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Teaching study skills in mathematics service courses—how to cope with students’ refusal?

Teaching study skills in mathematics service courses—how to cope with students’ refusal? Teaching Mathematics and Its Applications (2019) 38,20–42 doi:10.1093/teamat/hrx016 Advance Access publication 28 December 2017 Teaching study skills in mathematics service courses—how to cope with students’ refusal? †,∗ ‡ Frank Feudel and Hans M. Dietz Department of Mathematics, Humboldt-University of Berlin, Germany and Department of Mathematics, University of Paderborn, Germany Corresponding author. Email: [email protected] [Submitted 21 April 2017; accepted 07 November 2017] Many freshmen entering university have difficulties with finishing their mathematics courses successfully in the first year due to insufficient study skills and learning strategies, which on the one hand include general skills and strategies and on the other hand context-specific skills and strategies for university mathematics. In mathematics service courses for students of disciplines other than mathematics, for example engineering or economics, the effect of poor study skills is even worse due to large course sizes and great diversity regarding the students’ prior knowledge. Therefore, since 2010, first-year students of economics at the University of Paderborn are offered a methodological support program named CAT, which is integrated into the regular course context. The name CAT emerges from three important support-tools of the program (called instruments): Checklist, Ampel (English: Traffic Lights), Toolbox. Although some students claimed that CAT was helpful from the beginning (and a positive effect on academic achievement had also been proven in an experimental tutorial) the majority of students did not use the strategies provided by CAT at first. This was the initial point for a research project with the aim to raise the acceptance of CAT and thus to get more students to use the effective study skills provided by CAT. With the help of this research project, it was possible to identify reasons why students did not use the skills and strategies of CAT, to draw adequate conclusions concerning the implementation of CAT in the course and to raise the acceptance of CAT considerably in the following year. The results of the research project can be useful especially for developers of similar support programs and for teachers and lecturers, who also try to integrate study skills into mathematics courses at university. 1. Introduction Students face many problems when entering mathematics courses at university for the first time. One of the reasons for failure, as is widely believed by academic staff, is the lack of study skills and learning strategies (Anthony, 2000). This includes, on the one hand, general self-regulatory skills like time- management, self-control or effort regulation and on the other hand more context-specific skills and © The Author(s) 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 21 strategies like problem-solving, working with abstract mathematical concepts or rigorous reasoning based on formal definitions. There is a lot of empirical evidence that self-regulatory study skills influence the success in academic learning in general as well as in mathematics courses in particular. For example, Richardson et al. (2012) or Credé & Kuncel (2008) found by means of meta-analyses that self-regulatory study skills have a positive influence on academic achievement, represented by the students GPA (Grade- Point-Average). Furthermore, many interventions targeting students’ study skills were proven to be successful (Hattie et al., 1996), particularly if embedded in the standard course context of the discipline. In addition, several successful interventions promoting students’ study skills in tertiary mathematics education can be found in literature (Cornick et al., 2015; Taylor & Mander, 2003; Mireles et al., 2011; Wadsworth et al., 2007). Supporting study skills can even compensate lack of prior knowledge (Cardelle-Elawar, 1992). For students of economics the effects of poor study skills probably weigh even heavier than for students enrolled in mathematics due to their diverse prior knowledge in mathematics from school (e.g. Voßkamp & Laging (2014)) and due to large course sizes of over 1000 students. Thus, integrating study skills into a mathematics service course like ‘Mathematics for students of economics’ might be a promising approach to lower failure rates in exams and to raise academic achievement. This judgement coincides with the second author’s long-run teaching experience. Accordingly, first-year students attending the course ‘Mathematics for economists’ at the University of Paderborn are offered a special support program named ‘CAT’ (from Checklist, Ampel (English: Traffic Lights) and Toolbox) to improve their study skills. CAT’s main concern is to enable the students to achieve a full understanding of the taught matter on their own, in particular, by enabling them to properly ‘read’ and understand mathematics. CAT was initiated by the second author and fully integrated in the regular mathematics course context since 2010. Some students claimed CAT to be helpful from the very beginning. Furthermore, a positive effect of CAT on academic achievement was shown in an experimental tutorial group (Dietz & Rohde, 2012). However, the integration of the program into the large course faced one major problem: about 80% of the students did not use the provided strategies and did not change their learning behaviour (observation by the second author as course lecturer). Therefore, since 2011, a research project named ‘Support in Learning Strategies in Mathematics for Students of Economics’ was established at the ‘Centre for Higher Mathematics Education’, Germany (www.khdm.de). The project aimed to raise the acceptance of CAT and the use of its methods by the students, which would probably result in better exam results, just like it did in the experimental tutorial group. In particular, the purposes of the project were to identify reasons why many students did not use the strategies provided by CAT and to explore what activities can increase the number of students using these strategies. In the first—theoretical—part of the paper, a survey of the CAT concept and its implementation in the aforementioned course will be presented, as well as a literature review with regard to the question of why students do not change their learning behaviour (relevant for the question why students do not use CAT). The second part contains the results from the empirical research project. The emphasis lies on the reasons for the non-use of the strategies provided by CAT, the drawn conclusions and the resulting activities that have led to a greater acceptance of CAT in the following year. 2. The concept CAT—a course-integrated program to promote study skills and strategies The methodological support program CAT consists of several closely connected instruments: Checklists, Ampel (English: Traffic Lights), Toolbox, Vocabulary list and Concept Base, explained in more detail below (for a more thorough description see Dietz (2016)). Each of these instruments ought to support the students in several domains of their study process. 22 F. FEUDEL AND H. M. DIETZ Table 1. Steps of the Checklist ‘Reading’ Step Description Example: A × B 1. ‘Spell’ The students should assure themselves that they The students must know the meaning of every single know the meaning of every unit precisely. (These symbol, for example, ‘:=’ as a defining sign or ‘{}’ units can, for example, be symbols in a as set braces. They have to be aware particularly that mathematical expression or already known notions some symbols may have multiple meanings like ‘|’. in a new definition. As such, they can be found in the students’ Vocabulary list.) 2. ‘Play’ The students shall try to fit the units together in a ‘A times B is defined as the set of all ordered pairs spoken sentence, as if they were to read the (a,b) with a being an element of A and b being an expression out aloud. element of B.’ 3.‘Animate’ The students ought to provide examples (and Example: non-examples). A = {1, 2} , B = {3} ⇒ A × B = {(1, 3) , (2, 3)} 4.‘Visualize’ The students should try to create a visualization The given example in Step 3 would result in two fitting the definition, if appropriate. points in the plane. A good example for visualizing the cross product can also be the cross product of two closed intervals. 5. ‘Talk’ The students should try to explain the concept behind the expression to someone else (including possible questions). 2.1 A brief description of the six CAT instruments (1) A core instrument of CAT is the Checklist ‘Reading’, as reading mathematics has proven to be a big problem in tertiary mathematics education. Even students who have good reading skills in general and who are not bad at mathematics seem to have difficulties with reading mathematical texts (Shepherd et al., 2012). As a remedy, the Checklist ‘Reading’ provides a list of activities that students ought to perform in order to gain understanding when reading mathematics (e.g. a mathematical expression or a definition). When doing so, they ought to refer to their Vocabulary list,which contains all symbols introduced in the course (explained in detail below). The recommended steps of the Checklist ‘Reading’ are illustrated by the definition A × B := {(a, b) |a ∈ A, b ∈ B} (Table 1). The literature shows that the steps of the Checklist ‘Reading’ are consistent with the activities usually performed by mathematical experts when reading mathematics. In a case study, Shepherd & van de Sande (2014) documented the way in which three students at graduate level and three math- ematicians tried to read a mathematical text for understanding. They were asked to read aloud a chapter about manifolds in a differential geometry book with the aim to learn about the content. All steps in Table 1 except for the last step ‘Talk’ can be found in this study. The students, for example, asked for definitions of homeomorphism and topology which the text did not explain. They tried to ensure they understood the meanings of the units of the text (Step 1). The step ‘Play’ can mainly be observed in the reading process of the mathematicians, when they tried to skim familiar passages and replaced the names of symbols with the names of the underlying concepts (for example quotient space) to achieve higher fluency. A major difference between the students and the mathematicians was that the latter generated examples to ensure understanding (e.g. examples for charts to show that the projective plane given in the book really is a two-dimensional submanifold of R ) (Step 3). The students, however, were more likely to use illustrations, for example when trying to make sense of the theorem about the map composed of a chart and the inverse of another chart being a diffeomorphism (Step 4). TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 23 Österholm (2006) additionally showed that special attention has to be paid to mathematical symbols. In the study, two groups of students were asked to answer questions on a text about the group concept they just read. One group had a text with symbols; the other had a text without symbols. The students having read the text with symbols understood less while having the same prior knowledge as the students from the other group. This suggests that beginners should pay special attention to the first step ‘Spell’ in the Checklist ‘Reading’. (2) One reason why students cannot work with mathematical concepts when trying to solve mathe- matical problems is that they do not know a (verbalized) definition of the concepts (Moore, 1994). As mentioned before, the students are encouraged to maintain a Vocabulary list consisting of all concept and symbol definitions introduced in the course. (3) To support the development of adequate concept images in the sense of Tall & Vinner (1981) the students should extend the vocabulary entries with examples, counterexamples, visualizations, state- ments involving the concept like formulas, theorems or connections to other concepts and applications. This extension is called Concept Base. The students are told in the lecture that they must know these categories for each concept introduced in the course. They can download a template from the homepage of the course containing these categories (but they do not have to use it). A detailed description of the Concept Base is provided in Dietz (2015) or Dietz (2012). By connecting different Concept Bases, the students should develop a Concept Net. The category ‘statements involving the concept’ of the Concept Base should help to link a given concept to other concepts. However, to interconnect different concepts simultaneously, more activities like creating concept maps would be necessary (Novak & Cañas, 2008). An implementation of such activities in CAT has not taken place yet because many students did not even have a profound understanding of the individual concepts itself, which form the basis of the Concept Net. So the emphasis was put on the understanding of these first (it is intended to change this in the future). (4) The Toolbox is supposed to support the students in the process of problem-solving. It is well-known that many students do not know where to start when trying to solve problems (for example, where to start proofs (Moore, 1994)). The idea of the Toolbox is to collect all relevant information, which could be helpful in order to solve the problem. For a proof problem this would include especially relevant definitions, theorems and formulas as well as the relevant preconditions in the task. For example, if the T −1 −1 T task was to prove that for an invertible matrix A its transpose is also invertible with (A ) = (A ) (typical proof task in the exam at the end of the course), the Toolbox should contain the definition of the −1 −1 inverse matrix, the precondition AA = I = A A and formulas involving the transpose of a product of two matrices. The students should then try to use these tools (one after another) to solve the problem. This ‘Toolbox principle’ could at best become an intentional strategy in the development of the students’ own proofs. The advice of first collecting relevant information (especially theorems, definitions and formulas) is also mentioned in literature of problem-solving (Bruder & Collet, 2011; Polya, 2014). Such a collection can be useful in all phases of problem-solving: collecting the tools can help devising a plan, using these tools can help to structure the execution of the plan and checking the tools used to solve the problem can help to check the solution (e.g. by checking if the precondition in a proof was used). In the course the emphasis was put on using the Toolbox in the first two phases and especially the first phase of problem-solving because it was observed in the tutorials that students failed in problem-solving very often because they did not know how to start. (5) The Traffic Lights aim to support cognitive monitoring (Greene & Azevedo, 2007) because it was observed that students often failed in the exam although they had felt well prepared before. This phenomenon is documented in the literature as illusion of knowing (Glenberg et al., 1982). To prevent it, the students are asked to mark parts of their lecture notes according to three levels of understanding 24 F. FEUDEL AND H. M. DIETZ (fully understood = green; understood to some extent, but with a residual uncertainty = yellow; not understood = red). Of course, this is to be taken in a metaphoric sense and not literally. Controlling one’s own understanding is also important when reading mathematical texts (Shepherd & van de Sande, 2014), justifying steps in algebraic calculations (Cohors-Fresenborg et al., 2010) and problem-solving (Schoenfeld, 1992). So students are advised in the lecture to use the Traffic Lights in these settings, i.e. while working through the steps of the Checklist ‘Reading’, performing algebraic calculations or solving problems. To evaluate their own understanding, students are provided with standards in the lecture, which is important (Greene & Azevedo, 2007). For stepwise procedures like algebraic calculations, the students are told that they must be able to justify every step by ‘citable’ knowledge from the course. For mathematical concepts the standards are set in terms of the Concept Base: the students ought to be able to provide the definition, examples, visualizations and so on. The Concept Base can be used as a self-test as well, with the requirement that its elements should be recalled from memory. An easy test for conceptual understanding proposed in the course is to recite the definition from memory and compare the result with the source. If the test fails, the concept is not fully understood (a positive test, however, does not guarantee understanding). Cognitive monitoring also involves activities that deal with the results of these evaluations (Pintrich, 2004), which are also recommended in the lecture. If the result of the evaluation is not positive (i.e. in terms of the Traffic Lights that ‘the colour is not green’), the students are recommended to resolve their comprehension failures as soon as possible. Options for resolution proposed in the course are using other strategies of CAT to gain understanding (e.g. creating Concept Bases), reading the explanations in the corresponding chapter in the course textbook (Dietz, 2012) or asking the tutors. The students are advised to remediate ‘yellow lights’ first (yellow = ‘understood to some extent, but with a residual uncertainty’) because these issues can be sources of many subsequent problems, misunderstandings and misconceptions. A concept built on another concept, that has not been fully understood, cannot be fully understood either. Therefore, starting with aspects that are somehow, but not fully, understood is reasonable (this is the special benefit of using three degrees). 2.2 Implementation of CAT in the course CAT has been integrated into the regular course context as literature strongly recommends (Hattie et al., 1996; Wingate, 2006). Initially, in 2010, ‘integration’ meant that the methods of CAT became explicit parts of the lectures, during which they were explained and exemplified stepwise. But the research project presented below indicated clearly the need of an integration of CAT in all parts of the course (lecture, tutorials and written assignments) with the result that students can decide about using the methods of CAT or not on the basis of their own experience. As a result, since winter semester 2013/14, the instruments of CAT have been introduced in several consecutive steps. These are 0. Motivation of the instruments in the lecture by showing the students that they often fail with their own methods 1. Introduction of the instruments during the lecture and demonstration by the lecturer in which ways the instruments can lead to success 2. Activities during the tutorials in which the students are asked to solve tasks together with the tutors by using the instruments 3. Tasks on written assignments, in which the students are explicitly asked to use the instrument (and which the students should hand in weekly) 4. Tasks in the tutorials and on written assignments involving the instrument implicitly TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 25 Table 2. Steps of the implementation of the instruments of CAT for the example of the Checklist ‘Reading’ Step Realization for the Checklist ‘Reading’ in winter 2014/15 0. Motivation of the instruments The students are asked to state in own words what the elements of the set M := {x ∈ N|∃y ∈ N:y = 3x} are (week four in the semester). 1. Introduction of the instruments The steps of the checklist ‘Reading’ were introduced and demonstrated by the during the lecture example of the definition of the Cartesian product A × B := {(x, y) |x ∈ A, y ∈ B} (week four of the semester). 2. Activities with the instruments The students had to carry out the steps of the checklist for the definition of during the tutorials divisibility n|z :⇐⇒ ∃q ∈ N:z = qn with help of the tutors (week six of the semester). 3. Tasks on written assignments The students had to identify the set T := {x ∈ N| x|18} in a task structured by asking explicitly to employ the the steps of the Checklist ‘Reading’ (week seven of the semester). instruments 4. Tasks involving the instruments A sample task involving the Checklist ‘Reading’ implicitly was to find out if the implicitly in the following weeks relation a  b : ⇔ a|b ∨ b|a is reflexive, transitive, symmetric, antisymmetric or complete (week eight in the semester). To solve the task students have to try to make sense of the given definition first, which requires ‘mathematical reading’. These steps are similar for all the instruments (except the Traffic Lights, which are only motivated and introduced in the lecture so far). They are explained in detail for the Checklist ‘Reading’ in winter 2014/15 as a typical example in Table 2. The activities in the tutorials are, in the opinion of the authors, of major importance for the students’ recognition of the benefit of the instruments. Here they can observe how tutors and peers succeed in using them. Furthermore, they can experience themselves this success by solving tasks with the help of the tutors. As can be seen in Table 2 the degree of obligation concerning the use of the instruments in the course decreases over time. The introduction and the activities in the tutorials can be seen as mandatory for the students (passing the exam is impossible without attending the lecture and the tutorials). The tasks on the weekly written assignments are not mandatory because their completion is not a prerequisite for the exam. Tasks involving the instruments implicitly do not prompt to actually use the methods provided by CAT. Furthermore, the degree of explicitness also decreases (see Table 2). Although the degrees of obligation and explicitness with respect to the use of the instruments decrease with time after their introduction, the instruments are used or at least mentioned repeatedly by the lecturer and the tutors throughout the semester in order to remind the students of their presence. 3. Literature review concerning students’ resistance to change their study behaviour Although successful programs to promote study skills and strategies are offered in many universities, many students still do not participate in these support programs unless they are obliged to do so (Dembo & Seli, 2004). A useful framework for analysing this phenomenon is provided by Prochaska & Prochaska (1999). According to this framework, there are four reasons why people do not change their behaviour (Table 3). This framework was, for example, used by Dembo to identify reasons why some students did not change their learning behaviour after two thirds of a ‘learning-to-learn’ course given at a university in the USA in 2003. Similar reasons for not using learning strategies are given by Garner (1990). She 26 F. FEUDEL AND H. M. DIETZ Table 3. Reasons for students not to change their learning behaviour Reason Example They cannot change. Students are not aware of automatized strategies. They do not know what to change. Students recognize their difficulties (for example problems with beginning a proof), but cannot do anything about it. They do not know how to change. Students know that self-testing is useful for checking their knowledge but do not know which self-tests to use. They do not want to change. Students do not see the benefit of putting the effort into changing their learning behaviour, often because their strategies worked well at school. additionally mentions the learning environment as an important factor and also emphasizes that the integration of the strategies into the (course) context is important to get students to use them. Concrete reasons why students do not use strategies in the subject of mathematics were identified by Anthony (1996) in case studies with students in New Zealand during their last year at school. The reasons were: failure to realize that a problem exists, inadequate strategic knowledge regarding the identified problems, failure to apply the strategic knowledge although the students have it, the students’ decision not to apply learning strategies and a learning environment that is not supportive of strategy use. The first four reasons correspond well with the reasons in the framework of Prochaska & Prochaska (1999). In addition, Anthony again mentions the importance of the learning environment. As demonstrated by the aforementioned reasons, students basically need three types of knowledge to apply learning strategies successfully (Paris et al., 1983): they need to know what learning strategies they can use (declarative knowledge), how to use them (procedural knowledge) and when and why to use them (conditional knowledge). The conditional knowledge plays a particular role in adult learning (Schraw & Moshman, 1995). Justice & Weaver-Mcdougall (1989), for example, showed that adults mainly used strategies for memory tasks, which they judged to be effective. Anyway, if students lack in any of the three types of knowledge, they might choose not to employ the strategies taught. 4. Description of the project ‘Support in Learning Strategies in Mathematics for Students of Economics’ The research project started in 2011 with the aim of getting more students to use the strategies provided by CAT and thus to improve the students’ study skills. 4.1 Research questions of the project 1. How many students actually use the strategies provided by CAT regularly? 2. What are the reasons that students do not use the strategies provided by CAT? 3. What activities can increase the number of students using the provided strategies? 4.2 Chronology of the project The project consisted of several studies, which were conducted between 2011 and 2015. The results presented here originate from a pre-study that was conducted from 2011 till 2012 and the main study, conducted between 2013 and 2015. In this paper, the main study is presented in detail, while the presentation of the pre-study is restricted to major findings which influenced the main study. TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 27 Table 4. Answers to the item ‘I use CAT regularly’from a questionnaire administered in January 2012, 1 = ‘do not at all agree’, 4 = ‘totally agree’(N = 851) 1 2 3 4 Mean Median SD 57.9% 31.6% 9.3% 1.2% 1.54 1 0.711 Table 5. Possible reasons for the non-use of CAT ranked by the number of statements occurring in the answers to the question ‘What do you criticize about CAT?’(N = 851) Possible reason for the non-use of CAT Number of statements 1. (Time) effort of the provided methods 65 2. Use of own methods 33 3. Perceived unhelpfulness of the methods 28 4. Lack of understanding of the methods 17 5. Difficulties in working with the provided methods 15 4.3 Major findings from the pre-study of the project and consequences The pre-study (2011–2012) aimed to get a first estimation of the proportion of students who used CAT and to identify possible reasons for the non-use of the methods of CAT. Therefore, a short questionnaire was administered to the students in January 2012, in which the students were asked to rate on a Likert scale if they used CAT regularly and to respond to the open questions ‘What do you like about CAT?’ and ‘What do you criticize about CAT?’. The answers to the item ‘I use CAT regularly’ showed that most students did not use CAT regularly in January 2012 (Table 4). So the lecturer’s observation that many students did not use the methods provided by CAT was clearly confirmed. The responses to the open question ‘What do you criticize about CAT?’ were examined with respect to possible reasons why students might decide not to use CAT. 175 of the 229 answers given made corresponding statements. These statements were categorized inductively. An answer could also be assigned to more than one category. The five most common categories are shown in Table 5. In addition, many of the statements complained that the tutors did not use CAT (32 statements). By asking the tutors afterwards it was discovered that only a minority of them actually used CAT up to that point. In addition, the students’ statements suggested that the use frequency of CAT and the reasons for its non-use may vary between the instruments of CAT (sometimes only one instrument was actually criticized). This led to the decision to analyse the instruments separately from one another in the subsequent main study. The results of the pre-study led to initial changes in the implementation of CAT. The lecturer spent more time introducing the instruments; moreover, a detailed explanation of the instruments of CAT (and how to use them) was provided on the homepage of the course. But more importantly, from the winter semester 2013/14 onwards, CAT was integrated into the tutorials as well (p. 6). Therefore, particular training for tutors concerning the use of CAT in the tutorials was implemented. The training started with an initial mandatory workshop about the methods of CAT at the beginning of the semester, during which the methods of CAT were introduced to them. This was followed by weekly meetings between the lecturer and the tutors, where useful possibilities to practise CAT in the tutorials (adapted to the current topic) were discussed. 28 F. FEUDEL AND H. M. DIETZ Fig. 1. Example of a question asking if time effort is a reason for the non-use of an instrument of CAT. 5. Methodology of the main study of the project 5.1 Design and participants The research method used for the main study was design-based research. Therefore, two subsequent questionnaires were administered to the students in the course ‘Mathematics for economists’ at the Uni- versity of Paderborn at the end of their respective first semesters. The first one (the main questionnaire) was administered in January 2014, the second one (the follow-up questionnaire that consisted of key items from the main questionnaire) in January 2015. In between, changes regarding the implementation of CAT based on the results of the questionnaire in 2014 were made. 5.2 Description of the questionnaire administered in January 2014 The questionnaire administered in 2014 consisted of three different types of questions: I. Questions to investigate the frequency with which the methods of CAT were used II. Questions to detect reasons why students did not use the methods of CAT III. In-depth questions concerning specific reasons identified in the pre-study with the aim to draw direct conclusions concerning the implementation of CAT I. Questions to investigate the use frequency of the methods of CAT: Since the pre-study had shown that the use frequency might vary among the instruments, the following question was asked for each instrument individually: How often did you use the following instrument of CAT in the semester? The answer format was a Likert scale ranging from 1 = ‘never’ to 6 = ‘every week’. II. Questions to detect reasons for the non-use of CAT: The data of the pre-study showed five pos- sible reasons why students might not use methods of CAT (Table 5). In the subsequent main study the proportion of students who were not using the instruments of CAT for four of these reasons (‘time effort’, ‘difficulties in working with the methods’, ‘lack of understanding of the methods’ and ‘perceived unhelpfulness of the methods’) were investigated by asking questions of the type shown in Fig. 1.For the fifth reason—‘use of own methods’—a question of this type was judged to be not meaningful by the authors (and therefore not asked) because from the answer it cannot be seen if these ‘own methods’ correspond to the rejected instrument or have different aims instead. The reason ‘use of own methods’ was investigated in another study (Feudel, 2015). To find out other reasons for the non-use of the instruments of CAT that were not yet taken into account, students were also asked the following open question for each instrument: If you do not use the instrument regularly, please state one or several reasons for that. TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 29 III. In-depth questions that concerned specific reasons identified in the pre-study: Some of the five reasons for the non-use of CAT that had been identified in the pre-study (Table 5) were investigated in the subsequent main study in order to draw direct conclusions for teaching. The authors had to restrict themselves to only some of these five reasons because of time constraints regarding the completion of the survey during the lecture (30 minutes) due to course restrictions. Hence, the survey did not contain in-depth questions concerning ‘lack of understanding’ because this aspect was already addressed after the pre-study by extending the time amount used for the explanation of CAT in the lecture and by providing detailed information about the instruments on the homepage of the course (p. 9). The ‘use of own methods’ was also excluded here and investigated in a separate study instead (Feudel, 2015). So the questionnaire contained in-depth questions concerning the reasons ‘time effort’ (a), ‘difficulties in working with the methods’ (b) and ‘perceived unhelpfulness’ (c). III. a) In-depth questions concerning ‘time effort’: In the pre-study many students stated ‘time effort’ as criticism about CAT (Table 5), although only a minority did actually use CAT (Table 4). Therefore, the authors conjectured that ‘time effort’ was more the students’ apprehension than their experience. Students might overestimate the time needed to use the instruments of CAT. This hypothesis was examined for the Toolbox by comparing the students’ time estimations to create a Toolbox with the time they would actually need. So, in the first step, the students were asked: How much time do you need to create a Toolbox? In the second step, they actually had to create a Toolbox for the problem Prove that for all x, y ∈ R the inequality xy ≤ |x||y| holds. and afterwards, to write down the time needed to do so. Furthermore, the students had to state how much self-study time they invested per week. This is the time outside class they spent for the revision of the lecture, the preparation for the tutorials and for work on written assignments that were administered weekly. According to the course program, the students are obligated to spend 6 h per week on these activities because they received credit for 90 h of self-study time in the semester, which comes down to 6 h per week, as the semester counts 15 weeks with classes. However, there was no control mechanism to check if they actually invested the time required. So the authors wanted to find out if the students already spent the full required amount of self-study time without using CAT. Otherwise, they could be expected to invest some additional time, which might be necessary at the beginning of the semester to get used to CAT. III. b) In-depth questions concerning ‘difficulties in working with the methods’: The students were asked explicitly about difficulties which came up during the tutorials. These were 1. Checklist Reading: inability to carry out certain steps of the checklist correctly 2. Traffic Lights: difficulties in identifying the right stage of understanding 3. Toolbox: difficulties to create a Toolbox or to solve a problem with a given Toolbox 4. Concept Base: difficulties to ‘fill out’ one or more categories (e.g. examples, visualizations, etc.) The corresponding items/tasks of the questionnaire are given in Table 6. The Likert scales in Table 6 rating the degree of agreement were comprised of 6 points (an even number) in order to push the students to take sides (agreeing or disagreeing). III. c) In-depth questions concerning ‘perceived unhelpfulness’: The students were asked to rate the level of perceived helpfulness for each instrument on a Likert scale ranging from 1 = ‘not at all helpful’ to 6 = ‘very helpful’. 30 F. FEUDEL AND H. M. DIETZ Table 6. Items/tasks concerning difficulties with the methods of CAT Instrument Items/tasks Answer format Checklist Task: ‘State for each step of the Checklist ‘Reading’ if Multiple choice with the options ‘able to ‘Reading’ you are able to perform it or not.’ perform’, ‘not able to perform’ and ‘step not known’ Task: ‘Carry out the steps ‘Spell’, ‘Play’ and ‘Animate’ Open answer format for each step of the of the Checklist ‘Reading’ for the set Checklist ‘Reading’ M := {x ∈ N|∃k ∈ N:x = 3k} and state in your own words what its elements are.’ Traffic Lights 7 items concerning perceived difficulties in 6-point Likert scale ranging from ‘do not at self-assessment (formulated in terms of the Traffic all agree’ to ‘totally agree’ Lights) Toolbox 4 items concerning perceived difficulties in creating a 6-point Likert scale ranging from ‘do not at Toolbox and 4 items concerning perceived difficulties in all agree’ to ‘totally agree’ working with a given Toolbox Task: ‘Create a Toolbox for the problem to prove that Open answer format for all x, y ∈ R the inequality xy ≤ |x||y| holds.’ Concept Base 5 items concerning perceived problems in creating a 6-point Likert scale ranging from ‘do not at Concept Base all agree’ to ‘totally agree’ 5.3 Description of the questionnaire administered in January 2015 The follow-up questionnaire administered in January 2015 intended to evaluate the changes made in the implementation of CAT that had been realized as a consequence of the survey in January 2014. Hence, it consisted of key items from the January 2014 questionnaire. In January 2015 the authors wanted to find out to what extent the proportion of the students using the instruments of CAT changed and how many students now did not use instruments of CAT due to the reasons for non-use that had been identified in January 2014 and addressed afterwards (‘time effort’, ‘difficulties in working with the methods’ and ‘perceived unhelpfulness’). Therefore, the follow- up questionnaire contained the items asking for the use frequency for each instrument of CAT and the multiple-choice questions, in which the students were asked to select, which instruments were not used because of ‘time effort’, ‘difficulties in working with the methods’, ‘lack of understanding’ and ‘perceived unhelpfulness’ (p. 10). The open questions asking for reasons for irregular use of the instruments of CAT were omitted because the goal in January 2015 was to measure changes and not to explore previously undetected reasons. 5.4 Data collection The questionnaires were administered at the end of the winter semester in January 2014 and in January 2015, both before the respective final exam. Due to its size, the January 2014 questionnaire was split up in two versions, each administered to about half of the students (N = 376 and N = 381). One version contained the in-depth questions concerning the Traffic Lights and the Toolbox, the other version contained the in-depth questions concerning the Checklist ‘Reading’ and the Concept Base. The questions concerning the self-study time spent and the use frequency of the instruments of CAT were given to all students (N = 757). 6. Results of the survey administered in January 2014 6.1 Results concerning the frequency of use of CAT The frequency with which the students used the instruments of CAT can be seen in Fig. 2. As shown there, the use frequency varied among the instruments. Altogether, the degree with which the instruments of TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 31 Fig. 2. Use frequency of the instruments of CAT in January 2014, 1 = ‘never’, 6 = ‘every week’ (N = 757). CAT were used was higher in January 2014 than in January 2012, which can be seen from a comparison of Fig. 2 with Table 4: in January 2012 over 50% clearly disagreed to use CAT regularly, in January 2014 at least the Vocabulary list was used regularly by the majority of the students (and other instruments were at least used sometimes). The increase in the frequency of the application of CAT can be explained by the integration of CAT into the tutorials since the winter semester 2013/14. Except for the Vocabulary list, however, the acceptance was still not satisfactory. 6.2 Results concerning the reasons why students did not use the methods of CAT The students were asked to select which instruments of CAT they did not use due to ‘time effort’, ‘difficulties in working with the methods’, ‘lack of understanding’ and ‘perceived unhelpfulness’. The results are shown in Table 7 (numbers over 25% are highlighted). While ‘Time effort’ and ‘Perceived unhelpfulness’ were selected as reasons for the non-use for the majority of the instruments, ‘lack of understanding’ and ‘difficulties in working with the methods’ were only selected often for the Toolbox or the Concept Base. It was surprising for the authors that, although the instruments were carefully introduced during the lecture and detailed information about their use was provided on the homepage of the course in 2013/14, ‘lack of understanding’ was still selected so often. The answers to the open question ‘If you do not use the instrument regularly, please state one or several reasons for that’ were categorized inductively for each instrument (the intercoder-reliability after Table 7. Proportion of students, who selected ‘time effort’, ‘difficulties in working with the methods’, ‘lack of understanding’ or ‘perceived unhelpfulness’ as reason for the non-use of instruments of CAT in January 2014, multiple answers were possible (N = 757) Checklist ‘Reading’ Traffic Lights Toolbox Vocabulary list Concept Base ‘Time effort’ 23.2% 32.3% 30.1% 12.8% 50.5% ‘Difficulties with the methods’ 13.2% 18.4% 32.3% 3.5% 41.1% ‘Lack of understanding’ 19.6% 24.3% 30.9% 3.6% 31.7% ‘Perceived unhelpfulness’ 27.0% 48.2% 21.9% 8.1% 26.8% 32 F. FEUDEL AND H. M. DIETZ Table 8. Reasons for not using the instruments of CAT (answers to the open-ended question to state a reason if an instrument is not used regularly) Checklist ‘Reading’ (N = 158) Traffic Lights (N = 221) Toolbox (N = 94) Concept Base (N = 241) not necessary 23.4% own methods 35.7% (time) effort 26.6% (time) effort 37.3% (time) effort 18.4% not necessary 28.1% own methods 25.5% difficulties 18.3% own methods 17.1% (time) effort 15.8% difficulties 23.4% own methods 18.3% not helpful 10.1% own markings 13.1% not necessary 14.9% vocabulary list enough 11.6% not known 7.6% lack of benefit 7.2% not helpful 6.4% lack of benefit 10.8% recoding by a tutor of the course was above 0.75 for all instruments). The most common categories are shown in Table 8 (all other categories contained less than 7% of the answers given). Similar to the data in Table 7, the data in Table 8 also show that ‘time effort’ was an important reason for the non-use for each of the instruments of CAT and the reason ‘difficulties in working with the instruments’ was a major reason applied to the Toolbox and the Concept Base. The perception that an instrument did not help was less often explicitly mentioned in the open questions. ‘Lack of understand- ing’ was very seldom mentioned in the open questions (under 5% for all instruments), which indicates that other reasons for the non-use of the instruments of CAT were of greater importance to the students. Three additional reasons besides the four reasons in Table 7 were stated in the open questions: ‘use of own methods’, ‘lack of necessity’ and ‘lack of benefit’. It was surprising that ‘lack of necessity’ was stated that often—contradictory to the students’ low grades in the following exam in 2014. This indicates that the students may have overestimated themselves. What the students meant by ‘lack of benefit’ was not clear. It might mean that they perceived that the instrument did not help at all. It could also mean that their use did not provide enough benefit in comparison to the required time. The latter assumption is also supported by the data: 58% of the students who stated ‘lack of benefit’ as reason for the non-use of the Concept Base also stated ‘time effort’. Summary of the reasons why students do not use the methods of CAT: Altogether, the main study showed that there were seven important reasons for the non-use of one or more of the instruments of CAT (see Tables 7 and 8): (1) Time effort (2) Lack of understanding (3) Difficulties in working with the methods (4) Perceived unhelpfulness of the methods (5) Use of own methods (6) Lack of benefit (7) No necessity for the methods (perceived) The quantity with which the reasons were stated (stated explicitly in the open questions and selected in the multiple-choice questions) varied among the instruments. The reasons ‘lack of benefit’ and ‘lack of necessity’ were not mentioned in the pre-study, but were newly discovered in the main study. Comparing these seven reasons to the reasons why students did not change their learning behaviour mentioned by Dembo & Seli (2004) in Table 3 (they cannot change, they do not know how to change, they do not know what to change, they do not want to change), most of the identified reasons why students TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 33 Table 9. Estimated time and time actually needed to create a Toolbox (N = 91) Mean Median SD Estimated time 9:42 5 13.094 Time actually needed 2:36 2 2.138 did not use CAT indicated that the students either did not want to change their learning behaviour (reasons 4–7, maybe reason 1 if students were not willing to invest time) or did not know how to change their learning behaviour (reasons 2–3). This result coincides with Dembo & Seli (2004). 6.3 Results of the in-depth questions and resulting consequences for the implementation of CAT in the following year The questionnaire contained in-depth questions concerning the reasons ‘time effort’ (I.), ‘difficulties in working with the methods’ (II.) and ‘perceived unhelpfulness’ (III.) (p. 11–12). In addition, some results concerning the reason ‘no necessity’ (IV.) can be concluded from the data of the survey. I. Results concerning ‘time effort’: The students sometimes overestimated the time needed to work with the instruments of CAT. This was demonstrated by the data in the case of the Toolbox by comparing the students’ estimations and the time the students actually needed to create a Toolbox for the problem to prove that for all x, y ∈ R the inequality xy ≤ |x||y| holds (Table 9). Including the students who did not create the requested Toolbox (N = 164), the estimated time to create a Toolbox was even higher (median: 10 min). Consequence: In the following winter semester 2014/15, students were obligated to work with the instruments in the tutorials at least once in order to adjust false time estimates. The data of the study also showed that the students invested far too little self-study time outside classes (for revision of the lecture, preparing for the tutorials and completing written assignments). According to the course program they were required to spend 6 h per week on these activities (p. 11). However, on average, they just spent half of that required time (Table 10). This evidence indicates that for many students more time spent on self-study might lead to improved results, provided that the self-study time was spent effectively using appropriate study methods such as those provided by CAT. Consequence: At the beginning of the winter semester 2014/15, students were explicitly told how much self-study time they ought to spend outside class each week and it was communicated that some additional time spent for getting used to the methods of CAT first may very well pay off later during Table 10. The amount of time in minutes spent by the students on weekly occurring activities in their self-studies (N = 714) Mean Median SD Review the lecture and revise the lecture notes 55.69 30 55.22 Preparation for the tutorial 68.03 60 45.18 Completing written assignments 78.06 60 55.06 Altogether 201.78 180 119.26 34 F. FEUDEL AND H. M. DIETZ Table 11. Perceived difficulties in working with the Toolbox, 1 = ‘do not at all agree’, 6 = ‘totally agree’(N = 376) Mean Median SD Perceived difficulties in creating a Toolbox (4 items) 3.62 3.5 1.19 Perceived difficulties in working with a given Toolbox (4 Items) 3.51 3.5 0.94 Table 12. Answers to the item ‘I have problems in choosing suitable definitions and theorems for the Toolbox’, 1 = ‘do not at all agree’, 6 = ‘totally agree’(N = 376) 1 2 3 4 5 6 Mean Median SD 3.8% 11.7% 19.9% 26.8% 25.1% 12.7% 3.96 4 1.331 the preparation for the exam. The authors hoped that the students were then more willing to invest some additional time for getting used to CAT. II. Results concerning ‘difficulties in working with the methods’: The answers to the in-depth questions concerning difficulties with instruments of CAT (Table 6) showed that the students had several difficulties. The results are only presented for the Toolbox due to the constraints of this paper. The students were asked to rate perceived difficulties in creating a Toolbox or in working with a given Toolbox to solve a problem (Table 6). The results are shown in Table 11.By looking at the items individually, the creation of a suitable Toolbox for a given problem was judged as especially difficult (Table 12). This was also supported by the Toolboxes created by the students for the problem to prove xy ≤ |x||y| for all x, y ∈ R with the properties of the absolute value given in the lecture (Table 6). Only two of the 233 students who wrote down a Toolbox collected both of the following items needed to solve the | | | | | || | problem: x ≤ x ∀x ∈ R, and xy = x y ∀x, y ∈ R. Consequence: Since the winter semester 2014/15, the difficulties identified in January 2014 have been addressed explicitly in the tutorials (e.g. strategies to choose and collect suitable items for the Toolbox were explicitly discussed). III. Results concerning ‘perceived unhelpfulness’ The data showed that the Checklist ‘Reading’ and the Traffic Lights were judged to be less helpful (Table 13). The correlations between the perceived helpfulness and the use frequency were over 0.6 for all instruments, which indicates that perceived unhelpfulness was a major reason for not using it. So, it is important to find out reasons for perceived unhelpfulness. In the case of the Checklist ‘Reading’ a careful analysis of the data revealed that insufficient knowledge about the details of the instrument might have caused the notion of its unhelpfulness. The students were asked for each step of the Checklist ‘Reading’ if they were able to perform it or not, or if they did not know it (see Table 6). The data surprisingly showed that many students did not know the steps ‘Animate’ and ‘Visualize’, which are important for gaining understanding of a mathematical concept (Table 14). Of course, if one thinks that the Checklist ‘Reading’ consisted only of ‘Spell’ and ‘Play’, one would of course judge it as unhelpful because these steps only help during the decoding process and do not help to get the meaning of the content. TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 35 Table 13. Answers towards the extent of perceived helpfulness by the instruments of CAT, 1 = ‘not at all helpful’, 6 = ‘very helpful’(N = 757) Mean Median SD Checklist ‘Reading’ (N = 381) 3.27 3 1.515 Traffic Lights (N = 376) 3.07 3 1.528 Toolbox (N = 376) 4.18 4 1.318 Concept Base (N = 381) 3.77 4 1.360 Table 14. The students’ability to perform the steps of the Checklist ‘Reading’ Step Able to perform Not able to perform Step not known ‘Spell’ (N = 352) 61.4% 19.0% 19.6% ‘Play’ (N = 352) 80.4% 7.4% 12.2% ‘Animate’ (N = 347) 15.9% 45.0% 39.2% ‘Visualize’ (N = 346) 28.3% 41.6% 30.1% ‘Talk’ (N = 347) 40.1% 29.7% 30.3% Consequence: In the winter semester 2014/15, these two steps were explicitly emphasized during the tutorials and had to be carried out in several written assignments explicitly. In the case of the Traffic Lights, however, the cause for the belief of its unhelpfulness could not be identified. Maybe students failed to see the benefit of rating their own understanding with three rather than two levels (fully understood = green; understood to some extent, but with a residual uncertainty = yellow; not understood = red as opposed to just ‘understood’ vs. ‘not understood’). This assumption is also supported by students’ statements in the open question why they did not use the Traffic Lights regularly like ‘half understood=pointless’. Last but not least ‘perceived unhelpfulness’ was also often selected as a reason for the non-use of the Concept Base (Table 7). Furthermore, ‘lack of benefit’ was often stated in the open question as a reason why this instrument was not used regularly (Table 8). Here the authors conjecture that the students lacked conditional knowledge concerning possibilities of use of the Concept Base. This is supported by students’ statements in the tutorials (‘no benefit for problem solving’). However, the authors believe that the Concept Base can indeed be helpful during the process of problem solving, e.g. as a knowledge base for creating Toolboxes. Possible future consequence: The usefulness of Concept Bases for problem-solving should be made more explicit during the tutorials. IV. Results concerning ‘no necessity’: ‘Lack of necessity’ was stated often in the open question why an instrument of CAT was not used regularly in the case of the Traffic Lights and the Checklist ‘Reading’ (Table 8). This surprised the authors and was in contradiction to the lecturer’s long-term experience. However, our data showed that many students overestimated themselves. Only 21.6% of the 37 students that stated ‘not necessary’ as a reason for the non-use of the Checklist ‘Reading’ (Table 8) gave a correct description of the set M := {x ∈ N|∃k ∈ N: x = 3k} in their own words (task originally included to identify students’ difficulties with the Checklist ‘Reading’, Table 6). 36 F. FEUDEL AND H. M. DIETZ In the case of the Traffic Lights, the data showed a similar result. 46% agreed with the statement ‘I can recognize my difficulties in understanding without the Traffic Lights’ on at least level 4 on a Likert scale from 1 = ‘do not at all agree’ to 6 = ‘totally agree’ (N = 381). However, 25% of them later agreed on level 6 to the statement ‘I have recognized gaps during the exam, which I had not recognized before’ (in a short survey directly after the exam, N = 295). In the whole population, only 15.9% agreed on that statement on level 6. However, since the term ‘gaps’ was not specified in the latter statement, the students could have referred to gaps in knowledge, gaps in understanding or gaps in procedural calculations. So the data only show that overestimation occurred, but not in which area. Possible conclusion: One possible conclusion would be to provide learning environment in which the students recognize their overestimation, e.g. by giving them tasks which they have to grade by themselves first and that are later graded by the tutors. Summary of the results of the in-depth questions from January 2014 and consequences concerning the implementation of CAT in the following year: With the help of the in-depth questions, it was possible to identify aspects that could be addressed when changing the implementation of CAT in the course. The findings (based on the data) and the resulting consequences concerning the implementation of CAT in the next year (winter semester 2014/15) are summarized in Table 15. Moreover, students’ answers to the open question ‘If you do not use the instrument regularly, please state one or several reasons for that’ like ‘half understood=pointless’ in the case of the Traffic Lights, as well as students’ statements in the tutorials like ‘no benefit for problem solving’ in the case of the Concept Base, lead the authors to conjecture that students might lack conditional knowledge when and why to use these instruments of CAT. That could cause a perceived lack of benefit. This conditional knowledge should be provided in the course in the future, e.g. by using the Concept Base as knowledge base for the creation of Toolboxes during the process of problem-solving. In the results presented here, one can clearly see that a lack of any of the three knowledge types in thesenseof Paris et al. (1983) (declarative knowledge, e.g. knowledge of the steps of the Checklist Table 15. Findings on the basis of the in-depth questions and consequences for the implementation of CAT in the next year in 2014/15 Findings based on answers given to the in-depth questions Activities concerning the implementation of CAT in the winter semester 2014/2015 Overestimation of the time needed to work with the Obligation of students use the instruments in the tutorials at instruments of CAT (here shown for the Toolbox, Table 9) least once to reduce their apprehensions concerning time effort Not enough self-study time spent by the students for Clear communication of the curriculum’s workload learning activities outside classes (Table 10) demands concerning the self-study time at the beginning of the semester Problems of the students in working with the methods of Addressing these difficulties during the tutorials explicitly CAT (here proved for the example ‘creating a Toolbox’, (e.g. practising the creation of Toolboxes) Table 12) Insufficient knowledge about the details of the instruments Provision of exercises after the introduction of the (here shown for the steps ‘Animate’ and ‘Visualize’ for instruments addressing aspects that have been identified to Checklist ‘Reading’, Table 14) be not well known Students’ overestimation of their own abilities (here shown Not addressed yet for the Checklist ‘Reading’ and the Traffic Lights, p. 19) TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 37 Fig. 3. Use frequency of the instruments of CAT in January 2014 (N = 757) and January 2015 (N = 710), 1 = ‘never’, 6 = ‘every week’. ‘Reading’, procedural knowledge, e.g. knowledge about strategies to create a Toolbox and conditional knowledge, e.g. knowledge about the possibilities to use the Concept Base during problem-solving) can lead to the non-use of the methods provided by CAT. Students need to know about the strategies themselves in detail, they should learn how to apply them and have to know when and why to use them. 7. Results of the survey in January 2015 to evaluate changes in the implementation of CAT The changes in the implementation of CAT in the course that were based on the results of the survey in January 2014 raised the frequency with which the students used of most of the instruments of CAT (Fig. 3). The number of students who used the instruments was much higher in January 2015 than 1 year before. It was now mostly satisfactory, except for the Traffic Lights and the Concept Base (Fig. 3). The strongest positive impact on the acceptance of CAT was achieved by addressing the difficulties during the tutorials. This holds particularly for the Checklist ‘Reading’, where special emphasis was directed to the step ‘Animate’, and for the Toolbox, where the creation of Toolboxes was practised (Table 16). The authors conjecture that if the students have fewer problems when working with the methods, their work is more often successful. This would explain why the two mentioned instruments were also less often not used because of ‘perceived unhelpfulness’ (Table 17). Table 16. Amount of students not using instruments of CAT because of ‘difficulties when working with them’ in January 2014 (N = 757) and January 2015 (N = 710), multiple answers were possible Checklist ‘Reading’ Traffic Lights Toolbox Vocabulary list Concept Base 2014 (N = 757) 13.2% 18.4% 32.3% 3.5% 41.1% 2015 (N = 710) 5.7% 22.7% 10.2% 5.7% 34.0% 38 F. FEUDEL AND H. M. DIETZ Table 17. Amount of students not using instruments of CAT because of ‘perceived unhelpfulness’in January 2014 (N = 757) and January 2015 (N = 710), multiple answers were possible Checklist ‘Reading’ Traffic Lights Toolbox Vocabulary list Concept Base 2014 (N = 757) 27.0% 48.2% 21.9% 8.1% 26.8% 2015 (N = 710) 17.0% 61.3% 9.0% 9.8% 18.7% Table 18. Amount of students not using instruments of CAT because of ‘time effort’ in January 2014 (N = 757) and January 2015 (N = 710), multiple answers were possible Checklist ‘Reading’ Traffic Lights Toolbox Vocabulary list Concept Base 2014 (N = 757) 23.2% 32.3% 30.1% 12.2% 50.5% 2015 (N = 710) 13.9% 25.0% 13.9% 18.7% 44.5% Informing the students about the amount of self-study time they were required to spend also had a positive effect because, ‘time effort’ was less often selected as a reason for the non-use of the instruments in January 2015 except for the Vocabulary (Table 18). 8. Conclusion and discussion The research project aimed at raising the acceptance of CAT and at improving the students’ study skills by using CAT. Thus it focused on the following research questions: 1. How many students actually use the strategies provided by CAT regularly? 2. What are the reasons that students do not use the strategies provided by CAT? 3. What activities can increase the number of students using the provided strategies? As to 1.: At the beginning of the research project, the strategies provided by CAT were only used by a minority of students. After changes in the implementation of CAT based on the results of the surveys, the acceptance of CAT’s strategies was raised considerably. As to 2.: By asking the students directly for reasons why they did not use the instruments of CAT, seven reasons were discovered: (1) ‘time effort’, (2) ‘lack of understanding’, (3) ‘difficulties in working with the methods’, (4) ‘perceived unhelpfulness of the methods’, (5) ‘use of own methods’, (6) ‘no necessity of the methods’ and (7) ‘lack of benefit’ (p. 15). With the help of the in-depth questions it was possible to identify concrete problems causing the above mentioned reasons that could be addressed in the next year. These were: (a) overestimation of the time needed for working with instruments of CAT, (b) not enough time investment for self-studies outside class, (c) difficulties in working with instruments of CAT (that could be further specified), (d) insufficient knowledge about details of CAT and (e) students’ overestimation of their own abilities (Table 15). As to 3.: The problems (a)–(d) (based on the answers of the in-depth questions) led to changes in the implementation of CAT in the following year (Table 15). These changes raised the acceptance of some of the instruments of CAT considerably (Fig. 3). The most important consequence was to integrate the methods of CAT into all parts of the course, especially into the tutorials and written assignments in order to (1) provide detailed knowledge about the methods and to practise them, (2) to address difficulties the students had when working with the methods and (3) to adjust the students’ estimation of the time TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 39 needed to apply the methods by obligating them to use the instruments at least once. In the case of the Toolbox and the Checklist ‘Reading’ the positive impact of that integration can be seen in the data of the study. The exercise of the creation of Toolboxes (p. 17) and the exercise of the steps of the Checklist ‘Reading’ on written assignments (p. 18) lowered the proportion of students not using these instruments because of difficulties in working with them considerably (Table 16). The authors conjecture that these activities also lowered the number of students who perceive these two instruments as unhelpful because if students have fewer difficulties with the instruments their work with the instruments will more often be successful. For an effective integration of the methods of CAT into the tutorials, however, the tutors had to be trained carefully, though. The workshop for new tutors at the beginning of the semester and the weekly meetings during the semester, in which the practical implementation of CAT into the tutorials were discussed, were essential for that training. In addition, informing the students about how much self- study time they were required to spend for the course also helped to reach some students that had not been willing to spend much time on their self-studies beforehand (because ‘time effort’ was less often mentioned as reason for rejection for instruments of CAT in 2015, Table 18). Two major problems remain. One is the students’ overestimation of their own abilities, which was pro- ven by the data of our study. Students overestimating themselves do not feel a need to use strategies provided in support programs (Dembo & Seli, 2004). Nowell & Alston (2007) showed that overestima- tion is a general problem with students of economics across the whole curriculum. Grimes (2002),for example, pointed out that even after a test many students of economics assumed to have achieved a grade better than it actually was. Kruger & Dunning (1999) proved for several disciplines (not only economics) that students who perform poorly are the ones who overestimate their own abilities the most. A solution could be to provide a learning environment, in which self-testing is practised. The students should experience actively how they can monitor and address comprehension failures with the help of the Traffic Lights in the course in several settings: when reviewing the lecture, when justifying algebraic calculations and when reading mathematical texts. However, up to now, the students were advised to do so only in the lecture. There was no explicit training because the lecturer and second author thought before the study that the instructions he gave during the lecture concerning the use of the Traffic Lights were very clear to the students and the necessity to judge one’s own understanding carefully would be obvious to the students. But the present study shows clearly a need for an explicit and guided training regarding the use of the Traffic Lights for self-evaluation during the tutorials in order to get the students to realize the necessity of a careful self-evaluation and the benefit of the Traffic Light system. Providing an active training can be expected to improve self-evaluation strategies more than just giving advice (similar to Bielaczyc et al. (1995)). The training should involve the self-test to recall a definition from memory and then to compare it with the source (already mentioned on p. 5). This self-test provides an external standard, which increases the probability that the self-evaluation is done correctly (Rawson & Dunlosky, 2007). In this setting students can also recognize their tendency to overestimate their skills more easily. Later, one should proceed to more complicated self-tests like the creation of a Concept Base from memory. Besides these self-tests, activities to remediate comprehension failures should be included in the training as well (e.g. reading the explanations in the corresponding chapter in the course textbook (Dietz, 2012)). The second problem was that some students did not see the benefit of certain instruments of CAT. Concerning the Traffic Lights, for example, students commented in the January 2014 questionnaire that they did not see the benefit of rating their understanding in three, rather than in two, levels. Regarding the Concept Base, students stated in the tutorials that it did not yield any benefit for problem-solving. In both cases, the students did not see the benefit of applying these two instruments. If students do not see the benefit of changing their learning behaviour, they will not change it especially if the new learning 40 F. FEUDEL AND H. M. DIETZ behaviour is time consuming like creating a Concept Base (Jakubowski & Dembo, 2004). Hence, the tutorials should direct much more attention to making the benefit of the instruments clearly visible. One example would be to use the Concept Base as a knowledge base to create Toolboxes during the process of problem-solving. A limitation of the study is that it is not clear if the students who adopted the methods of CAT will keep using them after the exam or if they return to their old methods. Changing one’s learning behaviour is not completed after the new methods have been acquired. They have to be maintained until they are actually adopted (Jakubowski & Dembo, 2004; Prochaska & Prochaska, 1999). To investigate if the study skills program CAT has a long term effect, i.e. if the acquired methods are maintained by the students, further data has to be collected after the exam or at the end of the mathematics courses, which would be almost one year after the introduction of the methods of CAT. In conclusion, one can say that although the application of the strategies provided by CAT had a positive effect on academic achievement (Dietz, 2012), the implementation of CAT into the whole course faced the problem that the majority of students did not use the methods of CAT first. Many reasons for the non-use of the strategies (that could occur in similar study skills programs as well, see for example Dembo & Seli (2004)) were identified in the research project. With the careful introduction of the concept CAT and the integration of it into all components of the course (lecture, tutorials and written assignments) most of the problems were remediated (except for the problem of overestimation of their own abilities and a perceived lack of benefit by some students) and the provided methods became widely accepted among the students. So the study presented here does not only confirm that the embedding of taught methods and strategies into the learning environment is important (Anthony, 1996; Garner, 1990), but also provides practical tips, how to accomplish a successful implementation of a study skills program into a large first semester service mathematics courses (with the limitation that it is not clear if the acquired skills will be maintained by the students). 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Sci., 3, 425–453. Dembo, M. H. & Seli, H. P. (2004) Students’ resistance to change in learning strategies courses. J. Dev. Educ., 27, 2. Dietz, H. M. (2012) Mathematik für Wirtschaftswissenschaftler: Das ECOMath-Handbuch. Berlin, Heidelberg, Germany: Springer. Dietz, H. M. (2015) Semiotics in “Reading Maths”. Semiotische Perspektiven auf das Lernen von Mathematik (G. Kadunz, ed.) Berlin, Heidelberg: Springer. Dietz, H. M. (2016) CAT–ein modell für lehrintegrierte methodische unterstützung von studienanfängern. Lehren und Lernen von Mathematik in der Studieneingangsphase. Wiesbaden, Germany: Springer. Dietz, H. M. & Rohde, J. (2012) Studienmethodische unterstützung für erstsemester im mathematikservice. Beiträge zum Mathematikunterricht 2012, Münster, Germany: WTM. Feudel, F. (2015) Studienmethodische förderung in der mathematik für wirtschaftswissenschaftler – Chancen und Schwierigkeiten. Beiträge zum Mathematikunterricht. Münster, Germany: WTM. Garner, R. 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Y. & Wixson, K. K. (1983) Becoming a strategic reader. Contemp. Educ. Psychol.,8, 293–316. Pintrich, P. R. (2004) A conceptual framework for assessing motivation and self-regulated learning in college students. Educ. Psychol. Rev., 16, 385–407. 42 F. FEUDEL AND H. M. DIETZ Polya, G. (2014) How to Solve It: A New Aspect of Mathematical Method. Princeton, New Jersey, USA: Princeton University Press. Prochaska, J. & Prochaska, J. (1999) Why don’t continents move? Why don’t people change? J. Psychother. Integration, 9, 83–102. Rawson, K. A. & Dunlosky, J. (2007) Improving students’ self-evaluation of learning for key concepts in textbook materials. Eur. J. Cogn. Psychol., 19, 559–579. Richardson, M., Abraham, C. & Bond, R. (2012) Psychological correlates of university students’ academic performance: a systematic review and meta-analysis. Psychol. Bull., 138, 353–387. Schoenfeld, A. H. (1992) Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. Handbook of Research on Mathematics Teaching and Learning (D. A. Grouws, ed). New York, USA: Macmillian Publishing Co. Schraw, G. & Moshman, D. (1995) Metacognitive theories. Educ. Psychol. Rev., 7, 351–371. Shepherd, M. D., Selden, A. & Selden, J. (2012) University students’ reading of their first-year mathematics textbooks. Math. Thinking Learn., 14, 226–256. Shepherd, M. D. & Van De Sande, C. C. (2014) Reading mathematics for understanding-From novice to expert. J. Math. Behav., 35, 74–86. Tall, D. & Vinner, S. (1981) Concept image and concept definition in mathematics with particular reference to limits and continuity. Educ. Stud. Math., 12, 151–169. Taylor, J. A. & Mander, D. (2003) Developing study skills in a first year mathematics course. New Zealand J. Math., 32, 217–225. Voßkamp, R. & Laging, A.(2014) Teilnahmeentscheidungen und Erfolg. Mathematische Vor- und Brückenkurse: Konzepte, Probleme und Perspektiven (I. Bausch, R. Biehler, R. Bruder, R. P. Fischer, R. Hochmuth, W. Koepf, S. Schreiber & T. Wassong, eds). Wiesbaden: Springer Fachmedien Wiesbaden. Wadsworth, J., Husman, I., Duggan, M. A. & St Pennington, M. (2007) Online mathematics achievement: effects of learning strategies and self-efficacy. J. Dev. Educ., 30, 6–14. Wingate, U. (2006) Doing away with ‘study skills’. Teach. Higher Educ., 11, 457–469. Frank Feudel is a Ph.D. student in the field of higher mathematics education at the Centre for Higher Mathematics Education (khdm) in Germany since 2013. E-mail: [email protected]. Dr. Hans M. Dietz is a professor in mathematics and lecturer of the course ‘Mathematics for students of economics’ at the University of Paderborn for many years. E-mail: [email protected]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png :Teaching Mathematics and Its Applications: An International Journal of the IMA Oxford University Press

Teaching study skills in mathematics service courses—how to cope with students’ refusal?

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Teaching Mathematics and Its Applications (2019) 38,20–42 doi:10.1093/teamat/hrx016 Advance Access publication 28 December 2017 Teaching study skills in mathematics service courses—how to cope with students’ refusal? †,∗ ‡ Frank Feudel and Hans M. Dietz Department of Mathematics, Humboldt-University of Berlin, Germany and Department of Mathematics, University of Paderborn, Germany Corresponding author. Email: [email protected] [Submitted 21 April 2017; accepted 07 November 2017] Many freshmen entering university have difficulties with finishing their mathematics courses successfully in the first year due to insufficient study skills and learning strategies, which on the one hand include general skills and strategies and on the other hand context-specific skills and strategies for university mathematics. In mathematics service courses for students of disciplines other than mathematics, for example engineering or economics, the effect of poor study skills is even worse due to large course sizes and great diversity regarding the students’ prior knowledge. Therefore, since 2010, first-year students of economics at the University of Paderborn are offered a methodological support program named CAT, which is integrated into the regular course context. The name CAT emerges from three important support-tools of the program (called instruments): Checklist, Ampel (English: Traffic Lights), Toolbox. Although some students claimed that CAT was helpful from the beginning (and a positive effect on academic achievement had also been proven in an experimental tutorial) the majority of students did not use the strategies provided by CAT at first. This was the initial point for a research project with the aim to raise the acceptance of CAT and thus to get more students to use the effective study skills provided by CAT. With the help of this research project, it was possible to identify reasons why students did not use the skills and strategies of CAT, to draw adequate conclusions concerning the implementation of CAT in the course and to raise the acceptance of CAT considerably in the following year. The results of the research project can be useful especially for developers of similar support programs and for teachers and lecturers, who also try to integrate study skills into mathematics courses at university. 1. Introduction Students face many problems when entering mathematics courses at university for the first time. One of the reasons for failure, as is widely believed by academic staff, is the lack of study skills and learning strategies (Anthony, 2000). This includes, on the one hand, general self-regulatory skills like time- management, self-control or effort regulation and on the other hand more context-specific skills and © The Author(s) 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 21 strategies like problem-solving, working with abstract mathematical concepts or rigorous reasoning based on formal definitions. There is a lot of empirical evidence that self-regulatory study skills influence the success in academic learning in general as well as in mathematics courses in particular. For example, Richardson et al. (2012) or Credé & Kuncel (2008) found by means of meta-analyses that self-regulatory study skills have a positive influence on academic achievement, represented by the students GPA (Grade- Point-Average). Furthermore, many interventions targeting students’ study skills were proven to be successful (Hattie et al., 1996), particularly if embedded in the standard course context of the discipline. In addition, several successful interventions promoting students’ study skills in tertiary mathematics education can be found in literature (Cornick et al., 2015; Taylor & Mander, 2003; Mireles et al., 2011; Wadsworth et al., 2007). Supporting study skills can even compensate lack of prior knowledge (Cardelle-Elawar, 1992). For students of economics the effects of poor study skills probably weigh even heavier than for students enrolled in mathematics due to their diverse prior knowledge in mathematics from school (e.g. Voßkamp & Laging (2014)) and due to large course sizes of over 1000 students. Thus, integrating study skills into a mathematics service course like ‘Mathematics for students of economics’ might be a promising approach to lower failure rates in exams and to raise academic achievement. This judgement coincides with the second author’s long-run teaching experience. Accordingly, first-year students attending the course ‘Mathematics for economists’ at the University of Paderborn are offered a special support program named ‘CAT’ (from Checklist, Ampel (English: Traffic Lights) and Toolbox) to improve their study skills. CAT’s main concern is to enable the students to achieve a full understanding of the taught matter on their own, in particular, by enabling them to properly ‘read’ and understand mathematics. CAT was initiated by the second author and fully integrated in the regular mathematics course context since 2010. Some students claimed CAT to be helpful from the very beginning. Furthermore, a positive effect of CAT on academic achievement was shown in an experimental tutorial group (Dietz & Rohde, 2012). However, the integration of the program into the large course faced one major problem: about 80% of the students did not use the provided strategies and did not change their learning behaviour (observation by the second author as course lecturer). Therefore, since 2011, a research project named ‘Support in Learning Strategies in Mathematics for Students of Economics’ was established at the ‘Centre for Higher Mathematics Education’, Germany (www.khdm.de). The project aimed to raise the acceptance of CAT and the use of its methods by the students, which would probably result in better exam results, just like it did in the experimental tutorial group. In particular, the purposes of the project were to identify reasons why many students did not use the strategies provided by CAT and to explore what activities can increase the number of students using these strategies. In the first—theoretical—part of the paper, a survey of the CAT concept and its implementation in the aforementioned course will be presented, as well as a literature review with regard to the question of why students do not change their learning behaviour (relevant for the question why students do not use CAT). The second part contains the results from the empirical research project. The emphasis lies on the reasons for the non-use of the strategies provided by CAT, the drawn conclusions and the resulting activities that have led to a greater acceptance of CAT in the following year. 2. The concept CAT—a course-integrated program to promote study skills and strategies The methodological support program CAT consists of several closely connected instruments: Checklists, Ampel (English: Traffic Lights), Toolbox, Vocabulary list and Concept Base, explained in more detail below (for a more thorough description see Dietz (2016)). Each of these instruments ought to support the students in several domains of their study process. 22 F. FEUDEL AND H. M. DIETZ Table 1. Steps of the Checklist ‘Reading’ Step Description Example: A × B 1. ‘Spell’ The students should assure themselves that they The students must know the meaning of every single know the meaning of every unit precisely. (These symbol, for example, ‘:=’ as a defining sign or ‘{}’ units can, for example, be symbols in a as set braces. They have to be aware particularly that mathematical expression or already known notions some symbols may have multiple meanings like ‘|’. in a new definition. As such, they can be found in the students’ Vocabulary list.) 2. ‘Play’ The students shall try to fit the units together in a ‘A times B is defined as the set of all ordered pairs spoken sentence, as if they were to read the (a,b) with a being an element of A and b being an expression out aloud. element of B.’ 3.‘Animate’ The students ought to provide examples (and Example: non-examples). A = {1, 2} , B = {3} ⇒ A × B = {(1, 3) , (2, 3)} 4.‘Visualize’ The students should try to create a visualization The given example in Step 3 would result in two fitting the definition, if appropriate. points in the plane. A good example for visualizing the cross product can also be the cross product of two closed intervals. 5. ‘Talk’ The students should try to explain the concept behind the expression to someone else (including possible questions). 2.1 A brief description of the six CAT instruments (1) A core instrument of CAT is the Checklist ‘Reading’, as reading mathematics has proven to be a big problem in tertiary mathematics education. Even students who have good reading skills in general and who are not bad at mathematics seem to have difficulties with reading mathematical texts (Shepherd et al., 2012). As a remedy, the Checklist ‘Reading’ provides a list of activities that students ought to perform in order to gain understanding when reading mathematics (e.g. a mathematical expression or a definition). When doing so, they ought to refer to their Vocabulary list,which contains all symbols introduced in the course (explained in detail below). The recommended steps of the Checklist ‘Reading’ are illustrated by the definition A × B := {(a, b) |a ∈ A, b ∈ B} (Table 1). The literature shows that the steps of the Checklist ‘Reading’ are consistent with the activities usually performed by mathematical experts when reading mathematics. In a case study, Shepherd & van de Sande (2014) documented the way in which three students at graduate level and three math- ematicians tried to read a mathematical text for understanding. They were asked to read aloud a chapter about manifolds in a differential geometry book with the aim to learn about the content. All steps in Table 1 except for the last step ‘Talk’ can be found in this study. The students, for example, asked for definitions of homeomorphism and topology which the text did not explain. They tried to ensure they understood the meanings of the units of the text (Step 1). The step ‘Play’ can mainly be observed in the reading process of the mathematicians, when they tried to skim familiar passages and replaced the names of symbols with the names of the underlying concepts (for example quotient space) to achieve higher fluency. A major difference between the students and the mathematicians was that the latter generated examples to ensure understanding (e.g. examples for charts to show that the projective plane given in the book really is a two-dimensional submanifold of R ) (Step 3). The students, however, were more likely to use illustrations, for example when trying to make sense of the theorem about the map composed of a chart and the inverse of another chart being a diffeomorphism (Step 4). TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 23 Österholm (2006) additionally showed that special attention has to be paid to mathematical symbols. In the study, two groups of students were asked to answer questions on a text about the group concept they just read. One group had a text with symbols; the other had a text without symbols. The students having read the text with symbols understood less while having the same prior knowledge as the students from the other group. This suggests that beginners should pay special attention to the first step ‘Spell’ in the Checklist ‘Reading’. (2) One reason why students cannot work with mathematical concepts when trying to solve mathe- matical problems is that they do not know a (verbalized) definition of the concepts (Moore, 1994). As mentioned before, the students are encouraged to maintain a Vocabulary list consisting of all concept and symbol definitions introduced in the course. (3) To support the development of adequate concept images in the sense of Tall & Vinner (1981) the students should extend the vocabulary entries with examples, counterexamples, visualizations, state- ments involving the concept like formulas, theorems or connections to other concepts and applications. This extension is called Concept Base. The students are told in the lecture that they must know these categories for each concept introduced in the course. They can download a template from the homepage of the course containing these categories (but they do not have to use it). A detailed description of the Concept Base is provided in Dietz (2015) or Dietz (2012). By connecting different Concept Bases, the students should develop a Concept Net. The category ‘statements involving the concept’ of the Concept Base should help to link a given concept to other concepts. However, to interconnect different concepts simultaneously, more activities like creating concept maps would be necessary (Novak & Cañas, 2008). An implementation of such activities in CAT has not taken place yet because many students did not even have a profound understanding of the individual concepts itself, which form the basis of the Concept Net. So the emphasis was put on the understanding of these first (it is intended to change this in the future). (4) The Toolbox is supposed to support the students in the process of problem-solving. It is well-known that many students do not know where to start when trying to solve problems (for example, where to start proofs (Moore, 1994)). The idea of the Toolbox is to collect all relevant information, which could be helpful in order to solve the problem. For a proof problem this would include especially relevant definitions, theorems and formulas as well as the relevant preconditions in the task. For example, if the T −1 −1 T task was to prove that for an invertible matrix A its transpose is also invertible with (A ) = (A ) (typical proof task in the exam at the end of the course), the Toolbox should contain the definition of the −1 −1 inverse matrix, the precondition AA = I = A A and formulas involving the transpose of a product of two matrices. The students should then try to use these tools (one after another) to solve the problem. This ‘Toolbox principle’ could at best become an intentional strategy in the development of the students’ own proofs. The advice of first collecting relevant information (especially theorems, definitions and formulas) is also mentioned in literature of problem-solving (Bruder & Collet, 2011; Polya, 2014). Such a collection can be useful in all phases of problem-solving: collecting the tools can help devising a plan, using these tools can help to structure the execution of the plan and checking the tools used to solve the problem can help to check the solution (e.g. by checking if the precondition in a proof was used). In the course the emphasis was put on using the Toolbox in the first two phases and especially the first phase of problem-solving because it was observed in the tutorials that students failed in problem-solving very often because they did not know how to start. (5) The Traffic Lights aim to support cognitive monitoring (Greene & Azevedo, 2007) because it was observed that students often failed in the exam although they had felt well prepared before. This phenomenon is documented in the literature as illusion of knowing (Glenberg et al., 1982). To prevent it, the students are asked to mark parts of their lecture notes according to three levels of understanding 24 F. FEUDEL AND H. M. DIETZ (fully understood = green; understood to some extent, but with a residual uncertainty = yellow; not understood = red). Of course, this is to be taken in a metaphoric sense and not literally. Controlling one’s own understanding is also important when reading mathematical texts (Shepherd & van de Sande, 2014), justifying steps in algebraic calculations (Cohors-Fresenborg et al., 2010) and problem-solving (Schoenfeld, 1992). So students are advised in the lecture to use the Traffic Lights in these settings, i.e. while working through the steps of the Checklist ‘Reading’, performing algebraic calculations or solving problems. To evaluate their own understanding, students are provided with standards in the lecture, which is important (Greene & Azevedo, 2007). For stepwise procedures like algebraic calculations, the students are told that they must be able to justify every step by ‘citable’ knowledge from the course. For mathematical concepts the standards are set in terms of the Concept Base: the students ought to be able to provide the definition, examples, visualizations and so on. The Concept Base can be used as a self-test as well, with the requirement that its elements should be recalled from memory. An easy test for conceptual understanding proposed in the course is to recite the definition from memory and compare the result with the source. If the test fails, the concept is not fully understood (a positive test, however, does not guarantee understanding). Cognitive monitoring also involves activities that deal with the results of these evaluations (Pintrich, 2004), which are also recommended in the lecture. If the result of the evaluation is not positive (i.e. in terms of the Traffic Lights that ‘the colour is not green’), the students are recommended to resolve their comprehension failures as soon as possible. Options for resolution proposed in the course are using other strategies of CAT to gain understanding (e.g. creating Concept Bases), reading the explanations in the corresponding chapter in the course textbook (Dietz, 2012) or asking the tutors. The students are advised to remediate ‘yellow lights’ first (yellow = ‘understood to some extent, but with a residual uncertainty’) because these issues can be sources of many subsequent problems, misunderstandings and misconceptions. A concept built on another concept, that has not been fully understood, cannot be fully understood either. Therefore, starting with aspects that are somehow, but not fully, understood is reasonable (this is the special benefit of using three degrees). 2.2 Implementation of CAT in the course CAT has been integrated into the regular course context as literature strongly recommends (Hattie et al., 1996; Wingate, 2006). Initially, in 2010, ‘integration’ meant that the methods of CAT became explicit parts of the lectures, during which they were explained and exemplified stepwise. But the research project presented below indicated clearly the need of an integration of CAT in all parts of the course (lecture, tutorials and written assignments) with the result that students can decide about using the methods of CAT or not on the basis of their own experience. As a result, since winter semester 2013/14, the instruments of CAT have been introduced in several consecutive steps. These are 0. Motivation of the instruments in the lecture by showing the students that they often fail with their own methods 1. Introduction of the instruments during the lecture and demonstration by the lecturer in which ways the instruments can lead to success 2. Activities during the tutorials in which the students are asked to solve tasks together with the tutors by using the instruments 3. Tasks on written assignments, in which the students are explicitly asked to use the instrument (and which the students should hand in weekly) 4. Tasks in the tutorials and on written assignments involving the instrument implicitly TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 25 Table 2. Steps of the implementation of the instruments of CAT for the example of the Checklist ‘Reading’ Step Realization for the Checklist ‘Reading’ in winter 2014/15 0. Motivation of the instruments The students are asked to state in own words what the elements of the set M := {x ∈ N|∃y ∈ N:y = 3x} are (week four in the semester). 1. Introduction of the instruments The steps of the checklist ‘Reading’ were introduced and demonstrated by the during the lecture example of the definition of the Cartesian product A × B := {(x, y) |x ∈ A, y ∈ B} (week four of the semester). 2. Activities with the instruments The students had to carry out the steps of the checklist for the definition of during the tutorials divisibility n|z :⇐⇒ ∃q ∈ N:z = qn with help of the tutors (week six of the semester). 3. Tasks on written assignments The students had to identify the set T := {x ∈ N| x|18} in a task structured by asking explicitly to employ the the steps of the Checklist ‘Reading’ (week seven of the semester). instruments 4. Tasks involving the instruments A sample task involving the Checklist ‘Reading’ implicitly was to find out if the implicitly in the following weeks relation a  b : ⇔ a|b ∨ b|a is reflexive, transitive, symmetric, antisymmetric or complete (week eight in the semester). To solve the task students have to try to make sense of the given definition first, which requires ‘mathematical reading’. These steps are similar for all the instruments (except the Traffic Lights, which are only motivated and introduced in the lecture so far). They are explained in detail for the Checklist ‘Reading’ in winter 2014/15 as a typical example in Table 2. The activities in the tutorials are, in the opinion of the authors, of major importance for the students’ recognition of the benefit of the instruments. Here they can observe how tutors and peers succeed in using them. Furthermore, they can experience themselves this success by solving tasks with the help of the tutors. As can be seen in Table 2 the degree of obligation concerning the use of the instruments in the course decreases over time. The introduction and the activities in the tutorials can be seen as mandatory for the students (passing the exam is impossible without attending the lecture and the tutorials). The tasks on the weekly written assignments are not mandatory because their completion is not a prerequisite for the exam. Tasks involving the instruments implicitly do not prompt to actually use the methods provided by CAT. Furthermore, the degree of explicitness also decreases (see Table 2). Although the degrees of obligation and explicitness with respect to the use of the instruments decrease with time after their introduction, the instruments are used or at least mentioned repeatedly by the lecturer and the tutors throughout the semester in order to remind the students of their presence. 3. Literature review concerning students’ resistance to change their study behaviour Although successful programs to promote study skills and strategies are offered in many universities, many students still do not participate in these support programs unless they are obliged to do so (Dembo & Seli, 2004). A useful framework for analysing this phenomenon is provided by Prochaska & Prochaska (1999). According to this framework, there are four reasons why people do not change their behaviour (Table 3). This framework was, for example, used by Dembo to identify reasons why some students did not change their learning behaviour after two thirds of a ‘learning-to-learn’ course given at a university in the USA in 2003. Similar reasons for not using learning strategies are given by Garner (1990). She 26 F. FEUDEL AND H. M. DIETZ Table 3. Reasons for students not to change their learning behaviour Reason Example They cannot change. Students are not aware of automatized strategies. They do not know what to change. Students recognize their difficulties (for example problems with beginning a proof), but cannot do anything about it. They do not know how to change. Students know that self-testing is useful for checking their knowledge but do not know which self-tests to use. They do not want to change. Students do not see the benefit of putting the effort into changing their learning behaviour, often because their strategies worked well at school. additionally mentions the learning environment as an important factor and also emphasizes that the integration of the strategies into the (course) context is important to get students to use them. Concrete reasons why students do not use strategies in the subject of mathematics were identified by Anthony (1996) in case studies with students in New Zealand during their last year at school. The reasons were: failure to realize that a problem exists, inadequate strategic knowledge regarding the identified problems, failure to apply the strategic knowledge although the students have it, the students’ decision not to apply learning strategies and a learning environment that is not supportive of strategy use. The first four reasons correspond well with the reasons in the framework of Prochaska & Prochaska (1999). In addition, Anthony again mentions the importance of the learning environment. As demonstrated by the aforementioned reasons, students basically need three types of knowledge to apply learning strategies successfully (Paris et al., 1983): they need to know what learning strategies they can use (declarative knowledge), how to use them (procedural knowledge) and when and why to use them (conditional knowledge). The conditional knowledge plays a particular role in adult learning (Schraw & Moshman, 1995). Justice & Weaver-Mcdougall (1989), for example, showed that adults mainly used strategies for memory tasks, which they judged to be effective. Anyway, if students lack in any of the three types of knowledge, they might choose not to employ the strategies taught. 4. Description of the project ‘Support in Learning Strategies in Mathematics for Students of Economics’ The research project started in 2011 with the aim of getting more students to use the strategies provided by CAT and thus to improve the students’ study skills. 4.1 Research questions of the project 1. How many students actually use the strategies provided by CAT regularly? 2. What are the reasons that students do not use the strategies provided by CAT? 3. What activities can increase the number of students using the provided strategies? 4.2 Chronology of the project The project consisted of several studies, which were conducted between 2011 and 2015. The results presented here originate from a pre-study that was conducted from 2011 till 2012 and the main study, conducted between 2013 and 2015. In this paper, the main study is presented in detail, while the presentation of the pre-study is restricted to major findings which influenced the main study. TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 27 Table 4. Answers to the item ‘I use CAT regularly’from a questionnaire administered in January 2012, 1 = ‘do not at all agree’, 4 = ‘totally agree’(N = 851) 1 2 3 4 Mean Median SD 57.9% 31.6% 9.3% 1.2% 1.54 1 0.711 Table 5. Possible reasons for the non-use of CAT ranked by the number of statements occurring in the answers to the question ‘What do you criticize about CAT?’(N = 851) Possible reason for the non-use of CAT Number of statements 1. (Time) effort of the provided methods 65 2. Use of own methods 33 3. Perceived unhelpfulness of the methods 28 4. Lack of understanding of the methods 17 5. Difficulties in working with the provided methods 15 4.3 Major findings from the pre-study of the project and consequences The pre-study (2011–2012) aimed to get a first estimation of the proportion of students who used CAT and to identify possible reasons for the non-use of the methods of CAT. Therefore, a short questionnaire was administered to the students in January 2012, in which the students were asked to rate on a Likert scale if they used CAT regularly and to respond to the open questions ‘What do you like about CAT?’ and ‘What do you criticize about CAT?’. The answers to the item ‘I use CAT regularly’ showed that most students did not use CAT regularly in January 2012 (Table 4). So the lecturer’s observation that many students did not use the methods provided by CAT was clearly confirmed. The responses to the open question ‘What do you criticize about CAT?’ were examined with respect to possible reasons why students might decide not to use CAT. 175 of the 229 answers given made corresponding statements. These statements were categorized inductively. An answer could also be assigned to more than one category. The five most common categories are shown in Table 5. In addition, many of the statements complained that the tutors did not use CAT (32 statements). By asking the tutors afterwards it was discovered that only a minority of them actually used CAT up to that point. In addition, the students’ statements suggested that the use frequency of CAT and the reasons for its non-use may vary between the instruments of CAT (sometimes only one instrument was actually criticized). This led to the decision to analyse the instruments separately from one another in the subsequent main study. The results of the pre-study led to initial changes in the implementation of CAT. The lecturer spent more time introducing the instruments; moreover, a detailed explanation of the instruments of CAT (and how to use them) was provided on the homepage of the course. But more importantly, from the winter semester 2013/14 onwards, CAT was integrated into the tutorials as well (p. 6). Therefore, particular training for tutors concerning the use of CAT in the tutorials was implemented. The training started with an initial mandatory workshop about the methods of CAT at the beginning of the semester, during which the methods of CAT were introduced to them. This was followed by weekly meetings between the lecturer and the tutors, where useful possibilities to practise CAT in the tutorials (adapted to the current topic) were discussed. 28 F. FEUDEL AND H. M. DIETZ Fig. 1. Example of a question asking if time effort is a reason for the non-use of an instrument of CAT. 5. Methodology of the main study of the project 5.1 Design and participants The research method used for the main study was design-based research. Therefore, two subsequent questionnaires were administered to the students in the course ‘Mathematics for economists’ at the Uni- versity of Paderborn at the end of their respective first semesters. The first one (the main questionnaire) was administered in January 2014, the second one (the follow-up questionnaire that consisted of key items from the main questionnaire) in January 2015. In between, changes regarding the implementation of CAT based on the results of the questionnaire in 2014 were made. 5.2 Description of the questionnaire administered in January 2014 The questionnaire administered in 2014 consisted of three different types of questions: I. Questions to investigate the frequency with which the methods of CAT were used II. Questions to detect reasons why students did not use the methods of CAT III. In-depth questions concerning specific reasons identified in the pre-study with the aim to draw direct conclusions concerning the implementation of CAT I. Questions to investigate the use frequency of the methods of CAT: Since the pre-study had shown that the use frequency might vary among the instruments, the following question was asked for each instrument individually: How often did you use the following instrument of CAT in the semester? The answer format was a Likert scale ranging from 1 = ‘never’ to 6 = ‘every week’. II. Questions to detect reasons for the non-use of CAT: The data of the pre-study showed five pos- sible reasons why students might not use methods of CAT (Table 5). In the subsequent main study the proportion of students who were not using the instruments of CAT for four of these reasons (‘time effort’, ‘difficulties in working with the methods’, ‘lack of understanding of the methods’ and ‘perceived unhelpfulness of the methods’) were investigated by asking questions of the type shown in Fig. 1.For the fifth reason—‘use of own methods’—a question of this type was judged to be not meaningful by the authors (and therefore not asked) because from the answer it cannot be seen if these ‘own methods’ correspond to the rejected instrument or have different aims instead. The reason ‘use of own methods’ was investigated in another study (Feudel, 2015). To find out other reasons for the non-use of the instruments of CAT that were not yet taken into account, students were also asked the following open question for each instrument: If you do not use the instrument regularly, please state one or several reasons for that. TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 29 III. In-depth questions that concerned specific reasons identified in the pre-study: Some of the five reasons for the non-use of CAT that had been identified in the pre-study (Table 5) were investigated in the subsequent main study in order to draw direct conclusions for teaching. The authors had to restrict themselves to only some of these five reasons because of time constraints regarding the completion of the survey during the lecture (30 minutes) due to course restrictions. Hence, the survey did not contain in-depth questions concerning ‘lack of understanding’ because this aspect was already addressed after the pre-study by extending the time amount used for the explanation of CAT in the lecture and by providing detailed information about the instruments on the homepage of the course (p. 9). The ‘use of own methods’ was also excluded here and investigated in a separate study instead (Feudel, 2015). So the questionnaire contained in-depth questions concerning the reasons ‘time effort’ (a), ‘difficulties in working with the methods’ (b) and ‘perceived unhelpfulness’ (c). III. a) In-depth questions concerning ‘time effort’: In the pre-study many students stated ‘time effort’ as criticism about CAT (Table 5), although only a minority did actually use CAT (Table 4). Therefore, the authors conjectured that ‘time effort’ was more the students’ apprehension than their experience. Students might overestimate the time needed to use the instruments of CAT. This hypothesis was examined for the Toolbox by comparing the students’ time estimations to create a Toolbox with the time they would actually need. So, in the first step, the students were asked: How much time do you need to create a Toolbox? In the second step, they actually had to create a Toolbox for the problem Prove that for all x, y ∈ R the inequality xy ≤ |x||y| holds. and afterwards, to write down the time needed to do so. Furthermore, the students had to state how much self-study time they invested per week. This is the time outside class they spent for the revision of the lecture, the preparation for the tutorials and for work on written assignments that were administered weekly. According to the course program, the students are obligated to spend 6 h per week on these activities because they received credit for 90 h of self-study time in the semester, which comes down to 6 h per week, as the semester counts 15 weeks with classes. However, there was no control mechanism to check if they actually invested the time required. So the authors wanted to find out if the students already spent the full required amount of self-study time without using CAT. Otherwise, they could be expected to invest some additional time, which might be necessary at the beginning of the semester to get used to CAT. III. b) In-depth questions concerning ‘difficulties in working with the methods’: The students were asked explicitly about difficulties which came up during the tutorials. These were 1. Checklist Reading: inability to carry out certain steps of the checklist correctly 2. Traffic Lights: difficulties in identifying the right stage of understanding 3. Toolbox: difficulties to create a Toolbox or to solve a problem with a given Toolbox 4. Concept Base: difficulties to ‘fill out’ one or more categories (e.g. examples, visualizations, etc.) The corresponding items/tasks of the questionnaire are given in Table 6. The Likert scales in Table 6 rating the degree of agreement were comprised of 6 points (an even number) in order to push the students to take sides (agreeing or disagreeing). III. c) In-depth questions concerning ‘perceived unhelpfulness’: The students were asked to rate the level of perceived helpfulness for each instrument on a Likert scale ranging from 1 = ‘not at all helpful’ to 6 = ‘very helpful’. 30 F. FEUDEL AND H. M. DIETZ Table 6. Items/tasks concerning difficulties with the methods of CAT Instrument Items/tasks Answer format Checklist Task: ‘State for each step of the Checklist ‘Reading’ if Multiple choice with the options ‘able to ‘Reading’ you are able to perform it or not.’ perform’, ‘not able to perform’ and ‘step not known’ Task: ‘Carry out the steps ‘Spell’, ‘Play’ and ‘Animate’ Open answer format for each step of the of the Checklist ‘Reading’ for the set Checklist ‘Reading’ M := {x ∈ N|∃k ∈ N:x = 3k} and state in your own words what its elements are.’ Traffic Lights 7 items concerning perceived difficulties in 6-point Likert scale ranging from ‘do not at self-assessment (formulated in terms of the Traffic all agree’ to ‘totally agree’ Lights) Toolbox 4 items concerning perceived difficulties in creating a 6-point Likert scale ranging from ‘do not at Toolbox and 4 items concerning perceived difficulties in all agree’ to ‘totally agree’ working with a given Toolbox Task: ‘Create a Toolbox for the problem to prove that Open answer format for all x, y ∈ R the inequality xy ≤ |x||y| holds.’ Concept Base 5 items concerning perceived problems in creating a 6-point Likert scale ranging from ‘do not at Concept Base all agree’ to ‘totally agree’ 5.3 Description of the questionnaire administered in January 2015 The follow-up questionnaire administered in January 2015 intended to evaluate the changes made in the implementation of CAT that had been realized as a consequence of the survey in January 2014. Hence, it consisted of key items from the January 2014 questionnaire. In January 2015 the authors wanted to find out to what extent the proportion of the students using the instruments of CAT changed and how many students now did not use instruments of CAT due to the reasons for non-use that had been identified in January 2014 and addressed afterwards (‘time effort’, ‘difficulties in working with the methods’ and ‘perceived unhelpfulness’). Therefore, the follow- up questionnaire contained the items asking for the use frequency for each instrument of CAT and the multiple-choice questions, in which the students were asked to select, which instruments were not used because of ‘time effort’, ‘difficulties in working with the methods’, ‘lack of understanding’ and ‘perceived unhelpfulness’ (p. 10). The open questions asking for reasons for irregular use of the instruments of CAT were omitted because the goal in January 2015 was to measure changes and not to explore previously undetected reasons. 5.4 Data collection The questionnaires were administered at the end of the winter semester in January 2014 and in January 2015, both before the respective final exam. Due to its size, the January 2014 questionnaire was split up in two versions, each administered to about half of the students (N = 376 and N = 381). One version contained the in-depth questions concerning the Traffic Lights and the Toolbox, the other version contained the in-depth questions concerning the Checklist ‘Reading’ and the Concept Base. The questions concerning the self-study time spent and the use frequency of the instruments of CAT were given to all students (N = 757). 6. Results of the survey administered in January 2014 6.1 Results concerning the frequency of use of CAT The frequency with which the students used the instruments of CAT can be seen in Fig. 2. As shown there, the use frequency varied among the instruments. Altogether, the degree with which the instruments of TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 31 Fig. 2. Use frequency of the instruments of CAT in January 2014, 1 = ‘never’, 6 = ‘every week’ (N = 757). CAT were used was higher in January 2014 than in January 2012, which can be seen from a comparison of Fig. 2 with Table 4: in January 2012 over 50% clearly disagreed to use CAT regularly, in January 2014 at least the Vocabulary list was used regularly by the majority of the students (and other instruments were at least used sometimes). The increase in the frequency of the application of CAT can be explained by the integration of CAT into the tutorials since the winter semester 2013/14. Except for the Vocabulary list, however, the acceptance was still not satisfactory. 6.2 Results concerning the reasons why students did not use the methods of CAT The students were asked to select which instruments of CAT they did not use due to ‘time effort’, ‘difficulties in working with the methods’, ‘lack of understanding’ and ‘perceived unhelpfulness’. The results are shown in Table 7 (numbers over 25% are highlighted). While ‘Time effort’ and ‘Perceived unhelpfulness’ were selected as reasons for the non-use for the majority of the instruments, ‘lack of understanding’ and ‘difficulties in working with the methods’ were only selected often for the Toolbox or the Concept Base. It was surprising for the authors that, although the instruments were carefully introduced during the lecture and detailed information about their use was provided on the homepage of the course in 2013/14, ‘lack of understanding’ was still selected so often. The answers to the open question ‘If you do not use the instrument regularly, please state one or several reasons for that’ were categorized inductively for each instrument (the intercoder-reliability after Table 7. Proportion of students, who selected ‘time effort’, ‘difficulties in working with the methods’, ‘lack of understanding’ or ‘perceived unhelpfulness’ as reason for the non-use of instruments of CAT in January 2014, multiple answers were possible (N = 757) Checklist ‘Reading’ Traffic Lights Toolbox Vocabulary list Concept Base ‘Time effort’ 23.2% 32.3% 30.1% 12.8% 50.5% ‘Difficulties with the methods’ 13.2% 18.4% 32.3% 3.5% 41.1% ‘Lack of understanding’ 19.6% 24.3% 30.9% 3.6% 31.7% ‘Perceived unhelpfulness’ 27.0% 48.2% 21.9% 8.1% 26.8% 32 F. FEUDEL AND H. M. DIETZ Table 8. Reasons for not using the instruments of CAT (answers to the open-ended question to state a reason if an instrument is not used regularly) Checklist ‘Reading’ (N = 158) Traffic Lights (N = 221) Toolbox (N = 94) Concept Base (N = 241) not necessary 23.4% own methods 35.7% (time) effort 26.6% (time) effort 37.3% (time) effort 18.4% not necessary 28.1% own methods 25.5% difficulties 18.3% own methods 17.1% (time) effort 15.8% difficulties 23.4% own methods 18.3% not helpful 10.1% own markings 13.1% not necessary 14.9% vocabulary list enough 11.6% not known 7.6% lack of benefit 7.2% not helpful 6.4% lack of benefit 10.8% recoding by a tutor of the course was above 0.75 for all instruments). The most common categories are shown in Table 8 (all other categories contained less than 7% of the answers given). Similar to the data in Table 7, the data in Table 8 also show that ‘time effort’ was an important reason for the non-use for each of the instruments of CAT and the reason ‘difficulties in working with the instruments’ was a major reason applied to the Toolbox and the Concept Base. The perception that an instrument did not help was less often explicitly mentioned in the open questions. ‘Lack of understand- ing’ was very seldom mentioned in the open questions (under 5% for all instruments), which indicates that other reasons for the non-use of the instruments of CAT were of greater importance to the students. Three additional reasons besides the four reasons in Table 7 were stated in the open questions: ‘use of own methods’, ‘lack of necessity’ and ‘lack of benefit’. It was surprising that ‘lack of necessity’ was stated that often—contradictory to the students’ low grades in the following exam in 2014. This indicates that the students may have overestimated themselves. What the students meant by ‘lack of benefit’ was not clear. It might mean that they perceived that the instrument did not help at all. It could also mean that their use did not provide enough benefit in comparison to the required time. The latter assumption is also supported by the data: 58% of the students who stated ‘lack of benefit’ as reason for the non-use of the Concept Base also stated ‘time effort’. Summary of the reasons why students do not use the methods of CAT: Altogether, the main study showed that there were seven important reasons for the non-use of one or more of the instruments of CAT (see Tables 7 and 8): (1) Time effort (2) Lack of understanding (3) Difficulties in working with the methods (4) Perceived unhelpfulness of the methods (5) Use of own methods (6) Lack of benefit (7) No necessity for the methods (perceived) The quantity with which the reasons were stated (stated explicitly in the open questions and selected in the multiple-choice questions) varied among the instruments. The reasons ‘lack of benefit’ and ‘lack of necessity’ were not mentioned in the pre-study, but were newly discovered in the main study. Comparing these seven reasons to the reasons why students did not change their learning behaviour mentioned by Dembo & Seli (2004) in Table 3 (they cannot change, they do not know how to change, they do not know what to change, they do not want to change), most of the identified reasons why students TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 33 Table 9. Estimated time and time actually needed to create a Toolbox (N = 91) Mean Median SD Estimated time 9:42 5 13.094 Time actually needed 2:36 2 2.138 did not use CAT indicated that the students either did not want to change their learning behaviour (reasons 4–7, maybe reason 1 if students were not willing to invest time) or did not know how to change their learning behaviour (reasons 2–3). This result coincides with Dembo & Seli (2004). 6.3 Results of the in-depth questions and resulting consequences for the implementation of CAT in the following year The questionnaire contained in-depth questions concerning the reasons ‘time effort’ (I.), ‘difficulties in working with the methods’ (II.) and ‘perceived unhelpfulness’ (III.) (p. 11–12). In addition, some results concerning the reason ‘no necessity’ (IV.) can be concluded from the data of the survey. I. Results concerning ‘time effort’: The students sometimes overestimated the time needed to work with the instruments of CAT. This was demonstrated by the data in the case of the Toolbox by comparing the students’ estimations and the time the students actually needed to create a Toolbox for the problem to prove that for all x, y ∈ R the inequality xy ≤ |x||y| holds (Table 9). Including the students who did not create the requested Toolbox (N = 164), the estimated time to create a Toolbox was even higher (median: 10 min). Consequence: In the following winter semester 2014/15, students were obligated to work with the instruments in the tutorials at least once in order to adjust false time estimates. The data of the study also showed that the students invested far too little self-study time outside classes (for revision of the lecture, preparing for the tutorials and completing written assignments). According to the course program they were required to spend 6 h per week on these activities (p. 11). However, on average, they just spent half of that required time (Table 10). This evidence indicates that for many students more time spent on self-study might lead to improved results, provided that the self-study time was spent effectively using appropriate study methods such as those provided by CAT. Consequence: At the beginning of the winter semester 2014/15, students were explicitly told how much self-study time they ought to spend outside class each week and it was communicated that some additional time spent for getting used to the methods of CAT first may very well pay off later during Table 10. The amount of time in minutes spent by the students on weekly occurring activities in their self-studies (N = 714) Mean Median SD Review the lecture and revise the lecture notes 55.69 30 55.22 Preparation for the tutorial 68.03 60 45.18 Completing written assignments 78.06 60 55.06 Altogether 201.78 180 119.26 34 F. FEUDEL AND H. M. DIETZ Table 11. Perceived difficulties in working with the Toolbox, 1 = ‘do not at all agree’, 6 = ‘totally agree’(N = 376) Mean Median SD Perceived difficulties in creating a Toolbox (4 items) 3.62 3.5 1.19 Perceived difficulties in working with a given Toolbox (4 Items) 3.51 3.5 0.94 Table 12. Answers to the item ‘I have problems in choosing suitable definitions and theorems for the Toolbox’, 1 = ‘do not at all agree’, 6 = ‘totally agree’(N = 376) 1 2 3 4 5 6 Mean Median SD 3.8% 11.7% 19.9% 26.8% 25.1% 12.7% 3.96 4 1.331 the preparation for the exam. The authors hoped that the students were then more willing to invest some additional time for getting used to CAT. II. Results concerning ‘difficulties in working with the methods’: The answers to the in-depth questions concerning difficulties with instruments of CAT (Table 6) showed that the students had several difficulties. The results are only presented for the Toolbox due to the constraints of this paper. The students were asked to rate perceived difficulties in creating a Toolbox or in working with a given Toolbox to solve a problem (Table 6). The results are shown in Table 11.By looking at the items individually, the creation of a suitable Toolbox for a given problem was judged as especially difficult (Table 12). This was also supported by the Toolboxes created by the students for the problem to prove xy ≤ |x||y| for all x, y ∈ R with the properties of the absolute value given in the lecture (Table 6). Only two of the 233 students who wrote down a Toolbox collected both of the following items needed to solve the | | | | | || | problem: x ≤ x ∀x ∈ R, and xy = x y ∀x, y ∈ R. Consequence: Since the winter semester 2014/15, the difficulties identified in January 2014 have been addressed explicitly in the tutorials (e.g. strategies to choose and collect suitable items for the Toolbox were explicitly discussed). III. Results concerning ‘perceived unhelpfulness’ The data showed that the Checklist ‘Reading’ and the Traffic Lights were judged to be less helpful (Table 13). The correlations between the perceived helpfulness and the use frequency were over 0.6 for all instruments, which indicates that perceived unhelpfulness was a major reason for not using it. So, it is important to find out reasons for perceived unhelpfulness. In the case of the Checklist ‘Reading’ a careful analysis of the data revealed that insufficient knowledge about the details of the instrument might have caused the notion of its unhelpfulness. The students were asked for each step of the Checklist ‘Reading’ if they were able to perform it or not, or if they did not know it (see Table 6). The data surprisingly showed that many students did not know the steps ‘Animate’ and ‘Visualize’, which are important for gaining understanding of a mathematical concept (Table 14). Of course, if one thinks that the Checklist ‘Reading’ consisted only of ‘Spell’ and ‘Play’, one would of course judge it as unhelpful because these steps only help during the decoding process and do not help to get the meaning of the content. TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 35 Table 13. Answers towards the extent of perceived helpfulness by the instruments of CAT, 1 = ‘not at all helpful’, 6 = ‘very helpful’(N = 757) Mean Median SD Checklist ‘Reading’ (N = 381) 3.27 3 1.515 Traffic Lights (N = 376) 3.07 3 1.528 Toolbox (N = 376) 4.18 4 1.318 Concept Base (N = 381) 3.77 4 1.360 Table 14. The students’ability to perform the steps of the Checklist ‘Reading’ Step Able to perform Not able to perform Step not known ‘Spell’ (N = 352) 61.4% 19.0% 19.6% ‘Play’ (N = 352) 80.4% 7.4% 12.2% ‘Animate’ (N = 347) 15.9% 45.0% 39.2% ‘Visualize’ (N = 346) 28.3% 41.6% 30.1% ‘Talk’ (N = 347) 40.1% 29.7% 30.3% Consequence: In the winter semester 2014/15, these two steps were explicitly emphasized during the tutorials and had to be carried out in several written assignments explicitly. In the case of the Traffic Lights, however, the cause for the belief of its unhelpfulness could not be identified. Maybe students failed to see the benefit of rating their own understanding with three rather than two levels (fully understood = green; understood to some extent, but with a residual uncertainty = yellow; not understood = red as opposed to just ‘understood’ vs. ‘not understood’). This assumption is also supported by students’ statements in the open question why they did not use the Traffic Lights regularly like ‘half understood=pointless’. Last but not least ‘perceived unhelpfulness’ was also often selected as a reason for the non-use of the Concept Base (Table 7). Furthermore, ‘lack of benefit’ was often stated in the open question as a reason why this instrument was not used regularly (Table 8). Here the authors conjecture that the students lacked conditional knowledge concerning possibilities of use of the Concept Base. This is supported by students’ statements in the tutorials (‘no benefit for problem solving’). However, the authors believe that the Concept Base can indeed be helpful during the process of problem solving, e.g. as a knowledge base for creating Toolboxes. Possible future consequence: The usefulness of Concept Bases for problem-solving should be made more explicit during the tutorials. IV. Results concerning ‘no necessity’: ‘Lack of necessity’ was stated often in the open question why an instrument of CAT was not used regularly in the case of the Traffic Lights and the Checklist ‘Reading’ (Table 8). This surprised the authors and was in contradiction to the lecturer’s long-term experience. However, our data showed that many students overestimated themselves. Only 21.6% of the 37 students that stated ‘not necessary’ as a reason for the non-use of the Checklist ‘Reading’ (Table 8) gave a correct description of the set M := {x ∈ N|∃k ∈ N: x = 3k} in their own words (task originally included to identify students’ difficulties with the Checklist ‘Reading’, Table 6). 36 F. FEUDEL AND H. M. DIETZ In the case of the Traffic Lights, the data showed a similar result. 46% agreed with the statement ‘I can recognize my difficulties in understanding without the Traffic Lights’ on at least level 4 on a Likert scale from 1 = ‘do not at all agree’ to 6 = ‘totally agree’ (N = 381). However, 25% of them later agreed on level 6 to the statement ‘I have recognized gaps during the exam, which I had not recognized before’ (in a short survey directly after the exam, N = 295). In the whole population, only 15.9% agreed on that statement on level 6. However, since the term ‘gaps’ was not specified in the latter statement, the students could have referred to gaps in knowledge, gaps in understanding or gaps in procedural calculations. So the data only show that overestimation occurred, but not in which area. Possible conclusion: One possible conclusion would be to provide learning environment in which the students recognize their overestimation, e.g. by giving them tasks which they have to grade by themselves first and that are later graded by the tutors. Summary of the results of the in-depth questions from January 2014 and consequences concerning the implementation of CAT in the following year: With the help of the in-depth questions, it was possible to identify aspects that could be addressed when changing the implementation of CAT in the course. The findings (based on the data) and the resulting consequences concerning the implementation of CAT in the next year (winter semester 2014/15) are summarized in Table 15. Moreover, students’ answers to the open question ‘If you do not use the instrument regularly, please state one or several reasons for that’ like ‘half understood=pointless’ in the case of the Traffic Lights, as well as students’ statements in the tutorials like ‘no benefit for problem solving’ in the case of the Concept Base, lead the authors to conjecture that students might lack conditional knowledge when and why to use these instruments of CAT. That could cause a perceived lack of benefit. This conditional knowledge should be provided in the course in the future, e.g. by using the Concept Base as knowledge base for the creation of Toolboxes during the process of problem-solving. In the results presented here, one can clearly see that a lack of any of the three knowledge types in thesenseof Paris et al. (1983) (declarative knowledge, e.g. knowledge of the steps of the Checklist Table 15. Findings on the basis of the in-depth questions and consequences for the implementation of CAT in the next year in 2014/15 Findings based on answers given to the in-depth questions Activities concerning the implementation of CAT in the winter semester 2014/2015 Overestimation of the time needed to work with the Obligation of students use the instruments in the tutorials at instruments of CAT (here shown for the Toolbox, Table 9) least once to reduce their apprehensions concerning time effort Not enough self-study time spent by the students for Clear communication of the curriculum’s workload learning activities outside classes (Table 10) demands concerning the self-study time at the beginning of the semester Problems of the students in working with the methods of Addressing these difficulties during the tutorials explicitly CAT (here proved for the example ‘creating a Toolbox’, (e.g. practising the creation of Toolboxes) Table 12) Insufficient knowledge about the details of the instruments Provision of exercises after the introduction of the (here shown for the steps ‘Animate’ and ‘Visualize’ for instruments addressing aspects that have been identified to Checklist ‘Reading’, Table 14) be not well known Students’ overestimation of their own abilities (here shown Not addressed yet for the Checklist ‘Reading’ and the Traffic Lights, p. 19) TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 37 Fig. 3. Use frequency of the instruments of CAT in January 2014 (N = 757) and January 2015 (N = 710), 1 = ‘never’, 6 = ‘every week’. ‘Reading’, procedural knowledge, e.g. knowledge about strategies to create a Toolbox and conditional knowledge, e.g. knowledge about the possibilities to use the Concept Base during problem-solving) can lead to the non-use of the methods provided by CAT. Students need to know about the strategies themselves in detail, they should learn how to apply them and have to know when and why to use them. 7. Results of the survey in January 2015 to evaluate changes in the implementation of CAT The changes in the implementation of CAT in the course that were based on the results of the survey in January 2014 raised the frequency with which the students used of most of the instruments of CAT (Fig. 3). The number of students who used the instruments was much higher in January 2015 than 1 year before. It was now mostly satisfactory, except for the Traffic Lights and the Concept Base (Fig. 3). The strongest positive impact on the acceptance of CAT was achieved by addressing the difficulties during the tutorials. This holds particularly for the Checklist ‘Reading’, where special emphasis was directed to the step ‘Animate’, and for the Toolbox, where the creation of Toolboxes was practised (Table 16). The authors conjecture that if the students have fewer problems when working with the methods, their work is more often successful. This would explain why the two mentioned instruments were also less often not used because of ‘perceived unhelpfulness’ (Table 17). Table 16. Amount of students not using instruments of CAT because of ‘difficulties when working with them’ in January 2014 (N = 757) and January 2015 (N = 710), multiple answers were possible Checklist ‘Reading’ Traffic Lights Toolbox Vocabulary list Concept Base 2014 (N = 757) 13.2% 18.4% 32.3% 3.5% 41.1% 2015 (N = 710) 5.7% 22.7% 10.2% 5.7% 34.0% 38 F. FEUDEL AND H. M. DIETZ Table 17. Amount of students not using instruments of CAT because of ‘perceived unhelpfulness’in January 2014 (N = 757) and January 2015 (N = 710), multiple answers were possible Checklist ‘Reading’ Traffic Lights Toolbox Vocabulary list Concept Base 2014 (N = 757) 27.0% 48.2% 21.9% 8.1% 26.8% 2015 (N = 710) 17.0% 61.3% 9.0% 9.8% 18.7% Table 18. Amount of students not using instruments of CAT because of ‘time effort’ in January 2014 (N = 757) and January 2015 (N = 710), multiple answers were possible Checklist ‘Reading’ Traffic Lights Toolbox Vocabulary list Concept Base 2014 (N = 757) 23.2% 32.3% 30.1% 12.2% 50.5% 2015 (N = 710) 13.9% 25.0% 13.9% 18.7% 44.5% Informing the students about the amount of self-study time they were required to spend also had a positive effect because, ‘time effort’ was less often selected as a reason for the non-use of the instruments in January 2015 except for the Vocabulary (Table 18). 8. Conclusion and discussion The research project aimed at raising the acceptance of CAT and at improving the students’ study skills by using CAT. Thus it focused on the following research questions: 1. How many students actually use the strategies provided by CAT regularly? 2. What are the reasons that students do not use the strategies provided by CAT? 3. What activities can increase the number of students using the provided strategies? As to 1.: At the beginning of the research project, the strategies provided by CAT were only used by a minority of students. After changes in the implementation of CAT based on the results of the surveys, the acceptance of CAT’s strategies was raised considerably. As to 2.: By asking the students directly for reasons why they did not use the instruments of CAT, seven reasons were discovered: (1) ‘time effort’, (2) ‘lack of understanding’, (3) ‘difficulties in working with the methods’, (4) ‘perceived unhelpfulness of the methods’, (5) ‘use of own methods’, (6) ‘no necessity of the methods’ and (7) ‘lack of benefit’ (p. 15). With the help of the in-depth questions it was possible to identify concrete problems causing the above mentioned reasons that could be addressed in the next year. These were: (a) overestimation of the time needed for working with instruments of CAT, (b) not enough time investment for self-studies outside class, (c) difficulties in working with instruments of CAT (that could be further specified), (d) insufficient knowledge about details of CAT and (e) students’ overestimation of their own abilities (Table 15). As to 3.: The problems (a)–(d) (based on the answers of the in-depth questions) led to changes in the implementation of CAT in the following year (Table 15). These changes raised the acceptance of some of the instruments of CAT considerably (Fig. 3). The most important consequence was to integrate the methods of CAT into all parts of the course, especially into the tutorials and written assignments in order to (1) provide detailed knowledge about the methods and to practise them, (2) to address difficulties the students had when working with the methods and (3) to adjust the students’ estimation of the time TEACHING STUDY SKILLS IN MATHEMATICS SERVICE COURSES 39 needed to apply the methods by obligating them to use the instruments at least once. In the case of the Toolbox and the Checklist ‘Reading’ the positive impact of that integration can be seen in the data of the study. The exercise of the creation of Toolboxes (p. 17) and the exercise of the steps of the Checklist ‘Reading’ on written assignments (p. 18) lowered the proportion of students not using these instruments because of difficulties in working with them considerably (Table 16). The authors conjecture that these activities also lowered the number of students who perceive these two instruments as unhelpful because if students have fewer difficulties with the instruments their work with the instruments will more often be successful. For an effective integration of the methods of CAT into the tutorials, however, the tutors had to be trained carefully, though. The workshop for new tutors at the beginning of the semester and the weekly meetings during the semester, in which the practical implementation of CAT into the tutorials were discussed, were essential for that training. In addition, informing the students about how much self- study time they were required to spend for the course also helped to reach some students that had not been willing to spend much time on their self-studies beforehand (because ‘time effort’ was less often mentioned as reason for rejection for instruments of CAT in 2015, Table 18). Two major problems remain. One is the students’ overestimation of their own abilities, which was pro- ven by the data of our study. Students overestimating themselves do not feel a need to use strategies provided in support programs (Dembo & Seli, 2004). Nowell & Alston (2007) showed that overestima- tion is a general problem with students of economics across the whole curriculum. Grimes (2002),for example, pointed out that even after a test many students of economics assumed to have achieved a grade better than it actually was. Kruger & Dunning (1999) proved for several disciplines (not only economics) that students who perform poorly are the ones who overestimate their own abilities the most. A solution could be to provide a learning environment, in which self-testing is practised. The students should experience actively how they can monitor and address comprehension failures with the help of the Traffic Lights in the course in several settings: when reviewing the lecture, when justifying algebraic calculations and when reading mathematical texts. However, up to now, the students were advised to do so only in the lecture. There was no explicit training because the lecturer and second author thought before the study that the instructions he gave during the lecture concerning the use of the Traffic Lights were very clear to the students and the necessity to judge one’s own understanding carefully would be obvious to the students. But the present study shows clearly a need for an explicit and guided training regarding the use of the Traffic Lights for self-evaluation during the tutorials in order to get the students to realize the necessity of a careful self-evaluation and the benefit of the Traffic Light system. Providing an active training can be expected to improve self-evaluation strategies more than just giving advice (similar to Bielaczyc et al. (1995)). The training should involve the self-test to recall a definition from memory and then to compare it with the source (already mentioned on p. 5). This self-test provides an external standard, which increases the probability that the self-evaluation is done correctly (Rawson & Dunlosky, 2007). In this setting students can also recognize their tendency to overestimate their skills more easily. Later, one should proceed to more complicated self-tests like the creation of a Concept Base from memory. Besides these self-tests, activities to remediate comprehension failures should be included in the training as well (e.g. reading the explanations in the corresponding chapter in the course textbook (Dietz, 2012)). The second problem was that some students did not see the benefit of certain instruments of CAT. Concerning the Traffic Lights, for example, students commented in the January 2014 questionnaire that they did not see the benefit of rating their understanding in three, rather than in two, levels. Regarding the Concept Base, students stated in the tutorials that it did not yield any benefit for problem-solving. In both cases, the students did not see the benefit of applying these two instruments. If students do not see the benefit of changing their learning behaviour, they will not change it especially if the new learning 40 F. FEUDEL AND H. M. DIETZ behaviour is time consuming like creating a Concept Base (Jakubowski & Dembo, 2004). Hence, the tutorials should direct much more attention to making the benefit of the instruments clearly visible. One example would be to use the Concept Base as a knowledge base to create Toolboxes during the process of problem-solving. A limitation of the study is that it is not clear if the students who adopted the methods of CAT will keep using them after the exam or if they return to their old methods. Changing one’s learning behaviour is not completed after the new methods have been acquired. They have to be maintained until they are actually adopted (Jakubowski & Dembo, 2004; Prochaska & Prochaska, 1999). To investigate if the study skills program CAT has a long term effect, i.e. if the acquired methods are maintained by the students, further data has to be collected after the exam or at the end of the mathematics courses, which would be almost one year after the introduction of the methods of CAT. In conclusion, one can say that although the application of the strategies provided by CAT had a positive effect on academic achievement (Dietz, 2012), the implementation of CAT into the whole course faced the problem that the majority of students did not use the methods of CAT first. Many reasons for the non-use of the strategies (that could occur in similar study skills programs as well, see for example Dembo & Seli (2004)) were identified in the research project. With the careful introduction of the concept CAT and the integration of it into all components of the course (lecture, tutorials and written assignments) most of the problems were remediated (except for the problem of overestimation of their own abilities and a perceived lack of benefit by some students) and the provided methods became widely accepted among the students. So the study presented here does not only confirm that the embedding of taught methods and strategies into the learning environment is important (Anthony, 1996; Garner, 1990), but also provides practical tips, how to accomplish a successful implementation of a study skills program into a large first semester service mathematics courses (with the limitation that it is not clear if the acquired skills will be maintained by the students). 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Journal

:Teaching Mathematics and Its Applications: An International Journal of the IMA – Oxford University Press

Published: Mar 5, 2019

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