Mathematical reasoning in teachers’ presentations

Journal of Mathematical Behavior 31 (2012) 252–269

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Mathematical reasoning in teachers’ presentations Tomas Bergqvist ∗ , Johan Lithner Umeå Mathematics Education Research Center, Umeå University, Sweden

a r t i c l e

i n f o

Available online 30 December 2011

Keywords: Rote learning Task solving Imitative and creative reasoning Teaching

a b s t r a c t This paper presents a study of the opportunities presented to students that allow them to learn different types of mathematical reasoning during teachers’ ordinary task solving presentations. The characteristics of algorithmic and creative reasoning that are seen in the presentations are analyzed. We find that most task solutions are based on available algorithms, often without arguments that justify the reasoning, which may lead to rote learning. The students are given some opportunities to see aspects of creative reasoning, such as reflection and arguments that are anchored in the mathematical properties of the task components, but in relatively modest ways. © 2011 Elsevier Inc. All rights reserved.

1. Introduction One of the main causes of difficulties in learning mathematics is that mathematics is often reduced to a large set of isolated, incomprehensible facts and procedures to be memorized and recalled for written tests (Hiebert, 2003; Tall, 1996; Tirosh & Graeber, 1990; White & Mitchelmore, 1996). The immense complexity of mathematical learning (Niss, 2007) implies that such rote learning will be affected by many factors, including influences from school and from the individual’s home and community cultures (Brenner, 1998). This study focuses on how classroom practice can affect learning and uses the baseline conclusion from a summary of research on both traditional and alternative classroom practices as a starting point: “One of the most reliable findings from research on teaching and learning is that students learn what they are given opportunities to learn” (Hiebert, 2003, p. 10). According to Hiebert, being given the opportunity to learn means more than just receiving information. It means setting up conditions for learning that take into account the students’ initial knowledge, the nature and purpose of the tasks and activities, the type of engagement required, and so on. “Providing an opportunity to learn what is intended means providing the conditions in which students are likely to engage in tasks that involve the relevant content. Such engagement might include listening, talking, writing, reasoning, and a variety of other intellectual processes” (Hiebert, 2003, p. 10). Students’ opportunities to learn are related to how they reason when they are solving mathematical tasks in school. In particular, rote learning is related to students’ tendency to use inefficient and mathematically superficial imitative strategies rather than creating their own solution through reasoning (Bergqvist, Lithner, & Sumpter, 2008; Lithner, 2000a, 2000b, 2003, 2004). To learn both imitative and creative reasoning, we acknowledge that students must practice how to solve different types of tasks, for example, routine tasks and problems (Schoenfeld, 1985). In addition, students may also learn from seeing how other persons reason. The teacher is particularly important because how the teacher solves the problem can be seen by the students as a model or example of good reasoning. In Sweden, a mathematics lesson is normally introduced with a presentation and discussion by the teacher regarding any new content followed by the solution methods for the new

∗ Corresponding author. E-mail address: [email protected] (T. Bergqvist). 0732-3123/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jmathb.2011.12.002

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tasks. This introduction is followed by individual or small-group work for the students involving exercises on the presented methods (Bergqvist et al., 2009). One possible consequence of this focus on solution methods is that the students will assume that problems are solved by applying methods that are known in advance, particularly if this how the teacher makes his or her presentations (Schoenfeld, 1985). The research question for this study is as follows: what opportunities do students have to learn different types of mathematical reasoning from their teachers’ task-solving presentations? This study is not concerned with what learning actually takes place—which may be none—but is only concerned with whether the students’ are offered opportunities to learn. A more specific research question is formulated in Section 2.5. The framework section of this paper (Section 2) contains five parts: mathematical reasoning, the conditions and constraints under which teachers design their practice, the foundation of the analytic framework, the aspects of similarity that are used as the focus of the analysis and the research question. The method for data collection and analysis (Section 3) is followed by the analysis (Section 4), the results (Section 5), and a discussion about possible consequences for students’ learning (Section 6). 2. Framework 2.1. Learning difficulties and mathematical reasoning This section is a modified summary of selected sections of a research framework that was published in Lithner (2008) and was based on the outcomes of a series of empirical studies on the relationship between reasoning and learning difficulties in mathematics. “Mathematical reasoning is no less than a basic skill” (Ball & Bass, 2003, p. 28). Despite this pronouncement, the term ‘reasoning’ is often used by mathematics educators without being defined under the implicit assumption that there is universal agreement on its meaning (Yackel & Hanna, 2003). The purpose of this section is to provide three things: (1) abroad definition of reasoning that allows the inclusion (and comparison) of both low- and high-quality arguments; (2) the underlying notions that make it possible to define creative, mathematically founded reasoning by using the logical value of the arguments, by anchoring the arguments in mathematics, and by creativity; and (3) a characterization of imitative reasoning as the opposite of creative reasoning. 2.1.1. A broad definition of reasoning Reasoning is defined in this paper as the line of thought that is adopted to produce assertions and reach conclusions when solving tasks. Reasoning is not necessarily based on formal logic and is therefore not restricted to proof; it may even be incorrect as long as there are some sensible (to the reasoner) reasons supporting it. This example illustrates that “reasoning” is used in a broad sense in this framework to denote both high- and low-quality argumentation; the quality of the argument is characterized separately. Reasoning can be seen as thinking processes, as the product of these processes, or as both. The data for this investigation are behavioral; thus, we can only speculate about the underlying thought processes (Vinner, 1997). Because one purpose of this framework is to characterize data, we choose to see reasoning as a product that (primarily) appears in the form of written and oral data as a sequence of reasoning that starts in a task and ends in an answer. The term problem has many different meanings in the literature, but in this paper, it denotes a type of task that is intellectually difficult for an individual who has no access to a complete solution scheme at the beginning (Schoenfeld, 1985). In a task-solving situation (including sub-tasks) two types of argumentation are central. (1) Predictive argumentation (why will the strategy solve the task?) can support the strategy choice. The ‘strategy’ can vary from local procedures to general approaches, and ‘choice’ is defined in a broad sense (choose, recall, construct, discover, guess, and so forth). (2) Verificative argumentation (why did the strategy solve the task?) can support the strategy implementation. 2.1.2. The logical value of arguments The meaning of a statement is based on its content, status (premise, conclusion, or theorem), logical value (true, false, or undecidable) and epistemic value. The latter is the degree of trust (absurd, unreal, possible, likely, or obvious) that a person has for a statement as soon as s/he understands its content (Duval, 2002). Valid reasoning is based on the organization of several propositions into a deductive step and the organization of several deductive steps into a proof (Duval, 2002). In Toulmin’s model (Krummheuer, 1995), the argument contains four components: a conclusion, data, warrant and backing. Data are the facts that constitute the starting point of the mathematical task to solve. The warrant supports the conclusion by using the data to register the legitimacy of the deductive step taken by referring to a class of steps whose legitimacy is presupposed. The warrant is a specific reference to the data, and its authority can be supported by a more general type of support categorized as “global convictions and primary strategies that can be expressed in the form of categorical statements”(Krummheuer, 1995, p. 244). School tasks normally differ from the tasks addressed by professionals such as mathematicians, engineers and economists. Within the didactic contract (Brousseau, 1997) in the school context, it is allowed, and sometimes encouraged, to guess, to

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take chances, and to use reasoning without any strict requirements on the logical value of the reasoning. Even in examinations, it can be acceptable, as in Sweden for example, to have only 50% of the answers correct, while it would be absurd if mathematicians, engineers, or economists were satisfied in being correct in only 50% of their conclusions. This framework proposes a wider conception of logical value that is inspired by Pólya (1954): “In strict reasoning the principal thing is to distinguish a proof from a guess, [. . .] In plausible reasoning the principal thing is to distinguish a guess from a guess, a more reasonable guess from a less reasonable guess.” Thus, a plausible argument can be constructive without being logically valid (in contrast to, for example, a proof which must be logically true).

2.1.3. Anchoring arguments in mathematics Validation is a mental process that determines the correctness of a sequence of reasoning. This sequence is only partly conscious and can include asking and answering questions, assenting to claims, constructing subproofs, remembering and interpreting other theorems and definitions, complying with instructions, and assessing feelings of rightness or wrongness (Selden & Selden, 2003). The process is social (Krummheuer, 1995) and “comprises a set of practices and norms that are collective” (Ball & Bass, 2003, p. 29). In mathematics, the acceptability of an argument is determined by sociomathematical norms (Yackel & Cobb, 1996). For example, it is a social norm that students are expected to justify solutions but what counts as an acceptable justification is a sociomathematical norm. However, valid reasoning is still based on mathematics rather than on social status, such as the authority of the teacher or the intelligence of a peer. However, what does it mean for an argument to be based on mathematics? Schoenfeld (1985) found that novices judged that geometrical constructions were correct if they ‘looked good,’ whereas experts used more relevant properties (for example, congruence). Thus, the reference to the mathematical content is important: what are the arguments about? To address this question, we introduce the notion of anchoring (Lithner, 2008). Anchoring does not refer to the logical value of the warrant but refers to its fastening the relevant mathematical properties of the components one is reasoning about—objects, transformations, and concepts—to data. The object is the fundamental entity; it is the ‘thing’ that one is doing something with, for example, numbers, variables, functions, and diagrams. A transformation is what is being done to the object, and the outcome of the transformation is another object. A sequence of transformations, finding polynomial maxima for example, is a procedure. A concept is a central mathematical idea built on a set of objects, transformations, and their properties, such as the concept of a function or of infinity. The status of a component depends on the situation. f(x) = x3 can be seen as a transformation of the input object 2 into the output object 8. If f is differentiated, then the differentiation is the transformation; f(x) is encapsulated (Tall, 1991) into an input object, and f (x) is the output object. Arguments can be anchored in either surface or intrinsic properties, and the relevance of a mathematical property can depend on context. In deciding if 9/15 or 2/3 is largest, the size of the numbers (9, 15, 2, 3) is a surface property that is (Lithner, insufficient to resolve the problem, while the quotient captures the intrinsic property. Another example of anchoring ∞ 2003) is a student trying to determine if the same test for absolute convergence is applicable to the series n cos n/2n n=1 ∞ and cos n/((n + 1) ln(n + 1)). He decides that it can be used, based on the surface property that cos n appears in both n=1 numerators. This decision is not correct because the intrinsic property for this comparison lies in other parts of the fractions. The intrinsic/surface distinction was introduced because one of the reasons behind students’ difficulties was found to be the anchoring of arguments in surface properties (Lithner, 2003).

2.1.4. Creative thinking There is no single definition of ‘creative’ that is used in the research (Haylock, 1997), and there are two major uses of the term: (i) a thinking process that is divergent and overcomes fixation; and (ii) a product that is perceived as creative for some reason, such as works of art. Haylock sees two types of process fixation: “content universe fixation” limits the range of elements recognized as appropriate for application to a given problem. Algorithmic fixation is the repeated use of an initially successful algorithm that becomes an inappropriate fixation. Silver (1997) suggests a view of creativity in which the thinking processes are related to deep, flexible knowledge in content domains and associated with long periods of work and reflection instead of rapid and exceptional insights. Silver sees fluency, flexibility and novelty as the key qualities of creativity. Because this framework addresses ordinary students’ thinking, imputing creativity only to experts is not sufficient. “Although creativity is often viewed as being associated with the notion of ‘genius’ or exceptional ability, it can be productive 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” (Silver, 1997, p. 75). The aspect of creativity that is emphasized in this framework is not ‘genius’ or ‘exceptional novelty,’ but the creation of mathematical task solutions that can be modest but that are original to the individual who creates them. Thus, creative is the opposite of imitative. A consequence of this broad view of creativity and the choice to analyze not thinking processes themselves but their products (reasoning sequences, Section 2.1.1) is that the definition of creative reasoning presented below cannot incorporate fluency and flexibility as a necessary criterion. A construction of a new task solution may be straightforward (no explicit signs of flexibility) and be temporarily hindered by the lack of fluency. Instead, we propose the weaker condition for creativity,

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namely, that the reasoning is not (apart from perhaps temporarily) hindered by fixations (lacking flexibility) or a lack of fluency. 2.1.5. Creative reasoning Sections 2.1.2–2.1.4 lead to a definition of Creative Mathematically Founded Reasoning (CMR) that fulfills all of the following criteria. (i) Creativity. A new (to the reasoner) reasoning sequence is created, or a forgotten one is re-created, in a way that is sufficiently fluent and flexible to avoid fixations. (ii) Plausibility. There are arguments supporting the strategy choice and/or strategy implementation explaining why the conclusions are true or plausible. (iii) Anchoring. The arguments are anchored in the intrinsic mathematical properties of the components that are involved in the reasoning. 2.1.6. Imitative reasoning The empirical studies behind this framework have identified two main types of imitative reasoning: memorized and algorithmic. In Memorized Reasoning (MR), the strategy choice is founded on recalling an answer by memory, and the strategy implementation only consists of writing it down. This type of reasoning is useful as a complete solution method in only a relatively small proportion of tasks (Lithner, 2008), such as recalling every step of a proof or the fact that one liter equals 1000 cm3 . When school tasks ask for calculations, it is normally more appropriate to use Algorithmic Reasoning (AR) (Lithner, 2008), where the strategy choice is to recall an algorithm and the strategy implementation is to apply the algorithm to the task data. An ‘algorithm’ includes all pre-specified procedures (not only calculations), such as finding the zeros of a function by zooming in on its intersections with the x-axis with a graphing calculator. “An algorithm is a finite sequence of executable instructions which allows one to find a definite result for a given class of problems” (Brousseau, 1997, p. 129). The importance of an algorithm is that it can be determined in advance. The nth transition does not depend on any circumstance that was unforeseen in the (n − 1)st transition—not on finding new information, any new decision, any interpretation, or thus on any meaning that one could attribute to the transitions. Therefore, the execution of an algorithm has high reliability and speed (Brousseau, 1997), which is the strength of using an algorithm when the purpose is only to solve a task. However, if the purpose is to learn something from solving the task, the fact that an algorithm is independent of new decisions, interpretations or meaning implies that all of the conceptually difficult parts are taken care of by the algorithm, and thus only the easy parts are left to the student. This segmentation may lead to rote learning. In particular, the resultant argumentation is normally superficial and very limited, as seen in the main AR types that are found in studies. Familiar AR/MR includes a strategy choice that can be characterized by (perhaps superficial) attempts to identify a task as being of a familiar type with a corresponding known solution algorithm or a complete answer. Justifying a successful solution by simply describing the algorithm is an accepted sociomathematical norm in most practice and test situations studied (Lithner, 2008). In 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 relationships with the task. For example, if the task contains a second-degree polynomial p, the reasoner can choose to solve the corresponding equation as p = 0 even if the task asks for the maximum of the polynomial (Bergqvist et al., 2008). In Guided AR, the reasoning is mainly guided by two types of sources that are external to the task. In person-guided AR, a teacher or a peer pilots the student’s solution (see Section 4 for examples). In text-guided AR, the strategy choice is founded on identifying, in the task to be solved, similar surface properties to those in a text source (e.g., a textbook; see the ‘cos n’ example in Section 2.1.3). Argumentation may be present, but it is not necessary because the authority of the guide ensures that the strategy choice and the implementation are correct. In students’ attempts to resolve problematic task solving situations, the CMR criteria i–iii (see Section 2.1.5) were found to capture the main differences seen in reasoning characteristics between MR/AR (where i–iii are absent) and constructive CMR (Lithner, 2008). A task solution in MR is immediate through recollection, in AR, it follows a known algorithm and in CMR, it is created (although CMR normally includes elements of MR/AR). Furthermore, in CMR the epistemic value lies in the plausibility and in the logical value of the reasoning. In MR and AR, it is determined by the authority of the source of the imitated information. 2.2. Conditions and constraints in teaching To study how teacher presentations convey different aspects of reasoning, it is necessary to consider the context in which the presentations take place. When discussing the analytic framework notions (Section 2.3) and the reasons for the results of the study (Section 6), we will use the ideas of Chevallard (1992), who treats didactic relationships in terms of conditions and constraints: “In this perspective the question to put in the face of a given fact will no longer be, ‘What is the cause of it?’, but rather, ‘Why is it as it is, and not otherwise?”’ (Chevallard, 1992, p. 222). What Chevallard calls ecological analysis can be summed up in two closely connected questions. The first question concerns the conditions required for an event to take

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place or to be possible. The second question discusses the constraints that might hinder these conditions from becoming fulfilled. In analyzing conditions and constraints, it is useful to recognize that the axiomatic form of mathematics, including the presentations of task solutions, seems well-adapted to the needs of teaching by offering a method for ordering teaching activities that maximizes the number of knowledge items conveyed in the shortest time. However, it hides the true functioning of mathematics by isolating certain notions and properties, removing them from the network of activities that provide their origin, meaning, motivation and use (Brousseau, 1997). Thus, it is not sufficient to merely consider the descriptions of the notions and methods in the analysis of teaching activities, including presentations of task solutions. In this study, it is of particular interest to consider the motivating arguments behind the reasoning that connect the task, the solution method and the conclusion. We know that an important cause of learning difficulties is rote learning (Hiebert, 2003). In particular, we know that to learn problem-solving reasoning, students must be exposed to problem-solving reasoning. There is no automatic transfer from the rote learning of algorithmic procedures to problem-solving competence (Schoenfeld, 1985). Thus, an assumption in this study is that a teacher’s presentation is a better example of task solving reasoning when it is more similar to the reasoning that students can use when they are practicing and applying task solving reasoning (this assumption does not imply that we assume that higher similarity is better in all aspects of learning). For example, if the motivating arguments supporting the reasoning are absent or rare, then rote learning is more likely to occur. The conditions specifying the aspects of similarity used in this study are presented in Section 2.4. One constraint that may reduce similarity is that the general conditions under which teachers present task solutions to the students are different from the conditions under which students solve practice and test tasks. For example: (a) Normally, the teachers choose a task to present (including the context), while the students are given a task by the teacher. Thus, the teacher will not choose a task that is unsuitable for the intended reasoning, while the students will have to adopt the reasoning to the given task (and to their knowledge). (b) The teachers are normally more knowledgeable than their students. This implies that the teachers have more direct access to suitable forms of reasoning, both in the sense that more tasks are routine tasks to them and can be solved by time-efficient AR and in the sense that they are more proficient in CMR when AR is unsuitable. (c) Teachers’ presentations of task solutions are often prepared in advance and/or are well-known to the teacher, while students’ practice and test tasks are often met for the first time by the students or are not so well-learned that they are well-known. Thus, teachers do not have to construct the solution method while they present the task, and large parts of the solution construction (that took place in advance) may be omitted in the presentation. 2.3. The analytic framework for teachers’ presentations of task solutions To compare the similarities between teachers’ presentations and real (not prepared in advance) task solutions, we apply ideas from the analyses of simulations, even though the teacher may not have the intention of simulating anything. Because this study focuses on what students have the opportunity to learn, as opposed to what they see in common teacher presentations, we do not consider what the teachers’ implicit intentions with the presentations are. To capture the quality of the correlation between a simulated and a real situation, Fitzpatrick and Morrison (1971) use the term representativeness, which refers to a combination of comprehensiveness and fidelity. We use the same concept to compare the teachers’ presented solutions with actual first-time solutions. Comprehensiveness refers to the range of the different aspects of the situation that are simulated and fidelity to the degree to which each aspect approximates a fair representation of that same aspect as presented in the criterion situation. To analyze how the presented solution is similar to an actual first-time solution, it is tested against six aspects of similarity. The method of identifying which aspects to analyze follows two steps: - The first step is to adapt the first five comprehensive aspects, theoretically argued below as deriving from the reasoning framework (Section 2.1), to the conditions and constraints of the teacher presentations. - These five aspects are in the second step used in preliminary analyses that determine how well they capture the essentials of the data. This approach is inspired by grounded theory and leads to a reevaluation and redesign of the central aspects of the analysis (in several cycles). The outcome is a final, sixth aspect that makes it possible to consider the main similarities and differences between the teachers’ presentations and the imagined first-time solutions. The fidelity, or how well each aspect is fulfilled, is evaluated during the analysis of the teaching situations in Section 4. Because this study focuses on what the students see and not what the teacher thinks, the aspects are present only if they were judged to be explicitly visible to the students. 2.4. Aspects of similarity The main role of imitative reasoning is for use in routine tasks or exercises (Section 2.1), and it is fairly straightforward to present a solution to a routine task using imitative reasoning. Vinner (1997) proposes a diagrammatic model for routine analytical (in contrast to pseudo-analytical) task solving, where one needs a pool of algorithms—mental schemes that identify

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the task type and its structure—and mental schemes that can assign the suitable algorithm to the task type. Two cognitive faculties are involved: the ability to identify similarities and the ability to imitate (Vinner, 1997). These correspond in AR to strategy choice and to implementation, respectively. Thus, a teacher’s presentation of a solution to a routine task could reasonably focus on how the algorithm is identified and how to apply it, and therefore the analysis would include the following steps.

1. Identification of the task type. This aspect considers whether the task is explicitly identified by the teacher to be of a certain type or structure. This explicit identification would involve describing the general properties that are characteristic of the family of tasks to which the specific task belongs. 2. Recognizing a solution method. There are two sub-aspects to this step that are related to the strategy choice and to the implementation: whether the teacher describes principles that (a) connect the task type to the choice and (b) provide the main elements of the solution method. To ‘describe principles’ goes beyond just mentioning the choice (e.g., ‘this task is solved by division’) or just describing every step as the solution is carried out. It does not need to involve any arguments, but should describe some principles that are valid for the task type and not just for that particular task. One may note that, although 1 and 2 derive from the characteristics of routine task solving, they are also relevant for CMR, where they may appear in different forms than in AR. To study the similarity between the presented task solution and the central properties of actual first-time CMR solutions, the following three steps (related to the three defining criteria of CMR identified in Section 2.1.5) are introduced. 3. Creative reflection. It is not reasonable to assume that creative thinking is easily visible to the students. Neither is it reasonable to assume that the students will be able to judge whether the arguments and conclusions are creative in themselves. However, a central distinction between AR and CMR is that in an actual first-time solution, where the strategy choice and implementation are not evident from the start, metacognition (monitoring and control in terms of reflections and considerations about one’s own reasoning, Schoenfeld, 1985) may be required to avoid fixations and to guide the fluency and the flexibility of the reasoning. This use of metacognition may be visible as reflections (in a broad sense), including questions, analyses, explorations, corrections of mistakes or non-productive strategy choices, verifications, evaluations of alternative solution strategies. These reflections can occur, for example, in the form of explicit strategy choice questions. Creative reflection is not seen as present when it is only seen through reflections related to very local and elementary parts of the task solution such as, for example, asking “what is 12/4?” when solving a quadratic equation. 4. Argumentation. The plausibility of the choices and conclusions may be motivated through explicit arguments of two types: (a) Predictive argumentation, which is formulated before conclusions. Predictive argumentation is not present if the reasoning begins with a conclusion that is afterwards explained because it is never the case in an actual first-time CMR that one knows the conclusion before reasoning. (b) Verificative argumentation, which can be used in first-time CMR to verify conjectures and may appear in similar forms during teacher presentations as explanations presented after conclusions. Verificative argumentation may help the students to understand both CMR and AR/MR solutions. 5. Mathematical foundation. The reasoning may be anchored in the intrinsic mathematical properties of the components that are involved in the reasoning, in the same way as CMR. This aspect is seen as present if the conclusions are based on the explicit considerations of relevant properties. For example, the conclusion is drawn in a form that is, or can be, reformulated as ‘the statement is true because the components have these mathematical properties, which have these consequences.’ A final aspect was identified in preliminary analyses of our classroom data as the primary additional aspect that, when absent, leads to low representativeness for both creative and imitative reasoning. 6. Alignment. Alignment considers whether the teacher’s main task situations and solution goals are similar to those of the students. In some situations, there is evidence that the teacher’s task situations differ from that of the students (see Section 4). Another cause of non-alignment may be that the teacher’s reasoning is unrealistic, in the sense that it is much too difficult or is based on facts and knowledge that are inaccessible to the students.

2.5. Research question This study concerns how students learn to perceive mathematics as either a creative subject or as a subject where they are supposed to imitate algorithmic procedures provided by the teacher (or textbook). The teaching situations analyzed were not explicitly designed by the teachers to treat non-routine problem solving, and the students in the classes studied received no special lessons in non-routine problem solving. The tasks presented are tasks that are central to the mathematics course; they are often new to the students, and the teachers demonstrate methods for solving them. These types of presentations can differ in a range from narrow algorithmic manipulations to CMR. Therefore, it is valuable to study to what extent the students may be encountering and experiencing different aspects of MR, AR and CMR during their teachers’ presentations. This study is based on the following research question: During teachers’ regular presentations of task solutions, how are opportunities for the students to develop the capacity to use MR/AR and CMR provided with respect to the six aspects listed above?

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Because this is primarily a qualitative study and because a data sample that is representative of Swedish teaching is hard to obtain (see Section 3), the goal is not to produce a precise quantitative description of how many aspects were present. The main purpose of this study is instead to find qualitative information that indicates how the teachers present different types of reasoning. 3. Method Because teaching can take place in numerous ways, it is impossible (simply due to the huge variety of content, tasks, goals, students, and teaching styles, along with their different combinations) to try to capture even the most common types of teaching situations. Therefore, it is equally impossible to specify in a unique way what the ‘regular’ teaching mentioned in the research question really is. Due to this impossibility, the lessons visited for data collection were not selected as a representative sample from the Swedish school system. Instead, we visited several different teachers whose teaching was judged by us (as experienced teachers) to be of a common, or at least not an uncommon, type. The schools and the university visited were chosen out of convenience, but we have no indication that they used uncommon teaching or organization. The data collected consisted of extensive field-notes from three educational levels for mathematics education: two lessons from lower secondary school, six from upper secondary school and four from undergraduate university courses. One of the researchers took written notes that focused on the presentation and on the interaction between the teacher and the students; the field notes were complemented and corrected immediately after the lesson. One primary reason for not collecting video data was that some of the teachers did not accept video recording. Rather than adding a data collection bias (visiting only teachers that were comfortable with video recording), it was decided to accept the disadvantages of less precise data. In addition, there are several technical difficulties in classroom video recording, and taking field-notes is more flexible in allowing the teacher to be followed during dialogues with small groups of students in different parts of the classroom. The analysis of each teacher presentation was made in four steps: Interpretation. The transcribed data in the form of utterances and blackboard notes were interpreted, and the key parts of the reasoning were summarized. As mentioned above, the teachers’ actual thoughts and goals (e.g., if the task presentation aims at teaching the students something about conceptual understanding, problem solving or routine algorithmic solutions) are not considered in this analysis. This analysis only considers the presented explicit reasoning because this reasoning is what the students actually see. Identification of the central task situations. The data sequence was split into chunks and represented in terms of sequences of reasoning that begin with a (sub)task and conclude with an answer. Characterization. The sequence of reasoning was characterized with respect to the six aspects defined in Section 2.4. In addition, although impossible to fully capture in a simple measure, an aspect that was seen as present was marked by a Y (N otherwise) to support that ability to compare the different task situations in this study. In some situations, the alignment aspect (6) is difficult to determine from the data, and for the sake of simplicity, this aspect was marked with a Y if there were no indications that the teacher and students had different task situations or solution goals. Summary. The characterization was summarized with respect to the research question. 4. Data presentation and analysis In this study, data were collected from 12 lessons and, from these lessons, 23 teaching situations (one to four per lesson) were identified and analyzed. The five situations presented below were chosen to represent teacher presentations that differ with respect to the six aspects of similarity. The situations presented are summarized versions of the actual analyses. 4.1. Algebraic manipulations without guiding rules In a ninth grade classroom, the teacher began the mathematics lesson by briefly repeating some earlier lessons, such as “What is 34 ?” and “How do you write x·x·x·x·x?” The subsequent questions asked, described below, were more difficult. Although they were still clearly within the basic curricula, the students had great difficulty in answering them: The teacher wrote x(x + 5) = and asked for the answer. No student replied. 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 interrupted: “No,” and writes without discussion −(8x2 + 4xy). Teacher: “What does this become? Remove the parenthesis.” Eve: “−8x2 − 4xy.”

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Teacher: “What is 3x(2x + y) − (3x + 2y)(x − 2y)?” Ann seemed able to solve the task and started to answer, but the teacher did not let Ann speak herself. Instead, the teacher guided Ann by formulating all of the subtasks saying, for example, “What is 2·3?,” “What is x·x,” and so forth, without discussing why these particular steps should be taken. The teacher summarized by writing: =6x2 + 3xy − 3x2 + 6xy − 2xy + 4y2 = 3x2 . Bea: Bea interrupted 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 makes 3x2 . The teacher finishes the interrupted writing: +7xy + 4y2 . “Is it correct, Joe?” Joe: “Don’t know.” After this dialogue, the students worked individually or in pairs with their textbook exercises until the end of the lesson. The observer asked the students (individually) what they were doing. Most of them gave answers like “I don’t know” or “I haven’t got a clue.” 4.1.1. Interpretation and identification of central task situations Four (sub)task situations were identified: three tasks and Bea’s question. S1: How do you expand x(x + 5)? Max believed wrongly that “the first two” (whatever that meant to him) makes 2x. The teacher’s implicit strategy choice contained two parts: (i) the algorithm is to multiply the left factor by the two right terms, one by one. (ii) The first multiplication is performed using the definition of powers (x·x = x2 ), as in the earlier example (x·x·x·x·x). Max may have understood this simple, earlier example, as many of the other students seemed to, but not that the relationship a(b + c) = ab + ac should be applied. Therefore, Max was unable to use the teacher’s guidance and implemented a faulty algorithm to reach 5x2 . The teacher gave, without argumentation, the correct answer and proceeded to the next task. S2: How do you expand −4x(2x + y)? Jan either did not know the correct algorithm or made a careless mistake. The teacher just stated the correct answer. S3: How do you expand and simplify 3x(2x + y) − (3x + 2y)(x − 2y)? Instead of letting Ann try to reason herself, the teacher made all of strategy choices and left Ann to perform the elementary local transformations. S4: Why does 3x2 remain (as one of the terms) when simplifying 6x2 + 3xy − 3x2 + 6xy − 2xy + 4y2 ? The teacher’s strategy choice was to use the apple analogy. The intention was probably to demonstrate that you should add the terms that are of the same type (x2 ), but this connection was not explicit and there was no warrant anchoring this conclusion to the data. Bea surely knew that six apples minus three makes three, but the relevance of this information was not made visible to her. 4.1.2. Characterization (1) Identification of task type: (N) It was clear that the tasks concerned powers, but the teacher neither identified the tasks as belonging to a particular type nor discussed the characteristics of the tasks that could help to identify their type (that could, in step 2, be related to certain solution methods). (2) Recognizing a solution method: (N) The teacher neither described the principles of the solution methods nor their relationship to the tasks. (3) Creative reflection: (N) There was no reflection of any kind. The teacher described the algorithmic solution steps and quickly answered the questions posed if the students did not do so. A minor exception was when the teacher tried to help Max relate the solution to an earlier example. The teacher could have asked a reflecting question such as ‘What do we need to know?’ and ‘What relationships and rules can guide us?’ (4ab) Argumentation: (N) The teacher’s reasoning was essentially based on only short algebraic transformations without any predictive or verificative argumentation (e.g., explanations). One rudimentary exception was the verificative, but incomplete (see S4), argument that 6x2 − 3x2 = 3x2 because six apples minus three apples makes three apples. (5) Mathematical foundation: (N) There were essentially no references to the relevant intrinsic properties. There were a few exceptions, for example, when the rule am = a·a·a (with m factors a) was referred to as a guiding relationship. Several of the students seemed to know this basic power property. However, neither the teacher nor the students referred to the rule for multiplying parentheses, (a + b)(c + d) = ac + ad + bc + bd, which appeared to be the main intrinsic property required for the students’ problematic task situation. (6) Alignment: (N) The teacher focused on the basic meaning of the power expression am , but the students’ main difficulties appeared to be related to multiplication of parentheses, or perhaps the combination of these two and other rules. The teacher made no attempt to find out what the students’ problematic situations were and may have believed that the students’ problems were only related to the basic meaning of am .

4.1.3. Summary An advantage of this type of ‘pure algorithm’ presentation that only contains the calculation transformations is that several tasks can be solved in a short time, but this presentation also exemplifies the disadvantages mentioned in Section 2.2 that occur when the algorithms are isolated from their meaning and motivation. This isolation is made possible through the algorithm’s independence of meaning (Section 2.1.6). In addition, because aspects (1) and (2) are absent, the students are not given opportunities to see the principles behind the choice and the implementation of methods, which makes the memorization of solution methods difficult. Furthermore, the lack of reflection and arguments make it difficult for both the teacher and the students to discover that they are not aligned in their solution goals and means that there is no foundation to allow the teacher to flexibly adjust his or her reasoning to meet the needs of the students. The six aspects characterized demonstrate that this type of presentation provides limited opportunities to learn CMR and MR/AR, and the students may be restricted to rote learning attempts.

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4.2. Rules without explanations The second example is from the second year of the first mathematics course in the Hotel, Restaurant and Catering Program at an upper secondary school. The lesson was on the rules for solving linear equations. The teacher began the lesson by saying “You are good at finding x, but this lesson is more about the rules,” then followed with the solutions of three different linear equations. The first two were x − 11 = 32 and 2x + 12 = 28. The solution of the third equation will be discussed here. The teacher wrote 3x − 14 = 2 − x and said “Now we will make things even more complicated. This is almost outside the syllabus.” The teacher then said that all of the x’s should be on the left side and the numbers on the right side because “we want the x’s on the side where we have most of them, where they are positive. Here we must move x to the left and 14 to the right.” The teacher wrote +14 on both sides and then +x on both sides. One student protested and asked where the x came from. The teacher did not answer, but some other students tried to explain without success. One other student asked “Is it always like that, that they change signs?” The teacher answered “yes” and proceeded by simplifying the new expression to get 4x = 16, 4x/4 = 16/4, x = 4. Then, the presentation ended, and the students spent the rest of the lesson working with similar tasks. 4.2.1. Interpretation and identification of central task situations One task situation was analyzed: How do you solve the equation 3x − 14 = 2 − x through the use of the rules? This example appeared to be new to the students. The reasoning presented was highly connected to the use of the two central rules: A. “We want the x’s on the side where we have the most of them, where they are positive” and B. You should add the same number (or number of x’s), but with changed signs, to both sides. Neither rule was explained by the teacher. The second rule was also used in the solution of the two first equations. 4.2.2. Characterization (1) Identification of task type: (N) It was clear that the task was an equation, but the teacher neither identified the task as belonging to a particular type nor discussed any of the characteristics of the task that could have helped to identify its type. (2) Recognizing a solution method: (Y) There was no discussion regarding the connection between the task type and the solution method, but there was a description of one method that was assigned to the task (the rules A and B above). (3) Creative reflection: (N) There was no reflection. Some uncertainty can be found in the situation where a student questioned the adding of x to both sides but, because the teacher did not address the question, no reflection occurred. Reflection could have been present, for example, with open questions to the students such as “what rules can we use here?” and “why does this rule work?” (4ab) Argumentation: (N) No argumentation was offered; the teacher only described the solution to the equation. Argumentation could have been present if the teacher had started with the goals and principles of equation solving and then provided warrants to support the conclusion that the two rules used were suitable. (5) Mathematical foundation: (N) There were no references to any intrinsic mathematical properties of the components or references to other warrants involved in the reasoning. Rule A is not a mathematical necessity but is more of a practical rule of thumb, which was not clarified. Furthermore, the teacher said “move x to the left and 14 to the right” but actually had performed both moving and changing sign. This difference might have been a source of confusion, which may be the reason for the first student’s question. Rule B is mathematically correct, but none of its underlying properties were made explicit. For example, it was not explained why one cannot add different numbers to the left- and the right-hand sides of the equation. In addition, because the two rules were presented in the same manner, the students appeared to have difficulty understanding the difference in character between them. (6) Alignment: (Y) The task situation was how to use the rules to solve the equation; there are no indications that the students and the teacher had different task situations or solution goals.

4.2.3. Summary Compared to the example in Section 4.1, in this example the rules are articulated, which provides a better opportunity to learn how to carry out the method. However, because (1) is absent and because in (2) no connection is made between the method and the task type, the students may have difficulty learning what types of tasks are appropriate for the method. In addition, the teacher gives no mathematically grounded arguments or other explanations as to why the method works. Thus, there are several central properties of the solution method that the students are not given the opportunity to learn (unless they make their own analysis of these properties, which seems unlikely). As in Section 4.1, the students are primarily given opportunities to learn by rote, with the main difference that in this example the learning is easier because the algorithmic rules are made explicit. ‘Less difficult,’ however, does not imply that it is easy; it has been demonstrated that linear equations with unknowns on both sides are difficult for students (Filloy & Rojano, 1989). 4.3. Verificative explanations In the last lesson before the exam in the first-semester university course “Calculus 1,” the teacher presented solutions to three old exam tasks. The first task was to prove the inequality ln(1 + x2 ) < x2 , x = / 0. The transcript contains what the teacher said and wrote 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. . .”

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B: x2 − ln(1 + x2 ) > 0, for x = / 0 “The advantage of 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 (*) “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. . .” 2

3

3

B: f  (x) = 2x − 1 2 2x = 2x(1+x 2)−2x = 2x+x −2x = 2x 2 1+x 1+x 1+x2 1+x “. . . 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.”  B : f (x) is strictly decreasing on (−∞, 0) (**) f (x) is strictly increasing on (0, ∞) “The conclusion can be drawn by studying the sign of f (x) [The teacher sketched a curve that looked like y = x2 ] We have seen that f(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: (*) + (**) implies that f(x) > 0 for all x = / 0, that is x2 > ln(1 + x2 ) for x = / 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.” After this summary, the teacher turned to the next exam task. 4.3.1. Interpretation and identification of central task situations One global and three main local task situations are identified: S1: The global strategy choice (which was not articulated until the end of the presentation) is to rewrite the inequality to form a function to study. Then, rather than studying the function values explicitly, the next step is to find the minimum and use the derivative to show that the function is decreasing to the left and increasing to the right and thus lies above the minimum. S2: How is a function formed? Transform the inequality by rewriting it so that the right side is zero. Let f(x) = x2 − ln(1 + x2 ) (the left side). S3: How is the minima found? The teacher did not mention that the minima are sought, why the minima are sought, or why x = 0 was chosen as the point where f was evaluated. S4: Show that f is positive if x = / 0 by using derivatives to show that f is decreasing to the left and increasing to the right of its minimum. 4.3.2. Characterization (1) Identification of task type: (N) It was clear that the task was a proof, but the teacher neither identified the task as a particular type nor discussed the characteristics of the task that could have helped to identify its type. (2) Recognizing a solution method: (Y) The connection between the task type and the method was not characterized, but there was a description of the main principles of the solution method (S1 above). (3) Creative reflection: (N) There were no reflections or questions regarding the strategy choice and its implementation. For example, S1 could have been preceded by reflections on the different possible approaches to the task, in particular, the approach of proving the inequality indirectly by analyzing the function has an underlying flexibility that could have been explored by the teacher. (4a) Predictive argumentation: (N) The strategy choices were not explicitly constructed through predictive argumentation. (4b) Verificative argumentation: (Y) Most of the choices and conclusions were explained (the main exception is S3) after they were stated during the strategy implementations. Most of what the teacher said contained these explanations. (5) Mathematical foundation: (Y) The arguments were anchored in the following intrinsic mathematical properties: The inequality is true because f(x) > 0, / 0. The latter is true because f(0) = 0 and because the derivative shows that this is the function’s minimum. x = (6) Alignment: (Y) There were no indications that the students and the teacher had different task situations or solution goals (which is sufficient for the classification ‘Y’). In addition, it appeared that the students, like the teacher, were focused on solving the task, and most students could probably follow the well-structured reasoning. Because explanations were given after the conclusions were drawn, there were some situations where the students might not have realized until afterwards why things were performed the way they were, for example, why f(0) should be evaluated. Thus, there may have been some local situations where the teacher’s and the students’ reasoning were not aligned. If the teacher had not mentioned “the advantage of moving. . .” to form a function to study, it is more likely that some students would have been at a loss throughout the presentation.

4.3.3. Summary This example shows a well-structured description of a method known to the teacher where most of the statements are justified by warrants anchored in the intrinsic mathematical properties of the components of the reasoning. The strategy choices are explained after they have been carried out. In this respect, the reasoning cannot be seen as similar to CMR. The difference between explaining solution ideas in an effective way after the solution and providing predictive argumentation is that, in actual first-time CMR, the ideas are not known beforehand and have to be constructed as a part of the reasoning. Assuming that the students can themselves identify the task type as something such as ‘prove an inequality involving differentiable functions’ (which is not so difficult, even though (1) is absent), the presentation provides an opportunity to

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Fig. 1. Task 2.

learn an algorithmic method where the main principles are provided by S1. It is not clarified what functions the method will actually work for, but that will become apparent when trying to apply the method on specific inequalities. 4.4. Local creative questions The next example comes from the second year at an upper secondary natural science program. Consider a circle with center in O, and three points A, C and D on the circumference (see Fig. 1). The theorem that is the subject of this lesson then states that the angle AOC is twice the angle ADC (referred to below as T1). The teacher started by reminding the students of this theorem and discussing where in the figure the different angles were. After the initial discussion, she handed out a paper with four tasks. Toward the end of the lesson, the students were invited to go to the board and present their solutions to the tasks. When no one volunteered to do the second task, the teacher allowed the students to guide her instead. The task was to find the values of x and y in Fig. 1. One student said that y is 120◦ because the points A, B, C and D on the circle form an inscribed quadrilateral, which means that the sum of the opposite angles is 180◦ (the theorem will here be referred to as T2). The teacher agreed and said that method was appropriate if a student understands T2. Another student said that the angle at the center (x) would be twice as much as 60◦ [using T1]. The teacher then asked if anybody had found the value of y in another way. When no one answered she said: “If you turn the paper upside down? If y is the angle at the circumference, where is the angle at the center?” She marked the reflex angle at the center (opposite x). One student said that the angle the teacher marked was 240◦ because it was 360–120◦ [using that x = 120◦ ]. Another student asked whether you have to know x to find y using this method. The teacher answered “yes,” and the lesson ended shortly after this discussion. (Note that the answer to the task, the value of y, was never explicitly stated.) 4.4.1. Interpretation and identification of central task situations First, a student identified y using T2, and then another student identified x using T1. The task situation characterized below is the part of the teacher presentation that follows: how can you find the value of y by using theorem T1? The desired solution was based on T1, but was not recognized earlier by the students because the angle at the center in this task is a reflex angle (between 180◦ and 360◦ ). 4.4.2. Characterization (1) Identification of task type: (N) The teacher neither identified the task as belonging to a particular type nor discussed any characteristics of the task that could have helped to identify its type. (2) Recognizing a solution method: (Y) Because the teacher started the lesson by presenting T1 and reminded the students of T1 when handing out the task, it was likely that the students recognized that the task was of a type that could be solved using a method that has T1 as a main element. (3) Creative reflection: (Y) The task solution made flexible use of T1. T1 had so far been used only with a center angle less than 180◦ , but can also be used for larger angles. The flexibility and the reflection was not extensive, but the teacher asked two questions that did reflect on how to apply T1 in the new situation. (4a) Predictive argumentation: (Y) New knowledge was created by applying T1 in a new way (with a central angle larger than 180◦ ). The argumentation was, through reference to the figure, that if you turned the circle upside down, and if y was the angle at the circumference, the angle at the center was 360◦ − x. (4b) Verificative argumentation: (N) The teacher did not comment on the solution at all.

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(5) Mathematical foundation: (Y) There were warrants that anchored the conclusion in the intrinsic properties of T1 and in the relationship between the angles in the figure (see predictive argumentation). In this type of proof-like tasks (common in geometry and combinatorics), it is common to derive the conclusions from geometric properties. In the calculation tasks that are common in, for example, arithmetic, algebra and calculus, such references appear to be less common. (6) Alignment: (Y) There was a clear alignment on the general task situations because all of the students had been working with the tasks in advance. It was not obvious whether all of the students had the same task situation at a more local level, for instance, related to how to apply T1 and T2.

4.4.3. Summary Although predictive argumentation was present in a relatively elementary and limited way, no complete solution algorithm was given for this situation to fulfill aspects (3)–(6). Thus, it is locally (when using T1 in a new way) similar to a real first-time CMR solution. 4.5. Predictive argumentation without reflections 2”. The teacher started by repeating some This example is a lesson in the university course “Mathematics for Engineers ∞ basics on geometric series and then introduced the definition of a power series: a (x − c)n . After showing three examn=0 n ples, the teacher presented the proof that there is a number, R, such that the series converges if |x − c| < R and diverges if |x − c| > R and provided one example. Next, the teacher presented the following proof. The transcript contains what the teacher said and wrote on the blackboard (B). This proof can be found in undergraduate calculus textbooks. B: Theorem: Assume that limn→∞

an+1 an

= L exists,

∞

a (x − c)n is R = 1/L. B: then the radius of convergence for n=0 n “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.” convergence, that is, in the series B: Proof: ∞ We investigate absolute ∞ n B: |a (x − c) | = |a ||(x − c)|n n n n=0 n=0







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 rough that we can use the ratio test.” bn + 1 limn→∞ = bn |an+1 ||x − c|n+1 |an+1 | B: lim = lim |x − c| = L|x − c| n→∞ |an ||x − c|n n→∞ |an |







Assumed existence

tells us that:” “The ratio test ∞ B: The series converges on L|x − c| < 1, that is, if |x − c| < 1/L, n=0 B: and diverges if L|x − c| > 1, that is, if |x − c| > 1/L. “This means that the original series converges absolutely.” B: Therefore, R = 1/L The teacher then gave one example before providing a solution to a textbook exercise that required the application of the above theorem. 4.5.1. Interpretation and identification of central task situations One global and three main local task situations are identified: S1: The global strategy choice was presented at the start of the proof: Test convergence for the power series in question by using tests for ordinary series (without a variable, in this case x). This method for extending familiar tests was new to the students. S2: How are the familiar tests adapted? This strategy choice was implicit: (i) Test for absolute convergence, because, in this case, it is easier and because absolute convergence implies conditional convergence. (ii) Form the new term, bn , that is in a form that familiar tests can be applied to. S3: Which of the familiar tests should be applied? The use of the ratio test was prompted by the coarseness of the positive series. This argument was not elaborated, not explicitly founded on any mathematical properties, and probably difficult for

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the students to understand. The teacher could have meant that the ratio test is suitable when the terms decrease, at least b exponentially, quickly, that is, limn→∞ n+1 <1 bn S4: How do you interpret the outcome of the Ratio test in this new situation? The series converges if L|x − c| < 1, which implies that the radius of convergence is 1/L. 4.5.2. Characterization (1) Identification of task type: (N) The teacher said that the task was a proof, but neither identifies it as a particular type of proof nor discusses any characteristics of the task that could help to identify its type. (2) Recognizing a solution method: (N) It was mentioned that “We shall check if the series converges, and we will use the criteria we know for ordinary series.” However, this was not specific enough to provide the main elements of a complete algorithmic solution method (see Section 4.3 for an example of sufficient specificity). (3) Creative reflection: (N) This was a straightforward description of the proof and included no reflections. The teacher could have, for example, reflected on the strategy choice and asked “Can ordinary tests be used, and how?” or could have reflected on the flexible method for using known convergence tests in this new situation. (4a) Predictive argumentation: (Y) A convergence test that has not been used for power series earlier (only for series with constant terms) was used in a new way to prove the theorem. Some of the main strategy choices were explicitly motivated in advance: “We shall check if the series converges, and we will use the criteria we know for ordinary series” and “In a positive series, the convergence is often so coarse that we can use the ratio test.” (4b) Verificative argumentation: (N) The reasoning was not verified (apart from the predictive arguments). (5) Mathematical foundation: (Y) The reasoning was anchored in intrinsic properties for most of the conclusions. Using convergence tests to determine convergence for a power series was an intrinsic property in this task situation. That the sequence is positive was an intrinsic property when choosing the ratio test. The reasons for choosing this test were not supported by explicit intrinsic property considerations, as discussed in S3 above. (6) Alignment: (Y) The students and the teacher were focused on solving the task, and most students could probably follow the well-structured reasoning, apart from, perhaps, S3.

4.5.3. Summary Proving a mathematical theorem originally, from scratch, often means solving a difficult problem and may involve various methods of uncertain reasoning, including mistakes and extensive reflections. This presentation is an example of a traditional method for presenting a proof: an economic and structured way of summarizing the original reasoning as it was carried out. One may note that the main mathematical foundation and argumentation remain (although they are idealized), but the original uncertainty, reflections and flexibility are not explicit. Thus, the latter aspects of CMR are not visible to the students in this example.

5. Results The results are presented first as summaries of the six aspects of similarity (Section 2.4) and then in relation to the research question (Section 2.5).

5.1. Identification of task type The task type was identified in three out of 23 situations. In two of these situations, the teacher merely stated the task type: “It is a linear equation” and “we are going to make a table ∞ of2nvalues”. In a situation concerning the convergence of the series , the teacher mentioned some general characteristics of n=1 n! the task: it is a positive series, and it decreases rapidly. However, the teacher did not clarify how to identify a rapid decrease. There was no situation in which the task type was clearly identified by explicitly referring to properties of the task. In 20 of 23 situations, the teacher made no comments related to any general properties that characterized the type of task in question, even though most of the tasks could be solved by algorithmic reasoning (AR). Of course, the teacher might have made comments or remarks of this type in an earlier lesson, but the analysis in this study concerned only what actually occurred during the presentation.

5.2. Recognizing a solution method In nine situations, the teachers commented on the solution method. In seven situations, they only stated the name or main attribute of the method (e.g., “use the ratio test” or “solve the equation to the rules”). ∞ according 20,000 In one situation, in the discussion of the convergence of the series , the teacher said, “50 doesn’t matter n=100 3/2 n

−50

in comparison with n3/2 . Upon this realization, it becomes relatively easy to compare with 1/n3/2 .” Here, the teacher used properties of the task to relate to the solution method (the comparison test). Example 4.3 above was the only situation in which the teacher made relatively extensive comments on the solution method and its properties. One may note that while this teacher discussed the solution methods, he did not explicitly identify the task type.

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5.3. Creative reflection In most cases in this study (17 situations out of 23), there was no type of reflection, as in four of the examples above. In three of the six remaining situations, the teachers asked the students how to proceed, and in two other situations, the teachers made simple errors that had to be corrected. The criterion was most clearly fulfilled in the final situation, but still in a limited way (see Section 4.4), and the teacher asked questions about how to use a known method in a new situation. Overall, there was limited and modest reflection. 5.4. Argumentation Four situations contained explicit predictive argumentation. In three of these situations, the argumentation behind the novel reasoning was fairly modest, as in Section 4.4 where the teacher used questions to guide the students to a new way to use a known theorem. The more extensive argumentation was in the traditional proof in Section 4.5, where the teacher developed a solution by argumentation about how to use a series test in a new way. In all of these situations, the arguments had mathematical foundations. The situations in which the criterion was not fulfilled consisted most often of descriptions of algorithms (see Sections 4.1–4.3) by solving tasks as examples. The solutions to the tasks were then described without explicit argumentation to support the strategy choices. Six situations contained explicit verificative argumentation, all in the form of explanations after statements had been formulated. This argumentation ranged from relatively extensive (Section 4.3) to very short (e.g., explaining why 1 rad = 180/◦ by referring to the fact that  rad = 180◦ ). 5.5. Mathematical foundation In 12 situations, the intrinsic mathematical properties were made explicit to the students through discussion. One example of this, in a rather modest way, was when a teacher used a theorem as a basis for reasoning about a task (see Section 4.4). In another situation, the teacher discussed the meaning of the constant term in the function y = x2 + 2 and said, “It gives the intersection with the y-axis since x = 0 gives y = 2”. Here, the teacher used the intrinsic properties of the relationship between the function and the graph to justify his conclusion. If the teacher had only said, “The constant term gives the y-coordinate in the intersection with the x-axis”, the conclusion would be stated and not justified by explicit reference to any intrinsic properties. In both of these situations, a deeper discussion concerning the mathematical foundations of the conclusions would have been possible. For instance, in the latter case, the teacher could have discussed how the function behaves for other x-values. The most frequent reason for the absence of explicit mathematical foundation was that the teacher presented an algorithmic solution without providing any justification or reference to the mathematical foundation of the solution method. In one situation, the main argumentation was founded on nonmathematical properties when a teacher compared the shape of a graph of a quadratic function with a positive coefficient to the x2 -term with a happy mouth. The connection positive to happy had no mathematical foundation because no complementing justification was given. 5.6. Alignment In five situations, there were indications that the teacher’s reasoning and the students’ reasoning were not aligned. In one of these situations, the teacher presented very difficult reasoning that was based on knowledge clearly inaccessible to students. In one situation, the teacher focused on local steps, but the student’s difficulties concerned 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 yet another situation, the teacher asked a question about the kind of solution that the students would obtain when solving an equation of the type y = ax + b. The teacher stated that the equation x + 5 = 11 has a solution that is a number, and the equation x2 = 25 has a solution that is two numbers. After this explanation, the students guessed (i.e., “a table”, “a number“, “a new equation”), and the teacher stated which guess was correct. Here, the teacher was concerned with generalizing his examples while the students simply tried to guess. Several of the students’ guesses clearly had no connection to the teacher’s examples. The point is not that the guesses were wrong but that they largely lacked any relationship to the teacher’s reasoning. In most of the other 18 situations, the teacher and the students attempted to solve the same task, and no indications of differences in alignment were found. In some cases, the students had been working on the task in advance (Section 4.4). In other situations, the students’ activities indicated common task situations or solution goals by asking questions. Of course, there may have been differences in alignment that were not apparent. 5.7. Summary with respect to the research question This section summarizes the results with respect to the research question (Section 2.5). In teachers’ regular presentations of task solutions, in what ways are opportunities provided, from the six aspects above, for the students to develop the capacity to use MR/AR and CMR?

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Quantitatively, aspects 1 (identification of task type) and 3 (creative reflection) were visible in only a few task-solving situations, and aspects 2 (recognizing a solution method), 4 (argumentation) and 5 (mathematical foundation) were visible in 9–12 of the 23 situations. However, most of these aspects appeared in limited ways. In 18 of 23 situations, the teacher’s reasoning and the students’ reasoning were aligned (aspect 6), which is quantitatively high. However, while it could be argued that aspects 1–5 do not need to be present in teachers’ presentations, it seems unreasonable for aspect 6 to be absent. 5.7.1. Creative Mathematically Founded Reasoning (CMR) In some of the teachers’ presentations, the students were given opportunities to learn different aspects of CMR (indicated by the presence and character of aspects 3–5), but not all aspects simultaneously or extensively. The CMR aspects that were most common were mathematical foundations and argumentation. In these situations, the students could observe some (perhaps only partial and modest) evidence for the statements in the reasoning, potentially preventing rote learning. These situations can be seen as explaining what one already knows. However, CMR involves creating new knowledge, such as a task-solving method. The scarcity of the aspects of creative reflection and predictive argumentation makes the presentations different from actual creations of new (including elementary) task-solving methods; that is, they are less similar to first-time task solving by CMR. No clear connections were found between the CMR-related aspects 3–5 (e.g., between reflection and argumentation or between these and the other aspects (1, 2 and 6)), except for the fact that, in all situations except one (Section 5.5), the predictive and verificative argumentation had a mathematical foundation in accord with aspect 5. Thus, the problem in most of the situations studied was not that the students had to rely on the epistemic value instead of the logical value of the justification but that arguments were absent or limited. This situation requires that the students must rely on the authority of the teacher rather than any comprehensive argumentation. 5.7.2. Algorithmic reasoning (AR) Most of the presentations may be characterized as either a strict presentation of methods or algorithms or a process in which the teacher guided the students through an algorithm by posing leading questions. Thus, the students are given more extensive opportunities to learn AR than CMR, but often with two disadvantages. - In the majority of situations with no argumentation or mathematical foundation, the origin, meaning and motivation of the solution methods are not made explicit by, for example, justifications anchored in the mathematical properties of the components of the reasoning. Thus, in these situations, there are no opportunities to go beyond rote learning unless the students themselves can take the initiative and conduct an analysis of the method presented (which seems unlikely). - In Section 2.3, it was argued that two central aspects of analytical routine task solving are identification of the task type and recognition of a solution method. Even though the solution method is sometimes described (aspect 2), the nearly complete lack of characterization of the task type (aspect 1) or an explicit connection (aspect 2) between the solution method and the task type makes even rote learning difficult from the lack of explicit structure or guidance on how to connect a task type to a particular solution method. There is, of course, a possibility that this type of connection could be found in other sources (e.g., in the textbook, in other working materials or in previous lessons by the teacher). In this study, however, we have chosen to focus on the teacher’s presentations of the lessons. Our impression from other analyses of mathematical reasoning in teaching and textbooks is that such explicit connections are generally rare. From the results above, the algorithmic presentations be summarized in three types: (a) Conducting the algorithm without comments (Section 4.1). (b) Conducting the algorithm with descriptive comments (Section 4.2). (c) Conducting the algorithm with verificative arguments (Section 4.3). Because (a) and (b) do not include arguments or any other explanations, they are more likely to lead to rote learning. One may note that if (c) contains predictive instead of verificative argumentation and creative reflection, then the reasoning is more similar to CMR. 6. Discussion 6.1. Consequences for learning In one sense, the teachers’ presentations provide opportunities to learn algorithmic reasoning (AR) because most of the presented task solutions can be learned as algorithms. In another sense, the presentations provide inadequate opportunities to learn AR because there are no situations in which the identification of the task type and the recognition of a method are clear and systematic, which is central to analytical algorithmic routine task solving (Section 2.3). The teachers’ presentations are limited with respect to reflection, arguments and mathematical foundation and thus do not, by themselves, provide opportunities to learn creative reasoning (CMR). The analysis shows that this is the case in the absence of these aspects, their

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limited range when they are present and the lack of relationships between them. In this case, students may have to learn by rote through extensive experience or solve central aspects of AR and CMR independently. It could be argued that the latter is not negative because it requires the activation of metacognitive reflections. At the same time, if these aspects are never or seldom explicit in teaching, then (i) students may not be given guidance on this process, (ii) it may be too difficult for students to develop the process independently, and (iii) the students may form the belief that this process is not central in task solving (Schoenfeld, 1985). From a wider perspective, one may consider that learning mathematics is immensely complex (Niss, 2007), and the teachers’ presentations may be considered examples in which the reduction of complexity is taken too far. In general, the most frequent type of reduction of complexity seems to focus on actions (Asiala, Dubinsky, Mathews, Morics, & Oktac, 1997) in the form of routine procedures (Tall, 1996; Vinner, 1997; Wong, Marton, Wong, & Lam, 2002), which may prevent students from developing conceptual and relational understanding (Hiebert & Carpenter, 1992; Skemp, 1978). The theory of didactical situations (Brousseau, 1997) further clarifies this phenomenon in the teaching of problem solving. One of its main points is that the teacher must arrange not only the communication of knowledge but also the devolution of a good problem if the student is to develop new knowledge. The teacher may try to overcome obstacles and force learning by devolving less of the problem for the student, but telling the student that an automatic method exists relieves the student of the responsibility for her/his intellectual work, thus blocking the devolution of a problem. In Brousseau’s theory, the teacher expects the student to learn problem solving whereas the student expects that an algorithm should be provided. In conjunction with an algorithm being “designed to avoid meaning” (Brousseau, 1997, p. 130), this situation provides a way to understand the consequences of the outcomes of this study. The lack of devolution of a problem causes the obstacles to learning CMR to remain unaddressed, so they cannot be overcome. In turn, this situation can prevent the development of relevant reasoning, problem solving, and understanding competencies (including beliefs regarding the nature of the enterprise). Thus, the thinking processes that students manage to activate will be superficial and imitative. The lack of reflection and predictive argumentation has at least three negative consequences. First, students will not be shown what reflection can mean and how it can be used. Students will lack insight into why and how creative arguments are at the heart of mathematics, and they will fail to see mathematics as the creative domain it actually is (Silver, 1997). Second, students may form and strengthen the belief that reflection and creative reasoning are not useful, not expected and/or not possible for ordinary students. Students exposed only to verificative argumentation receive the message that one must always know exactly what to do from the start and that the only relevant argumentation involves explaining what one already knows. This message may lead to the belief that mathematical expertise consists only of knowing the complete solution in advance (Schoenfeld, 1985). Third, there is a risk that the teacher and the students will be unaware that their reasoning is not aligned. Nonalignment is more common in superficial reasoning whereas reflection and argumentation that refer to the meaning and goals of the specific activity will enhance alignment. One may argue that the teachers’ presentations were not designed to display aspects 1–5 and that these aspects should be present in other parts of the learning environment, such as through exercises, while the teachers’ presentations focus on other aspects. However, the emphasis on algorithmic teaching in this study supports similar outcomes from studies of textbooks and tests (Bergqvist, 2007; Boesen, Lithner, & Palm, 2010; Lithner, 2000b, 2003, 2004; Palm, Boesen, & Lithner, 2005) and thus provides further evidence that the students’ AR focus was, to a large extent, promoted by insufficiencies in the learning environment. The problem is not that imitative reasoning exists; it is a natural part of mathematical learning. The problem is its domination.

6.2. Why is the situation like it is? One of the main goals of mathematics education is to help students develop competencies that can activate thinking processes that are useful in constructing new reasoning in nonroutine problem solving. This goal has not been achieved (Hiebert, 2003). Schoenfeld (1985) argues that many of the counterproductive behaviors that we see in students are unintended byproducts of their mathematics instruction that result from a strong classroom emphasis on performance, memorizing, and practicing, which ultimately causes students to lose sight of rational reasons. If this is the case, under what conditions and constraints do such unintended by-products appear? One elementary constraint is that the teachers’ presentations usually do not contain any first-time problem solving because the task is routine for the teachers or the presentation is prepared in advance. Thus, it is simply not necessary to display aspects 1–5 because the task will be solved anyway. In real analytic routine task solving, it is necessary to invoke aspects 1 and 2, and real mathematically founded problem solving requires aspects 3–5. In addition, because the absence of reflection and argumentation makes it difficult for the teacher and students to discover differences in their main task situations or solution goals (aspect 6), it is difficult for the teacher to modify the presentation to better suit the students’ needs. The scarcity of arguments may be understood from the perspective that arguments are not strictly necessary to convince students that the solution method is correctly chosen and implemented; it is likely that the authority of the teacher is sufficient and seldom questioned. The authority of the teacher may thus become an implicit argument, with no logical value in itself but with high epistemic value. In addition, the algorithm’s built-in avoidance of meaning (Section 2.1.6) reduces the need for argumentation.

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The constraints that are present when teachers engage in the process of didactic transposition (transforming a body of knowledge from a tool into something to be taught, Chevallard, 1992), when interpreting the national curricula and turning this interpretation into teaching, are important for the outcome of the final task-solving presentations. Some constraints come from the demands of society, such as the requirement that a limited number of students should receive a failing course grade. Chevallard interprets teachers’ situation as follows: “Within the didactic order, nobody is free to do as he or she likes” (Chevallard, 1992, p. 230). This situation affects the teacher, who must balance between short-term goals (solving the task) and long-term goals (students’ development of mathematical competency). The AR focus leads to short-term gains, such as textbook exercises and exams that are often adapted to AR (Bergqvist, 2007; Lithner, 2004; Palm et al., 2005), but it may also lead to long-term losses, such as weak conceptual understanding and weak problem-solving competence. The short-term gains seem related to the reduction of complexity and the focus on imitative reasoning. There are indications that teachers believe that only high-performing students can use and benefit from creative reasoning (Bergqvist, 2005; Boesen, 2006) and that too many students will receive failing course grades if CMR requirements are increased. Teachers say that the lack of CMR requirements in teacher-made tests is conscious (Boesen, 2006). Thus, both the choice of tasks and the ways of presenting them contribute to the low appearance of CMR in the task-solving presentations. One may note that this view of creativity as something that is important for high-performing students only is not in line with the view advocated by Silver (1997), in which creativity is seen as a central aspect of teaching for all students. The teachers’ beliefs concerning mathematical task solving are likely affected by the influential components of the learning environment. Ongoing studies indicate that Swedish mathematics teaching is little affected by national curricular goals and nationally mandated tests (which emphasize reasoning and problem solving) and instead is guided by textbooks (which are not under official control) that largely emphasize AR (Lithner, 2008). 6.3. What may be done to develop teaching? The Swedish national curriculum and modern frameworks of learning goals in mathematics (e.g., NCTM, 2000) emphasize conceptual understanding, reasoning, argumentation, and problem solving for students at all levels. One way to help students develop these abilities and to reduce rote learning could be to let creative reasoning and argumentation be a natural part of most activities in the classroom. In the discussion above, some difficulties in this endeavor have been highlighted. There seem to be no simple solutions, but the outcomes of this study, including the research framework, may be of assistance in the development of teaching. Specifically, a basic condition is that creative, nonroutine problem solving is seen as useful and central for all students, not only for the most proficient ones. For example, textbooks could include tasks that enhance CMR but are not conceptually and/or technically very difficult, but such tasks are currently rare (Lithner, 2008). In addition, more extensive, systematic and explicit reflections in teachers’ presentations (on the properties of the task and the possible solution methods) are encouraged to support students’ learning of routine tasks and problem solving. Such reflections can be used as starting points for reasoning based on the argumentation founded in intrinsic mathematical properties. Such argumentation may be seen as the key to teaching and learning all fundamental aspects of mathematics through task solving. One may question why argumentation is not present in all situations to reduce rote learning. The examples in this study indicate that argumentation does not have to be extensive or advanced. However, it is not sufficient to simply increase justifications through any argument: “There is a major gap between ‘argumentation’ and ‘mathematical reasoning’ that, if not understood, could lead us to advance mostly argumentation skills and little or no mathematical reasoning” (Harel, 2006, p. 60). The framework of this paper may be used to focus and communicate particular aspects of reasoning in teaching. It may thus complement more comprehensive frameworks (NCTM, 2000) that aim to develop the central aspects of mathematics education. Because educational development is complex, it seems essential that teachers be given reasonable conditions to develop competence, including time, resources and in-service education. References Asiala, M., Dubinsky, E., Mathews, D., Morics, S., & Oktac, A. (1997). Development of students’ understanding of cosets, normality, and quotient groups. The Journal of Mathematical Behavior, 16, 241–309. Ball, D., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27–44). Reston, VA: National Council of Teachers of Mathematics. Bergqvist, T. (2005). How students verify conjectures: Teachers’ expectations. 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