other aids, for example peers or textbooks, the reasoning may be radically different, see ..... believes that the two values are the answers to the task, but she is still uncertain when ... She has no difficulties in solving the quadratic equation. ...... When Adam tries to verify his result (x = 3.5), he gets severe difficulties since the.
Reasoning Characteristics in Upper Secondary School Students’ Task Solving. Tomas Bergqvist, Johan Lithner, Lovisa Sumpter Abstract. Upper secondary students’ task solving reasoning was analysed, with a focus on what grounds they had for different strategy choices and implementations. The results indicate that mathematically well-founded considerations were rare. The dominating reasoning types were algorithmic reasoning, where students tried to remember a suitable algorithm, sometimes in a random way, and piloted reasoning, where essentially all important strategy choices were made by the interviewer.
Contents 1. Introduction 2. Theoretical framework 2.1. Reasoning structure 2.2. Plausible Reasoning (PR) 2.3. Definitions 2.4. Established Experiences (EE) 2.5. Algorithmic Reasoning (AR) 2.6. Piloted Reasoning (PdR) 3. Research questions 4. Method 5. Analysis: The Natural Science Programme 5.1. Sally 5.2. Ellen 5.3. Eric 6. Analysis: The Social Science Programme 6.1. Eva 6.2. Julia 6.3. Analysis: The Hotel, Restaurant and Catering Programme 6.4. Helen 6.5. Adam 7. Discussion 7.1. Summary 7.2. Reasoning types and possible implications Appendix A. Mathematical tasks References
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1. Introduction As we see it, the Swedish school system1 cannot help sufficiently many students to reach a desired level of mathematical competence. The syllabus for mathematics at upper secondary school in Sweden mentions the ability to analyse and solve problems as one of the central purposes of the subject2. Reasoning is a central component in mathematics and especially in problem solving. Ross (1998) claims that “It should be emphasised that the foundation of mathematics is reasoning. [. . . ] If reasoning ability is not developed in the students, then mathematics simply becomes a matter of following a set of procedures and mimicking examples without thought as to why they make sense.” Earlier research (Lithner, 2000) indicates that in problematic situations undergraduate students tend to rely heavily on their, often mathematically superficial, experiences from school situations. Their strategies are rarely grounded in relevant mathematical concepts, and in the few cases where the students’ reasoning is grounded in consistent mathematics, it is still dominated by the individuals memory images and familiar routines. Our concern is the possibility or risk that this kind of behaviour also is common among upper secondary school students. If this is the case, it might be one important cause to the problems with the mathematics education at upper secondary school in Sweden. The purpose here is to make a study similar to Lithner (2000), focusing on upper secondary school students. It seems safe to claim that students in upper secondary school in Sweden spend a large part of their time solving different types of mathematical tasks, and mainly textbook exercises. In this activity, the students often meet problematic situations where it is not obvious how to proceed. When a student faces this kind of difficulty she must make a strategy choice concerning what to do in order to solve the problem, and then implement this strategy. What we are interested in is the bases on which the choices and implementations are made. Why does the student choose to deal with the problematic situation in a certain way? Why does the student belive that a certain strategy has resolved the difficulty? What kinds of explicit or implicit reasons guides the student? All this is studied in relation to the question What makes students succeed or fail in a problematic situation? We investigate this when students work with mathematical tasks alone and without other aid than a calculator, a setting similar to the one in exams. When students have other aids, for example peers or textbooks, the reasoning may be radically different, see e.g. (Lithner, 2003).
2. Theoretical framework The presentation of the theoretical framework in this section is to a large extent from (Lithner, 2002b) and (Lithner, 2003). A student asks a teacher: “Is a5 · a3 = 2a15 correct? I think it has something to do with adding or multiplying bases and exponents, but I can’t remember how.” The teacher is a bit puzzled, thinking that if the student fails to recall the rules why not just consider the fundamental property of powers: am is just an economic way of writing repeated multiplication a · a · a · · · a with m factors. So a5 · a3 just means a · a · a · a · a · a · a · a, which can be rewritten as a8 . 1
including, among others, teachers at all levels, teacher educators and researchers “Utbildningen i matematik i gymnasieskolan syftar ocks˚ a till att eleverna skall kunna analysera, kritiskt bed¨ oma och l¨ osa problem f¨ or at sj¨ alvst¨ andigt kunna ta st¨ allning i fr˚ agor, som a ¨r viktiga b˚ ade f¨ or dem sj¨ alva och samh¨ allet, som t.ex. etiska fr˚ agor och milj¨ ofr˚ agor.” (Skolverket, 2001) 2
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Previous research indicates that this type of thinking, also in more complex settings, is common among undergraduate students while the reasoning wished by the teacher is rare (Lithner, 2000). This seems to be the case even when the students know the relevant basic facts, e.g. that a3 = a · a · a. Strategies focusing on what is superficially familiar and remembered lead to difficulties and dominate over reasoning based on ‘deeper’ mathematics, even when the latter can lead to considerable progress. Mathematical reasoning and thinking can be well-founded or superficial in different meanings. The ‘Keyword strategy’ is an elementary form of superficial reasoning (e.g., Hegarty et al. 1995). In its simplest version there are only two solution procedures, addition or subtraction of two given numbers, to choose from, and the choice is completely determined by the appearance of one of the keywords ‘more’ or ‘less’ in the task. Vinner (1997) suggests a theoretical framework where “pseudo-conceptual” and “pseudoanalytical” are defined as thought processes that are not conceptual and analytical respectively, but which in routine task solving might give the impression of being so and could even produce correct solutions. Because of the didactic contract (Brousseau, 1997) students may, consciously or not, try to please the educational system with behaviour that, perhaps only superficially, is considered acceptable by the system. Skemp (1978) distinguishes between “instrumental understanding” and “relational understanding” of mathematical procedures. The former can be apprehended by a person as true (relational) understanding, but is only the mastering of a rule or procedure without any insight into the reasons that make it work. A similar distinction, with respect to procedures, is made between “action” and “processes” (Asiala et al., 1997). Investigating students’ conviction of validity, Balacheff (1988) singled out two reasoning types: Pragmatic ‘proofs’ are about showing that the result is true because it works. Conceptual proofs concern establishing the necessary nature of the truth by giving reasons. Schoenfeld (1985) obtained similar results, that novices used “naive empiricism” to verify geometrical constructions: the constructions are correct if and only if they look good. This reasoning often led to wrong conclusions, while the experts to a larger extent based their reasoning on fundamental properties of geometrical constructions and had a more stable base for proper conclusions. The focus of the framework, indicated by the introductory example, is: What is the reasoner’s (implicit or explicit) argumentation supporting a particular task solving strategy, and in what ways is this argumentation mathematically well-founded or not? The framework consists of three parts: - A reasoning structure that divides a sequence of task solving reasoning into substeps that are easier to handle. It also defines mathematical reasoning. - Definitions of the basic components and properties that one reasons about. - Empirically based definitions of reasoning characteristics. 2.1. Reasoning structure. When studying a solution attempt of a mathematical (sub)task that is not routine, one way to structure the reasoning is: (1) A problematic situation is met where it is not obvious how to proceed. (2) Strategy choice: Try to choose (in a wide sense: choose, recall, construct, discover, etc.) a strategy that can solve the difficulty. This choice can be supported by predictive argumentation: Will the strategy solve the difficulty? (3) Strategy implementation: This can be supported by verificative argumentation: Did the strategy solve the difficulty?
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(4) Conclusion: A result is obtained. Dictionaries define different aspects of ‘reasoning’ somewhat differently. Some of the aspects (not the full definitions) that are relevant for this paper are as follows (OALD 2000): Thinking is “to use your mind to consider something, to form connected ideas, to try to solve problems, etc” and “to have ideas, words or images in your mind”. Logic is “a way of thinking or explaining something” and “sensible reasons for doing something”. Reasoning is “the process of thinking about things in a logical way; opinions and ideas that are based on logical thinking”. Thus ‘reasoning’ in this paper is the line of thought, the way of thinking, adopted to produce assertions and reach conclusions. It is not necessarily based on formal deductive logic, and may even be incorrect as long as there are some kind of sensible (to the reasoner) reasons that guide the thinking. Argumentation is the substantiation, the part of the reasoning that aims at convincing oneself, or someone else, that the reasoning is appropriate. In the literature, the term mathematical reasoning is often implicitly used to indicate that the reasoning taking place is of high deductive-logical quality. In the present paper this is not the case, the term is used here for any type of reasoning that concerns mathematical task solving. The quality of the reasoning is described separately according to the following reasoning characteristics. A sequence of reasoning analysed in this paper is always the part of a person’s reasoning that is accessible as a set of empirical data that can be represented in a documentary form (text, symbols, figures, pictures, video recordings, etc.), and not necessarily the actual reasoning that took place in the person’s mind. This means that a task solving attempt can be classified in different ways depending on what parts of the actual reasoning that are accessible as data. 2.2. Plausible Reasoning (PR). In most tasks normally encountered by students it is allowed to guess and use ideas that are not completely firmly founded. Even in exams it is accepted to answer only, for example, 50% of the problems correctly, and, failing this, to be given another chance later. It would be absurd if in working life the mathematician, the engineer, or the economist are correct only in 50% of the cases they claim to be true. The following is an attempt to capture a type of reasoning that includes deductive logic, but can also be less strict: A sequence of mathematical reasoning is classified as plausible reasoning if the argumentation in step (2) or (3) in the reasoning structure (i) is founded on intrinsic mathematical properties of the components involved in the reasoning, and (ii) is meant to guide towards what probably is the truth, without necessarily having to be complete or correct. For example, the solution of a calculus maximisation task may be supported by (not formally complete) PR: “If one sees the graph of a function as hills and valleys, a maximum is found at the top of a hill. There the slope, described by the derivative, is zero, so the task is solved by examining where f (x) = 0.” The term plausible reasoning, but not the PR definition above, is adopted from P´ olya (1954), who uses plausible reasoning to “distinguish [-] a more reasonable guess from a less reasonable guess” while “in strict reasoning the principal thing is to distinguish a proof from a guess”.
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2.3. Definitions. Before introducing other reasoning types it is necessary to specify the terms component, mathematical property and intrinsic property used in Section 2.2. 2.3.1. Components. The object is the fundamental entity, the ‘thing’ that one is doing something with or the result of doing something. Examples are numbers, variables, functions, graphs, diagrams, matrices, etc. A transformation is what is being done to an object (or several), and the outcome is another object (or several). Counting apples is a transformation applied to real-world objects and the outcome is a number. To calculate a determinant is a transformation on a matrix. A well-defined sequence of transformations, e.g. finding a local maxima of a third degree polynomial, will be called a procedure. A concept is here seen as a central mathematical idea built on a related set of objects, transformations, and their properties. For example the concept of function or the concept of infinity. The status of a component depends on the situation. The function defined by f (x) = x3 can be seen as a transformation of the input object 2 into the output object 8. If this function shall be differentiated, then the differentiation is the transformation, f (x) is encapsulated to be an input object, and the derivative f (x) is the output object (Asiala et al., 1997). All component definitions concern the individual’s conceptions, including misinterpretations, of the actual component in a specific reasoning situation. For example, it is the reasoner’s concept image (see Tall and Vinner, 1981) rather that the actual concept that is considered. A component is accepted if it is an object, transformation, or concept that is accepted by the mathematical society as correct and relevant. 2.3.2. Properties. A property of a component is an accepted property if it is an objective property (independent of social context) of an accepted component that could be verified by deductive reasoning. For example, that a whole number is divisible by 3 if the sum of its digits is divisible by three. A plausible property of a component is a subjectively apprehended property, not necessarily correct, that could be made plausible by PR. In order to avoid a circular definition (like ‘PR can be founded on plausible properties of the components involved in the reasoning, and these components and properties can be constructed by PR’) the plausible property should be traceable through a chain of PR where the first is founded on accepted properties. Mathematical modelling can be seen as being done in three phases: 1) A real (including simulated) situation and question is translated into a pure mathematical task. 2) The task is solved by mathematical methods. 3) The result is translated back to the real situation and interpreted. A modelling property is a property of the transformation in phase 1 or 3 that correctly translates between the relevant aspects of the real situation and a suitable mathematical solution in phase 2. A property of a component is defined as a mathematical property if it is an accepted, plausible or modelling property. 2.3.3. Intrinsic property. Since a property of a component may be more or less relevant, depending on the task solving situation, it is necessary to distinguish between intrinsic and surface properties. An intrinsic property is central in a particular context and problematic situation. A surface property has no or little true relevance. For example, a student is trying if the same test for absolute convergence can be applied to the series ∞ ∞ to determine cos nπ n cos nπ and n=1 (n+1) ln(n+1) n=1 2n . He decides it can, based on the surface mathematical property that cos nπ appears in both numerators, while the intrinsic mathematical property in this comparison lies in the behaviour of the denominators. Another example
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is the “naive empiricism” mentioned earlier: To try to determine if an attempt to bisect an angle is correct (in Euclidean geometry) or not by considering the visual appearance of the drawn angles’ magnitude, is a surface property, while the intrinsic property here is the formal congruency of the triangles in the construction. 2.4. Established Experiences (EE). A sequence of mathematical reasoning is classified as reasoning based on established experiences if the argumentation: (i) is founded on notions and procedures established on the basis of the individual’s previous experiences from the learning environment, and (ii) is meant to guide towards what probably is the truth, without necessarily having to be complete or correct. Here the attempt to solve the problematic situation is based on trying to transfer and combine solution procedures from familiar situations, perhaps based only on surface considerations. It is important to stress that EE does not only include rote learning and simple keyword strategies, since these are often not applicable in more complex settings such as calculus. For example, the solution to the same calculus maximisation task mentioned under PR above may be supported by EE instead: “The solutions to the many maximisation tasks I have managed have all been found where f (x) = 0.” In previous research students’ reasoning when solving exam-like tasks (no aids at hand) was studied (Lithner, 2000). One of the main causes behind their difficulties was that EE dominated while PR was rare and of limited range. 2.5. Algorithmic Reasoning (AR). The solving of routine tasks, where the solution is carried out without any problematic situations occurring, is not within the main focus of this paper. The characteristics of routine reasoning, with respect to strategy choice and strategy implementation is trivial: One knows exactly what to do. Still, since it is a very common type of reasoning it deserves to be mentioned. Exactly the same routine reasoning is possible to carry out with different types of underlying understanding. For example, finding the maximum of the second degree polynomial f (x) = 2x2 − 2x + 3 (a common exercise type) can be done with the following reasoning: “I know that a second degree polynomial has one minimum and no maximum if the x2 -coefficient is positive. This maximum is found where f (x) = 0, which is given by 4x − 2 = 0 so x = 2. The coordinates of the maximum is (2, f (2)) = (2, 7). This routine reasoning could be carried out by memorised reasoning without any understanding of the components and properties involved in the reasoning. It could also be carried out by a highly competent person with full understanding of the relevant mathematics. In neither of these situations is there any need to use deeper understanding or PR to solve the task. Often routine reasoning is purely algorithmic: A procedure is defined above as a sequence of transformations. An algorithm is a step-by-step procedure that will solve a certain task where all the transformations and their order are given. If the complete algorithm is known to the task solver and also that it will solve the task, then it is a routine task.
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The reasoning in a task solution attempt will be called algorithmic reasoning (abbreviated AR) if the reasoning fulfils both of the following two conditions: (i) The strategy choice is founded on recalling that a certain algorithm will (probably) solve a certain task type. The connection between the task type and the algorithm may be based on surface considerations but not on PR (no constructive predictive argumentation is required). (ii) The strategy implementation is carried through by following the algorithm. No verificative argumentation is required. 2.6. Piloted Reasoning (PdR). This type of reasoning is introduced by an example (Lithner, 2002a). The 9:th grade student Moa does not know how to calculate 15 % of 90, and asks the teacher for help. The teacher starts to help Moa by writing in her book: 90 · 0.15
(*)
Teacher: “What is 5 · 0?” Moa: “0.” [The teacher writes 0 under (*)] Teacher: “What is 5 · 9?” Moa: “45.” [The teacher writes 45 under (*)] Teacher: “What is 1 · 0?” Moa: “0.” [The teacher writes 0 under (*)] Teacher: “What is 1 · 9?” Moa: “9.” [The teacher writes 9 under (*)]
The teacher makes no attempt to find out what Moa’s difficulties are, but just steers her through the procedure.
Then the addition that the teacher wrote under (*) shall be carried out: 450 (**) + 90
Teacher: “What is 5+0?” Moa: “5.” [The teacher writes 0 under (**)] Teacher: “What is 4+9?” Moa: “13.” [The teacher writes 13 under (**)] Teacher: Where shall the decimal point be placed?” Moa: [Silence]
Moa carries out several trivial primary schools calculations, but has not learnt anything new and still does not understand the procedure when the teacher leaves.
Moa hesitates, and the teacher quickly writes the decimal point in the correct position and leaves. Reasoning characteristics: Moa does not participate in any other activities than adding and multiplying one-digit numbers (which is primary school mathematics and irrelevant for Moa’s difficulties). Moa’s reasoning is completely controlled by the teacher’s strict guidance. It can be seen as a piloting that provides the complete structure for a reasoner who is unable to do so. There are no opportunities for Moa to make, or even think about, any strategy choices, or to consider the relevant intrinsic mathematical properties. The teacher does not try to find out what Moa’s difficulties really are, does not discuss
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the fundamental principles behind the algorithm, does not help Moa to by herself reflect over these principles, does not help Moa to consider the strategy choices, etc. The teacher only describes the algorithm. After the teacher has left, Moa is probably still unable to solve a similar task by herself: Even if each individual solution step is easy it is hard to remember them all and their order. The discussion is not mathematically meaningful to Moa. If she is to understand anything of the algorithm (which is Moa’s main problem here) she has to, by herself, find out its properties, which is a tremendously difficult task for her. The more realistic alternative is that Moa sees the algorithm repeated so many times that she can memorise it. The teacher would probably have helped Moa better if they had been reasoning about the concepts of percentage and multiplication, and perhaps about problem solving and strategy choices (but with another terminology). It must be stressed that the latter is not easy, and that the example above does not describe a bad teacher but indicates that teaching with meaning is difficult. The reasoning in a task solution attempt will be called piloted reasoning (abbreviated PdR) if the reasoning fulfils both of the following two conditions: (i) All, or essentially all, local and global strategy choices that could have been problematic for the solver are controlled by someone else. The solver is not able to make these strategy choices without extensive guidance, and does not consider the intrinsic mathematical properties of the components involved. The guider gives no or little explicit predictive argumentation supporting the local and global strategy choices. (ii) The strategy implementations are carried through by following the guidance and executing the remaining routine transformations based on (in the context) surface properties of the components involved. The success of the solution is guaranteed by the guider’s, in relation to the solver’s, superior competence and authority, but not explicitly supported by verificative argumentation. The term ‘piloting’ is chosen in analogy with the pilot that boards a ship that is entering a harbour, and takes control of the helm. Other researchers have discussed similar phenomena, but with other terminology. Brousseau (1997) calls it the Topaze effect (after a scene in a play) when a teacher, instead of asking for understanding, lets the teaching act collapse by taking responsibility for the essential part of the work that should have been done by the student and letting the target knowledge disappear. Bauersfeld (1988) describes the funnel pattern of interaction, when the teacher stepwise reduces an open question to a closed one.
3. Research questions This study is based on the following research questions: Q1: In what ways do students manage or fail to engage in PR as a means of making progress in solving tasks? Q2: What are the roles of EE, AR, PdR or other types of reasoning in these situations?
4. Method Three different upper secondary school study programmes were represented in the study. The students were in their eleventh school year (age 17 - 18 years). Five students from
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each of the natural science programme (NV), the social science programme (SP) and the hotel, restaurant and catering programme (HR), the latter a vocational programme. NV is the most mathematically intense program in Swedish upper secondary, and HR is among the least intense. The study was presented to the teachers at an upper secondary school. Three interested teachers were then asked to suggest five students from each of their classes. The instruction to the teachers was to avoid students with extremely good or extremely poor results. The students were then asked to participate, and all selected students agreed to attend the project on their spare time. Written information concerning the study was presented to the headmaster, the teachers and the students. The students were also asked to approve the use of the data by signing a copy of the information sheet. They were also encouraged to inform their parents about the study. The sessions were video taped, using a camera placed directly above a sheet of paper. The recordings show the students written work and use of a calculator. The students were asked to “think aloud”, so the recorded voices was also an important source of information. The only restriction was that each session was limited to 40 minutes. To each programme three to six mathematical tasks were selected (see Appendix A). The tasks dealt with mathematics recently covered by the teachers in each class. Similar tasks could in most cases be found in the students’ textbooks. After each session we tried to interpret the work by the students. The goal was to make a first description of what was taking place, and also to speculate why this was happening and how the students were thinking. A second meeting with the students took place three or four days after the first session. The aim was to increase the reliability of the interpretations and speculations. The students were invited to comment on their own performance and we also suggested possible reasons why they were acting in a certain way. Our impression was that the students commented our suggestions in a clear and honest way. In all three groups a selection of students and tasks will be described and analysed extensively. Several task solving attempts will be analysed with the goal to give the reader a both detailed and general picture of how the students were working. Seven out of approximately fifty tasks were chosen. The choice was made to be a representative selection of all problematic situations. The analysis will, for each student, be conducted in three steps: First a description of the data from the video recording and the post-interviews. Then an interpretation of the data (sometimes supported by the post-interview), with the aim to understand the central parts of the reasoning that is not explicit in the data. Finally, in the analysis the reasoning is characterised with the help of the framework in Section 2, or perhaps classified as a new (not already in the framework) reasoning type. The purpose is to clarify what kind of reasoning the students used in each problematic situation. What is treated in the analysis is the students’ reasoning, which sometimes is mathematically incorrect. All students will be referred to by the use of invented names. In quotes . . . will be used to indicate a pause, and [. . . ] to indicate omitted passages. All quotes are direct translations from Swedish. The natural science programme is the most advanced Swedish programme concerning mathematics. The students from this programme that participated in this study can be described as students who, to a large extent, try to figure out or remember the appropriate
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method. They are rather good in the carrying out of the attempted procedures, but they have problems in reviewing their work. The social science programme is also a programme meant to prepare the students for higher studies, but focus is not on mathematics. The students from this programme that participated in this study are very much focused on finding correct algorithms, but they all have severe difficulties in analysing the mathematics involved in the algorithms. The hotel, restaurant and catering programme is a vocational programme with a very small amount of mathematics. The students we studied from this programme have a tendency to guess what algorithm should be used, but they seem not to remember the algorithms very well.
5. Analysis: The Natural Science Programme Three students’ work on Task 1 will be analyzed. The task looks like this: Task 1 Find the largest and smallest values of the function y = 7 + 3x − x2 on the interval [−1, 5]. All tasks can be found in Appendix A. 5.1. Sally. 5.1.1. Sally: description. The description of Sally’s work will be separated into four parts. In each part a new method is introduced and used. Part 1. Sally reads the task and immediately says “Here you are supposed to differentiate, so that you can find maximum and minimum points.” She appears very confident in this, even if she adds “at least I think so”. She starts in a correct way and carries out the differentiation without any problems. Sally finds the zero of the derivative (x = 1.5) and evaluates y(1.5) to obtain y = 9.25. Now she stops (I stands for interviewer and S for Sally, a notation that will be used throughout the paper): S Hum. . . I wonder why. . . I What is making you uncertain? S I think that I should have got two values, and I don’t know why I didn’t, what I have done wrong. Sally tries to remember someway to get two values and proposes that she could put ± in front of the value “or something like that”. When the interviewer asks her about the value she already has, she becomes very uncertain: I The values you have written there, x = 1.5 and y = 9.25, what are those? S This [y] is the value itself, and this [x] is where it is in the interval. At least I think so. I Is that the largest or smallest value? Or is it something else? S Well. . . It should be. . . The largest? Maybe? I Because? S You might get the smallest if you take minus that [points at the x-value]. Or else I am completely wrong. No, I really don’t know. . .
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Sally leaves the task for a while and returns 20 minutes later. During this time she has been working with two other tasks (for Task 2 and 3, see Appendix A). Part 2. When she gets back to the Task 1 Sally decides to use the graphing calculator. She draws the graph on the calculator (see Figure 1) and tries to apply the built in minimumfunction. This function asks for Left bound, Right bound and Guess and then returns the smallest value between these limits. The user is supposed to move the cursor along the curve to give the left bound, right bound and guess, but it is also possible to enter values from the number pad. The function is mainly used by marking an interval around a local minimum. [If Sally had entered −1 as left bound and 5 as right bound, the calculator would have returned (5, −3) or (−1, 3) as the minimum point, depending on where the guess was marked.] For Sally, this doesn’t work. She says “There isn’t any of those to look at” probably meaning that there is no local minimum to put the interval around. When the method doesn’t work, she tries another one.
Figure 1. Sally’s calculator
Part 3. Sally’s next step is to use the table-function, which renders a table with x- and y-values (see Figure 2). The calculator is set for integer steps so Sally finds −3 as the smallest value and 9 as the largest value for x from −1 to 5. The largest value is at x = 1.5 but she doesn’t realise this. She says Between −1 and 5. . . Then the smallest value should be −3 and the largest value 9. Here [in the differentiation] I got 9.25. That’s. . . clever. The contradiction between 9 and 9.25 as two different largest values from the two methods makes Sally believe that the table cannot be used to solve the task, and she tries to find another acceptable algorithm.
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X −1. 0. 1. 2. 3. 4. 5.
Y1 3. 7. 9. 9. 7. 3. −3. Figure 2. Sally’s table
Part 4. Sally now tries to solve the task by setting the function equal to zero. She uses the familiar formula to solve the equation and gets two values, x1 ≈ 4.54 and x2 ≈ −1.54. She believes that the two values are the answers to the task, but she is still uncertain when the interviewer asks her about the values and the method: I Why do you get them [the largest and smallest values] when you do like this? S Because. . . I don’t know. I think I get them. But I’m still uncertain because of that differentiation stuff. . . Sally cannot answer a direct question why she would get the largest and smallest value by solving the quadratic equation. From this point the interviewer guides Sally towards a more correct solution to the task. 5.1.2. Sally: interpretation. Sally is very flexible in her work, she tries four different methods in less than ten minutes. Each method gives her one or two problematic situations which she cannot resolve in a, to her, satisfactory way, and this causes her to repeatedly change method: 1. Differentiation: Sally only got one value while the task is about two values, which is a conflict to her. Perhaps she is not used to this kind of trouble when she solves tasks in the textbook, since then there are often guiding examples. When Sally is asked if 9.25 is largest or smallest, she becomes very uncertain. She seems to recall that in some cases you obtain two values from one by inserting ± in front (e.g, when solving some second degree equations, but Sally does not explicitly refer to this), and wrongly applies this procedure to her task. When Sally looks at her values, x = 1.5 gives one value and x = −1.5 gives the other. If that is the case, then her vague idea is that x = 1.5 probably gives a larger value and x = −1.5 a smaller value. She is hesitant and does not seem satisfied with her conclusion. 2. Using the calculator, the minimum-function: The second situation is when Sally 20 minutes later, comes back to Task 1, and decides to use the calculator (see Figure 1). The reason why she starts using the calculator is not clear, but she appears to remember that you can examine a derivative that way. Sally normally finds a stationary point on the calculator by using the built-in minimum-function, and here she intends to work in her normal way. Since the minimum value is at the endpoint of the interval, and not at a stationary point, the method doesn’t work. The problematic situation that there is no stationary point is not solved, simply avoided by abandoning the method. 3. Using the calculator, the table-function: The next method, looking at a table, is also carried out using the calculator (see Figure 2). The problematic situation, that
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she gets a different maximum value compared to when she used differentiation, is again avoided by abandoning the method. 4. Solving the equation y = 0: Sally remembers something else she can do with a quadratic polynomial (but does not consider how this is related to finding maxima and minima): solve the corresponding quadratic equation, and tries to use that method instead. She has no difficulties in solving the quadratic equation. She finds two x-values and claims hesitantly that they are the x-values for the largest and smallest values. Sally still wants to include the differentiation procedure, but is unsure of its role. 5.1.3. Sally: analysis. The analysis will structure the interpretation in problematic situations (PS). For each PS a strategy choice (SC), a strategy implementation (SI) and a conclusion(C) may occur. Each problematic situation could be subdivided into several smaller problematic situations, but the grain size chosen below is sufficient for this paper. This structure will be used for all analyses in the paper. PS1 One x-value is obtained when the differentiation procedure is used, but the task asks for two values. Is 9.25 the largest or the smallest value? How can the other be found? SC1 Search in the memory for some procedure that has some connections to the components in the task and the solution work. In some procedures related to second degree polynomials you put plus/minus in front of a value. SI1 Trivial. C1 It is possible that 1.5 gives the largest value and −1.5 the smallest value. She is hesitant but does not try to verify this faulty conclusion. PS2 Is there any other way to find the largest and smallest values? SC2 Search in the memory for some procedure that has some connections to the components in the task and the solution work. One familiar procedure in the calculus area of graphs, differentiation, and optimisation is to use the calculator to examine the graph. SI2 It is straightforward to draw the graph, but she is unable to interpret it in relation to PS2. She is unable to make the minimum-function work. C2 She can see from the graph where the largest value is, but not the smallest. She abandons the method since it does not lead to an acceptable answer, without trying to analyse why. PS3 Is there any other way to find the largest and smallest values? SC3 Search in the memory for some procedure that has some connections to the components in the task and the solution work. Another familiar procedure in the same area as in SC2 is to make a table. SI3 It is straightforward to make the table, but she is unable to interpret it in relation to PS3. C3 The table gives 9 as the largest value (wrong), while differentiation gives 9.25 (correct). Since there is a contradiction she abandons the method, without attempting to analyse the contradiction. PS4 What can be done to get two values?
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SC4 Search in the memory for some procedure that has some connections to the components in the task and the solution work. A familiar procedure related to second degree polynomials is to solve the equation p(x) = 0. SI4 Straightforward. C4 She obtains two values but is uncertain, especially since the interviewer questions the result.
Reasoning characteristics. Sally’s initial difficulty is caused by her inability to correctly recall the complete procedure for this type of optimisation task: To investigate both critical points and endpoints of the interval. Her next difficulty, in trying to resolve this difficulty, seems to be that her conceptual understanding is too weak to help her construct or reconstruct a suitable solution method. She may also be hindered by a belief that she is unable to even try to construct her own solution procedure, a belief that is common among students (Schoenfeld, 1992). Sally makes five major strategy choices: First to find the critical point(s), and then SC1SC4. Characteristic of these choices are: i) All choices concern trying familiar algorithms. ii) There are no signs that any of these choices are based on intrinsic properties of the task or the procedures. There is no explicit motivation behind neither the reasonable nor the faulty choices. It seems that only surface properties are considered, e.g., that the task concerns second degree polynomials (SC4). iii) It seems likely that all have some connection to her prior learning: She has surely solved many optimisation tasks by solving f (x) = 0, and some of them using graphing calculator functions. She has also solved many tasks about second degree polynomials using the procedures in SC1 and SC4, but here the connection to the present task is extremely superficial. The general characteristics of her strategy implementations are: iv) The implementation of each algorithm is carried out by following it step by step. There is no (successful) analysis, evaluation or some other consideration whether the algorithms are suitable and generate any useful knowledge or not. v) When progress does not occur as expected (too slow, contradicting results or something else), Sally is quick to abandon the procedure and search for another one. Because of i) and iv), each of the five strategies can be characterised as algorithmic reasoning (AR): to follow an already existing algorithm stepwise without considering its intrinsic properties. In AR, the predictive argumentation, if existing, is rather limited and concern identifying (perhaps surface) similarities between the task and the algorithm. There is no verificative argumentation involved. ii), iii), and v) together implies then her general strategy is a repeated AR that will be called Repeated Algorithmic Reasoning (RAR), with the following definition: Repeated Algorithmic Reasoning (RAR) The reasoning in a task solution attempt will be called repeated algorithmic reasoning (abbreviated RAR) if the reasoning fulfils both of the following two conditions:
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(i) The general strategy choice is to repeatedly apply algorithms, where each local strategy choice is founded on recalling that a certain algorithm will (probably) solve a certain task type. The algorithms are chosen from a set of (to the reasoner) available algorithms that are (to the reasoner) related to the task type by surface properties only. (ii) The strategy implementation is carried through by following the algorithms. No verificative argumentation is required. If an implemented algorithm is stalled or does not lead to a (to the reasoner) reasonable conclusion, then the implementation is not evaluated but simply terminated and a new algorithm is chosen. There is nothing wrong in itself with trying different procedures, it is a quite reasonable approach, but Sally’s problem is that her weak conceptual and procedural understanding is too incomplete to help her make proper choices and evaluate them, the choices become mathematically random and the evaluations are simply never done. RAR does not differ from AR (Section 2.5) only in the sense that it is repeated. The main difference is that to be classified as RAR, the chosen algorithms must chosen on surface property considerations only. It must be stressed that, especially if the set of algorithms to choose from is small and each one is relatively simple, then RAR often works well. It is possible that this is a well-established experience with Sally, a kind of EE on a general strategy level. This may be the reason that she is so very limited to RAR and does not consider other approaches, e.g., PR. Relatively elementary PR could have been applied in order to make progress, for example: Sally could have considered some of the circumstances under which second degree polynomials have both a minimum and a maximum. She could have evaluated y(−1.5) and tried to understand its meaning. She could have tried to relate the table values or the function or derivative zeros to the graph. In this way she may have come to realise that a second degree polynomial can have only one critical point, so the other extremum must be found elsewhere: at an endpoint. Summary: Sally’s reasoning seems to be a clear example of RAR. She is unable and/or reluctant to make intrinsic property considerations supporting the strategy choices and implementations. The chosen familiar algorithms are related to the task by surface properties, and they are simply abandoned without reflection as soon as they are stalled. Elementary PR could have led to good progress. 5.2. Ellen. 5.2.1. Ellen: description. Ellen’s work on Task 1 will be divided into three parts, where Part 1 and 2 are most important. Part 1. Ellen first action is to rewrite the function expression into y = −x2 + 3x + 7. She then proceeds to solve the corresponding equation, −x2 + 3x + 7 = 0. However, she uses the solution formula without taking into account the negative sign in front of the x2 -term: 9 3 +7 x=− ± 2 4 3 √ x = − ± 8.5 2 15
Since she also makes an error with the sign of the constant term she avoids complex solutions to the equation, and finally she fails in adding 94 + 7. At the point where Ellen has arrived at x ≈ −1.5 ± 2.9 the interviewer asks her: I What have you done so far? E I have found out what the square-root of 8.5 is, and from that I will try to find out what x is. I What is the x-value you are looking for? E First I will take the positive x, minus one point five plus two point nine. The questions from the interviewer was intended to cover the whole solution of the quadratic equation and its relation to the task, but Ellen’s answers are on a very local level. After a minor calculation error she ends up with the two solutions x1 = 1.4 and x2 = −4.4 [correct zeroes to the equation y = 0 are x1 ≈ 4.54 and x2 ≈ −1.54]. Now Ellen compares the two values with the interval given in the task. Her conclusion is that x1 is inside while x2 is outside the interval, and therefore is x1 = 1.4 the answer. She also makes a written comment that x = −4.4 is outside the interval. Part 2. Now the following discussion takes place: I E I E I E I E I E
So, what is valid here is 1.4 as you have written? Yes, that’s right. Okay. What is it then? [points at the value 1.4] It’s the x-value. For this function [points at the hole solution]. Since it said in the task largest and smallest, I’m becoming very uncertain. What kind of x-value is it that you have found here? Do you know that? No. Now I’m totally stuck. Why do you become uncertain? I don’t know. . . What is it that makes you uncertain? Is it the largest and smallest values? Yes. Yes it is. Because I know that I have made correct calculations. But now that I’m going to think about what I have calculated it becomes much more difficult.
Ellen’s problem is that she doesn’t really understand the task. She is also concerned with the fact that she only gets one valid answer when the task asks for two values. Her last comment above indicates that Ellen has serious difficulties when it comes to reflect over her own work. The discussion continues: I Can you see how it [x = 1.4] is connected to the issue of largest and smallest values? E Well. . . I would think that it’s the largest value for this function. . . [points at y = −x2 + 3x + 7] . . . and that this is the smallest value [points at x = −4.4], but it’s not in the interval. I So, we are talking about largest and smallest x-values? E Yes. I Is it possible to. . . can you look the other direction too? [indicating looking at y-values] E Yes, you can see what y becomes, the function values. Ellen is very quick in adapting the idea of looking at the y-values. She calculates the function values for x1 and x2 , getting y1 = 9.24 and y2 = −1.8. She notices that both
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values are outside the interval [−1, 5] (which in fact is the interval for the x-values), but she doesn’t seem to be bothered by this. On a direct question whether the interval is for the x-values or the y-values, Ellen (sounding very convinced) answers “for the y-values”. She claims that she didn’t understand the task correctly from the beginning, and consequently she crosses over her comment on x = −4.4 (see end of Part 1). At this point the work on Task 1 is left for a while. Part 3. On the return to Task 1 nothing of any significance occurs. There is a dialogue between Ellen and the interviewer, but this mostly consists of clear guidance, or piloting, from the interviewer. Ellen tries to interpret what the interviewer says, and what she is supposed to answer on the various questions. Ellen makes no progress in this part of the task. 5.2.2. Ellen: interpretation. Ellen works and answers questions at a very local level. When she encounters something problematic, she appears to make guesses or to rely on what the interviewer says. There are no signs that she considers the intrinsic properties of the mathematical components involved, and also no signs of reflection over what she is doing. That might be one reason why she makes three basic errors when she tries to solve the quadratic equation. Ellen’s first difficulty is to choose between the two algorithms she is familiar with concerning quadratic polynomials: differentiation and solving the quadratic equation. She chooses to solve the equation. In the post-interview she said I can do two things with a second degree polynomial, I can differentiate or I can solve the equation. Since I am better at using the solution formula for quadratic equations I chose that method. The next clear problematic situation is when she is asked about the value 1.4. She says “Since it said in the task largest and smallest, I’m becoming very uncertain.” This is really the only problematic situation (apart from the task as a whole) where the difficulty is verbalised by Ellen. In all other situations the interviewer mentions the difficulty in some kind of question or comment. This can be found several times in the discussion in the description, Part 2, for example the first question in the later part of the discussion: “Can you see how it [x = 1.4] is connected to the issue of largest and smallest values?”. Another example is when she is asked if it is possible to look at y-values instead of x-values. Ellen has severe difficulties when it comes to analyse the task or reflect over her own work. She even mentions this herself: “But now that I’m going to think about what I have calculated it becomes much more difficult.” The reason why Ellen makes any progress at all seems to be that she is very quick in adapting ideas from the interviewer. It is possible that this is a well developed strategy, which she uses in her everyday mathematics work in school, where she can ask the teacher or a peer. 5.2.3. Ellen: analysis. Four problematic situations will be discussed here. The first is Ellen’s initial choice to use the formula for solving quadratic equations. The second situation is when she herself discovers that she only has one solution, the third when the teacher asks about the value x = 1.4, and the fourth when she is encouraged to look at y-values instead of x-values:
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PS1 What algorithm can be applied to the polynomial? Note that the primary problematic situation is (to Ellen) not to consider what algorithm or method that may relate mathematically to what is asked for in the task. SC1 There are two possibilities, one is to differentiate the polynomial and the other is to solve the equation p(x) = 0. Since Ellen is better at using the solution formula, the second algorithm is chosen. SI1 Applying the solution algorithm to solve −x2 + 3x + 7 = 0 without taking the negative sign into account, gives incorrect zeroes. C1 x ≈ −1.5 ± 2.9. Since x = −4.4 is outside the interval, x = 1.4 is regarded as the answer. PS2 Initiated by the interviewer: What is x = 1.4 and (implicitly) what is its role in the task? SC2 Consider (parts of) what is asked for in the task. SI2 There is only one x-value in the interval, but the task is about largest and smallest value, two values. C2 Something is wrong. PS3 Question from the interviewer: Is the value 1.4 the largest or the smallest value? SC3 Look at the possibilities. There are two values to compare, 1.4 and −4.4. SI3 Since the other possible value [x = −4.4] is smaller, x = 1.4 should be the largest value. As a consequence of this, x = −4.4 is the smallest value, but it is outside the interval. C3 x = 1.4 is the largest value. PS4 Question from the interviewer: Is it possible to look at y-values instead of x-values? SC4 Adapt the suggestion from the interviewer and calculate y-values. This would also mean that the interval is valid for the y-values. SI4 The function values for x = 1.4 and x = −4.4 is correctly calculated by use of the calculator, resulting in y1 = 9.24 and y2 = −1.8 respectively. C4 The conclusion is that both y-values are outside the given interval. Reasoning characteristics Ellen has very weak understanding about most mathematical components in her work. The most important indication of this is that she has no intrinsic property foundations for her strategy choices, especially SC1 above. One other indication of this is that she answers the questions from the interviewer on a very local level, describing small procedural steps in the algorithm at hand but says nothing about general ideas. Yet another indication is when she first is very convinced that the interval concerns the x-values, and later equally convinced that it concerns the y-values without even trying to motivate either standpoints. Similar to Sally, Ellen is worried by the situation that she only gets one answer, but while Sally could try several other methods, Ellen seem unable to do anything by herself when her method fails: “Now I’m totally stuck”. Ellen makes four strategy choices, the first when she decides to solve the quadratic equation (SC1), and the other three as response to questions from the interviewer (SC2, SC3 and SC4). The first choice appears to be based on AR without mathematical considerations. It is possible that this on a global level is EE, an established experience that
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there is a better chance of success if she chooses an algorithm based on how well she master it, instead of trying to understand more of the task and then try to find or construct a suitable solution method. The second strategy choice, SC2, is a reasonable action to take when encouraged to evaluate a solution, and the evaluation it based on considerations of mathematical properties. The problem is that Ellen considers only a minor part of the task: how many values are asked for? This is not sufficient for making a stable strategy choice. The three last strategy choices Ellen make can be described as Piloted Reasoning (PdR), an acceptance of a suggestion from the interviewer. In fact, almost everything that occurs in PS3 and PS4 can be seen as PdR, where the interviewer guides the student in the strategy choices. One clear example is when the interviewer suggests looking at y-values, and Ellen immediately calculates these values and claims that the interval concerns the y-values, and not (as she said in the beginning of the session) the x-values. However, Ellen can in most cases not benefit from the guidance given by the interviewer. If Ellen’s reasoning is so affected by the interviewer’s guidance, why is not the interviewer simply quiet? First of all, in the methodology used the interviewer does not provide any guidance until the student has been stuck in a problematic situation for a longer period of time and when there are no signs that progress will be made. Secondly, it is of interest to see what use the student can make of the guidance. Thirdly, it is also important to see how dependent the student is of guidance, and to what extent it is complemented by the students own constructive reasoning (e.g. PR). It is clear that that Ellen, in her reasoning above, prioritises PdR before the latter. The strategy implementations Ellen makes are on a very basic level. She uses the solution formula for the quadratic equation, she compares values to an interval and she calculates function values. Ellen claims to master the solution formula better than differentiation, but she still makes several errors in the process. It is possible that a strategy where a student chooses an algorithm that he or she masters, in some situations could lead to success in solving a mathematical tasks. If the student knows how to carry out the algorithm, he or she will solve the task all the times when the algorithm is appropriate. In school mathematics the amount of algorithms in a certain area is rather limited, e.g. by examples preceding textbook exercises, so a student might very well make progress in enough situations to pass a certain test. Of course, it is not a desirable strategy to use a method where you choose an algorithm based on how well it can be carried out. As soon as you get into a situation where there is no suitable algorithm to use, or where you choose the wrong algorithm, the possibility to solve the task is almost absent. In the case of Ellen, appropriate but still rather elementary PR may be unreachable, since her difficulties are on such a basic level. The first step in plausible reasoning (PR) is to consider the meaning of the situation, but Ellen seems reluctant or unable to do even this. Summary: It is, in a mathematical perspective, remarkably unreasonable to choose one of two methods simply because you are better in carrying it out. Ellen’s reasoning is based on AR and PdR, and even with the relatively extensive guidance from the interviewer Ellen is not able to really consider the relevant intrinsic properties of the components involved in her reasoning. Her weak conceptual understanding may hinder (even elementary) PR.
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5.3. Eric. 5.3.1. Eric: description. Eric’s work on Task 1 will be divided into three parts. Part 1. Eric reads the task and is silent for half a minute. Then he says Well. . . this. . . should be solved by. . . differentiate this, and then use. . . no, first differentiation. He performs the differentiation correctly (getting y = 3 − 2x), but then he stops and is silent for almost a minute. Eric says that he is uncertain if this is the correct method, and that maybe he chose the method because he has used differentiation a lot lately. He decides change method and instead solve the corresponding quadratic equation, and he writes: y = 7 + 3x − x2 y = −x2 + 3x + 7 After this he says “I think I will check, for each value, make a table” and he writes 3 3 2 − ± −7 2 2 [Note that Eric makes a sign-error, similar to one of the errors that Ellen made, in the use of the solution formula.] “But wait, what am I doing? No, I’m so lost on this task.” He continues without a clear goal or general idea for a few minutes. Part 2. Now the following discussion takes place between the interviewer and Eric: I Could you try to explain the task to yourself? What are you supposed to do? E What I’m thinking. . . I don’t know what it looks like, that’s the main problem. I You mean the function? E Yeah. . . if it is kind of. . . I Can you find out? E Yes, I can look at the calculator. Eric looks at the graph and points at the limits for the interval, and then he says that he will find where the derivative is zero in order to find the maximum point. He finds the maximum value without trouble and writes Largest value = [1.5 , 9.25] Eric talks about the largest value, but answers with the maximum point. Part 3. Eric is not certain that x = 1.5 is the only zero for the derivative (y = 3 − 2x), so he tries to see if x = −1.5 could be a zero too. To find y (−1.5) he uses the calculator and gets 3 − (2 ∗ −1.5) = 6, and then claims that the derivative has no zeroes other than x = 1.5. Now Eric tries to find the smallest value, and after a few minutes he finds it, but not without trouble. First he says [correctly] that the smallest value is either in x = −1 or in x = 5. Then he changes his mind and claims that there is no smallest value, which he modifies and says that since the graph is symmetric there are two equal smallest values [incorrect]. He tries to verify his faulty assumption by using the calculator to show that the intersections with the x-axis are at the two interval endpoints. When he zooms in on
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the right hand zero, he finds that the intersection is not at x = 5, and that x = 5 gives a negative value. Now he checks x = −1 and finds that it gives a positive value. From this he draws the conclusion that the smallest value is for x = 5. He calculates the value and finishes the session by writing smallest value = [5 , −3] 5.3.2. Eric: interpretation. Eric tries both differentiation and solving the equation in the first part of his work. He cannot really figure out how this should work, and therefore he goes back and forth, mainly between differentiation and using the equation. Eric’s main objective in the first part of his work is that he tries to figure out how to solve the task, or rather, what algorithm he should use. There are no obvious signs that he considers the intrinsic mathematical properties of the components involved when he tries to decide what to do. However, in the interview three days later, he said that he wanted to solve the equation in order to find the symmetry line of the parabola. In the second part, Eric says that his problem really is that he doesn’t know what the graph looks like. When the interviewer asks if he can find this out, it seems like all his difficulties are more or less gone. He uses the calculator to look at the graph, and then determines the largest value using differentiation. It is interesting to note that Eric in the interview says that he would not have used the calculator if the interviewer had not asked if he could find the shape of the graph. He said “I’m not supposed to use the calculator, it is not accepted by the school or society”. In the third part Eric finds the smallest value by reasoning about the appearance of the graph, and the endpoints of the interval. He also checks his answers by looking at the graph on the calculator. Apart from the test to see if x = −1.5 was a zero, Eric’s work was based on reasoning about the properties of the quadratic function, and the appearance of the graph. 5.3.3. Eric: analysis. Five problematic situations are identified in Eric’s work. The first two from Part 1, the third from Part 2 and the two last from Part 3. PS1 What method should I use here? SC1 Search in the memory for an appropriate method in this situation. Differentiation is the normal strategy choice when dealing with optimization tasks. SI1 Starts working but stops. PS2 Am I using differentiation because I have differentiated a lot lately? SC2 If that is the case, solving the quadratic equation in order to find the symmetry line, could be a way to make progress. Change to the quadratic equation. SI2 Starts working but stops. PS3 What does the graph look like? SC3 The question from the interviewer concerning the appearance of the graph makes it acceptable to use the calculator to study the graph. SI3 No problem. C3 When the graph looks like that, differentiation will give the maximum value. PS4 Is it possible that the derivative has another zero?
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SC4 Search in the memory for a situation where several values are answers to one equation. Sometimes you put plus/minus in front of the solution. In that case the zeroes are x = ±1.5. Check x = −1.5 also. SI4 Calculates 3 − 2 · (−1.5) = 6 C4 No, x = −1.5 is not a zero. x = 1.5 is the only zero to the derivative. PS5 How do I find the smallest value? SC5 Look at the graph and try to see if the endpoint of the interval gives y = 0. If both endpoints are zeroes, there are two x-values that gives the smallest y-value. SI5 x = 5 is not a zero, it gives a negative value, and x = −1 gives a positive value. Therefore the smallest value must be at x = 5. C5 The smallest value is calculated. Reasoning characteristics To Eric, the most important reason behind his initial difficulties is that he doesn’t know the shape of the graph to the function, and that he for some reason is unable to initiate an investigation of the shape. Because of this he tries to remember what algorithm to use in order to solve the task, instead of trying to analyse where in the interval the largest point could be found. An interesting situation is when Eric changes from differentiation to solving the quadratic equation because he thinks that he made the choice of differentiation only since he had used differentiation som much lately. This is a strong indication that the strategy choice is not based on intrinsic considerations. When Eric is encouraged to use the calculator to look at the graph the hole situation is changed. Suddenly Eric knows exactly where the largest value is, and finds the maximum point without any trouble. The sudden shift from uncertainty to confidence is striking. In the interview Eric claims that he is not supposed to use the calculator in that way, “It is not approved by the school or society” he says. The fact that he makes this claim is interesting, since calculators according to the curriculum should be an integrated part of the Swedish school mathematics. Eric makes five strategy choices in his work: The two first, together with the fourth choice, seem to be based on a search among available algorithms, without consideration of the intrinsic mathematical properties of the components. These parts of Eric’s work can be categorized as repeated algorithmic reasoning (RAR) since he tries to test if he can find an algorithm that will solve the task. The two first strategy implementations Eric makes are aborted at an early stage, possibly because of lack of progress. There are no clear indications of mathematical property considerations when he decides to stop working with the algorithms. There is of course a possibility that Eric in fact is considering the mathematical properties behind the use of for example the symmetry line, but since we cannot see any clear indications of this, it is not possible to categorize the situation as anything but RAR. The third strategy choice is made after a discussion with the interviewer, a discussion which seem to remove some kind of restriction from using the calculator. It seems clear that Eric would not have made the choice of using the calculator to examine the graph, if not the interviewer had asked him the question if he could find out the shape of the graph. This is Piloted Reasoning (PdR), the interviewer helps the student to take a crucial step in the solution process. The fourth strategy choice is not a standard method altogether, but the core is to apply known (faulty) AR based on surface considerations. The third and fourth implementations, looking at the graph on the calculator and then finding the local maximum by differentiation, and checking if x = −1.5 is a zero to the derivative, are
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made step by step without hesitation or errors. In SC5 Eric discusses the properties of the mathematical components in the task, mainly the appearance of the graph and the meaning of the interval. This is carried out carefully, but is not always straight forward. He reflects over his work and manages to draw correct conclusions about the smallest value of the function. This is PR since he uses the intrinsic properties of the components to construct a solution which he also verifies, and it is not a familiar standard method. The combination of surface reasoning and reasoning based on intrinsic mathematical properties, makes Eric a special case. This gives rise to some questions: - Why does he think that there might be a second zero to the derivative, which in that case would be at x = −1.5 (and no other place)? This idea, to use ± in front of a calculated value was also present in Sally’s work (see Section 5.1.1). - Eric’s behaviour changed when he was asked if he could find out what the graph looked like. What kind of restrictions stopped him from trying to do this by himself? Why didn’t he pose the question himself when he could formulate what his main difficulty was? Summary: Eric’s work includes examples of several reasoning types: AR, RAR, PdR, and PR. Though most parts are not based on intrinsic property considerations, he is not really stalled and can, partly thanks to active control and metacognition (Schoenfeld, 1985), make progress in problematic situations. His work is far from straightforward but he is, unlike most other task solving attempts presented in this paper, not restricted to one or two superficial reasoning types.
6. Analysis: The Social Science Programme The work by two students, Eva and Julia, on Task 1 will be analysed. Both students also worked on Task 2. The tasks looked like this: Task 1 y = 3x − 2 is the equation for a straight line. a) What is the slope of the line? b) Find where the line cuts the y-axis. c) Sketch the line in a coordinate system. Task 2 Find the equation for the straight line passing through the points (2, 3) and (5, 9). 6.1. Eva. 6.1.1. Eva: description. The description of Eva’s work will be presented i three parts, where each part covers a problematic situation for her. Part 1. After reading the task Eva says “there is nothing to calculate. . . ” Then she directly states the answers on a) and b) and writes them on the paper: a) 3x and b) −2. [The answer on a) should be 3, not 3x.] When starting on question c), Eva hesitates after drawing a coordinate system:
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“Well. . . It’s. . . It intersects at minus two on the y-axis. And. . . yes, that’s right. . . then you are supposed to move like this [moves the pencil first to the right and then up in the coordinate system]. But how did you do that? [. . . ] I don’t really know how it was.” She knows that the line goes through (0, −2) and that one solution algorithm involves marking points moving step-by-step. Eva can’t remember how the method was carried out, if it was one step in x and three in y or three in x and one in y. She writes: 3 1 or 3 1 “One is x and one is y, but I can’t remember which is which.” k=
Part 2. Eva finds that the method should result in either the point (1, 1) or the point (3, −1). “I think that x is 1 and y is 3 [meaning x2 − x1 = 1 and y2 − y1 = 3]. But I’m uncertain”. [This guess should implicate that (1, 1) is the correct point to use.] Since Eva so clearly expresses her uncertainty, the interviewer asks her two questions: a) Is it possible to draw the line in another way? b) Can you check both points to decide which one is correct? The questions make her try to calculate something using the equation: E You can put a number instead of x, right? And then calculate what you get. [. . . ] I know I will get the point (1, 1) or the point (3, −1). I Can you check both points somehow? Silence E If I take this dot, [points at (3, −1) in Figure 3] and se what it has. . . it’s on three. . . minus one. . . right? So if I put it into this [points at y = 3x − 2] I should see if it works. I think. Eva takes the values from the point (3, −1) and inserts them into the equation y = 3x − 2, and writes −1 = 3 · 3 − 2. After a few seconds of silence she says “but now I don’t have a clue what I am calculating”. Part 3. Eva gives up on the question and moves on to Task 2 (see above). This task is solved 1 straight forward with only a few arithmetic mistakes. She uses the procedure xy22 −y −x1 when she calculates the k-value for the equation of the line. Now the interviewer asks her if she can use the result on Task 2 when trying to figure out how to solve Task 1. She answers “I wonder if I can take this dot [points at (0, −2)], where it is, and do like this one [points at the first point in Task 2], and then I can use that one [points at (3, −1)] and use it like that one [points at the second point in Task 2]. And then calculate if it. . . what answer I get.” Her idea is to copy the procedure of finding k from Task 2, and try to find the point that gives an expected k-value, k = 13 or 31 . [This process, if carried out correctly, would not lead to a correct conclusion, since the point (3, −1) is found using k = 13 and (1, 1) by using −3 k = 3.] Eva first tries the points (0, −2) and (3, −1) and gets k = −2−1 0−3 = −3 = 0 [this is 1 −3 = −1 incorrect in two ways, she should get −2−(−1) 0−3 −3 = 3 , and −3 = 1.] Since her result is not equal to any of the two possibilities she wanted to find [k = 13 or 31 ], she tries the other
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3 point. Now she uses (0, −2) and (1, 1) and gets k = 1−(−2) 1−0 = 1 . Eva concludes that (1, 1) is the right point, and she draws a correct line. [If she had made correct calculations both points would have satisfied her faulty algorithm, and her method would not have worked. This time she was ‘lucky’ to make the calculation error while working with the incorrect point.] Figure 3 is a copy of Eva’s paperwork.
Figure 3. Eva’s paperwork on Task 1
6.1.2. Eva: interpretation. Three problematic situations (corresponding to the three parts of the description) dominate Eva’s work: (i) How do you use k = 3? (ii) When searching for the next point, should the pen be moved three steps in x and one in y or one step in x and three in y? (iii) How can I use the method from Task 2 in solving Task 1? The first situation is solved by trying to remember. Eva has a vague idea about a method where she marks points in the graph by moving step-by-step (see quote in 6.1.1, part 1). She doesn’t really know how the method works, only that she is supposed to move step-by-step, starting from the intersection on the y-axis. Eva does not seem to reflect over her choice of method, it is more as if she is satisfied to have found something to use at all. The choice is made almost immediately when she remembers the method, and then
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her focus is entirely on situation (ii), to find out where to move the point. To resolve the second situation, Eva tries to remember how it was done. She marks the two possible points [(1, 1) and (3, −1)] in the coordinate system, and then she makes a guess that x2 − x1 = 1 and y2 − y1 = 3 which would mean that (1, 1) is the correct point. Since Eva hesitates, the interviewer asks her two questions: a) Is it possible to draw the line in another way? b) Can you check both points to decide which one is correct? Eva never abandons the step-by-step method, both questions seem to be interpreted in relation to that method. This means that the questions are interpreted as something that has to do with situation (ii), a way to decide which point should be used. The second question makes her try to calculate something using the equation (see quote in 6.1.1, part 2), she takes the point (3, −1) and puts it into the equation y = 3x − 2. She ends up with −1 = 3 · 3 − 2, something she cannot interpret, or at least cannot use. The third problematic situation occurs when Eva, after completing Task 2, comes back to the first task. She is encouraged to use her work on Task 2 when trying to decide what to do on Task 1. Eva is still trying to find out which point to use in the step-by-step method, so she decides to test the two points together with the point (0, −2) (see quote in 6.1.1, part 3). Eva just picks one of her two possible points and applies the method of finding k. Because of errors in her calculations, her faulty algorithm gives her a correct answer. The identification of the two similar situations helps her to find a match, and then she draws the line. 6.1.3. Eva: analysis. The four problematic situations described in the Section 6.1.2 will here be referred to as PS1, PS2, PS3 and PS4. PS1 How do you use the step-by-step method? SC1 Try to remember how it was done. The algorithm deals with marking points one at a time in the coordinate system. SI1 One step in x and three in y gives one possibility, and three steps in x and one in y gives another. C1 Two possibilities are found. k = 13 or 31 , where the numerator is the change in x and the denominator the change in y, or the other way around. PS2 Should the point be moved three steps in x and one in y, or one step in x and three in y? SC2 Unable to remember which is which leads to making a guess. SI2 Guessing that y is in the denominator and x is in the numerator. C2 x2 − x1 = 1 and y2 − y1 = 3. PS3 SC3 SI3 C3
Question from the interviewer: Which point is correct? Test one point (e.g. (3, −1)) by inserting the values in the equation y = 3x − 2. The result is −1 = 3 · 3 − 2. Something is wrong but no further conclusion is drawn.
PS4 Question from the interviewer: How can the method from Task 2 be used in solving Task 1?
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SC4 Imitate the method by using each of the two possible points together with the 1 known point (0, −2), and calculate k = xy22 −y −x1 . The correct point should give 1 3 k = 3 or 1 . SI4 Because of some arithmetic errors, the point (3, −1) gives an incorrect answer. However, the point (1, 1) (correctly) gives k = 31 . C4 (1, 1) is the correct point, which can be used to draw the line. Reasoning characteristics One of the main causes behind Eva’s difficulties is that the connection between the fol1 lowing expressions is incomplete: k, xy22 −y −x1 , slope, the number in front of x, and the step-by-step method. She seems to know that k is the number in front of x in the equa1 tion and that it is the same as the slope. She also uses the procedure xy22 −y −x1 in order to find k. She does not know how the slope is defined or how this is connected to the step-by-step method. When Eva tries to resolve her problematic situations, she is very focused on finding algorithms that might somehow help her. Eva makes four strategy choices in her work. In the first two she tries to find a suitable algorithm to solve the task. The first time she remembers the step-by-step method which she tries to use, and the second time she fails to recall any algorithm at all, so the alternative she sees is to make a guess. Since her strategy implementation in the first situation is carried out step by step, without verificative argumentation apart from the observation that there are two possibilities, Eva’s reasoning in the first part of her work can be classified as algorithmic reasoning (AR). The third and fourth strategy choices are made mainly based on communication with the interviewer. When she is asked if she can check both points, she puts the values from one point into the equation, and when she is asked if she can use the method from Task 2, she copies the algorithm. This second part of her work is therefore classified as piloted reasoning (PdR). The first problematic situation is solved by trying to remember the correct method. When she has come up with the step-by-step method, no signs of questioning the method can be found in her work. This AR does not work since she can not recall the complete procedure and there is no PR to structure her reasoning. In this situation, Eva relies on her experience and memory from the school situation, where she has encountered tasks of this kind before. To hold on to the first (superficially) chosen strategy, without questioning the choice, is often a main cause behind students’ problem solving difficulties (Schoenfeld (1985)). A trace of PR might be found in strategy choice SC3, when Eva tries to test a point. If Eva had carried out her attempt, it might have resulted in PR of this kind: PS* SC* SI* C*
Is (3, −1) a point on the line? In that case x = 3 and y = −1 should satisfy y = 3x − 2. 3 · 3 − 2 = 7 = −1 (3, −1) is not on the line.
However, to do this, she must know what it means to satisfy an equation, and that this is the same as a point being on the line. In the interview three days after the session, Eva claimed that she knows that it is possible to plot points when y = f (x) is expressed as an algebraic expression. She also
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said the she knows how to do it, but the method didn’t come to her mind. If her claim is true, i.e. if she knows how to plot points, and if she understands the method, it is not a lack of resources (Schoenfeld, 1985) that makes her fail when she tries to understand the result in her strategy implementation SI3. She could also have managed to use a PR resembling the reasoning in the star marked example above. However, she failed to use the contradiction in the expression −1 = 3 · 3 − 2. One possible explanation of this is that Eva can’t plot points (y = f (x)), in contradiction to what she said in the interview. Then her failure is caused by a lack of resources . Another possibility is that she can use the method of plotting a point, but doesn’t understand what it means for a point to be on the line, to satisfy the equation, or the relationship between these properties. This can also be seen as a resource issue, the concept of equation is essential when it comes to test a point in this manner. Yet another possibility is that an attempt to figure out what −1 = 3 · 3 − 2 means, is not an option, due to her beliefs about mathematics. To Eva, mathematics might always be a question of recalling the correct method, and since she can’t remember any method connected to the calculation, she can’t interpret the expression. Schoenfeld (1985) showed that students often have the belief that ordinary students can not construct their own reasoning, that mathematics is only about recalling methods provided by others. Summary: Eva’s reasoning is mainly based on AR and PdR. There are few intrinsic property considerations, probably partly because her conceptual understanding is insufficient and partly because she does not try. She puts in a lot of effort in attempting to recall the steps in the algorithms, but does not really try to understand how they work and how they relate to the task. By using fairly elementary PR she could have made good progress in short time. 6.2. Julia. 6.2.1. Julia: description. The two parts in the description of Julia’s work corresponds to the subtasks 1a and 1c, in reversed order. Part 1, Task 1c. Julia starts by saying “Oh, I can’t do stuff like this. The slope of the line. . . Am I supposed to draw something or?” The interviewer says that it is OK to start with task 1c if she wants to, and Julia draws a coordinate system. After some moments of silence, she describes her idea to the interviewer. “I wonder if this is right, if you should move step-by-step, counting two ahead and one up or something.” She sketches a small picture (see Figure 4) showing what she means. The interviewer then asks her how she would use that method on Task 1. Julia seems very uncertain about the whole situation, but after a few moments she uses the indicated method and marks some points: (0, 0), (3, −2) and (6, −4). She then draws a line through the three points (see Figure 5). “I don’t know, no idea. . . Something like this maybe.” Now Julia solves Task 1b by looking at the graph, and states that the answer is zero [this is an incorrect answer to the task, but correct according to her faulty graph]. She also marks the origin in the graph. Part 2, Task 1a. When it comes to Task 1a, Julia can’t do anything. She says “Are you supposed to count. . . no maybe not. . . I don’t know.” The interviewer moves on to Task 3 and 4.
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Figure 4. Julia’s sketch.
Figure 5. Julia’s graph on Task 1c Twenty-five minutes later Julia comes back to Task 1. When she reads the task she says: “What is the slope. . . Are you supposed to calculate k maybe? [. . . ] k is that funny. . . gradient. I don’t know.” When the interviewer wants to discuss the k-value more she says ∆y ].” The interviewer then gives her Task “Yes. . . because that is delta y delta x [writing ∆x 2, to calculate the equation of a line through two given points, which she solves almost without problems. When the interviewer returns to Task 1a, the following conversation takes place: I Here [Task 1a] you were supposed to find the slope of the line. J Yes. Well, I didn’t like that question anyhow. I If you compare the task to that one [Task 2], where you calculate a slope. J Hmm. I Which was? J What? I What slope did you calculate there? J That one. [points at k = 2] Err. . . Is that. . . No. Yes, then that [underlines 3 in the equation in Task 1a] must be the slope. I The slope is 3 in that task. J Yes, because if you write y = kx + m, k is the slope. I Right. And it is 3. J It’s 3. But that is too easy. . . I Too easy? Just to look at the expression?
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J Yes, but it should be possible, maybe. ∆y was presented by Julia, but the fact that she had One may note that the notation ∆x calculated a slope in Task 2 was mentioned by the interviewer.
6.2.2. Julia: interpretation. Julia shows weak understanding of the involved components and their properties, and she has severe difficulties when she is trying to remember facts and procedures. She is throughout the task in need of clear guidance from the interviewer. It seems clear that ∆y she does not understand the connection between slope, k and ∆x . The subtasks a) and c) given to Julia, each becomes a problematic situation: (i) Task 1c, to draw the graph. (ii) Task 1a, find the slope of the line. After encouragement from the interviewer, Julia presents a graph on Task 1c where she uses a step-by-step method (similar to Eva). Her main problem is that she can only remember fragments of the procedure. She only knows that you are supposed to mark dots by moving first horizontally and then vertically. To solve this problematic situation she simply takes some numbers from the equation and put them into her very superficial method. Julia constructs a graph by starting at the origin (without motivating discussion), marking a point three steps to the right and two steps down. It appears obvious that she used the numbers from the equation y = 3x − 2, but in the interview she denies that she used the numbers in that way. Julia can’t explain where the three and minus two came from, but it wasn’t from the equation she said. Still, it is likely that she used the numbers from the equation, since the numbers (3 and −2) cannot be found anywhere else. Task 1b is now easy, Julia just looks at the [faulty] graph she has constructed and finds that the line cuts the y-axis at the origin. It is not clear, but she is probably aware of the difference between intersecting the y-axis and intersecting the x-axis, since she marked the axes (x and y) in the coordinate system, and also wrote y = 0 as answer on Task 1b. Julia appears to be rather skillful when she solves Task 2, she finds the equation of the line in about three minutes. She appears much more at ease with the situation, probably because it is a more familiar procedure to her. Although Julia has created an equation of a straight line she is not clear how this can be used to solve Task 1a. When the interviewer now tells her to compare the result from Task 2 with Task 1a, she says “Then it [the slope] is 3. But that is too easy. . . ” However, she is satisfied when the interviewer questions the statement. 6.2.3. Julia: analysis. The are five problematic situations in Julia’s work: PS1 How do you draw the graph? SC1 Drawing a graph has something to do with “moving step-by-step”. Maybe that can be used. Sketch an example first. SI1 Julia sketches an example (see figure 4). C1 It is possible to do it like that. PS2 Question from the interviewer: Can you use the step-by-step method on Task 1c? SC2 Find numbers from the equation to decide how many steps in each direction the points should be moved. There is a 3 and a −2 in the equation.
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SI2 Points are marked, starting in the origin and then moving three steps to the right and two steps down. C2 A graph is drawn (see Figure 5). PS3 SC3 SI3 C3
What is the slope of the line? Recalling that k has something to do with lines. Suggest calculating k. “Are you supposed to calculate k maybe?” It is probably correct since the interviewer want’s to discuss the k-value.
PS4 How can I proceed? SC4 When trying to find the slope of the line, the interviewer suggests to compare the equation in Task 1 with the equation constructed in Task 2. The interviewer also connects the slope with the k-value. Julia decides to compare the numbers in the two tasks. ∆y ∆y . So k = ∆x . In Task 1 that number SI4 The number in front of x is the same as ∆x is 3. C4 The slope is 3. PS5 SC5 SI5 C5
Can it really be that easy? Tell the interviewer that it looks too easy. “But that is too easy. . . ” No further conclusion is drawn.
Reasoning characteristics Julia’s major problem is that she knows bits and pieces of the mathematics in the task, but there is no connection between the different parts. A clear example is the lack of ∆y and the concept of slope. In some way she knows that k and connection between k, ∆x ∆y ∆x is connected (and maybe equal), but she cannot use this in her work. Throughout her work, Julia relies heavily on the comments from, and the discussion with the interviewer. Julia makes five strategy choices in her work. There are only very little information in the process that actually comes from Julia. All the important choices are made, or at least supported, by the interviewer. This indicates that the reasoning can be characterised as PdR, Piloted Reasoning. In PS1 and PS2 Julia is encouraged to try her idea, in PS3 Julia suggests an algorithm and the interviewer tells her to try it. In PS4 the interviewer makes the connection between slope and the k-value, and finally in PS5 a conclusion is drawn based on the interviewers comment. The strategy implementations are made in a very superficial way. In PS1, Julia remembers something about moving step-by-step. When she has made the strategy choice in PS2, she applies the information in the Task the same way as in PS1. There is no attempt to analyse how the algorithm works, or why she should start at the origin. When the graph is drawn, she can probably not decide whether it is correct or not, at least there is no attempt on verifying the result. This lack of reflection may be one of the more important reasons behind Julia’s difficulties. The last three strategy choices mainly consists of following the interviewers lead.
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The way the strategy choices are made and the implementations are carried out leads to a characterisation of Julia’s work as PdR. It is not likely that she would have solved the task without clear guidance from the interviewer. The lack of both predicative and verificative argumentation also supports the characterisation. What is it Julia really does in Task 1a? In the end she carries out a very basic comparison, when she compares y = 2x + 1 in Task 2 and y = 3x − 2 in Task 1. In Task 2, Julia looks skillful, something that might depend on the fact that there is a complete and familiar procedure she is using. Since she has no clear method or procedure to use in Task 1, she gets in trouble. It is interesting to see that even though the interviewer has guided her a great deal, she still doubts the result since the method was so easy (PS5). This seems to be rather common EE, methods that are too easy or too difficult are seldom correct. Summary: Julia recalls only fragments of a suitable solution method. She tries to work with these fragments, but neither to reach a deeper understanding of the components and their properties nor to evaluate her reasoning. She is dependent on AR and PdR in the problematic situations, and does (similarly to Eva) not invoke PR that could have led to considerable progress. 6.3. Analysis: The Hotel, Restaurant and Catering Programme. Two students’ work with Task 6 will be analysed. Before Task 6, both students had been working with Task 3 and Task 5. The first student, Helen, solved both problems, while the second student, Adam, did well on task 5, but had some difficulties on task 3. Task 3 Solve the equation 3x − 2 = 7 Task 5 Is x = 7 a solution to the equation 2x + 3 = 15? Task 6 Solve the equation 4 − x = 3x + 14 6.4. Helen. 6.4.1. Helen: description. A copy of Helen’s written work can be found in Figure 6. Her work is here divided into three parts: Part 1 Helen writes down the task, 4 − x = 3x + 14, and then she says Ok, now we are going to move all x:es to one side, and the numbers to the other side. It will be like this: She writes 3x − x = 14 − 4, and continues to get x = 5 (which is incorrect, she should get −3x − x = 14 − 4 to get x = −2.5). The calculation takes about 25 seconds. Part 2 The interviewer now asks her if she can check if the answer is correct. “Well, yes I suppose. . . ” she says, and enters 4 − 5 = 3 · 5 + 14 = into the calculator [she first uses the =-button to enter the whole equation, which can not be done, and then as some kind of operation that should reveal if x = 5 is correct and says:
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I got 29! But what am I supposed to check really? I just calculated as it said here [points at the equation], and I got 29. Part 3 The Interviewer asks Helen to solve the equation once again. “Ok, let’s move the x:es” she says with a little sigh. This time she makes another error and gets 3x + x = 14 + 4, and then 2x = 18 and finally x = 6. Her reason to get 14 + 4 is that “it is minus, so you get plus 4”. [Note that according to Helen the negative sign is connected to the 4, so the x becomes positive.] H You can do it differently if you want. I Will you then get different answers? H Yeah, ha ha.
Figure 6. Helen’s paperwork on Task 6.
6.4.2. Helen: interpretation. Helen knows that she can solve the equation by moving all the x:es to one side and all the numbers to the other side of the equal sign. When she starts moving the numbers and the x:es, most transformations are carried out correctly. She adds up 3x − x to get 2x on the left hand side, and 14 − 4 = 10 on the right hand side. There is in effect only one error in her solution, a sign error. When Helen is encouraged to check the answer, she shows a severe lack of understanding of the equal sign. Since her calculator is of a very simple design (it has a button with an equal sign working as a button for execution of a calculation), Helen uses the equal sign
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as an operator and simply presses = as one in a long row of buttons. She cannot interpret the result 29 (which is the right hand side result only). When the interviewer asks Helen to do the task again, she makes a lot more errors compared to the first time. It is possible that she tries to solve it differently, since she was asked to solve the task again. This time Helen uses 4 − x the same way as x − 4, she makes an incorrect addition 3x + x = 2x, and an incorrect division 18/2 = 6. It is all done rather quickly, and without indications of reflection. 6.4.3. Helen: analysis. Three problematic situations will be discussed here, one for each part in the description: PS1 How do I solve this equation? SC1 The algorithm is about moving all numbers to one side and all the x:es to the other side, and then finding out what x is. SI1 3x is moved from left to right without change, the number 4 is moved from right to left and the sign is changed. C1 x = 5 PS2 Question from the interviewer: Can you check the answer x = 5? SC2 Checking an answer means that you replace the x in the equation with the value 5. Then you calculate the result. SI2 Using the equal sign as an operation gives only the right hand side value as the result. C2 The result is 29. No conclusion. PS3 The interviewer wants me to solve the equation again. SC3 Use the same method, but something in the first solution was probably wrong, so it should be done a little more carefully. SI3 Several errors, both algebraic and arithmetic, are made in solving the task. C x=6 Reasoning characteristics Helen’s main problem is that she only knows how she is supposed to do, not why or how it works, and that she is unable to evaluate her work. There are also strong indications that she has a weak understanding of the concept of equations. The idea of moving variables or numbers from one side to the other is often used, but several errors are made in the procedure. The errors are of a type that indicate that Helen does not know how the method works, or when to apply different rules. One example of this can be found in her first attempt, when Helen moves the 3x without change of sign, but the 4 with a change of sign. To understand the solution of an equation of this type it is important to realise that it can be seen as a statement which is true for one specific x-value. To solve an equation can be seen as rewriting the equation stepwise in simpler forms, and keeping the equality valid for the same x in each step by doing the same transformations to the left- and right-hand sides, until the correct x-value can be determined by observation. This understanding does not seam to be present in Helen’s work. Helen makes three strategy choices and implementations: The first and the third choice is to use an algorithm where you move all variables to one side and all constants to the other side. Carried out correctly this is a valid algorithm for
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this kind of task. The second strategy choice is to apply an algorithm where you replace the variable with the supposed root of the equation. This algorithm will, if used in a proper way, solve the problematic situation. There are no signs of predictive argumentation in Helen’s work, Since the strategy implementation in all three cases are carried out by following the algorithm without reflecting over neither the algorithm nor the result, all three situations are classified as algorithmic reasoning (AR). Helen does not know how the methods works, and therefore she is vulnerable to arithmetic and algebraic errors. It is interesting to notice that if she had mastered the algorithms, she would have solved the task without difficulty. Helen has learned, from the book or from the teacher, that she is supposed to separate the variable from the numbers in an equation, and check an answer by replacing the variable with the assumed root. Since she gets correct answers in many situations (for example in Task 3, which she solves without problems), the use of AR may be a well established reasoning type. However, Helen does not master the algorithms, and since she cannot analyse the reasons why she gets incorrect results, she can only solve tasks where the algorithm is used in a very basic way. There is nothing in Helen’s work that indicates the presence of PR. Summary: Helen does not know or does not try to consider what an equation is, she knows the procedures only as moving not their origin, and she does not know what the calculator does. She is highly dependent on AR, to recall complete algorithms and carrying them out without careless mistakes, otherwise she seems rather lost. Helen also seems unable to evaluate and understand the background to her problematic situations. 6.5. Adam. 6.5.1. Adam: description. Task 3 Solve the equation 3x − 2 = 7 Task 5 Is x = 7 a solution to the equation 2x + 3 = 15? Task 6 Solve the equation 4 − x = 3x + 14 Part 1 After reading the task, Adam starts to solve the equation. He uses small figures to indicate changes on both sides of the equal sign: 4−4 − x x x+x 4x
= = = = 14 = 4 x =
3x + 14−4 3x + 14 3x+x + 14 14 3.5 3.5
There are several errors in Adams work, but he proceeds methodically through the process. Part 2 Adam’s next step is to try to verify the answer. He uses the calculator and calculates
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4 − 3.5, but then he stops. He sits quiet, looking at the equation and the calculator. The interviewer poses a question to him: I You calculated 4 − 3.5 and then you stopped. Why? A I didn’t know if I should use plus or minus 3x. The interviewer takes out an earlier task, where Adam successfully checked an answer. He verified that x = 3 was a solution to the equation 3x − 2 = 7. On a question about the difference between the tasks, Adam says: It is easier if you have. . . if you have the final. . . the final result alone, and you don’t have that here. Here you have. . . [points at Task 6]. Part 3 Throughout the last part of the session with Adam, the interviewer tries to guide him in order to solve the equation and check the answer, but in spite of clear guidance Adam makes no progress. In the interview a few days later Adam describes his way of working in mathematics as trying to remember algorithms to solve tasks in the textbook. He indicates that for him mathematics consists of algorithms and methods. 6.5.2. Adam: interpretation. Adam starts out by using a method he believes is correct. The interview (three days later) also indicates that Adam to a large extent tries to remember how to solve mathematical tasks in school. A large part of his errors comes from trouble with the signs, for example the first and the third row of his calculations (in Part 1 above), but serious algebraic errors also occurs. When Adam tries to verify his result (x = 3.5), he gets severe difficulties since the equation has x present both also on the right hand side. If the meaning of an equation is clear (see page 34), this is a trivial transformation. To Adam, the equal sign indicates an operation, that something should be calculated. In this case, he cannot get the result on the left hand side to become equal to the right hand side. This interpretation is supported by Adam’s last statement (in Part 2 above), that it is easier to verify the result if you only have a number on the right hand side, and not an algebraic expression. 6.5.3. Adam: analysis. Two problematic situations are identified in Adam’s work, one for each of the first two parts of the description: PS1 How do I solve the equation 4 − x = 3x + 14? SC1 Try to remember the correct algorithm and then apply it to the equation. The idea is to move all numbers to the right side and all the x:es to the left side. SI1 Several errors are made, ending up in 4x = 14. C1 x = 3.5 PS2 Is x = 3.5 a correct answer? SC2 Try to verify the result by putting x = 3 into the equation and then calculating the result. SI2 Since there is not only a numerical value on the right hand side, this method will not work [incorrect]. Stop here. C2 No conclusion. Reasoning characteristics Adam relies on his memory when he tries to decide what method to use. He said in the
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interview that he goes to class, listens to the teacher presenting methods, and tries to remember how to solve different types of tasks. Following this line of thought would mean that he makes errors because he cannot remember the algorithm good enough. The two problematic situations described above are both examples of AR (algorithmic reasoning). Adam chooses an algorithm (from memory) which he believes will solve the task. Then he carries out the algorithm step by step, without trying to verify what he has done or the result he has found. When he cannot finish the second algorithm, he just stops, unable to modify or adjust the method to the situation at hand. There are no signs of initiatives in order to understand the situation. Adam views the equal sign as an operation, in the same way as Helen. A clear indication of this is when he says that he cannot check his answer because the right hand side of the equation is not a number, which would mean that the left hand side should be calculated and directly compared to the result in the equation. When the equal sign represent an operation (something is supposed to be calculated), then an equation where the variable is present on both sides might be impossible to solve. When Adam’s method doesn’t work, he appears to have no means and/or does not to analyse why it is not working, or how it could be changed to work in the present situation. It is possible that Adam has a toolbox of methods which he can use in specific situations, but that he has no understanding of the mathematics behind them. In the interview a few days later Adam describes his way of working in mathematics as trying to remember algorithms the teacher introduces in order to solve the tasks in the textbook. He indicates that he doesn’t reflect over the mathematics behind the methods. Summary: Adam’s only strategy is AR and he needs to recall complete algorithms, otherwise he makes several mistakes and has no means and/or does not try to (e.g. by PR) reconstruct the missing parts. As with Helen, the procedures seems to be a set of almost meaningless algebraic transformations. He show really no insights into what the transformations really do and why they are done.
7. Discussion 7.1. Summary. In this study we analysed seven situations. Many other situations were also considered, but these seven can be seen as a good enough representation of all situations where students met some kind of difficulty. In many situations the students simply solved the task by applying a correct algorithm. In the chosen situations a problematic situation was encountered, where it was not clear to the students how to proceed. In the data almost all students failed or had severe difficulties, but many of the chosen students solved other tasks correctly. Those tasks that were not chosen were rejected because the students didn’t meet any problematic situations after choosing an algorithm. In the analysis we found that six out of the seven students chose their strategies on only or mainly surface property considerations, and they focused on using more or less well mastered algorithms. The dependence on finding relevant complete algorithms, or at least algorithms possible to use, was very high. The choice of method was in most situations made by remembering something related to the situation at hand. This may sometimes be a reasonable strategy, but it is often insufficient when meeting different kinds of problematic situations.
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When the students for some reason failed to carry out the chosen algorithm, two main different approaches was found. One was to quickly change to another algorithm chosen from a ‘toolbox’ of possible alternatives, and the decision whether an algorithm was appropriate or not was based on surface considerations. The other was to simply stop working. Both approaches were often combined with questions or comments to the interviewer in order to get some kind of hint or guidance about what to do next. There were almost no situations where the students tried to evaluate the chosen algorithm, reconstruct it, or tried to modify the algorithm to the situation at hand. Only one clear example of PR was found among the chosen situations (see Section 5.3.3). However, there were several situations where possible PR seemed close. A desirable PR could be of a very elementary (in relation to the courses taken) kind, for example asking one self about the appearance of a graph or the meaning of an equal sign. Even limited amounts of reflection or afterthought could have lead the students to far more positive results. One purpose in our work was to make a study similar to Lithner (2000). Both studies dealt with only a limited amount of students, but the following comparison can still be made: • The same lack of PR was found, but when in Lithner’s study it was replaced by EE (reasoning based on established experiences), it was here replaced by AR (algorithmic reasoning) and RAR (repeated algorithmic reasoning). One reason behind this difference might be that AR and RAR are not effective choices in undergraduate mathematics. Due to the much larger range of available algorithms and procedures, some other means to guide the superficial strategies must be found. • In both studies there were a dominance of superficial reasoning strategies. This was in both studies found to be on of the main reasons behind the students’ difficulties. 7.2. Reasoning types and possible implications. The research questions that served as a basis for the analysis were Q1: In what ways do students manage or fail to engage in PR as a means of making progress in solving tasks in school? Q2: What are the roles of EE or other types of reasoning in these situations? To answer the questions we will summarise and discuss the types of reasoning that were found: Algorithmic reasoning The students in the Social Science programme (SP) and the Hotel, Restaurant and Catering programme (HR) to a large extent used algorithmic reasoning (AR) in their work (see for example Section 6.4.3). They tried to find a suitable algorithm, often by trying to remember or on other superficial grounds. The algorithm was chosen because it had something to do with the situation at hand, and (occasionally) that it seemed possible that it would solve the task. Then the algorithm was carried out, step by step, mostly without attempts to verify or evaluate neither algorithms nor results. It is possible that AR is an effective method in school mathematics, at least on lower levels (year 6–10), where the number of possible algorithms in each area is very small. If there are two alternatives, the chance of picking the right one is rather good, especially if you
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take surface ‘keyword-like’ characteristics into consideration. Repeated algorithmic reasoning Repeated algorithmic reasoning (RAR) where new algorithms, more or less chosen by surface considerations, are tested when no progress is experienced, was found in the work of a few students in the Natural Science programme. They changed quickly between algorithms, and appeared rather skillful in their work. The problem is that they rarely have any intrinsic property considerations as a ground for their choices of algorithms, and therefore the possible success depends on two things: if the chosen algorithm will solve the task, and if the algorithm is mastered good enough by the student. In the most clear example of RAR (see Section 5.1.3), Sally tries four different algorithms in only a few minutes. The first choice would have solved the task if she had mastered the whole algorithm and not only the first part. RAR can maybe need a slightly higher level of understanding compared to AR, since it requires some kind of evaluation of the algorithm at hand, even if this evaluation can be made on very superficial grounds. If this is the case, RAR would have had a better chance of success than AR, since almost all available algorithms could be tested. RAR should also be more valuable at upper secondary school (year 10–12) where each mathematical area contains several algorithms, and not only two or three as in earlier years. The definition of RAR is a result of this study, see Section 5.1.3. Among the situations where the students used algorithms that failed, the most notable part was the almost total lack of attempting to understand why the algorithm failed, or if it could be modified to the situation at hand. This lack also seemed to be one major reason behind their difficulties. Piloted reasoning There are also several situations, mainly in the Social Science programme and in the Natural Science programme, where students rely heavily on their interaction with the interviewer. This is classified as piloted reasoning (PdR), since all important strategy choices are either made by the interviewer, or as a result of a question or a comment from the interviewer. One example of this can be found in Section 6.2.3. Piloted reasoning is in line with the concept of the didactical contract (Brousseau, 1997). The students listen carefully to what the teacher says, and acts according to the conversation. For the students, this is a way to get correct answers in a very large part of the tasks. To the teacher, it means a quick and manageable way to guide almost a whole class through the textbook. Brousseau claims that as long as both teacher and student follow the didactical contract, no learning occurs. However, piloted reasoning can also be something positive, a way to help a student to reach understanding in an area. If the student is uncertain on a specific level, it is possible for the teacher to, by piloted work on a higher level, help the student to strengthen his understanding on the lower level. In that case, it is crucial that the guidance is not to extensive so that the guider does not resolve all problematic situations for the student as in the example in Section 2.6. Plausible reasoning One student in the Natural Science programme used PR in his work (see Section 5.3.3). He analysed the shape of a graph and could from that decide where to find the smallest value. Traces of PR was also found in a few situations, closest was maybe when Eva tried
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to test if a point was on a line (see Section 6.1.3). In several situations plausible reasoning seemed near at hand to the students, but for various reasons, they refrained from using it. It appears that sometimes students do not attempt PR, and sometimes their conceptual understanding is not sufficient for PR. The two competences conceptual understanding and PR ability are probably connected, since PR requires basic conceptual understanding, and the latter may not be possible to develop by only solving routine, non-PR, exercises where the main goal is to practice algorithms (Lithner, 2003). It should be stressed that the conceptual understanding that is indicated in the analyses as missing are relatively elementary in relation to the courses the students have taken. Students who are encouraged and able to use plausible reasoning in their mathematics learning, will stand a better chance to avoid what Ross (1998) calls “a matter of following a set of procedures and mimicking examples without thought as to why they make sense”.
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Appendix A. Mathematical tasks The following tasks were used: For the natural science programme: (1) Find the largest and smallest values of the function y = 7 + 3x − x2 on the interval [−1, 5]. (2) Determine the equation for the tangent line y = x2 + x − 1 at the point (1, 1). (3) Is there a constant a so that f (x) = ax3 + 4x has a local maximum at x = 1? For the social science programme: (1) y = 3x − 2 is the equation for a straight line. a) What is the slope of the line? b) Find where the line cuts the y-axis. c) Sketch the line in a coordinate system. (2) Find the equation for the straight line passingthrough the points (2, 3) and (5, 9). 2x + 3y = 1 (3) Solve the following system of equations: y = x−y =3 (4) The two lines y = 4 and y = 2x − 4 enclose together with the positive coordinate axes a region in the first quadrant. Find the area of the region. For the hotel, restaurant and catering programme: (1) How much candy will you get for 70 kronor if the price is 8 kronor/hg? (2) A chair costs x kronor and a lamp y kronor. a) What is the meaning of the expression 3x? b) What is the meaning of the expression x + 2y? c) Write an expression that shows the price of four chairs and one lamp. (3) Solve the equation 3x − 2 = 7. (4) Find the value of x if the circumference is 57 cm.
(5) Is x = 7 a solution to the equation 2x + 3 = 15? (6) Solve the equation 4 − x = 3x + 14. (7) A certain time of the year the day is 5.5 hours shorter than the night. How long is the day?
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