” In this framework, well-founded arguments are anchored in intrinsic properties of components involved in the reasoning. The components one is reasoning about consist of objects, transformations, and concepts. The object is the fundamental entity, the ‘thing’ that one is doing something with or the result of doing something. E.g. numbers, variables, functions, graphs, diagrams, matrices, etc. A transformation is what is being done to an object (or several), and the outcome is another object (or several). Counting apples is a transformation applied to real-world objects and the outcome is a number. To calculate a determinant is a transformation on a matrix. A concept is a central mathematical idea built on a related set of objects, transformations, and their properties. For example the concept of function or the concept of infinity. Since a property of a component may be more or less relevant in a particular context and problematic situation, it is necessary to distinguish between intrinsic properties that are central and surface properties that have no or little relevance. In deciding which of the fractions 99/120 and 3/2 that is largest, the size of the numbers (99, 120, 3, and 2) is a surface property that is insufficient to consider in this particular task while the quotient captures the intrinsic property.
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Creative reasoning (CR) fulfils the following conditions: i) Novelty: A new (to the reasoner) sequence of solution reasoning is created, or a forgotten sequence is re-created. To imitate an answer or a solution procedure is not included in CR. ii) Flexibility: It admits different approaches and adaptations to the situation. It does not suffer from fixation that hinders the progress. iii) Plausibility: There are reasons supporting the argumentation in the strategy choice and/or strategy implementation, motivating why the conclusions are true or plausible. This means that pure guesses or vague intuitions are not considered, neither are affective reasons. iv) Mathematical foundation: The argumentation is founded on intrinsic mathematical properties of the components involved in the reasoning. 2.2. Imitative reasoning. More frequent than CR are different versions of imitative reasoning, that is, copying or following a model or example without any attempt at originality. (Hiebert, 2003) finds massive amounts of converging data showing that students know some basic elementary skills but there is not much depth and understanding. Students are more proficient in processes like calculating, labelling, and defining then reasoning, communicating, conjecturing, and justifying. Learning difficulties are partly related to a reduction of complexity that appears as a procedural focus on facts and algorithms and a lack of relational understanding Lithner (2003a). The framework should also capture reasoning that is not justified by mathematical reasons, but has other origins Vinner (1997). The definitions below aims at characterising imitative reasoning that may be based on surface clues in non- or semi-cognitive attempts to cope. Memorised reasoning (MR) is when: (i) The strategy choice is founded on recalling by memory an answer. (ii) The strategy implementation consists only of writing it down. One can describe any part of the answer without having considered the preceding parts. An example is to recall every step of a proof. An algorithm is a set of rules that will if followed solve a particular task type. The most common algorithms consist of procedures (sequences of transformations of mathematical objects).
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Algorithmic reasoning (AR) is when: (i) The strategy choice is founded on recalling by memory, not the whole answer in detail as in MR, but a set of rules that will guarantee that a correct solution will be reached. (ii) After this set of rules is given or recalled the reasoning parts that remain in the strategy implementation are trivial for the reasoner and only a careless mistake can hinder that an answer to the task is reached. Fundamental in AR is how to identify a suitable algorithm. If this can be done, the rest is straightforward. AR based on surface property considerations is common, often dominating, and the studies mentioned in Section 1 have distinguished three (partly overlapping) families of common reasoning: Familiar AR This reasoning consists of strategy choice attempts to identify a task as being of a familiar type and with a corresponding known solution algorithm. The simplest example is a version of the Key word strategy where the word ‘more’ in a text task is connected to the addition algorithm and ‘less’ to subtraction Hegarty et al. (1995). Another example is in Lithner (2000a) where students make a holistic but superficial interpretation of the task text and reach a clear but faulty image that it is of a particular familiar type. Delimiting AR The algorithm is chosen from a set of algorithms that are available to the reasoner, and the set is delimited by the reasoner through the included algorithms’ surface property relations to the task. E.g. if the task contains a second degree polynomial the reasoner can chose to solve the corresponding equation p(x) = 0 even if the task asks for the maximum of the polynomial Bergqvist et al. (2003). Here, the reasoner do not have to see the task as a familiar one. Guided AR An individual’s reasoning can be guided by a source external to the task. The two main types empirically found are: (i) Piloted reasoning, when someone (e.g. a teacher) pilots a student’s solution. (ii) Identification of similarities, where the strategy choice is founded on identifying similar surface properties in an example, definition, theorem, or some other situation in a text source connected to the task.
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2.3. Simulation of creative reasoning. In order to capture the quality of the correlation between the simulated and the real situation, (Fitzpatrick and Morrison, 1971) uses the term representativeness which refers to the combination of comprehensiveness and fidelity. Comprehensiveness refers to “the range of different aspects of the situation that are simulated” and fidelity to “the degree to which each aspect approximates a fair representation of that aspect in the criterion situation”. In order to analyse whether the simulated problematic situation solution is of high representativeness or not, it is tested against the following CR simulation criteria: 1. Mathematical foundation: The reasoning is founded on intrinsic mathematical properties of the components involved in the reasoning, in the same way as CR. This criteria is seen as fulfilled if the conclusions are based on explicit considerations of relevant properties. The properties need to be discussed, and this is often done in a form that is, or can be reformulated as, ’the statement is true since the components has these mathematical properties which has these consequences’. If this is not present, e.g. in a pure algorithmic description, then the criteria is seen as not fulfilled. 2. Creativity: The reasoning is similar to CR in the sense that a new (to the students) sequence of reasoning is created, that starts from a (simulated) problematic situation and, through explicit predictive argumentation that in advance supports the strategy choices, terminates in a conclusion. In the examples analysed in this study the CR criteria ii) is always fulfilled since the teacher’s control over the reasoning guarantees that a solution is reached without being hindered by fixations. The criterion is not fulfilled if the reasoning starts with a conclusion that is afterwards explained. 3. Reflection: Reflections and/or (simulated) uncertainty is present. In routine task solving there is no uncertainty since the solver knows from the start exactly what to do. In a real problematic situation, the strategy choices are not all evident and metacognitive control Schoenfeld (1985) may be required in the strategy implementation. This can appear as reflections, hesitations or openness for alternative solution strategies, e.g. in the form of explicit strategy choice questions. It could also be fulfilled by the presence of (simulated) mistakes. It is not fulfilled by questions or mistakes that concern very local and elementary parts of algorithmic procedures, e.g. asking ”what is 12/4?” when using the standard formula 10
to solve a second degree equation. 4. Focus: The teacher’s main simulated problematic situations and solution goals are similar to the students’. In some situations there is clear evidence that the teacher’s (simulated) problematic situations differ from the students’, then the focus criteria is seen as not fulfilled. Another cause of different focus may be that the teacher’s reasoning is unrealistic in the sense that it is much too difficult or based on facts and knowledge that is clearly not accessible to the students. If there are no indications of a different focus, it is categorised as fulfilled. The comprehensiveness, that the criteria are the proper ones, originates from the CR definition (if not 1 and 2 are fulfilled there is no CR, and from preliminary analyses of empirical data: 3 and 4 were found to be the main additional aspects were low fidelity lead to simulations with low representativeness. In analysing a specific reasoning situation, a criteria is classified as Y if it is fulfilled and as N otherwise. Since this is mainly a qualitative study and a data sample that is representative of Swedish teaching is hard to reach (see Section 3), the goal is not to produce a quantitative description of how many criteria that were fulfilled. The purpose is instead to find in what sense, and why, the teachers’ reasoning can be seen as simulations of creative problem solving and in what ways it is not. 2.4. Research question. The main teacher–student interaction in Swedish upper secondary mathematics classrooms is through teacher-lead presentations and dialogues. In essentially all of them the teachers do not present real problematic situations, but ones (e.g. exercise solutions) where the teacher has prepared a solution. Thus they are not real problematic situations, and the teacher’s intention may not be to simulate one. There are two types of teaching situations where simulated CR can be studied: Teaching that is designed specifically with the purpose to emphasise CR or regular teaching. Since there are indications (Section 1) that the former is uncommon it is important to study how teachers’ ordinary presentations involve elements of creative, mathematically well-founded, reasoning. This study is based on the following research question: • In what ways do teachers in regular teaching manage or fail to engage themselves and students in simulating CR as a means of making progress in solving tasks? 11
3. Method Since teaching can take place in numerous ways it is impossible, (simply due to the huge variety of contents, tasks, goals, students, teaching styles, etc., and their different combinations) to try to capture even the most common types of teaching situations and therefore also to specify in a unique way what the ’regular teaching’ mentioned in the research question really is. The lessons visited for data collection were due to this complexity not possible to choose as a representative selection from the Swedish school. Instead, we visited several different teachers whose teaching were judged by us to be of common, or at least not uncommon, types. The data collected consisted of field-notes from three educational levels: two lessons from lower secondary school, six from upper secondary school and four from undergraduate mathematics education. An observer took notes focusing on the presentation and on the interaction between the teacher and the students. After each lesson a brief discussion with the teacher was initiated. The teacher was asked questions about the lesson, e.g. if the general goal with the lesson was met. The analysis of each transcribed teacher presentation was done in three steps: 1. Interpretation.: A description of the researchers’ interpretation of the reasoning as situated in the classroom context, and of how the groups of students may apprehend it. 2. Identification of (simulated) problematic situations.: The notion of problematic situation (Section 2.1) is extended to include simulations (Section 2.4). The central actual and simulated problematic situations are identified, and the argumentation is characterised. The teachers’ actual thought and goals, e.g. conceptual understanding, algorithm description, or problem solving, are not considered in this analysis, only the presented explicit reasoning. 3. Characterisation: Characterise the reasoning from the previous step according to the framework in Sections 2.1, 2.2, and 2.3.
4. Data presentation and local analysis From 23 teaching situations analysed the five examples below represent different types of reasoning. Summarised versions of the actual analyses are presented. 12
4.1. Insufficient explanations. In a ninth grade classroom the teacher starts by briefly repeating some earlier items like “What is 34 ?” and “How do you write x · x · x · x · x?”. The questions below are more difficult, still clearly within the basic curricula, but the students have great difficulties in answering them Lithner (2002): The teacher writes x(x + 5) = and asks what this is. No student replies. Teacher: “We start with the first two. What will that be, Max?” Max: “2x.” Teacher: “No, what was x · x · x · x · x?” Max: “Could it be 5x2 ?” Teacher: “No. It is x2 + 5x. What is −4x(2x + y)?” Jan: “8x−” The teacher interrupts: “No”, and writes without discussion −(8x2 +4xy). Teacher: “What does this become? Remove the parenthesis.” Eve: “−8x2 − 4xy.” Teacher: “What is 3x(2x + y) − (3x + 2y)(x − 2y)?” Ann seems able to solve the task and starts to answer, but the teacher does not let Ann speak herself. Instead, the teacher leads Ann by formulating all sub tasks, for example “What is 2 · 3?”, “What is x · x, etc., without discussing why these particular steps should be taken. The teacher summarises by writing: = 6x2 + 3xy − 3x2 + 6xy − 2xy + 4y 2 = 3x2 Bea: Bea interrupts the teacher’s writing: “Does it become 3x2 ?” Teacher: “If you have six apples and remove three apples, what remains?” Bea: “Three apples.” Teacher: “Yes, therefore it gets 3x2 ” The teacher finishes the interrupted writing: = 3x2 + 7xy + 4y 2 . “Is it correct, Joe?” Joe: “Don’t know.” After this dialogue the students turn to their textbook exercises. The observer asks the students (individually) what they are doing. Most of them give answers like “I don’t know” or “I haven’t got a clue”. 4.1.1. Interpretation and identification of problematic situations. Four problematic situations are identified: three tasks and Bea’s question: PS1: How do you expand x(x + 5)? Max believes wrongly that ”the first two” (whatever that means to him) becomes 2x. The teacher’s implicit strategy choice contains two parts: i) The algorithm is to multiply the left factor with the two right terms one by one. ii) The first multiplication is done using the definition of powers (x · x = x2 ), as in the earlier example 13
(x · x · x · x · x = x5 ). Max may understand this simple earlier example, as many of the other students seem to do, but not that the algorithm a(b + c) = ab + ac should be applied. Therefore Max is unable to use the teacher’s guidance, and implements some faulty algorithm to reach 5x2 . The teacher gives without argumentation the correct answer and proceeds to the next task. PS2: How do you expand −4x(2x + y)? Jan does not know the correct algorithm or makes a careless mistake. The teacher just states the correct answer. PS3: How do you expand and simplify 3x(2x + y) − (3x + 2y)(x − 2y)? Instead of letting Ann try to reason herself, the teacher makes all strategy choices and leaves only the elementary local transformations to her. PS4: How come 3x2 remains (as one of the terms) when simplifying 6x2 + 3xy − 3x2 + 6xy − 2xy + 4y 2 ? The teacher’s strategy choice is to use the apple analogy. The intention is probably to demonstrate that you should add the terms that are of the same type (x2 ), but there is no explicit connection made between this and Bea’s problematic situation, which concern the mathematical foundation for simplifying the whole sixterm expression. Bea surely knows that six apples take away three makes three, but that does not explain the connection to the task. 4.1.2. Characterisation. For each of the four criteria a letter N will mean that the criterion is not fulfilled and a letter Y will mean that the criterion is fulfilled. (1) Mathematical foundation: (N) Several of the students seem to know the basic power property that am = a · a · · · a (with m factors a), but not the rule for multiplying parentheses, (a + b)(c + d) = ac + ad + bc + bd, which seems to be the intrinsic property in their problematic situations. This key rule is never articulated by the teacher. (2) Creativity: (N) Most of the teacher’s reasoning is based on short algebraic transformations without predictive argumentation, explanations, or references to relevant intrinsic properties. There are a few exceptions, where the rule am = a · a · · · a is referred to. (3) Reflection: (N) There are no signs of uncertainty and no time for reflection. The teacher describes the algorithmic solution steps and quickly answers the questions posed when the students can not do so. One minor exception is when the teacher tries to help Max relate the solution to an earlier example. (4) Focus: (N) The teacher focuses on the basic power property but the 14
students’ main difficulties seem related to multiplication of parentheses, and perhaps also the combination of these two and other rules. The teacher makes no attempt to find out what the students’ problematic situations are, and may believe that the students problems are only related to the basic power property. Summary: None of the CR simulation criteria are fulfilled. The presentation is AR based on algorithmic transformations that are not motivated through arguments founded in intrinsic mathematical properties, and not even through descriptions of the relevant rules. 4.2. Solving equations: Presentation of algebraic rules. An example from the first mathematics course, the second year, at the Hotel, Restaurant and Catering programme at upper secondary school. The lesson was about rules for solving linear equations. “You are good at finding x, but this lesson is more about the rules” the teacher said, and then she started show the solutions of three different equations. The two first were the equations x − 11 = 32 and 2x + 12 = 28. The solution of the third equation, 3x − 14 = 2 − x, will be discussed here. The teacher writes the third example on the board, 3x − 14 = 2 − x, and says “Now we will make things even more complicated. This is almost outside the syllabus.” She then says that she wants all the x:es on the left side and the numbers on the right side, because “we want the x:es on the side where we have the most of them, where they are positive. Here we must move x to the left and 14 to the right.” The teacher writes +14 on both sides, and then +x on both sides. One student protests and asks where the x comes from. She gets no answer from the teacher, but some other students try to explain, but without success. One other student asks “Is it always like that, that they change sign?”. The teacher answers “yes”. The teacher proceeds by simplifying the new expression and gets 16 4x = 16, 4x 4 = 4 , x = 4, and then the presentation ends. 4.2.1. Interpretation and identification of problematic situations. The third example appeared to be new to the students. The reasoning presented here was highly connected to the use of the rules, e.g. “we want the x on the side where we have the most of them, where they are positive”. The problematic situation was how to solve the equation by use of the rules. The two central rules seemed to be that you should put the x:es on the side where there are the most of them, and that you should add the same number (or amount of x:es), but with changed sign, to both sides. 15
4.2.2. Characterisation. (1) Mathematical foundation: (N) The presented reasoning is based on the use of two rules. The first, that you should put all the x:es on the side where we have the most of them, is not based on intrinsic properties of equations but on practical experiences. The second rule, that you should add the same number to both sides, is based on intrinsic properties of equations. The criterion is not fulfilled since none of these properties are made explicit. Since the two rules are presented in the same manner, it would be hard for the students to realise the difference in character between the two rules. (2) Creativity: (N) No argumentation is offered, the teacher only describes the solution of the equation. In an algorithmic perspective, new knowledge is presented to the students, i.e. a way to solve a new kind of equation. (3) Reflection: (N) There is no uncertainty or reflection in the teacher’s presentation of the method. Some uncertainty may be found in the situation where a student questions the adding of x to both sides, but since the teacher doesn’t address the question, there is no reflection. (4) Focus: (Y) The problematic situation is how to use the rules to solve the equation. Summary: No simulated CR is used, the reasoning is a presentation of a method, how to follow the rules, without creative argumentation of why or how they work. The students can learn the method this way, but only for equations of exactly this type (ax + b = cx + d or maybe only for ax + b = d − x). The teaching in this example does probably not help the students to learn how to create their own solution strategies if new types of equations are encountered. 4.3. An inequality: Non-creative explanations. The task is to prove the inequality ln(1 + x2 ) < x2 , x 6= 0. The transcript contains what the teacher says and writes on the blackboard (B) when presenting a solution to an undergraduate class. ”When showing this inequality, it is easier if everything is moved to one side of the inequality sign. If we move ln(1 + x2 ) we obtain” B: x2 − ln(1 + x2 ) > 0, for x 6= 0 ”The advantage with moving over is that we can study the left part as a function, interpret the graph and see if the function lies above the x-axis. So define” B: f (x) = x2 − ln(1 + x2 ) ”What happens at x = 0?” B: f (0) = 02 − ln(1 + 02 ) = 0 − ln 1 = 0 (1) 16
”f (0) says that the inequality is not true for x = 0. Now it has to be shown for other x-values. The derivative is” 2x(1+x2 )−2x 1 2x+x3 −2x 2x3 B: f 0 (x) = 2x − 1+x 2x = = = 1+x 2 2 1+x2 1+x2 ”which is less than 0 if x < 0 and greater than 0 if x > 0. We have earlier seen that the sign of the derivative determines if the function is increasing or decreasing. If x < 0 the nominator is negative and the denominator is positive, then the quotient is negative. If x is positive a positive nominator and denominator is obtained, and the quotient becomes positive.” f (x) is strictly decreasing on (−∞, 0) B: (2) f (x) is strictly increasing on (0, ∞) ”The conclusion can be drawn by studying the sign of f 0 (x) [He sketches a curve that looks like y = x2 ] We have seen that f 0 (0) = 0, that if x < 0 then f (x) is decreasing, and that if x > 0 then f (x) is increasing. We do not need to know exactly what the function looks like. If we combine this, we get that the graph lies above the x-axis.” B: (1)+(2) implies that f (x) > 0 for all x 6= 0, that is x2 > ln(1 + x2 ) for x 6= 0 ”We see that we have an application of the derivative, we are studying where the function is increasing and decreasing, and we can draw conclusions about inequalities. Sometimes the derivative is difficult to handle, but then we have seen that we can differentiate again.” 4.3.1. Interpretation and identification of problematic situations. One global and three main local problematic situations are identified: PS1: The global strategy choice, which is not fully articulated until the end of the presentation is: First, rewrite the inequality to form a function to study. Then, instead of studying the function values explicitly, find the local minimum and use the derivative to show that the function is deceasing to the left and increasing to the right and thus lies above the minimum. PS2: How to form a function? Transform the inequality by rewriting it so that the right side is zero. Let f (x) = x2 − ln(1 + x2 ) (the left side). PS3: How to find the minima? The teacher does not mention that or why the minima are sought, or why x=0 is chosen as the point were f is evaluated. PS4: Show that f is positive if x 6= 0 by using derivatives to show that f is decreasing to the left and increasing to the right of its minimum. 4.3.2. Characterisation. (1) Intrinsic properties: (Y) The argumentation is in summary: The inequality is true since f (x) > 0, x 6= 0. The latter 17
is true since f (0) = 0 and the derivative shows that this is the function’s minimum. (2) Creative: (N) The strategy choices are not constructed through predictive argumentation, only afterwards (partly) explained in the strategy implementations. (3) Reflection: (N) No uncertainty, no strategy choice reflections or questions. (4) Focus: (Y) The students are, as the teacher, focused on solving the task, and most students can probably follow the well structured reasoning. Since explanations are given after conclusions there are some situations were the students may not realise why things are done the way they are, for example why f (0) should be evaluated, until afterwards. Thus there may be some situations where the teacher and the students have different focus temporarily, but the criterion is essentially fulfilled. Summary: A well structured description of a (to the teacher) known method were most of the strategy choices are explained after they have been done, and thus can not be seen as a simulation of a creative solution to a problematic situation. 4.4. Finding Angles: Creative questions. The example comes from the second year at the upper secondary Natural science programme. Consider a circle with centre in O, and three points A, B and C on the circumference. The theorem which this lesson concerns then states that the angle AOB is twice the angle ACB (below referred to as T1). The teacher started by reminding the students of the theorem, and then she handed out a paper with four tasks. Towards the end of the lesson the students were invited to go to the board and present solutions of the tasks. When nobody wanted to do the second task, the teacher let the students guide her instead. The task was to find the values of x and y in Figure 1. One student said that y is 120◦ since the four points A, B, C and D on the circle form an inscribed quadrilateral, and then opposite angles are 180◦ together (the theorem will here be referred to as T2). The teacher agreed and said that you can do like that if you have understood that theorem (T2). Another student said that the angle at the centre (x) is twice as much as 60◦ [uses T1]. Now the teacher asked if anybody had found the value of y in another way. When she didn’t get any answer she said: “If you turn the paper upside down? If y is the angle at the circumference, where is the angle at the centre?” She marked the reflex angle at the centre (opposite x). 18
Figure 1. Task 2 Now one student says that the angle the teacher marked is 240◦ , since it is 360◦ − 120◦ [using that x = 120◦ ]. Another student asked a question if you have to know x in order to get y using this method. The teacher answered “yes”. 4.4.1. Interpretation and identification of problematic situations. The angle x was found by a direct use of T1, the angle at the centre (x) is two times the angle at the circumference (D). Two different ways of finding the angle y was presented, one by a student and one by the teacher. The first way was based on a theorem which has not been mentioned during the lesson, that opposite angles in an inscribed quadrilateral sum up to 180◦ (T2). A student not familiar with T2 would find the solution very hard to understand or construct. The second solution was based on the central theorem of the lesson, T1. The theorem was used in a not obvious way, since the angle at the centre is a reflex angle (between 180◦ and 360◦ ). The argumentation for the second method is that if you turn the circle upside down, and if y is the angle at the circumference, the angle at the centre is 360◦ − x. 4.4.2. Characterisation. (1) Mathematical foundation: (Y) The reasoning in all three problematic situations was based on the properties of T1 or T2. (2) Creativity: (Y) New knowledge was created by applying T1 in a new way (with a central angle larger than 180◦ ). The creative argumentation 19
was limited to identifying the angles in relation to the T1. The arguments were indicated by questions from the teacher. (3) Reflection: (Y) Since the task of finding y was solved in two different ways, and the second version in a not directly obvious way, the criterion was met. (4) Focus: (Y) Good correspondence of the general problematic situations since all students had been working with the tasks in advance. It is not obvious that all students had the same problematic situation at a more local level, for instance how to apply the theorem. Summary: Although the creativity criterion was met in a relatively simple and limited way, the reasoning is an example of a simulated CR. The discussion concerning the angles, and the questions from the teacher, lead up to a solution of the problematic situation. The students were given opportunities to understand and learn something new, as well as learning more about using theorems since direct application of the theorems also was a part of the lesson. 4.5. Proving a theorem: Creative reasoning without reflections. The transcript contains what the teacher says and writes on the blackboard (B) when presenting a proof to an undergraduate class. This proof can be found in most undergraduate calculus textbooks. B: Theorem: Assume that limn→∞ an+1 anP = L exists, 1 n B: then the radius of convergence for ∞ n=0 an (x − c) is R = L . ”Then we know, except for two points, where it converges and diverges.” B: L = 0 is interpreted as if R = ∞. ”That is, the interval is infinite on both sides.” B: L = ∞ is interpreted as if R = 0. Convergence only for c. ”We will prove this, it will be the proof of this book section. We shall check if the series converges, and we will use the criteria we know for ordinary series.” B: Proof: We investigatePabsolute convergence, that is, in the series P ∞ n B: ∞ |a |an ||(x − c)|n n=0 n (x − c) | = n=0 | {z } bn
”We can take the absolute value term-wise in a product. This is a positive series, we have studied this in book Section 10.3. In a positive series, the convergence is often so coarse that we can use the ratio test.” |an+1 | |an+1 ||x−c|n+1 B: limn→∞ bn+1 = lim |x − c| = bn = limn→∞ |an ||x−c|n n→∞ |an | | {z } Assumed existence 20
L|x − c| ”The ratio testPtells us that:” B: The series ∞ n=0 bn converges on L|x − c| < 1, that is if |x − c| < B: and diverges if L|x − c| > 1, that is if |x − c| > L1 . ”This means that the original series converges absolutely.” B: Therefore R = L1
1 L,
4.5.1. Interpretation and identification of problematic situations. One global and three main local problematic situations are identified: PS1: The global strategy choice is presented at the start of the proof: Test convergence for the power series in question by using tests for ordinary series (without a variable, x in this case). This way of extending familiar tests is new to the students. PS2: How adapt the familiar tests? This strategy choice is not articulated: i) Test for absolute convergence, since it it easier in this case and absolute convergence implies conditional convergence. ii) Form the new term bn which is in a form that familiar tests can be applied to. PS3: Which of the familiar tests should be applied? The use of the Ratio test is motivated by the coarseness of positive series. This argument is not elaborated, and not explicitly founded on any mathematical properties, and probably difficult for the students to understand. The teacher probably means that the ratio test is suitable when the terms decrease at least exponentially fast, that is, limn→∞ bn+1 bn < 1. PS4: How interpret the outcome of the Ratio test in this new situation? The series converges if |x − c| < 1 which implies that the radius of convergence is 1/L. 4.5.2. Characterisation. (1) Mathematical foundation: (Y) The criterion is fulfilled in most of the conclusions. That convergence tests can be used to determine convergence for power series is an intrinsic property in this (simulated) problematic situation. That the sequence is positive is an intrinsic property when choosing the ratio test. The reasons for choosing this test is not supported by explicit intrinsic property considerations, as discussed above. (2) Creativity: (Y) A series test that have not been used for power series earlier (only for series with constant terms) is used in a new way to prove the theorem. The strategy choices are explicitly motivated in advance. (3) Reflection: (N) No uncertainty, no strategy choice reflections or questions. The teacher could, for example, have asked ”Can ordinary tests be 21
used, and how?” (4) Focus: (Y) The students are, as the teacher, focused on solving the task, and most students can probably follow the well structured reasoning, perhaps apart from PS3. Summary: This is a traditional way of presenting a mathematical proof: It starts in a set of conditions (properties of a mathematical phenomena) and through an unbroken chain of arguments terminates in a conclusion. The traditional proof is normally a very economic and structured way of summarising the main arguments and key ideas that lay behind the original proof of the theorem, the first time it was proved. The mathematical foundation and the predictive argumentation remain (though idealised), but the reflections and uncertainty are missing.
5. Global analysis and results In this study 23 teaching situations have been analysed, five of them are presented in Section 4. In this section the outcomes of the 23 analyses will be summarised, one simulation criterion at a time. 5.1. Mathematical foundation. The criterion was fulfilled in 13 situations, that is, the intrinsic mathematical properties were made explicit to the students by being discussed in some way. Advanced discussions were not required for the criterion to be fulfilled. One example of a situation classified as fulfilled is when a teacher uses a theorem as a base for the reasoning around a task (see Section 4.4). In another situation the teacher discusses the meaning of the constant term in the function y = x2 + 2 and says that ”it gives the intersection with the y-axis since x = 0 gives y = 2”. Here the teacher uses the intrinsic properties of the relationship between the function and the graph when he motivates his answer. If the teacher would have said only that ”the constant term gives the y-coordinate in the intersection with the x-axis”, the conclusion is just stated and not justified by explicit reference to any intrinsic properties. In both these situations a deeper discussion concerning the underlying mathematics would have been possible, for instance in the latter the teacher could have discussed how the function behave for other x-values. Where the criterion wasn’t fulfilled, the most frequent reason for this was that the teacher presented a solution of a task or an algorithm of some kind, without discussing the underlying mathematical properties. 22
We found one situation where the presented reasoning was founded on nonmathematical properties, when a teacher compared the shape of a graph to a quadratic equation with a positive coefficient to the x2 -term with a happy mouth. The connection positive – happy has no mathematical ground when no other motivation is given. 5.2. Creativity. We found three situations where the reasoning was classified as creative. Our demands on creativity were not set very high. In one of these three examples (see Section 4.4) the teacher uses questions to guide the students into a new way to use a theorem. However, one more advanced situation was also found (see Section 4.5). Here the teacher builds up a solution by careful argumentation how to use a series test in a new way. The situations where the criterion wasn’t fulfilled consisted most often of descriptions of algorithms (see Sections 4.1, 4.2 and 4.3) by solving tasks as examples. The solutions to the tasks was then described without explicit argumentation supporting the strategy choices. One of the main findings in this study concerns the absence of predictive argumentation as a contrast to the (at least occasional) presence of verifying argumentation. The message to a listening student exposed only to verifying argumentation is that you always have to know exactly what to do from the start, and the only relevant argumentation concerns explaining what you already know. This is a reality transparent in the classroom situations of this study, the teacher is always in full control and is never in a real problematic situation, and consistent with findings on students beliefs about what they are supposed to accomplish Schoenfeld (1992). 5.3. Reflection. In most cases in this study (18 situations out of 23) there were no reflection or uncertainty. The reasoning was guided by the teacher, and it was clear that the goals would be met. Extensive uncertainty simulations were not required in order for the criterion to be fulfilled. In two of the five situations where the criterion was fulfilled, the teacher asked questions to the student on how to proceed, and in two other situations the teacher made an error that had to be corrected. This means that in the eighteen situations without uncertainty or reflection, the teacher asked no questions to the students and posed no reflective statements. In the remaining situation where the criterion was fulfilled, the task was solved in more than one way (see Section 4.4). 23
5.4. Focus. In this study we found eleven situations where the criterion wasn’t fulfilled. In five of these there were clear differences concerning the problematic situations. In the other six situations the teacher presented very difficult creative reasoning, or reasoning that was based on knowledge clearly inaccessible to students. These are other aspects that can lead to different focus. The criterion was often fulfilled since the the teacher and the students were trying to solve the same task and no indications of differences in focus could be found. Sometimes the students had been working on the task in advance which made the fulfilment of the criterion very obvious (see Section 4.4). In other situations student activity indicated common focus, most often by asking questions. The nature of the difference in problematic situations varied. In one situation the teacher worked with very local steps, but the student’s problematic situation was the global strategy choice. In the example in Section 4.1 the teacher focused on one mathematical property while the students’ difficulties concerned other, more basic, properties. In another situation the teacher asked a question concerning what kind of solution you will get when you solve an equation of the type y = ax + b. The teacher discussed that the equation x + 5 = 11 has a solution which is a number and the equation x2 = 25 has a solution which is two numbers. After this the students made guesses (i.e. ”a table”, ”a number”, ”a new equation”) and the teacher said which guess was correct. Here the teacher’s focus concerned how to generalise his examples, while the students focused of more coming up with more guesses. Several of the students’ guesses has clearly no connection to the teacher’s examples. The point is not that the guesses are wrong, but that they to such a large extent lack any relationship to the teachers reasoning. It essential in all teaching that the teacher and the students have the same focus. If, in a teaching situation, the students have one problematic situation and the teacher addresses another, there is a risk that no learning will take place, or at least not the learning the teacher intended. 5.5. Summary. There were two significant ways that the teachers managed to meet CR simulation criteria. The first was by discussing mathematical properties with students, letting them come up with ideas and solutions, helping them along by asking questions or posing statements concerning the mathematical properties involved (see Section 4.4). The other was by using motivational argumentation when presenting a new 24
way to use a theorem (see Section 4.5). In all cases where the reasoning was classified as creative, there was an argumentation of some kind present. When the teachers failed to use simulated CR, several reasons for this was found in this study. The lack of motivating as well as explaining argumentation was one important reason. Another was that the presented reasoning was free from reflection and uncertainty, in which cases it often was a straight forward presentation of an algorithm. We also found situations where the teacher and the students had different problematic situations, and situations where it was clearly unrealistic for the students to carry out the presented reasoning themselves. When the analysed reasoning wasn’t CR, it was either strict presentations of methods or algorithms, or a teacher guided process where the teacher guided the students by posing leading questions (often on a very basic level) which the students tried to answer, sometimes (we believe) by guessing. In these cases, the possible learning gains for the students was how to solve tasks very similar to those presented by the teacher. The algorithm presentations were of three types: (1) to carry out the algorithm without comments. (2) to carry out the algorithm with comments on exactly what is done. (3) to carry out the algorithm with arguments on what rules or methods are used. One may note that if 3 contains predictive argumentation concerning the use of the rules, then the reasoning would be classified as CR.
6. Discussion The focus of this study was not on lessons that were specifically designed to enhance CR. Thus the lessons may have many other qualities that does not appear in the analysis above. Some situations in this study fulfils several of our criteria for simulation of creative reasoning. However, in these cases the criteria are with few exceptions met in rather modest ways as is described in Section 5. The results indicate that algebraic methods for solving specific tasks often is what students may learn from a presentation. In most situations in this study there is no creative argumentation. It is hard to find situations where the teachers explain what they are doing during presentations, and we especially find the lack of predicting argumentation troublesome. If the students almost never see CR, simulated CR, or some other type of 25
creative mathematical reasoning which builds on the intrinsic mathematical properties, how can we expect them to learn mathematical reasoning and become problem solvers even in an elementary way? If the teachers seldom uses predictive argumentation during presentations, how can we expect students learn creative reasoning? In the Swedish national curriculum, logical reasoning is pointed out as a central competence for students at all levels. In the syllabus for the compulsory school (year 1 to 9) it says: The school in its teaching of mathematics should aim to ensure that pupils develop their ability to understand, carry out and use logical reasoning, draw conclusions and generalise, as well as orally and in writing explain and provide the arguments for their thinking. Swedish National Agency for Education (2000) One possible way to help students develop this ability could be to let creative reasoning and argumentation be a normal part of the activities in the classroom. In this, simulated CR according to our definitions could be one of the ways to introduce a higher presence of creative mathematical reasoning in everyday teaching.
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