Facilitating note-taking with guided notes – students’ preferences regarding positions for blanks and preprinted parts

Abstract

In traditional mathematics lectures given on the board, students are often busy taking notes and have trouble listening to the instructor simultaneously. One approach to address this problem is the use of guided notes. These are parts of the instructor’s notes containing blanks at certain positions. Such guided notes can help students process information presented during the lecture, because they have to write less and can be more attentive towards the instructor’s explanations. Furthermore, the blanks provide opportunities for fostering students’ active engagement with the content presented. But which positions of a mathematics lecture might be suitable for blanks in guided notes? While there exist some proposals from instructors, students’ perspectives have been rarely considered, although they are the ones who need to actively process the information presented in the lecture during their note-taking. We therefore investigated at which positions of a mathematics lecture students appreciate blanks within guided notes, and why they perceive the blanks at these positions as helpful for their note-taking. The results provide some indications about suitable positions for blanks or preprinted parts in guided notes, and explain why students might benefit from these during their note-taking.

KEYWORDS:

• Guided notes
• note-taking
• mathematics lectures
• number theory

1. Introduction and motivation of the research

In mathematics lectures, one important student activity is usually note-taking (Weber, 2004). This can help students process the material presented, and store important information, which are two major functions of note-taking (Di Vesta & Gray, 1972). These two benefits of note-taking are backed up by numerous empirical studies showing that students can recall more information from a lecture after having taken notes – with or without later review (Kiewra, 1989; Kobayashi, 2005, 2006). Some studies even show that students who have taken notes in a lecture or while reading a text perform better in tasks that require a deep understanding of the topic covered, for instance, in transfer tasks (Peper & Mayer, 1978, 1986; Slotte & Lonka, 2001; Titsworth, 2001). A theoretical explanation for this result was provided by Peper and Mayer (1978) who argued that note-taking stimulates encoding activities, namely, organising and aligning the information presented with past experiences to integrate them into their own cognitive structure.

However, concerning mathematics lectures, there are indications in the literature that many students can or do not encode the information presented during their note-taking. Since they have to concentrate on copying from the board, they have trouble being attentive towards the instructor’s explanations (Freitag, 2020). Furthermore, they mostly just make a verbatim copy of what the lecturer has written on the board (Fukawa-Connelly et al., 2017), which indicates that they do not carry out activities that foster encoding, such as paraphrasing or making connections to their prior knowledge. In addition, the note-taking behaviour Fukawa-Connelly et al. (2017) observed shows that students rarely store the instructors’ oral comments, which are, however, often essential for making sense of the formal content.

One approach to address these problems while still keeping the abovementioned potentials of note-taking is 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). If parts of the information conveyed during a lecture are already provided to students in print, they do not have to concentrate on writing exclusively anymore and can be more attentive towards the instructor’s explanations (Feudel & Panse, 2022; Iannone & Miller, 2019; Krapf & Pfefferkorn, 2022). Furthermore, the blanks provide opportunities for student activities that foster an active engagement with the content (Alcock, 2018; Feudel & Panse, 2022; Krapf & Pfefferkorn, 2022). Hence, guided notes can help students in multiple ways in the encoding process during their note-taking, which can support them in their learning as described above. Compared to other teaching innovations, guided notes have the advantage that instructors who are used to lecturing do not need to change their whole mode of teaching, as required in many other innovative teaching approaches, for instance, a flipped classroom. Guided notes are therefore ‘low in costs’ for many instructors.

However, past research suggests that the effect of handouts like guided notes on students’ note-taking and learning depends on their format (e.g. Hartley, 1976; Kiewra et al., 1995). Regarding guided notes, this especially refers to the positions of the blanks, as indicated in our own past research on perceived benefits of guided notes, in which, for instance, one student claimed that a theorem is less comprehensible if only the proof needs to be written while the statement to be proven is already printed (Feudel & Panse, 2022). Therefore, a fundamental question is where to put the blanks within guided notes so that many students benefit from them during their note-taking.

While there exist some proposals of instructors on where to put the blanks in guided notes (Alcock, 2018; Kotsopoulos et al., 2022; Krapf & Pfefferkorn, 2022; Montis, 2007; Tonkes et al., 2009), students’ views on this issue have been rarely considered, although they are the ones who need to process the information presented in the lecture actively while taking notes. We therefore conducted a mixed-methods study with the research question ‘At which positions of a mathematics lecture do students appreciate blanks in guided notes, and how might blanks at these positions support them in their note-taking according to their perceptions?’. The study especially enhances the discussion about suitable positions for blanks in guided notes with students’ perspectives, which can provide further ideas regarding the design of guided notes that might help students encode and store the information presented. Furthermore, the findings are practically useful for instructors who wish to implement guided notes.

2. Literature review and embedding of our research

2.1. Effects of guided notes in tertiary mathematics courses

There are some studies that suggest a positive effect of guided notes on students’ achievement (Cardetti et al., 2010; Kotsopoulos et al., 2022; Montis, 2007). All these studies compared classes with guided notes and similar classes without guided notes. The classes with guided notes always performed better – even if the ‘control class’ scored lower in an entry test (Cardetti et al., 2010).

As highlighted by Kotsopoulos et al. (2022), it has not yet been extensively studied how guided notes help students learn mathematics (p. 1633). However, the three studies above yield some ideas about how the guided notes provided supported the students, as the authors asked them what they liked or disliked about guided notes or about class in general (Cardetti et al., 2010; Kotsopoulos et al., 2022; Montis, 2007). The students then especially mentioned the following benefits of the guided notes:

• With the help of the guided notes, they could be more attentive to the instructor and think about what the instructor said instead of having to concentrate on writing permanently.

• The guided notes were interactive, i.e. the students worked together with the instructor on them.

• The organisation and neatness of the guided notes helped them create notes that were useful in a later review phase.

These statements indicate that the guided notes provided helped the students especially in two ways compared to traditional lectures:

1. They supported them in encoding the information presented during the lecture by allowing them to listen besides writing, and by requiring them to work actively at the blanks.

2. They provided a neat and structured basis for storing the information presented adequately.

This suggests that the guided notes especially helped the students in the two functions of note-taking according to Di Vesta and Gray (1972): encoding and storing the information presented, which are both known to have a positive impact on students’ learning (see Section 1).

Similar results have been found also in further studies investigating explicitly perceived benefits of guided notes in mathematics lectures (Feudel & Panse, 2022; Iannone & Miller, 2019; Krapf & Pfefferkorn, 2022). In the study by Krapf and Pfefferkorn (2022), for instance, 42.1% of the 209 participants perceived that guided notes help to keep the attention up, 27.8% said that these allow simultaneous thinking and writing, and 20.6% claimed that these help to memorise the content. However, a few students also criticised that they cannot follow the lecturer even if guided notes were provided due to a high presentation speed. Furthermore, in our study and Iannone and Miller (2019), there were students who perceived that guided notes might also facilitate losing attention. Nevertheless, the studies just mentioned overall suggest that guided notes can aid many students in the encoding aspect of note-taking.

However, these studies also indicate that guided notes can help students store the information presented on the paper in a convenient way. In our study, for instance, many participants claimed that the printed parts of the guided notes provided a good basic structure to them, or that their notes based on a guided notes script were neater than notes taken in a traditional lecture (Feudel & Panse, 2022). Furthermore, Iannone and Miller (2019) found in a qualitative study of eight students in a course with guided notes that some of them noted down additional oral comments from the instructor. Such additional comments might help students much to make sense of the content again in a later review phase. However, the studies just mentioned also point out some issues that can hinder students in storing the information presented adequately. One is that the size of the blanks might be too small (Feudel & Panse, 2022; Krapf & Pfefferkorn, 2022). Another one is that the parts students have to write themselves can contain mistakes or might be incomplete (Krapf & Pfefferkorn, 2022). Nevertheless, the studies just cited overall indicate that guided notes can help students create notes that might be useful for later study activities such as exam preparation.

Hence, overall, guided notes can help students encode the information presented during the note-taking process in a mathematics lecture, but also store the information in a way that makes the notes taken useful for later study. Both issues can contribute to students’ learning – either during the lecture or in a subsequent review phase.

2.2. Contributions to the question of how to design guided notes for tertiary mathematics courses

As already mentioned in the introduction and indicated in our own prior research (Feudel & Panse, 2022), the extent to which students benefit from guided notes in their note-taking probably depends on the positions of the blanks. However, there only exist few contributions to the question of where to put blanks in guided notes accompanying mathematics lectures thus far.

The first proposals were formulated by Montis (2007) who implemented guided notes into a college algebra course. He especially recommended to leave blanks for procedural exercises. He asked the students of his classes to complete such exercises by themselves first and then to discuss the solutions with classmates. He also left blanks for a justification of steps in calculations or for the final result.

Further proposals regarding the design of guided notes in tertiary mathematics courses were put forward by Tonkes et al. (2009), who implemented guided notes into two core mathematics courses at a university in Australia. They provided in print foundations that needed to be accurate such as definitions, but also narrative passages and linking material. They left blank all examples and calculations because an unfolding of the solution in real-time can initiate an active learning process. The instructor in the study by Kotsopoulos et al. (2022) already mentioned also followed these suggestions. He printed the definitions, the framing of the theorems, and statements of problems, which he named ‘low-level-content’. On the other hand, he left positions blank at which the students were required to work with this content, for instance, proofs and derivations, connections to new ideas, and all examples. Krapf from Krapf and Pfefferkorn (2022), who implemented guided notes into a course ‘Mathematics from a higher standpoint’ for future teachers at a German university, also basically followed these ideas. However, concerning proofs, she proceeded differently, as one of the course’s goals was that students learn how to write proofs. She therefore provided new methods of proof such as induction in print first, then provided in print proof frameworks for the first proof using this new method, and later left blank whole proofs of this kind. Hence, the guidance decreased.

Of particular relevance for this article are proposals for suitable positions of blanks by Alcock (2018), because the instructor in our study followed her ideas. Alcock stated that she always uses guided notes in her mathematics lectures, as these provide lots of opportunities for active participation. She especially leaves blanks at positions at which students are likely to be able to fill these in by themselves using their prior knowledge or analogies. Furthermore, she leaves blanks at claims that students might be able to conjecture by themselves, for instance, the values for $x$ such that $\mathrm{li}{m}_{n\to \mathrm{\infty }}{x}^n$ equals $0$, $1$, or $\mathrm{\infty }$. She also emphasises that the dependence of this limit on $x$ might be not considered important if it is just printed. Sometimes, Alcock also offers options for filling in the blanks and initiates discussions about the right choice. She especially does so in positions in which many students might intuitively have wrong ideas and conceptions about the content.

But at which positions do students consider blanks as useful and why? This question is important because, according to cognitive models of learning (Mayer, 1996; Yilmaz, 2011), they are the ones who need to process the information presented in the lecture actively – during their note-taking and/or in a later review phase using the notes taken. We only found one study taking this question into account (Tonkes et al., 2009). As just said, Tonkes et al. implemented guided notes into two mathematics courses at an Australian university and asked the participants in a survey about whether they appreciated the guided notes provided, filled in the blanks for themselves, and if they would like blanks in definitions, examples, and when introducing new techniques. The answer format was a Likert scale ranging from $1=$ strongly agree to $5=$ strongly disagree. While the students mostly appreciated blanks in examples (mean: 1.84), there was no clear tendency for definitions and introductions of new techniques (means: 3.13 and 2.77). However, Tonkes et al. did not systematically investigate at which positions of a mathematics lecture students appreciate blanks, and their study did not provide reasons for students’ preferences that might help to understand how blanks at these positions might support students in their note-taking. The research we present here attempts to fill this gap.

3. Theoretical framing

The first theoretical framework used in research on note-taking was put forward by Di Vesta and Gray (1972). According to this framework, note-taking has two functions: an external storage function and an encoding function. The external storage function means that the notes a learner takes have the purpose of an external memory that can be used as a reference for later review. The encoding function ‘allows the learner to transcribe whatever subjective associations, inferences, and interpretations occurred to him while listening’ (p. 8). Peper and Mayer (1978) later provided a theoretical explanation on how this encoding process might support learning – the so-called generative theory, according to which ‘note taking encourages learners to actively integrate new information within their own past experiences because these subjects are required to paraphrase, organise, and make sense out of the presented material’ (p. 515). These activities are similar to mental activities individuals need to carry out in learning processes according to cognitive models of learning, namely to select, organise, and integrate new information into their own cognitive structure (Mayer, 1996; Yilmaz, 2011). Hence, note-taking can facilitate learning during lectures by stimulating exactly such encoding activities.

Although the note-taking framework by Di Vesta and Gray (1972) has been used in many empirical studies for interpreting note-takers’ better performance on later tests (Kiewra, 1985, 1989), it ignores that note-taking is often embedded into a whole study process. Therefore, Van Meter et al. (1994) developed a more sophisticated data-grounded framework – the so-called college students’ theory of note-taking – that puts the students as self-regulated learners into the centre (Zimmerman, 1989). For this, they interviewed 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. This interview process consisted of five consecutive phases in which Van Meter et al. (1994) developed and elaborated categories describing different aspects of students’ note-taking.

The first of these aspects (aspect I.) comprises students’ goals of note-taking. Besides the overarching goal of passing the exam, students pursue the following five goals when taking notes (p. 338):

1. Increase attention,

2. Increase comprehension and memory of the lecture content,

3. Organise the material presented,

4. Inform about the content of the exam, and

5. Inform about solutions to practice problems and provide information relevant for written assignments.

In the fifth goal, Van Meter et al. (1994) especially emphasised homework assignments – they also called it ‘homework aid’. We want to view the term ‘written assignment’ broader as any assignment students are required to solve, including the exam, which also fits the data Van Meter et al. presented.

Besides students’ goals of note-taking, the college students’ theory of note-taking contains the following three aspects II.–IV.:

• II. Content-structure of students’ notes: It varies from noting just key terms to verbatim copies of what the lecturer has presented.

• III. Contextual factors affecting students’ note-taking: These include aspects of the lecture style like speed or the lecture’s structure, the content covered, students’ characteristics like their know-how in note-taking or their prior knowledge of the content, and the course demands.

• IV. Students’ post-class processing of the notes: It varies from not using the notes at all after class over reviewing them to creating a newly written revision.

We used the college students’ theory of note-taking in our study as the main perspective for analysing our data, because the framework has been shown to describe students’ note-taking behaviour in mathematics lectures adequately (Freitag, 2020). Furthermore, the notions used there are based on students’ perceptions, which are also the focus of our study on why students perceive blanks or preprinted parts at certain positions within guided notes as beneficial for their note-taking. However, to also get some ideas about how guided notes might aid students cognitively during their learning, we will additionally reflect on our results from the perspective of the encoding/storage framework by Di Vesta and Gray (1972) in the discussion section.

4. Context of the study

The study took place at a midsize university in Germany in a proof-oriented course named ‘Introduction to mathematical thinking and working’. Its participants were future mathematics teachers for upper secondary level. The topics covered were: elementary number theory (divisibility, prime factorisation, residue classes, Euclidean algorithm), sets and relations, basic algebraic structures (groups, rings, fields), and the construction of the number systems. The course lasted 14 weeks. It consisted of one 90-minute lecture and one 90-minute tutorial per week. Furthermore, the students were obliged to solve written assignments each week and to take part in three tests. For admission to the final exam, they needed a certain number of points in these two components.

Inspired by Lara Alcock from Loughborough University in the UK, the instructor used guided notes in the lecture. This means that she distributed a set of notes with blanks at certain positions before each lecture that she filled in during the lecture live on a visualiser. After the lecture, she made the completed notes available to the students. A sample page of these guided notes can be seen in Figure 1¹.

Figure 1. Sample page of the guided notes used (see also Feudel & Panse, 2022, p. 15).

For choosing the positions of the blanks, the instructor followed the ideas presented in Alcock (2018) mentioned in Section 2.2. She especially left blanks at positions

1. at which she assumed that the students could fill in the blanks by themselves, for instance, the properties from proposition 7.5. in Figure 1, which the students might be able to conjecture themselves by drawing upon their prior knowledge on integers from school,

2. that contained information she regarded as particularly important, for instance, defining properties in a definition such as the ones of a commutative ring in Figure 1.

3. at which she suspected particular difficulties or that required additional explanations, for instance, at certain steps in a proof.

Small proofs that the students were expected to carry out by themselves were even left completely blank (see Figure 1).

Similar to Alcock (2018), the instructor also initiated in-class activities at the blanks. She, for example, let the students fill in blanks alone by themselves first on the basis of their prior knowledge, initiated peer discussions about how to fill in certain blanks, or she let the students carry out additional activities that can help them read formal mathematics presented at the blanks, such as generating examples.

5. Methodology of the study

5.1. Description of the instruments

For investigating at which positions of a mathematics lecture students appreciate blanks in guided notes and why, we designed a survey with two parts. The first part consisted of the following task:

The instructor of the course ‘Introduction to mathematical thinking and working’ wants to use guided notes in the next semester. She informs you about this idea and adds that she is not sure about where to put the blanks. A mathematics lecture consists of different elements (definitions, theorems, proofs, explanations, and examples). Please give the instructor an advice on where to put the blanks on the basis of your experience with guided notes. Justify your advice.

With this task, we wanted to get first ideas about the positions of a mathematics lecture at which our participants appreciated blanks and why.

The second part of the survey contained a text on elementary number theory covering EULER’s $\phi$-function. It was designed by the second author as a part of her dissertation project (Panse, 2024), and was based on paragraphs introducing EULER’s $\phi$-function that she found in a mathematics textbook for future teachers (Scheid & Schwarz, 2016). The text is shown in Figure 2.

Figure 2. Text on EULER's $\phi$-function that we used to explore at which positions students appreciate blanks and why.

She chose the text for this study for two reasons:

1. It contains all the different typical elements of a mathematics lecture: a definition, examples, a theorem, a proof, and some additional remarks (here: remarks about EULER).

2. The topic ‘EULER’s $\phi$-function’ is rather basic, and the text especially does not require much prior knowledge except the basics of divisibility and modulo calculation in order to understand it.

The students then received the following task:

The instructor now asks for concrete help for the preparation of the guided notes. For this, she gives you a mathematical text. She asks you to mark the passages that should be kept in print with a highlighter and to cross out the passages that should be left blank. She also asks you to justify your choices.

This task had the aim to substantiate our participants’ general ideas on where to put blanks in guided notes with specific examples. For reducing heterogeneity in the data, they additionally got the request to structure their response with ‘blank in line … , because … ’ and ‘text should remain in line … , because … ’.

5.2. Data collection

The survey was administered in paper form in the course described in Section 4 in a lecture three weeks before the final exam. Hence, the students had already gained some experience with guided notes during the semester. No further personal data was collected to guarantee anonymity. In addition, the students were informed about the research purpose, that participation was voluntary, and were asked to consent to the publication of the data in an anonymous form. 61 submitted a completed survey including consent. This covered all but one participant of the lecture on the day of survey. Since the time of survey was rather late in the semester, the students have been probably already influenced (much) by the teaching style of the course. Hence, one has to be careful with generalising our results.

5.3. Data analysis

The data analysis consisted of three steps. The first two steps can be considered as predominantly quantitative, while the last one was a pure qualitative analysis.

5.3.1. Description of the first two quantitative steps

Analysis in which lecture elements the participants appreciated blanks

In the responses to the general task to decide in which of the lecture elements ‘definition, theorem, proof, example, explanation’ to put blanks in guided notes, we coded for each of these elements and each participant whether she/he wished blanks in it, liked it to be printed or did not mention it. For the text on EULER’s $\phi$-function, we proceeded similarly. We coded for each of its elements ‘remarks about EULER, definition of the $\phi$-function, example 1, example 2, example 3, theorem on the multiplicativity, proof of the theorem’ (see Figure 2) whether the students highlighted it as to be printed or crossed something in it out as a blank.

Analysis of the extent, to which the students appreciated blanks in the different lecture elements

We first categorised the extent to which the students appreciated blanks in the elements ‘definition, theorem, proof, example, explanation’ in the first general task on where to put blanks. For this, we developed a category system on the basis of the first ten cases (Table 1). We chose such an inductive approach for categorising the extent, because we had not asked for an extent in the task explicitly, and did not know if and how students might mention such an extent. After the development of the categories, the two authors coded the whole data with this system separately and resolved disagreements in a discussion.

Table 1. Coding categories for the extent to which the students appreciated blanks in the different lecture elements.

Category	Description of the category
Leave the element completely blank	A student mentions explicitly that the whole element (e.g. whole definitions) should be left blank.
Leave blanks at some positions in the element	A student mentions that parts of the lecture element (e.g. parts of definitions) should be left blank, while other parts should be printed.
Leave blanks in some elements of this type	A student mentions that blanks should be left in some elements of a certain type (e.g. in some definitions).
Leave blanks, but no extent mentioned	A student states that the element should contain blanks, but does not mention an extent.
Print the element fully	A student mentions that the element should be printed.

In the text on EULER’s $\phi$-function (see Figure 2), we pursued a deductive approach. In an a-priori analysis, we split the elements ‘remarks about EULER, definition, example 1, example 2, example 3, theorem, proof’ into contextual submodules (see Table 2). We then coded for each submodule whether the students crossed it fully out, crossed parts of it out, or highlighted it as to be fully printed.

Table 2. Splitting of the elements in the text on EULER’s $\phi$-function (see Fig. 2) into different contextual submodules.

Element	Contextual submodule	Location in the text (Fig. 2)
Remarks about EULER	(No splitting)	ll. 1–8
Definition	Framing condition	‘n$\in \mathbb{N}$’ in l. 9
	Denotation	‘$\phi (n)$’ in l. 9 and ‘$\phi$ is called EULER’s $\phi$-function’ in l. 10
	Defining property	‘number of positive integers from the set $\{1,2,\dots ,n\}$ that are relatively prime to $n$’ in l. 9
Example 1	Framing condition	‘$p$ is a prime number’ in l. 11
	Result	‘$\phi (p)=p-1$’ in l. 12
	Derivation of the result	‘all numbers from $1$ to $p$ except for $p$ are relatively prime to $p$’ in l. 13
Example 2	Framing condition	‘$p^{\alpha }$ is a power of a prime number’ in l. 14
	Result	‘$\phi (p^{\alpha })=p^{\alpha }-p^{\alpha -1}=p^{\alpha }(1-1/p)$’ in l. 15
	Derivation of the result	‘$p^{\alpha }$ numbers from 1 to $p^{\alpha }$, every p-th number is divisible by $p$’ in l. 16
Example 3	Framing condition	‘$p$ and $q$ are different prime numbers’ in l. 17
	Result	‘$\phi (\mathrm{pq})=(p-1)(q-1)$’ in l. 18
	Derivation of the result	‘$p$ of the numbers $\mathrm{pq}$ from 1 to are divisible by $q$, $q$ of these numbers are divisible by $p$, one number is divisible by $p$ and $q$’, and ‘$\phi (\mathrm{pq})=\mathrm{pq}-p-q+1$’ in ll. 19–20
Theorem	Prerequisite	‘$\gcd (a,b)=1$’ in l. 21
	Claim	‘$\phi (\mathrm{ab})=\phi (a)\phi (b)$’ in l. 21
Proof	Idea of the proof	‘We arrange the numbers from 1 to $\mathrm{ab}$ in a rectangle’ and the subsequent rectangle in ll. 22–24
	Line of argumentation	All the arguments in ll. 25–30
	Final conclusion	‘$\phi (a)\phi (b)$ numbers from 1 to $\mathrm{ab}$ that are relatively prime to $a$ and $b$ (and hence to $\mathrm{ab}$)’ in ll. 31–32

5.3.2. Description of the third step investigating students’ reasons for their choices

This step was a pure qualitative analysis, in which we interpreted the reasons for blanks/preprinted parts in guided notes the students mentioned using the college students’ theory of note-taking framework (see Section 3). For this, we categorised their justifications for blanks or preprinted parts at certain positions with content analysis (Mayring, 2015), and used the different aspects of note-taking from the framework – especially the goals of note-taking – as categories. If a student referred to one of these aspects, we assigned the corresponding part of the response to the respective category. Furthermore, we coded the lecture element the student referred to. This shall be illustrated with the following sample response:

Definitions and examples should be left blank → writing these helps to gain an understanding and to memorize.

This answer was assigned to the note-taking goals ‘memorization’ and ‘understanding’ from the framework (see Section 3), and to the lecture elements ‘definition’ and ‘example’. Just as in the quote above, a student’s response could be assigned to multiple aspects of the college students’ theory of note-taking, and to more than one lecture element, if the student referred to more than one simultaneously.

As students’ responses were rather heterogeneous, the aim of this qualitative analysis was to shed light on what possible reasons for blanks at certain positions might be, and not to determine an extent of dissemination of these reasons within our sample.

6. Results of the study

6.1. Results of the quantitative analyses on the preferred positions for blanks

6.1.1. The elements of a mathematics lecture in which our participants appreciated blanks

We first start with the results from the analysis of our participants’ responses in the general task to decide in which of the lecture elements ‘definition, theorem, proof, example, explanation’ to put blanks in guided notes. Table 3 shows for each of these elements the proportion appreciating blanks in it. This shows that the majority liked to have blanks in definitions, theorems, proofs, and examples, while only less than a quarter wished for blanks in explanations. Although the similarity of the numbers in the first four rows might indicate that it was the same students who wanted blanks in all these elements, this was not the case. Only 29.5% wished to have blanks in all four elements. Even in the case of the similar elements ‘definition’ and ‘theorem’, only 50.8% wanted blanks in both. Hence, the students were selective about the elements in which they liked to have blanks.

Table 3. Proportion of students appreciating blanks in the different elements of a mathematics lecture ($N=61$).

Element	Leave blanks	Fully print	Not mentioned
Definitions	67.2%	21.3%	11.5%
Theorems	62.3%	21.3%	16.4%
Proofs	73.8%	9.8%	16.4%
Examples	67.2%	11.5%	21.3%
Explanations	23.0%	39.3%	37.7%

The proportion of students who wished blanks in the different elements of the text on EULER’s $\phi$-function is shown in Table 4. In the element ‘Remarks about EULER’, the percentages do not add up to 100%, as one further category appeared in this element: students who stated that this passage or parts thereof should be excluded from the script entirely (14.8%).

Table 4. Proportion of students wishing blanks in the different elements of the text on EULER’s $\phi$-function ($N=61$).

Element	Leave blanks	Fully print
Remarks about EULER	3.3%	82.0%
Definition of EULER’s $\phi$-function	65.6%	34.4%
Example 1 ($\phi (p)$ with $p$ being prime)	93.4%	6.6%
Example 2 ($\phi (p^{\alpha })$ with $p^{\alpha }$ being a power of a prime)	90.2%	9.8%
Example 3 ($\phi (\mathrm{pq})$ with $p,q$ being different primes)	91.8%	8.2%
Theorem on the multiplicativity of EULER’s $\phi$-function	62.3%	37.7%
Proof of the theorem	86.9%	13.1%

Table 4 elaborates some of the results from Table 3. It especially suggests that the proportion appreciating blanks in examples is higher than Tab 3 indicated, as over 20% did not refer to examples in the general task. Concerning the remarks about EULER, on the contrary, the proportion wishing blanks in it was much lower than for explanations in general. This could be due to the fact that these explanations had the special character of ‘additional information’. Nevertheless, our participants’ choices for the blanks in the text on EULER’s $\phi$-function basically backed up their responses to the general question on where to put the blanks.

Again, different students appreciated blanks in different elements. For instance, only 42.6% liked to have blanks in all elements except the remarks about EULER, and only 50.8% wished for blanks in the definition and the theorem (not always the same participants as in the general question). However, concerning the examples in the text, the students were very consistent: 88.5% wanted blanks in all three examples.

6.1.2. The extent to which our participants appreciated blanks in the different lecture elements

The results of the categorisation of the extent to which our participants wished blanks in the different lecture elements ‘definition, theorem, proof, example, explanation’ in the general task on where to put blanks in guided notes are shown in Table 5. Interesting numbers are marked in bold. The proportions do not add up to 100%, as not all students referred in their response to every element.

Table 5. Extent of the appreciation of blanks in the different elements of a mathematics lecture ($N=61$).

Element	Leave completely blank	Leave blanks at some positions in the element	Leave blanks in some elements of this type	Leave blanks, but no extent mentioned	Fully print
Definitions	23.0%	9.8%	11.5%	23.0%	21.3%
Theorems	8.2%	23.0%	14.8%	16.4%	21.3%
Proofs	11.5%	13.1%	23.0%	26.2%	9.8%
Examples	8.2%	6.6%	13.1%	39.3%	11.5%
Explanations	3.3%	3.3%	–	16.4%	39.7%

Table 5 indicates that the extent to which our participants appreciated blanks in the different elements of a mathematics lecture varied between the elements. For instance, while a substantial proportion of our participants stated explicitly that definitions should be left completely blank, this was not the case for the other elements. In the case of theorems, many wished blanks only at some positions. Concerning proofs, a substantial proportion preferred blanks just in some proofs.

More details regarding the extent to which our participants appreciated blanks at specific positions within the different elements of a mathematics lecture became apparent from the analysis of their markings in the text on EULER’s $\phi$-function from Figure 2. The results are shown in Table 6. Values over 45% are marked in bold.

Table 6. Proportion of our participants who wished for blanks in the different elements of the text on EULER’s $\phi$-function ($N=61$).

Element	Contextual submodule	Leave completely blank	Leave partly blank	Fully print
Remarks about EULER	–	–	3.3%	82.0%
Definition	Overall	26.2%	39.6%	34.4%
	Framing condition	29.5%	–	70.5%
	Denotation	34.4%	24.6%	41.0%
	Defining property	55.7%	9.8%	34.4%
Theorem	Overall	16.4%	45.9%	37.7%
	Prerequisite	19.7%	3.3%	77.0%
	Claim	44.3%	18.0%	37.7%
Example 1	Overall	6.6%	86.9%	6.6%
	Framing condition	8.2%	8.2%	83.6%
	Result	72.1%	11.5%	16.4%
	Derivation of the result	39.3%	31.1%	29.5%
Example 2	Overall	3.3%	86.9%	9.8%
	Framing condition	6.6%	3.3%	90.2%
	Result	63.9%	21.3%	14.8%
	Derivation of the result	36.1%	23.0%	41.0%
Example 3	Overall	3.3%	88.5%	8.2%
	Framing condition	4.9%	3.3%	91.8%
	Result	68.9%	21.3%	9.8%
	Derivation of the result	29.5%	41.0%	29.5%
Proof	Overall	8.2%	78.7%	13.1%
	Idea of the proof	13.1%	44.3%	42.6%
	Line of argumentation	13.1%	67.2%	19.7%
	Final conclusion	24.6%	21.3%	54.1%

Regarding the definition in the text on EULER’s $\phi$-function, Table 6 shows that our participants mostly wished the defining property to be left blank, while they wished the prerequisite to be preprinted. A similar result can be seen for the theorem in the text. Concerning the proof of this theorem, Table 6 shows that a vast majority wished at least some blanks within it. Over 50%, for instance, proposed to leave the equation in line 28 blank (see Figure 2). Finally, concerning the examples in the text, Table 6 shows similar preferences as for the elements mentioned before: they mostly liked the framing conditions printed, the final results blank, and some blanks within the derivations of the results.

Overall, Table 6 suggests that many of our participants wished preconditions in a definition, a theorem, and an example to be printed, while the defining property in a definition/ claim of a theorem/ numerical result of an example should be left blank. Concerning proofs and derivations, Table 6 indicates that many students liked to have some blanks within these, but did not want to write these completely for themselves.

6.2. Results of the qualitative analysis on reasons for the blanks at the positions mentioned

We will now present for the different elements of a mathematics lecture reasons why our participants appreciated blanks or preprinted parts in them, and relate these reasons to the different aspects of the note-taking framework by Van Meter et al. (1994) – the college students’ theory of note-taking. We especially present here reasons that point out well how blanks in a specific element can affect the note-taking aspect.

Definitions

The aspect of our theoretical framework that was mentioned most often as a reason for blanks in definitions was the note-taking goal ‘memorization’, for example, because having to write definitions for oneself helps to memorise them:

Definitions should be written for yourself, as students should memorize these. And if you write something for yourself, it better remains in your memory than if you only read it.

In their responses to the question on where to put blanks in the text on EULER’s $\phi$-function, there were also participants who made explicit, which parts of the definition they wanted to memorise, for instance, the name and the defining property:

Blanks in line 9-10, as the definition and the name are important, and the student is more likely to be able to memorize these by writing. (see Figure 2)

Other goals of note-taking several students referred to when justifying why blanks should be left in definitions were ‘understanding’ and ‘attention’, for instance, because having to write for yourself encourages thinking or being attentive. However, there were also participants who perceived blanks in definitions as a hindrance for gaining an understanding, for instance, due to writing mistakes:

Definitions should be printed, because these are very important, and you can be guided by these. If you must write them for yourself, you might make mistakes.

This quote also indicates why mistakes especially in definitions might hinder understanding: definitions are the basis, and mistakes affect all that follows (one student used the term ‘chain reaction’ for this).

Theorems

Similar to definitions, several participants also mentioned ‘memorization’ as a reason for blanks in theorems in the general question on where to put blanks in guided notes – but much less often than for definitions. In the case of theorems, our participants’ justifications why these should contain blanks were more often related to the note-taking goal ‘understanding’, as the following quote illustrates:

Theorems and proofs are elements of the lecture that need to be understood. Therefore, going through them step by step makes sense and is helpful.

Tables 5 and 6 furthermore suggested that many of our participants preferred the precondition and the framing to be printed, while the claim should be left blank. Some of their justifications also indicate a reason for this, namely that the latter can be conjectured by the students themselves:

You should leave blanks for the core assertion of a theorem (a mathematical formula, a deduction, an equivalence, …), since with the help of the remaining part of the theorem, students could try to deduce this core, so that the idea behind it becomes tangible.

Similarly, our participants’ justifications why the theorem in the text on EULER’s $\phi$-function – especially the claim – should be left blank were often related to the note-taking goal ‘understanding’, for instance, that writing it for yourself fosters gaining an understanding.

A further reason why only the claim should be left blank while the precondition should be printed, can be seen in the following quote:

The beginning can be printed, so that you have a structure of the page. The essential part, however, should be noted.

Hence, printing the precondition might help students structure their notes (note-taking goal ‘organization’).

A considerable number of our participants also wished theorems to be fully printed (see Tables 5 and 6). The reasons mentioned were rather diverse, for example, that not having to write allows listening or that theorems are just facts one needs to acknowledge.

Proofs

Similar to theorems, many of our participants referred to the note-taking goal ‘understanding’ as a reason for leaving blanks in proofs in the general question on where to put blanks in guided notes. However, in connection with proofs, a further goal of note-taking from Van Meter et al. (1994) was mentioned – the note-taking goal ‘homework aid’:

Perhaps, it [filling in the blanks during the lecture] helps for other assignments. You get used to proving because you never did this at school.

However, we have already seen that many of our participants did not want to write all proofs for themselves (see Tables 5). In their justifications, several participants became more explicit here – similar to the following quote:

It depends on the situation. Long and complicated proofs should be printed, but with blanks for additional comments. Small proofs that are also useful for practicing, and which are required by the students, should contain blanks/ be left blank. This helps to internalize the proof structure.

Reasons why our participants did not want to write long proofs completely for themselves were, for instance, that they could then be more attentive or that writing down everything is too much effort.

Concerning the proof in the text on EULER’s $\phi$-function, it was also only a minority that liked to have it completely blank (see Table 6). The majority wished blanks just at some positions, for instance, in formulae as in line 28 (see Figure 2). The reasons mentioned were often related to the note-taking goal ‘understanding’. A student who wished for blanks in the rectangle arrangement of the numbers from 1 to $\mathrm{ab}$, for example, stated:

Kept text in line 22, as this is the beginning of the proof; blanks in line 23, because you can check after the first two steps whether you have understood the proof step. (see Figure 2)

Finally, some of our participants demanded additional space for own comments within or under a proof, so that they can individualise the content-structure of their notes.

Examples

Due to confines of the paper, we refrain from a detailed description of the reasons why our participants wished blanks in examples because the reasons they mentioned were similar to the ones for theorems and proofs, for instance, that the blanks encourage thinking or that the numerical results can be deduced by the students themselves. The most important reasons for blanks/preprinted parts in examples are listed in Table 7 that summarises such reasons for all lecture elements.

Table 7. Summary of important reasons our participants mentioned for blanks and preprinted parts in the different elements of a mathematics lecture.

Lecture Element	Possible influence of blanks or printed parts on students’ note-taking that our participants mentioned	Aspect of note-taking addressed
Definitions	Having to write definitions or parts thereof helps to memorize these, as definitions are something that many of our participants considered as necessary to know.	Note-taking goal ‘memorization’
	Having to write in definitions encourages thinking about them, which may help to be attentive and to reach an understanding of them.	Note-taking goals ‘attention’ and ‘understanding’
Theorems	Leaving the claim of a theorem blank might encourage thinking for oneself about what to fill in, which can help to gain an understanding of the theorem.	Note-taking goal ‘understanding’
	Printing the beginning of theorems can help to structure the notes.	Note-taking goal ‘organization’
Proofs	Leaving blanks in proofs, especially for deductions that students can conjecture themselves, can help to gain an understanding of the proofs.	Note-taking goal ‘understanding’
	Having to write small proofs for oneself helps to get practice for assignments.	Note-taking goal ‘homework aid’
	Printing parts of longer proofs allows students to listen to the instructor besides writing, which can help to be attentive and gain an understanding of such proofs.	Note-taking goals ‘attention’ and ‘understanding’
	Providing additional blanks in or after proofs might be useful, so that students can write extra comments, which can help to make sense of the proofs again in a later review phase.	Content-structure of students’ notes, note-taking goal ‘understanding’
Examples	Leaving blanks in examples gives opportunities for practice, which might be an aid for later assignments.	Note-taking goal ‘homework aid’
	Leaving blanks in the numerical result of an example and its derivation – especially at positions at which students might be able to fill these in themselves by applying a definition or by calculation – can help to gain an understanding of the mathematical concept the example refers to.	Note-taking goal ‘understanding’
	Printing the framing conditions of examples can help to structure the notes.	Note-taking goal ‘organization’
Explanations	Printing explanations can help to be more attentive towards them.	Note-taking goal ‘attention’
	Printing explanations reduces the writing effort and saves time.	–

Explanations

As Table 5 suggested, many of our participants wished explanations to be printed. The justifications mentioned in the responses to the general question on where to put blanks in guided notes were rather diverse, for instance, that having to write these down for oneself makes it hard to be attentive, or that this requires too much time or effort. Concerning the remarks about EULER in the text on EULER’s $\phi$-function, however, our participants were pretty much in agreement about why these remarks should be printed, as illustrated in the following quote:

Lines 1–8 to be printed, because this is just interesting from a historical point of view but unimportant for gaining an understanding.

Others even suggested to erase these remarks entirely from the script (14.8%). A possible reason for this consensus could be that these remarks were of a special type: additional information which is not connected to the mathematics covered (some participants also stated this explicitly). In general, the way students like to proceed with explanations provided within a mathematics lecture probably depends much on the type of explanation.

Summary

Table 7 summarises important reasons why our participants liked to have blanks or preprinted parts in the different elements of a mathematics lecture within guided notes, and relates them to the corresponding aspect of the college students’ theory of note-taking by Van Meter et al. (1994).

7. Summary and discussion

7.1. Discussion of the results

With our mixed-methods study investigating the positions of a mathematics lecture at which students consider blanks in guided notes as useful and why, we provided a relevant new perspective to the question of where to put blanks in guided notes, which extends the proposals made by instructors thus far (Alcock, 2018; Kotsopoulos et al., 2022; Krapf & Pfefferkorn, 2022; Tonkes et al., 2009).

Our quantitative data point out positions in mathematics lectures at which many students might appreciate blanks. These are definitions, claims of theorems, numerical results of examples, short proofs, and some positions in longer proofs or derivations (Tables 5 and 6). On the other hand, many of our participants liked explanations and extra information as well as framing conditions of theorems/examples to be printed.

Our qualitative data then indicate how blanks or preprinted parts at these positions might support students during their note-taking (see Table 7), which – in turn – can then foster their learning. Table 7 especially shows that the reasons our participants mentioned for blanks/preprinted parts in guided notes differed between the elements of a mathematics lecture. Nevertheless, we can synthesizes these if we look at the results from the encoding/storage framework by Di Vesta and Gray (1972).

Most of the reasons our participants mentioned could be assigned to one of the following note-taking goals according to Van Meter et al. (1994): ‘attention’, ‘understanding’, or ‘memorization’ (see Table 7). All of them are related to the encoding function of note-taking by Di Vesta and Gray (1972). This suggests that leaving blanks at the corresponding positions in Table 7 – namely in definitions, for claims of theorems/results of examples and at certain positions in proofs, while printing parts of longer proofs and extra explanations can support students in encoding the information presented during their note-taking process in the lecture. This can then help them integrate this information into their own cognitive structure and therefore foster their learning from a cognitive point of view (Mayer, 1996; Yilmaz, 2011). Especially blanks at positions at which students can conjecture themselves what to fill in, for instance, the claim of a theorem or certain deductions in a proof, can help them connect the content covered to their prior knowledge, as they then have to build this connection for themselves. Furthermore, also the positions at which our participants wished to have blanks for practice, e.g. in short proofs or examples, may help students integrate the corresponding information into their cognitive structure, as practicing can help them develop a kind of procedural scheme in their minds. Hence, blanks/preprinted parts at most of the positions in Table 7 can help students in the encoding function of note-taking, which can foster their learning during the lecture.

But also the external storage function of note-taking by Di Vesta and Gray (1972) occurred in our participants’ reasoning for blanks/preprinted parts in guided notes at two occasions. First, printing the framing conditions of theorems/examples can help students structure their notes. Second, providing additional blanks in longer proofs might foster them to write additional comments that can be helpful in a later review phase (see Table 7). Hence, leaving blanks or not at certain positions can also help students store the information in a way that might facilitate their learning in a later review phase.

A comparison of the positions at which our participants appreciated blanks/preprinted parts within guided notes with the ones suggested in the literature by instructors (see Section 2.2) shows many matchings. Our participants, for instance, especially valued blanks at positions at which they might be able to fill these in by themselves – either by calculation or by thinking, for instance, blanks in examples or a blank for the claim of a theorem. This fits to what Alcock (2018), Tonkes et al. (2009), and Kotsopoulos et al. (2022) proposed for initiating active learning processes during a lecture. Furthermore, our participants liked to have blanks in short proofs for practicing, which fits to the proposal of Krapf and Pfefferkorn (2022) to leave proofs blank for fostering students’ ability to write proofs.

However, there were also two major differences. While many of our participants liked to have blanks in proofs just for ‘important arguments’ or conclusions that they can deduce themselves – at least in longer proofs, several of the instructors’ proposals from Section 2 suggested to leave proofs completely blank. A second difference between our participants’ wishes and what instructors have proposed thus far concerns definitions. These had been printed in most of the proposals mentioned in Section 2, while many of our participants liked to have blanks in them (Table 5). The reasons our participants mentioned for blanks/preprinted parts then explain why – in these cases – students might also benefit from different positions of blanks than in the existing instructors’ proposals. Blanks in definitions, for example, can help students memorise the definitions covered. However, students’ perspectives should certainly not be used as the only guideline for the design of guided notes, as they are not experts in the subject and might, for instance, not yet know what it means to actually understand mathematics.

7.2. Limitations and short outlook

Finally, we want to mention two important limitations of our study. The first one is that our study was restricted to one particular course. This generates two important problems. Firstly, the sample is not that large, so that the quantitative results can only yield tendencies. Secondly, the students were probably influenced by the design of the guided notes in their course, and might just have accepted what they experienced. For instance, several of them liked to have blanks at positions at which they were able to fill these in by themselves, which was also a typical activity in the course (see Section 4). However, only two of our participants mentioned explicitly that guided notes should be designed as in their course. Nevertheless, we need more research from other courses to find out to what extent our results are generalisable.

The second important limitation we want to highlight is that we did not investigate how blanks or preprinted parts at certain positions within guided notes affect students’ learning of the content directly. We chose to investigate how these might support students in their note-taking, as note-taking is a central activity in mathematics lectures and is in general known to have a positive effect on students’ learning (Kiewra, 1989; Kobayashi, 2005, 2006). However, a more direct approach investigating the effect of the blanks on students’ learning might be desirable in the future.

Nevertheless, our results give valuable suggestions for suitable positions of blanks in guided notes, so that many students might benefit from them – even though students’ wishes should certainly not be the only guideline for the design of guided notes.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

We acknowledge support by the Open Access Publication Fund of Humboldt-Universität zu Berlin.

Notes

1 The guided notes and all the material of the study have been translated by the authors from German to English for this paper.