Can Guided Notes Support Students’ Note-taking in Mathematics Lectures?

In traditional mathematics lectures the instructor normally writes the definitions, theorems, and proofs covered on the board, and gives informal oral explanations that help to make sense of them. The students have to take notes. However, there are serious problems concerning students’ note-taking in traditional mathematics lectures. Students often cannot think about the information presented during the lecture as they are busy writing. Making sense of the content later is also difficult because many students do not include the lecturer’s oral explanations in their notes. One approach to addressing these problems can be the use of guided notes: a modified version of the instructor’s notes with certain blanks the students have to fill in during the lecture. We investigated to what extent guided notes can support students in their note-taking in mathematics lectures in a study using a mixed-method design. This study provides on the one hand quantitative data suggesting that guided notes are perceived as beneficial by many students for several aspects of their note-taking. On the other hand, it offers qualitative data illustrating how the use of guided notes can influence these aspects. The results indicate in particular that the use of guided notes can address some of the problems concerning students’ note-taking in traditional mathematics lectures, while it can also lead to new problems that one needs to be aware of.


Introduction and Embedding of the Research
In mathematics at university level new content is often delivered in lectures. These lectures are often held as so-called chalk talk: The lecturer writes all definitions, theorems, and proofs on the blackboard while reading them out aloud, and makes additional explanatory comments orally (Artemeva & Fox, 2011;Fukawa-Connelly et al., 2017). Note-taking plays a major role in such lectures because students are often required to make sense of the content by themselves after the lecture (Bauer, 2015;Weber, 2004). This places a high demand on students as note-taking requires a range of skills (Lew et al., 2016;Williams & Eggert, 2002a): 1) Listening and paying attention 2) Cognitive processing 3) Noting important content 4) Post-class processing of the notes.
However, the literature suggests that in mathematics lectures students' note-taking is often limited to writing down the definitions, theorems, and proofs the instructor has written on the board (Weber, 2004). Students are often just busy copying everything correctly instead of paying attention to the lecturer's explanations (Freitag, 2020). This can lead to serious problems for their ability to make sense of the content presented during a lecture because the instructor's explanations often contain information crucial for students' ability to comprehend the content, like informal representations of a concept or ideas in a proof (Fukawa-Connelly et al., 2017;Lew et al., 2016). The problem worsens if the lecturer proceeds through the content relatively quickly (Freitag, 2020), which is often the case in traditional mathematics lecturers at university (Harris & Pampaka, 2016). Furthermore, it has been documented in the literature that students often do not write down the lecturer's oral explanations to the definitions, theorems, and proofs (Fukawa-Connelly et al., 2017). Hence, it might also be difficult for them to gain an understanding of the content later at home on the basis of the notes taken during the lecture.
One approach to addressing these problems might be the use of guided notes. These are preprinted lecture notes with blanks at certain places that the students are required to fill in as the lecture progresses (Austin et al., 2004). According to Iannone and Miller (2019), this is a relatively new mode of teaching for university mathematics, which is supported also empirically by Artemeva and Fox (2011), who found that the traditional 'chalk talk' practice described above is widespread in mathematics lectures around the world. The use of guided notes in mathematics lectures has one major potential: since the lecturer and the students no longer have to write everything down, they can really focus on the blanks during the lecture. This might save time and thus allow students also to follow the lecturer attentively besides writing, which is important for being able to make sense of the content during the lecture. This claim is also supported empirically for university mathematics lectures in recent qualitative studies with small samples (Cardetti et al., 2010;Iannone & Miller, 2019). However, as yet there exist no data from larger samples. Furthermore, the two studies just mentioned did not systematically investigate the effects of guided notes on the different aspects of students' note-taking that are documented in the literature (Freitag, 2020;Van Meter et al., 1994). Our research attempts to fill these gaps.
We will first present a literature review about guided notes illustrating its potentials for students' note-taking, and its actual effects on their note-taking behavior and on their performance. We will then present our study investigating the effect of the use of guided notes on students' note-taking: the theoretical perspective, the methodology and the results. Finally, we will discuss our results and give an outlook for further research on the basis of issues that have been left open in our study.

The Idea of Guided Notes and Its Potentials
Providing students with parts of the lecturer's notes as a compromise between requiring them to take their own notes and providing them with the lecturer's full notes has been suggested in the literature since the'70 s (Collingwood & Hughes, 1978). Collingwood and Hughes (1978) used the term partial notes, a phrase that still appears in more recent studies (Cardetti et al., 2010;Cornelius & Owen-DeSchryver, 2008). The term guided notes was first defined by Heward (1994) as "teacher prepared handouts that guide a student through a lecture with standard cues and specific spaces in which to write key facts, concepts, and relationships" (p. 304). Important here is that all blanks in the handouts are left by the lecturer intentionally. However, Heward's definition seems a bit narrow as blanks can also be left at other important spaces, e.g. for examples of concepts (Austin et al., 2004). A broader definition of guided notes that we used in our research was formulated by Austin et al. (2004), p. 314: "Guided notes are modified versions of the instructor's notes or slides that require students to fill in missing information as the lecture progresses." Several potentials of guided notes in university lectures in general have been hypothesized in the literature. Barbetta and Skaruppa (1995) claimed, for example, that guided notes might help students organize their writing because the blanks already give cues about key points and concepts presented in a lecture. They also argued that, as only the key points need to be written down during a lecture, students have more opportunities to listen to the lecturer's explanations, and that this might improve the accuracy of students' note-taking, which is an important predictor for their performance (Peverly et al., 2007;Williams & Eggert, 2002b). Being able to listen and follow the lecturer's explanations attentively is also necessary for processing the information presented (Williams & Eggert, 2002a) -that is, to understand the main points of the lecture and connect them to one's existing knowledge. Finally, Barbetta and Skaruppa (1995) claimed that blanks can provide prompts for interaction in class, which might lead to more engagement of students with the content during a class.

Empirical Studies on the Effect of Guided Notes on Students' Performance and Their Note-taking Behavior
There is limited empirical research investigating whether the use of guided notes genuinely improves students' note-taking and their academic performance. Konrad et al. (2009) found, in a meta-analysis for students at secondary level, that the use of guided notes has a positive effect on their learning outcomes. Concerning postsecondary level, however, the results are inconclusive. Nevertheless, some empirical studies indicate that using guided notes compared to requiring students to take their own notes has a positive effect on their performance (Annis, 1981;Austin et al., 2004;Cardetti et al., 2010;Chen et al., 2017;Konrad et al., 2009). However, only the study by Cardetti et al. (2010) specifically took place in a mathematics course. They investigated the effect of guided notes on students' performance in a calculus course in 2006 and 2007 by comparing the grades of the three examinations of the course in 2006 with the ones in 2007. In both years the course was given by the same instructor, and the same content was covered. In 2006 the students had to take notes solely for themselves. In 2007 the instructor instead provided portions of her own notes, and the students merely needed to fill in certain parts left as blanks (e.g. for definitions, names of concepts, and properties). The result was that the guided notes group performed better in all three examinations, and significantly outperformed the students in the control group in two of the three examinations ( p < 0.05 ), although the guided notes group scored lower in the mathematical section of the SAT entry test.
There is also some research indicating that the use of guided notes can improve students' notes. Chen et al. (2017) showed in a study with a quasi-experimental design in an introductory psychology course that the notes of students in an experimental group with guided notes had a higher quality than the notes of students in a control group, in which they had to take their own notes (and where only an outline of the course was provided). The quality of the notes was measured for each content area on a scale from 0 to 3, where 0 was given for missing or incorrect information, 1 for the naming of a topic without explanation, 2 for an incomplete explanation of a topic, and 3 for a topic mentioned with a complete explanation. The values for all content areas were then summed up for each participant to generate a score. The result was that the quality of the notes of the students with guided notes was significantly higher ( p < 0.05 ), although their initial note-taking skills did not differ from those of the students in the control group (for the first two teaching units, both groups were only provided with the outline). Furthermore, participants in the guided notes group also produced higher quality notes for the final unit of the course, in which they again only received an outline. This indicates that the use of the guided notes not only resulted in higher quality notes in the respective sessions with guided notes but also improved students' note-taking skills in general. Another study showing that the use of guided notes can improve the quality of students' notes was conducted by Austin et al. (2004). They investigated students' note-taking behavior under different teaching conditions that were used several times throughout the semester: a traditional lecture without the provision of notes, a lecture with the provision of critical points on slides, and a lecture with guided notes. In the latter condition, the lecturer provided slides with the key points of the lecture, and the students received the slides as handouts with blanks at certain spaces. The first result of the study was not surprising: students in the slide condition and the guided notes condition wrote down almost every key point whereas the others missed almost 40% of the key points in the lecture. The more interesting result is, however, that the students in the guided notes condition also noted down many more examples and extra explanatory comments that were presented orally compared with students in the slide condition (and the lectures did not differ in the difficulty and in the number of key points presented). This raises hopes that a similar change in the note-taking behavior might also occur in mathematics lectures at university.
However, there exists little research concerning the effects of the use of guided notes in mathematics lectures at tertiary level. We only found two studies focusing on this issue. One was the intervention study by Cardetti et al. (2010) already mentioned, in which students of a calculus course with guided notes attained better results in their exams than students in the same course held as a traditional lecture. In that study, some students also stated their appreciation of the guided notes because they could pay more attention to what the lecturer said, and did not have to concentrate exclusively on writing. The other study on the effect of the use of guided notes in a university mathematics lecture was a qualitative study by Iannone and Miller (2019), with eight students of a number theory course at a UK university. They investigated students' attitudes towards guided notes and the impact of guided notes on students' note-taking. Also in that study, some students stated that the provision of guided notes helped them follow the lecturer better. However, one student mentioned instead that the preprinted material was a stimulus for losing attention. Furthermore, Iannone and Miller (2019) found some indications that the use of guided notes can result in a different note-taking behavior than in traditional lectures. Some students wrote down comments just presented orally by the lecturer or noted links between different parts of the lecture, which is uncommon in traditional mathematics lectures (Fukawa-Connelly et al., 2017). There was a consensus among the students that the use of guided notes allowed them to do so because they were also able to pay attention to the lecturer's oral comments besides writing. This is some indication that the hope expressed from the results of the study by Austin et al. (2004) mentioned above could also be fulfilled in mathematics courses at university: students might also note down additional oral comments. However, the sample in the study of Iannone and Miller (2019) was very small. Our study will now present data from a larger sample with respect to the question: To what extent can guided notes support students' note-taking in mathematics lectures?

Theoretical Framework of the Study
Since note-taking requires students to perform several activities by themselves (listening, cognitive processing, recording, post-class processing Lew et al., 2016;Williams & Eggert, 2002a), as outlined in the introduction, we used a theoretical framework that describes students' note-taking from a self-regulated learning perspective (Zimmerman, 1990): the college students' theory of note-taking by Van Meter et al. (1994). By interviewing 252 students from different disciplines about their perceptions of the functions of note-taking, their regulations of note-taking, and their beliefs regarding factors that influence their note-taking, Van Meter et al. (1994) developed a framework that describes different aspects of students' notetaking behavior. These are manifested as different parts in the framework: I) Goals of students' note-taking, II) Content-structure of notes, III) Contextual factors affecting note-taking, and IV) Post-class processing of notes. We will now describe these parts in detail.
I. Goals of Note-taking: Although the students interviewed by Van Meter et al. (1994) stated as the overarching goal their wish to pass the exam, they formulated more sophisticated subgoals (p. 338): 1) Increase attention 2) Gain understanding: Increase student comprehension and memory of the material presented in the lecture 3) Organize material presented: Opportunity to connect ideas, provide structure, or generate a holistic representation of the lecture content 4) Study aid: Inform about the content of the exam 5) Homework aid: Inform about solutions to practice problems and provide information relevant for written assignments.
II. Content-structure of the Notes: Van Meter et al. (1994) found that students are selective about what they note, and favor information that they feel might be important to know. This in particular includes definitions, main points, and important concepts. But students also note what they do not understand. Information they already know is not noted. Furthermore, students more likely tend to note what the lecturer emphasizes or writes on the board. The structure of students' notes varies from just noting key terms to verbatim 'copies' of what the lecturer has presented. In traditional mathematics lectures at university, students' notes are mostly a copy of what the instructor has written on the board (Fukawa-Connelly et al., 2017).
III. Contextual Factors Affecting Note-taking: Van Meter et al. (1994) found different factors affecting students' note-taking behavior: 1) The lecturer's style (in particular the speed and the structure provided in the lecture) 2) Students' knowledge and characteristics (prior knowledge of content, know-how in note-taking, effort students are willing to make in note-taking) 3) Type of content and course demands.
IV Post-class Processing of the Notes: According to Van Meter et al. (1994) lecture notes are not always used outside class (they are not used if they are not considered as good or if material is already known). If they are used, they are used for different purposes -for example for homework or exam preparation. Furthermore, the method varies from just reviewing to creating a complete, newly written revision.
In our study, we used the framework by Van Meter et al. (1994) to investigate the influence of guided notes on students' note-taking. Of course, this begs the question of whether this general framework also fits to mathematics courses. Freitag (2020) has shown, in a recent qualitative research study with eight students at a large research university in the U.S., that this is indeed the case.

Description of the Participants
The study took place at a medium-sized university in Germany. The participants were students who aimed at becoming mathematics teachers at grammar schools. Since these students will later teach mathematics up to the end of secondary education, they are required to attend the same courses as mathematics majors in their first year (in later years they have separate courses). Normally mathematics teacher students are less interested in university mathematics than mathematics majors (Ufer et al., 2017), and have less prior knowledge (Rach, 2019), which might make the transition to university more difficult for these students.

Description of the Students' Courses
The students had to take two courses in their first year: the course "Introduction to mathematical thinking and working", in which this study took place, and a course in linear algebra.
The Course "Introduction to Mathematical Thinking and Working": The course consisted of one 90-min lecture and one 90-min tutorial each week. Furthermore, students had to submit written assignments each week (prerequisite for the final exam: 50% of the points over the semester).
The topics of the course were the following: 1) Elementary number theory (problems of divisibility, prime factorization, residue classes, Euclidean algorithm) 2) Sets and relations 3) Basic algebraic structures (groups, rings, fields) 4) Construction of the number systems (from natural via integer, rational to real numbers).
The lecture was given by a mathematician on the basis of guided notes following the idea of Alcock (2018). Before each session, the lecturer distributed a paper script with blanks at certain positions. A typical sample page is shown in Fig. 1. The choices for the positions of the blanks were made by the instructor on the basis of her teaching experience from previous years, during which she had taught the course traditionally. She especially left blanks at positions, at which 1) she judged that the students would be able to fill in the missing information by themselves on the basis of previously gained knowledge (example: the basic properties of the elements in a commutative ring from proposition 7.5 in Fig. 1 that students might be able to conjecture on the basis of their knowledge about properties of numbers from school), 2) she considered points or statements as particularly important (example: the defining properties of mathematical concepts like the ones for a commutative ring in Fig. 1), or 3) she suspected particular problems in making sense of the content (example: blanks at positions in formal proofs, at which she considered additional informal explanations as necessary, for instance visualizations or numerical illustrations of formulas, or the overall proof strategy).
For small proofs, which she expected students to be able to write for themselves, she even left the whole proof blank (for example the proof of the proposition 7.5 in Fig. 1).
During the lecture, the instructor completed the blanks with the help of a visualizer. Furthermore, at some blanks, the instructor initiated in-class activities. As just said, if she suspected that students could fill in blanks by themselves, she asked them to do so first (see also Alcock (2018)). In this way, she hoped especially that the students would not only copy what she wrote on the visualizer, but use her written contributions just as an aid for complementing their own thoughts. Even more, she hoped that letting the students write their own answers at the blanks first might stimulate them to complement also preprinted parts of the script. At other blanks, she presented several options for filling in the missing information, and initiated a discussion between the students about the right choice (example: at a blank for the definition of |a| the instructor offered the options a and −a , and the students had to decide which option was correct for a ≥ 0 and which for a < 0 ). After the lecture, the instructor distributed an electronic version of the guided notes whose blanks she had filled in herself.
Despite the greater degree of interaction, the lecture was given in a very traditional manner, particularly with respect to the following features: 1) The local structure followed a DTP (definition, theorem, proof) format. 2) By filling in the blanks on the visualizer, the lecturer provided all the formal content (definitions, theorems, proofs) in written form.
3) The lecturer communicated explanatory comments often just orally (exception: blanks that were intentionally left for comments at certain steps in a proof).
The Linear Algebra Course: It consisted of two 90-min lectures and one 90-min tutorial per week. The students were also obliged to submit written assignments weekly (prerequisite for the final exam: 50% of the points). The topics covered in this course were systems of linear equations, finite-dimensional vector spaces, matrices and linear mappings, and determinants. The lecturer was a mathematician who really cared about the learning of his students. His style was a classic 'chalk talk' lecture (see introduction).
Some students had also attended further courses given as 'chalk talk', like Analysis I or Geometry.

Methodology of the Study
To address our research question "To what extent can guided notes support students' note-taking in mathematics lectures?", we constructed a questionnaire asking for students' perceptions concerning their note-taking on the basis of the college students' theory of note-taking by Van Meter et al. (1994). This questionnaire consisted of closed items with a Likert scale, and open questions. The former aimed at focusing directly on different aspects of the theoretical framework. The latter had the goal of elaborating the results of the Likert items (for example to find out how the use of guided notes can influence students' note-taking in detail), and to serve as a validity criterion because qualitative data are valuable for either supporting or questioning the results of the Likert items.
Concerning the construction of the items, we followed recommendations given in handbooks on questionnaire construction for psychologists like Moosbrugger and Kelava (2012) in order to reduce mismatches between students' answers and their 'true' perceptions, for example by using inverted items. Furthermore, to achieve reliable results, we did not merely generate a single Likert item for each aspect represented in the framework, but rather a block of items (Moosbrugger & Kelava, 2012). Nevertheless, one needs to keep in mind that our data can only yield students' perceptions.

Detailed Description of the Questionnaire
Since the college students' theory of note-taking consists of different parts (see Sect. 3), the questionnaire consisted of different item blocks -each focusing on one part of the framework.
Items on the Goals of Note-taking: Concerning each of the five goals students have when taking notes (see Sect. 3), we constructed three items to find out if the goal is more likely to be achieved in a lecture with guided notes or in a traditional 'chalk talk' lecture. One item asked if the goal is achieved in a lecture with guided notes, one if it is achieved in a traditional lecture, and the third one directly compared the two lecture types. Concerning the first goal of note-taking -"attention" -the three items were, for example, the following: a) When taking notes in a lecture with guided notes, I pay attention. b) When taking notes in a traditional lecture, I pay attention. c) I pay more attention during note-taking in a lecture based on guided notes than in a traditional lecture.
The answer format was a Likert scale ranging from 1 = "Totally disagree" to 6 = "Totally agree".
Concerning the other goals mentioned in the college students' theory of note-taking, the items of type a) focusing on lectures with guided notes are shown in Table 1 (the ones of type b) and c) are analogous to the "attention" example). Concerning the goal organization, we only focused on the issue "holistic representation of the content" because the local structure that is given via the 'definition, theorem, proof' style of teaching was identical in both lecture types.
Items on the Content-structure of Notes: As mentioned in Sect. 1, the literature suggests that a crucial issue regarding students' notes from traditional mathematics lectures is that the notes are mostly verbatim copies of what the lecturer has written on the board. As previous research has suggested that the use of guided notes might address this problem (see Sect. 2.2), we constructed 2 × 3 items focusing on this issue (3 for guided notes and analogously 3 items for a traditional lecture). The three items for a lecture with guided notes are shown in Fig. 2.
Items on Contextual Factors Affecting Note-Taking: According to the college students' theory of note-taking, an important factor influencing note-taking is the speed of the lecture (see Sect. 3). As mentioned in Sect. 1, the literature suggests that the speed in mathematics lectures is rather high. This can lead to the problem that students feel rushed, merely concentrate on writing everything down correctly, and can no longer think about the information presented (Freitag, 2020). Since students have to write down less in a lecture with guided notes, the problem might be less common in this case. We checked this with the 5 items in Fig. 3 (analogously 5 items were constructed for a traditional lecture).
Post-class Processing of the Notes: To find out if the students post-class processed their notes taken in a lecture with guided notes differently, we constructed the following item for both lecture types: During the semester, I usually use my notes outside class as follows (multiple answers are possible): a) I don't use them. b) I read through the notes again. c) I copy the notes verbatim again. d) I complete my notes by adding additional comments.  We did not ask explicitly for note-taking in the questions 1) to 4) because we wanted to get broader answers which also refer to factors of the lecture that influence students' note-taking, for example the speed (we suspected that students might only associate the process of writing down with the term 'note-taking'). The term 'traditional lecture' was defined on the questionnaire as a lecture in which the instructor writes all the definitions, theorems, and proofs on the board while reading them out aloud, and gives additional oral comments.

Data Collection
The questionnaire was administered to the prospective grammar school mathematics teacher students described in Sect. 4.1 ( N = 70 ) in the 11th week of the semester (of 15 weeks). These students had experienced a lecture based on guided notes in the course "Introduction to mathematical thinking and working" and had attended at least one traditional chalk talk lecture (linear algebra).

Data Analysis
Reliability Analysis for the Likert Items: We checked the reliability of our measurements by grouping together on a scale those items that focused on the same issue (in this case, the same aspect of the college students' theory of note-taking) and by checking their consistency, as recommended in the psychology literature (Moosbrugger & Kelava, 2012). For this, researchers often calculate the quantity "Cronbach's " for each scale, which yields a lower bound for the reliability of the scale under the assumptions of the classical test theory (see Krauth (1995) for mathematical details). However, if a scale consists of just two items, tends to underestimate the reliability, as Eisinga et al. (2013) have shown in simulations, in particular if the variances of the two items differ. They therefore suggest another estimate for the reliability of two-item scales, which has been shown to be more accurate in their simulations: the Spearman-Brown coefficient. Therefore, we will report this coefficient instead of Cronbach's when the scale consists of only two items.
The scales we considered to check the reliability resulted from the construction of the questionnaire, which consisted of three blocks of items focusing on different aspects of the college students' theory of note-taking. The first block consisted of 5 × 3 ( = 15 ) items investigating students' perceptions of whether they were more likely to achieve the five goals mentioned in the college students' theory of notetaking in a lecture with guided notes or in a traditional lecture. For each goal, one item focused on a lecture with guided notes, one on a traditional lecture and one on a direct comparison (see Sect. 5.1). To check the reliability of the measurements obtained with these items, we calculated for each goal the difference between the item focusing on guided notes and the item focusing on a traditional lecture, and checked if the results were consistent with the results of the item asking for a direct comparison between the two lecture types (considering these as a two-item scale). The corresponding Spearman-Brown coefficients that yield estimates for the reliability of these scales are shown in see Table 2. All coefficients are above 0.7, which can be viewed as acceptable (Moosbrugger & Kelava, 2012).
The second block consisted of 2 × 3 items focusing on the question of the extent to which students considered oral comments: 3 for lectures with guided notes and 3 for traditional lectures (see Fig. 2). The values of Cronbach's for the resulting 3-item scales were = 0.639 for lectures with guided notes and = 0.798 for traditional lectures (after inverting the third item, see Fig. 2). The final block of Likert items consisted of 2 × 5 items (see Fig. 3) focusing on the question of whether there was enough time for students in the lecture to listen and think about the information presented besides writing. The values of Cronbach's for the resulting 5-item scales were = 0.905 for lectures with guided notes and = 0.895 for traditional lectures (after inverting items 2-5, see Fig. 3).

Analysis of Responses to the Open Questions:
We categorized these by means of content analysis (Mayring, 2015), used to analyze the content of communications systematically on the basis of clear coding rules (Mayring, 2015, p. 13). In this method, the communication is broken down into parts that are assigned to certain categories. The categories can be defined on the basis either of a theory or of parts of the data. After the categories have been defined, criteria are formulated under which a certain part of the communication is assigned to a particular category. These criteria form the coding instructions with which the whole material is then analyzed. If it becomes clear during the analysis that the categories did not capture all the information represented in the material that contributes to answering the research question, the category system is refined, and the analysis is conducted again.
We used this method to analyze the written responses to our open questions. Since we wanted to investigate with these how the use of guided notes influenced students' note-taking on the basis of the college students' theory of note-taking, we used the different aspects of the framework (see Sect. 3) as categories. A description of the resulting coding rules can be found in Table. 3. The authors coded the data separately first, then discussed their results, and finally resolved disagreements. The student refers to characteristics of the lecture (like speed) that can influence note-taking Post-class processing of the notes The student mentions how the notes are used after class Figure 4 shows the distributions of the 5 × 2 Likert-items investigating students' perceptions of whether they could achieve the five goals of note-taking mentioned in the college students' theory of note-taking in the two lecture types (for the formulation of the 5 items in the case of guided notes, see Table 1). Wilcoxon signed rank tests showed that these distributions as well as the distributions of the average over the five goals differed significantly, with p < 0.01 (Table 4). This indicates that students feel they are more likely to be able to reach these five goals of note-taking in a lecture with guided notes than in a traditional lecture. Nevertheless, one has to be cautious about generalizing from these results because the participants only experienced one specific course using guided notes. Figure 4 furthermore indicates that the use of guided notes supported our participants in particular to maintain attention and to gain an understanding of the content during the lecture. Possible reasons have already been described in Sect. 2.1: guided notes allow students to write less, so that they can concentrate more on the lecturer's explanations. Furthermore, the blanks can be used as prompts for interaction in class (and were used for this in our participants' guided notes course; see Sect. 4.2). Both issues could have raised our participants' attention and fostered an understanding of the material, for example because they were already able to engage with the content during the lecture.

Results of the Items Focusing on Students' Content-structure of the Notes
We focused our investigation on the issue of whether the students considered oral comments during note-taking more in a lecture with guided notes. The results are shown in Fig. 5 (for the formulation of the items in the case of guided notes, see Fig. 2). Wilcoxon signed rank tests again showed that the distributions differed significantly between the two lecture types with p < 0.01 (Table 5). Table 5 indicates that students tend to consider oral comments in a lecture with guided notes more than in a traditional lecture. However, Fig. 5 illustrates that many of our participants still just noted what the instructor had written down. One important reason for this was already found by Iannone and Miller (2019): students often think that only what the instructor writes is relevant for the exam. Other reasons might be that students may be unsure about which of the comments they should note, or have doubts about whether their own comments are really correct.

Results of the Items Focusing on Contextual Factors that Influence Students' Note-taking
We investigated if the use of guided notes might help overcome the problem that there is often not enough time in mathematics lectures to listen to the lecturer's explanations and to think about the information presented besides writing. The results of the corresponding items are shown in Fig. 6. Again, Wilcoxon signed rank tests showed that the distributions between the two lecture types differed  significantly, with p < 0.01 for the 5 individual items and for the average over all these items (after inverting items 2-5). Figure 6 and the corresponding Wilcoxon tests indicate that the use of guided notes can address the problem that mathematics lectures proceed so fast that students can only concentrate on copying down correctly everything the instructor has written on the board (at least according to students' perceptions). A possible explanation for these results could be that in a lecture using guided notes students have to write less, and are therefore not busy only with writing. Instead, they have more time to grasp and think about the information the instructor presents. The responses to the open questions will provide more detailed explanations.

Results Concerning Post-class Processing of the Notes
The results for the question of which activities the students engaged in with their notes at home are shown in Table. 6 (the question was given with answer options; see Sect. 5.1). As can be seen there, fewer students claimed to engage in the post-class processing activities listed on the questionnaire in the case of a lecture with guided notes. The biggest difference can be observed for the activity of copying the notes verbatim again. This suggests that the use of guided notes makes it unnecessary for some students to create a new version of the notes at home. Overall, the lower percentages for all activities and the higher proportion of students not using their notes at home in the case of guided notes suggest that students tend to process their notes at home less, or at least less intensively, in a lecture with guided notes. An explanation might be that students are more likely to feel they have gained an understanding of the content already during the lecture (see Fig. 3 for our participants). Other explanations will become evident in the responses to the open questions.

Results of the Open Questions
We categorized the responses according to the aspects of the college students' theory of note-taking the students referred to in their responses (Table 3). The results can be seen in Fig. 7. We will now present in detail students' responses to the aspects, to which at least 10% referred, and which can explain how the use of guided notes influenced the corresponding aspect of students' note-taking.

Goals of Students' Note-taking
According to the college students' theory of note-taking, students pursue the following five goals when taking notes: 1) Increase attention, 2) Gain an understanding of and memorize the material presented, 3) Organize the material presented, 4) Study aid, and 5) Homework aid (see Sect. 3).
To the Goal "Attention": Firstly, Fig. 7 illustrates that many students referred to this aspect in the questions on what they liked about guided notes or disliked about a traditional lecture. A possible explanation is that in a lecture with guided notes Table 6 Activities that students engaged in with their notes at home; answers were given as options, and multiple answers were possible (N = 70)

Activity
Lecture with guided notes students were not only busy with writing, but could also follow the lecturer's explanations, as the following statement about guided notes suggests: "You are not only busy with writing. You have time to follow the lecturer's explanations attentively." Nevertheless, they still had to write something, which could have helped to keep the attention up: "Since you have to write for yourself, your attention remains, but you have enough time to gain an understanding of the issues." However, Fig. 7 suggests that the use of guided notes also impacted the attention of some participants negatively, for example because they had time to get distracted, as the following response to "What do you dislike about the use of guided notes in lectures?" illustrates: "The attention can decrease because there remains time for other things like gossip, mobile phone, etc." To the Goal "Understanding": Besides the potential for enhancing students' attention, Fig. 7 shows that many students believed that guided notes helped them gain an understanding of the content. One explanation is that their usage allowed them to really follow the instructor during the lecture: "You do not have to write so much, and have time to follow the instructor, and to understand the content." Furthermore, students claimed to have time to think about the content in a lecture with guided notes: "You save time for being able to recapitulate and make sense of the content already during the lecture." A third issue relating to guided notes mentioned by the students, which can help them reach the goal of "understanding", refers to mistakes. Students mentioned that in a lecture with guided notes fewer mistakes occur (either made by the instructor or while copying).
However, our data suggest that also with regard to the goal "understanding", guided notes can be disadvantageous for some students. First, four students mentioned explicitly that the use of guided notes made it more difficult to make sense of content, as the following sample response illustrates:

"You only write the proof by yourself, but not the statement to be proven. = > sometimes less comprehensible what the statement is, and what needs to be shown."
Hence, the provision of information in preprinted form that is crucial for comprehension might hinder students from gaining an understanding of the material.
Furthermore, nine students (12.3%) claimed, in their responses to question 3 asking what they liked about traditional lectures, that having to write helped them memorize the content (memorization was not mentioned in questions the 1 and 2 focusing on the guided notes): "You have written down all the content for yourself. This helps to memorize and later to remember it." Hence, although the use of guided notes might be beneficial for students to make sense of the content already during the lecture, it might also hinder them from recalling the material already preprinted.
To the Goal "Organization": The responses to the open questions indicate that the use of guided notes helped many students organize the material presented (37.1%, for example, referred to this goal in question 1: "What do you like about the use of guided notes?"; see Fig. 7). The students either referred to the "good structure" of the guided notes script and/or claimed that their notes were neater than in a traditional lecture, as the following example illustrates: "The notes based on a guided notes script are more structured and wellarranged because the notes taken in a traditional lecture are more like a continuously written text." However, the responses of 15.7% of the students to the question of what students disliked about guided notes suggest that the presence of preprinted parts can also make it harder to organize notes, for example due to not enough space provided by the blanks, as the following response illustrates:

"Too little space for spontaneous additional notes or examples."
Furthermore, three students emphasized that what they liked about traditional lectures was that they could organize the material exactly how they wished, as the following quote makes clear: "If you have to take only notes for your own, you can construct your 'own' script, for example with comments or similar." This indicates that the size and the arrangement of the blanks play an important role, so that guided notes might help or hinder students in organizing the information presented.
To the Goals "Study Aid" and "Homework Aid": The students rarely referred to these goals in their responses (see Fig. 7), so we refrain from a detailed discussion of the corresponding responses.
Overall, the responses to the open questions suggest that the use of guided notes in particular helps students achieve the note-taking goals "attention", "understanding" and "organization", but how to arrange the blanks in the script needs to be carefully considered beforehand. Figure 7 shows that only a few students referred in their responses to the content and the structure of their notes (more than 10% referred to this aspect only in question 1). Except for three students mentioning that in a lecture with guided notes they did not write down everything the instructor said, the other students referring to this aspect stated that the guided notes allowed them also to note down some of the instructor's oral comments or to add own comments, as the following quotes show:

Content-structure of Students' Notes
"Time to add further notes and comments of the instructor." "There remains time to think, and you have time to add additional comments that might help to gain an understanding." Overall, the low number of students referring to the aspect "content-structure of notes" in their responses to the open questions suggests that either the influence of the guided notes on this aspect was small, or that our students did not consider this influence important enough to mention. Figure 7 suggests that many of the students perceived an effect of guided notes on contextual factors that influence note-taking. First, students often referred to the factor "speed" that is also mentioned explicitly in the college students' theory of notetaking. 30.0%, for example, perceived the speed of traditional lectures as too high or felt rushed in such lectures, as the following quote illustrates:

Contextual Factors Affecting Students' Note-taking
"Quicker advancing [in traditional lectures]. Less time to follow explanations. More post-class processing necessary." On the other hand, 11.4% referred to the factor "speed" positively in question 1: "What do you like about the use of guided notes in lectures?" Nevertheless, some students also criticized the speed in a lecture with guided notes, and claimed that even more content was covered there within one lecture.
However, our data indicate that there were also other relevant characteristics of the lecture influencing students' note-taking that were affected by the use of guided notes. One factor is "the amount to write", which in mathematics lectures is mainly determined by the instructor, because the students mostly copy everything the instructor writes (see Sect. 1). If this amount is too great, students might be busy only with writing, as the following answer to "What do you dislike about traditional lectures?" indicates: "Too much is written on the board, so that you cannot follow anymore. Mostly, also the speed is too high (much information requires much time for copying)." Guided notes have the potential to address this problem. If students need to write less during the lecture, they could use the time to listen to the lecturer and/or think about the content, as the following answer to "What do you like about the use of guided notes in lectures?" indicates: "You do not have to write that much, and can more concentrate on the lecturer's explanations." Our data suggest that the impact of guided notes on the factor "amount to write" was even more relevant to the students than the impact on the actual lecture speed because they referred to this factor more often in their responses to what they liked about guided notes (see Table 7).
Several students also referred in their responses to the teacher-student interaction. For example, 5.7% referred to it in the open question on what they liked about the use of guided notes. These students either appreciated that there was interaction in the guided notes course or emphasized the opportunity to pose questions there. On Table 7 Proportion of students referring to the factors "speed" and "amount to write" in their answers to the open questions 1-4 (N = 70) the other hand, 11.4% criticized a lack thereof in traditional lectures. However, no student mentioned explicitly if and how the interaction actually influenced his or her note-taking. Nevertheless, there can be an influence because interaction can lead to a higher engagement with the content in class, which might, for example, stimulate students to also note own additional comments. Finally, some participants mentioned further factors of the lecture that can influence students' note-taking. These were: • The instructor's ability to handle the media used, for example to produce readable notes on the visualizer in the case of guided notes • The number of explanations provided besides the written content.
But since only few participants referred to these in their responses (less than 5%), we will not discuss these in detail here.
Overall, the responses to the open questions indicate that the use of guided notes affects contextual factors of the lecture that influence note-taking like "speed" or "amount to write". In particular, the amount to write is strongly affected by the presence of guided notes because students do not have to write down all the content themselves. Figure 7 indicates that the need for an intensive post-class processing of the notes is a critical issue in traditional lectures (10% of the students referred to this aspect in their answers to the question "What do you dislike about traditional lectures?"). Students, for example, perceived this necessity in order to be able to make sense of the content or to produce a "clean" version of the notes:

Post-class Processing of the Notes
"You are only busy with writing and need to invest much time for post-class processing to gain an understanding." "Partly messy notes, so that you need to invest time to copy the notes again." As already mentioned, the use of guided notes could address these problems. Since students have to write less, they can use some of the lecture time to already make sense of the content in class and take more accurate notes, as the following response to question 5, asking for differences in students' note-taking between a lecture with guided notes and a traditional lecture, illustrates: "You have fewer mistakes in your notes because you have time to think about the content already during the lecture. Furthermore, you can write more neatly and do not need to produce a second version of the notes at home." Furthermore, three students mentioned that the use of guided notes allowed them to also post-class process a lecture if they had been absent.
Overall, the responses to the open questions indicate that the use of guided notes also has an influence on students' post-class processing of the notes. The responses in particular suggest that some students might post-class process a lecture with guided notes less intensively than a traditional lecture.

Summary
As mentioned in the literature review, previous research with small samples has indicated that the use of guided notes can address some problems concerning students' note-taking in mathematics lectures. The studies conducted especially suggested that in mathematics lectures with guided notes, students do not only have to concentrate on writing, but instead can also follow the instructor attentively during the lecture. Furthermore, the use of guided notes can affect students' note-taking behavior in such a way that they might be more likely also to note the instructor's oral comments, which often contain information that is crucial for making sense of the formal content.
We investigated with a larger sample the degree to which guided notes can support students' note-taking in mathematics lectures. For this, we examined to what extent and how the use of guided notes affects the different aspects of students' notetaking as presented in the college students' theory of note-taking (Van Meter et al., 1994). A summary of the most important results is shown in Table 8.
Overall, our study indicates that the use of guided notes can in particular address the problem that, in traditional mathematics lectures, students are often only busy with writing and cannot follow the lecturer's oral explanations. Furthermore, it suggests that significantly more students also note down the instructor's oral comments in a lecture with guided notes compared to a traditional lecture (but still not the majority). However, the use of guided notes can also lead to new problems, which instructors should be aware of: an inappropriate choice of the positions and sizes of the blanks can make it harder for students to organize their notes, and if students overestimate the understanding acquired during the lecture, they might invest too little time for post-class processing.

Discussion and Outlook
Although our study illustrates important potentials of guided notes concerning students' note-taking, it has several limitations that need to be discussed, and that can be starting points for future research.
First, our data only yield students' perceptions. Although many students stated that the use of guided notes helped them gain an understanding of the lecture content already during the lecture, it is not clear if this actually was the case. Hence, in future research, more objective measures need to be used to investigate this issue. This is especially important because some of our students reported that they processed their notes less intensively at home after class in a course using guided notes. Guided notes can particularly help students reach the goals "attention" and "understanding" The distributions of the Likert items on these two aspects differed substantially between the two lecture types, and many students referred to these aspects in the open question on what they liked about guided notes. In particular, students claimed to be better able to follow the lecturer and make sense of the content during a lecture with guided notes because they did not have to concentrate on writing continuously Guided notes can help, but also hinder students in organizing their notes Students claimed that the preprinted parts of the script provided a good basic structure. However, some claimed that the blanks were sometimes too small to personalize their notes adequately Contextual factors affecting students' note-taking Guided notes can affect the speed of a mathematics lecture According to the Likert items, the students tended to perceive that they had enough time for listening and thinking about the content in a lecture with guided notes, and many students referred to "speed" in the open questions on what they liked about guided notes or disliked about traditional lectures Guided notes affect the amount to write in a mathematics lecture, which influences students' note-taking as well Many students claimed that since the amount to write (which is usually determined by the amount the instructor writes) is less in a lecture with guided notes, they could also focus on other issues of the lecture that are relevant for their note-taking, for example the instructor's oral explanations Content-structure of students' notes The use of guided notes affects the content-structure of the notes taken for some students In the open questions, only a few students referred to this aspect by mentioning that they could also note down the instructor's oral comments in a lecture with guided notes. In the Likert items directly focusing on this issue, however, the corresponding distributions differed significantly between the two lecture types Post-class processing of the notes Guided notes can lead to less intensive post-class processing of the notes Some students claimed that they needed to invest less time for the post-class processing of a lecture with guided notes because they could already gain an understanding of the content during the lecture or that they could already produce neat notes in class A similar limitation concerns the issue "consideration of oral comments". Although more students said that they had also considered and noted the instructor's oral comments, it is not at all clear to what extent this was the case. Here, it would be important to determine, in a lecture with guided notes, at which points in the lecture students actually note oral comments and of what type these are, in order to find out if the use of guided notes can really help students to capture essential information just presented orally. The second point concerns the role of the interaction in the course. The responses to the open questions indicated that there was an influence of the guided notes on the teacher-student interaction in class, which can again influence students' note-taking. However, since our data cannot specify this influence, an interesting issue for future research might be to investigate the effect of guided notes on the interaction in a mathematics course, and the influence of interaction on students' note-taking more deeply. But it is not only teacher-student interaction that can be affected by the use of guided notes. Their usage might also influence other teachers' practices, for example the accuracy of their lecture planning. Hence, investigating the influence of guided notes on teachers' practices in general might be a fruitful path for future research.
Finally, although our participants rarely referred to the content, some responses to the open questions indicated that the usefulness of guided notes may depend on the content to be learned. One student, for example, claimed to have understood a proof less because the assumption had already been preprinted. It would therefore be important to explore the influence of guided notes on the learning of specific content, for example for finding out how preprinted material and blanks should be chosen within a lecture, so that such problems would be less likely to arise.
Nevertheless, our study illustrates that many students would be likely to benefit from the use of guided notes in mathematics lectures for their note-taking, in particular because they do not only have to concentrate on writing, and have instead time to follow the lecturer's explanations and to consider her/his oral comments, which might help them gain an understanding of the content already during the lecture.
Funding Open Access funding enabled and organized by Projekt DEAL.

Conflict of Interest There is no conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.