Strengthening mathematical orientation: how university mathematics courses can gain relevance for pre-service teachers

Multiple reports suggest that pre-service teachers (PSTs) commonly criticize their university studies in mathematics as lacking practical applicability to their future professional work (Cooney & Wiegel, 2003; Hefendehl-Hebeker, 2013). Many decades ago, Felix Klein (1932/2016) referred to this issue as the problem of a “double discontinuity” between school mathematics and university-level courses. Since the observation was first made, many teacher educators and researchers have been tasked to convey university mathematical knowledge to PSTs in a way that renders it applicable to their later professional practice as mathematics teachers and meets their mathematical “needs” for teaching (Ableitinger et al., 2013; Ball & Bass, 2000; Gueudet et al., 2016; Winsløw & Grønbæk, 2014; Wasserman, 2018). Following Klein (1932/2016), many approaches to teaching university mathematics to PSTs aim to enable this cohort to take a higher standpoint towards school mathematics. Within the mathematics education community, two distinct research strands have thus emerged in recent decades: (1) designing university learning opportunities that enable PSTs to take a higher standpoint (i.e., course development); and (2) identifying and analyzing the higher standpoint that PSTs (and teachers) have achieved (i.e., analytical and evaluation studies).

In the tradition of the first research strand, many researchers follow Felix Klein. Klein’s (1932/2016) approach, namely of considering “elementary mathematics from a higher standpoint,” refers to a series of lectures specifically designed to help PSTs see the connections between problems in various mathematical fields and emphasizes the relations between these problems and those of school mathematics (Kilpatrick, 2008; Allmendinger, 2019; Weigand et al., 2019). Thus, the key to overcoming the double discontinuity in designing university learning opportunities seems to be twofold: making connections between school mathematics and university mathematics visible in university courses, on the one hand, and showing the relevance of university mathematics to future teachers, on the other. Many researchers in mathematics education have worked on these connections, either by trying to collect good examples of them for teacher education practices (e.g., Usiskin et al., 2003; Heid et al., 2015) or by developing guidelines for improving university secondary mathematics teacher education, such as by integrating historical or didactical aspects in mathematical lectures (e.g., Beutelspacher et al., 2011; Wasserman et al., 2021; Murray & Star, 2013).

On the other hand, empirical approaches within the second research strand have suggested several analytical frameworks to assess PSTs’ professional knowledge base in school-related mathematics and the higher standpoint, for example to follow up changed practices in university teacher education with empirical evidence (e.g., Buchholtz et al., 2013; Dreher et al., 2018; Ball & Bass, 2009).

In this article, we present an approach that can contribute to both research strands and to the discussion of this Special Issue by empirically investigating the benefits for PSTs of addressing explicit connections to school mathematics as well as to teaching situations in university mathematics courses. In doing so, we first conceptualize an analytical framework, with reference to relevant existing frameworks in the tradition of Felix Klein, which describes the mathematical needs of PSTs in teaching situations. For this, we introduce the concept of mathematical orientation. Subsequently, from a reconstructive analytic perspective, we take a particular view of the mathematical orientation which emerges in the written reflections of PSTs in two university mathematics courses, one in Switzerland and the other in Norway. Overall, our results give insight into the contribution of mathematical university courses to developing mathematical orientation.

Despite progress in the research field toward overcoming double discontinuity, there seems to be no unifying consensus on what the “higher standpoint” is, how to describe the mathematical needs of PSTs regarding teaching, and what kinds of connections university mathematics courses should have with school mathematics and teaching practice. Multiple theoretical conceptualizations have therefore been introduced, among them mathematical sophistication (Seaman & Szydlik, 2007), mathematical literacy (Bauer & Hefendehl-Hebeker, 2019), and mathematical understanding (Kilpatrick et al., 2015). These conceptualizations are often built on discussions of a specific cognitive knowledge base held by secondary teachers, certain abilities, or even affective aspects (Blum-Barkmin, 2022).

We notice that some of these conceptualizations, as well as their respective analytical frameworks, can be described as oriented towards ontological differences: They arose from the aim of overcoming the problem of double discontinuity by creating awareness among PSTs of the perceived ontological gap and the differences between university mathematics and school mathematics (Ableitinger et al., 2013; Dreher et al., 2018; Blum-Barkmin, 2022). Due to the underlying assumption in these difference-oriented frameworks, mathematics is often presented as divided into two different “worlds”, namely a school mathematical and a university mathematical world. The notion of a higher standpoint in addressing connections between university mathematics and school mathematics implies a hierarchical ordering, with university mathematics being understood as the “higher” world. We claim that this hierarchical model is not adequate to illustrate this unified awareness. Rather than sewing together two pieces that have been ripped apart, we want secondary teachers to understand school mathematics as part of mathematics within one world (see also Wasserman 2016, 2018). This was already supported by Toeplitz (1932, p. 1, translated by the authors), who went as far as to demand a “unified awareness of the two mathematics.”

Furthermore, we note a tendency in several conceptualizations on the periphery of the higher standpoint, as well as in descriptions of the necessary teacher knowledge, to use conceptual metaphorical language (Toh & Choy, 2021; Scheiner et al., 2022; Wasserman, 2016). Similarly, Felix Klein long ago claimed that a teacher “must be familiar with the cliffs and the whirlpools in order to guide his students safely past them” (Klein, 1932/2016, p. 162) and thus introduced the term “higher standpoint” to address the mathematical needs of PSTs in teaching situations. Several mathematics educational researchers have followed this notion. Ball and Bass (2009, p. 6), for example, introduce “horizon knowledge,” which consists of a kind of mathematical “peripheral vision” or view of the larger mathematical landscape. Zazkis and Mamolo (2011) extend and differentiate this geographical metaphor and use a philosophical interpretation from Husserl to characterize the term “horizon” within the teaching context. We follow Lakoff and Johnson (2008), who declare that metaphors are not only used as a prosaic or illustrative device, but contribute significantly to conceptualizations by establishing analogies to systems of reference that are more familiar. Thus, we also use a metaphorical approach for our conceptualization. However, we use this approach systematically: We consider not an isolated metaphor, but a set of interrelated geographical metaphors, and we interpret these within the teaching context with the aid of Stegmaier’s (2019) philosophical investigation of the term “orientation.” In so doing, we aim to introduce a specific terminology used to describe the mathematical needs of PSTs in specific teaching situations.

2.1 Conceptualization of mathematical orientation

Mathematics can be understood as a landscape, following Davis and Hersh (1998) or Wasserman (2016). Locations in this landscape can be interpreted as pieces of mathematical content, related to each other through logical connections and axiomatic relationships. In this landscape, we can locate school mathematics as those pieces of mathematical content that can principally be taught and learnt in school. Even if its boundaries are blurry, we can still obtain the one-world model we seek (cf. Figure 1). When students learn mathematics at school, they discover these pieces of content with the help of their teacher.

School mathematics within the mathematical landscape

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Following the practical philosopher Stegmaier, moving through the mathematical landscape involves meeting situations that require an orientation, which means the “achievement of finding one’s way in a situation and making out promising opportunities for actions to master the situation” (Stegmaier, 2019, p. 25). In his monography What is orientation? A philosophical investigation, Stegmaier (2019) clarifies the conditions and structures of such an orientation and establishes a new philosophical terminology that can be transferred to and interpreted in various contexts. He introduces the different (geographical) sub-concepts on which orientation will depend, such as different points of reference in the landscape, a set of perspectives, and individual standpoints. When we face a specific situation and start to orient ourselves, our eyes will wander around to gain an overview, and we will concentrate on certain points of reference for, as Stegmaier (2019, p. 56) observes, “what a new situation brings is not yet certain; it must first be made surveyable […] by means of such points.” In our case, these are connected locations within the mathematical landscape that will lead the way from one spot to the next. To identify relevant points of reference and make decisions concerning the given situation, one will have to use different perspectives, which will help us to concentrate on relevant aspects in order to “differentiate the sight” (Stegmaier, 2019, p. 47).

These perspectives will be taken from different standpoints, a concept Stegmaier understands in two separate ways. On the one hand, a standpoint is the momentary position in the landscape that can and will change over time. On the other hand, he claims that everyone has a “global” personal standpoint “that one cannot ‘adopt’ or ‘abandon’ […]. It is the reference and starting point for any kind of orientation” (Stegmaier, 2019, p. 44). All standpoints (local and global) adopted can be understood as opinions on or attitude towards a specific topic, as “one does not only ‘stand’ in a situation, but one’s mood, the psychophysical state of one’s orientation, also depends on it” (Stegmaier, 2019, p. 44). This perception is known from other contexts as well, for example when considering a political or moral standpoint.

Interpreting Stegmaier’s (2019) notion of orientation and the related sub-concepts within the context of teaching mathematics, we conceptualize mathematical orientation as the orientation teachers need to master mathematical situations in the classroom. Mathematical orientation therefore consists of the four aspects outlined below.

(1) Teachers will need a selection of points of reference, that is, pieces of mathematical content from school mathematics and university mathematics, with connecting paths that will help them guide students securely around the mathematical landscape. These points are captured on an individual reference map, which teachers carry around with them mentally.

Ideally, university mathematics courses provide PSTs with a reference map that will contain all topics covered in school with connecting paths. PSTs should be able to see connections within the relevant topics but also paths that for example connect algebraic and geometrical pieces of mathematical content. Additionally, the map should contain pieces outside of the school mathematical boundary and PSTs should learn from university mathematics courses to connect those to pieces in school mathematics with paths. When being confronted with a teaching situation (e.g., in planning a lesson or reacting to a students’ question) we assume teachers to mentally consult this map before deciding how to (re)act.

(2) They will need the ability to take different mathematical perspectives, that is, be able to focus on various aspects of mathematical content, such as underlying principles or its presentation. These perspectives will be used to evaluate different points of reference and paths within the reference map. We will refer to this collection of mathematical perspectives that a teacher has at their disposal as an individual’s perspective toolbox.

A perspective toolbox should contain a variety of perspectives. In Kleins (1932/2016) lectures one can distinguish between e.g., a mathematical perspective, where mathematical content is solemnly analyzed with regard to the mathematical facts, or a historical perspective, which focuses on the historical background. A PST should be able to similarly take different and adequate perspectives to analyze different pieces of mathematical content or paths when being confronted with a teaching situation. For example, in a teaching situation, teachers must choose the degree of formal rigor with which a mathematical content can be presented, or a student’s argument is evaluated. We believe university mathematics courses should enable PSTs to use adequate perspectives on mathematical content.

(3) In addition, orientation will require the will and courage to make decisions in each situation based on observations made using different perspectives. Whether a teacher is willing to do this will depend on their standpoint towards mathematics as a whole and towards specific pieces of content in school mathematics or university mathematics, namely certain beliefs, values, preferences, and feelings, such as being confident about understanding a topic or seeing its relevance for students’ mathematical understanding. These standpoints characterize the individual’s mathematical attitude.

A PST will take certain standpoints to each point of reference or path on his or her map for example with regard to it’s usefulness in teaching situations and how confident the PST feels with the content. These standpoints will influence how a PST will react in certain situations. For example, a teacher might value the importance of applications and problem solving for learning a specific mathematical content higher if his or her corresponding attitude was fostered in university mathematics courses by a special emphasis on examples of applications (Buchholtz & Behrens, 2014). Being confident with certain applications, the teacher may be more willing to emphasize corresponding aspects in his or her own teaching as well.

(4) Finally, these three components are used to link pieces of mathematical content to teaching situations. The nature of these links will depend on the requirements a teacher is facing in a concrete situation (cf. Figure 2).

Conceptualization of mathematical orientation

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For example, when a student asks a question about a piece of mathematical content in class, the teacher should locate it on the reference map, assess it from a certain perspective and decide if he or she wants to follow up on this content with the student or the whole class.

We are aware that mathematical aspects are not the only ones to be considered when decisions are made in specific teaching situations, but that didactical or pedagogical aspects, among others, will also be involved. However, our conceptualization of mathematical orientation focuses on mathematical needs for teaching, as was the case in concepts such as mathematical understanding (cf. Kilpatrick et al., 2015), mathematical sophistication (cf. Seaman & Szydlik, 2007), and mathematical literacy (cf. Bauer & Hefendehl-Hebeker, 2019).

2.2 Contribution to understanding Klein’s higher standpoint

Regarding the research strands on the higher standpoint, our conceptualization of mathematical orientation leads to a slight adjustment of the perception of the Kleinian term “higher standpoint,” which pervades ongoing discussion in the scientific community and is often related to specific knowledge or abilities. The presented conceptualization replaces the hierarchical understanding of mathematics as divided into two different worlds with a unified awareness of mathematics, where standing higher is understood as standing above the entire mathematical landscape (school and university) and having an overview. Additionally, standpoints as part of mathematical orientation describe a personal attitude, rather than specific knowledge or abilities. This conceptualization, however, aligns with Klein’s understanding. He claims to “want the future teacher to stand above his subject, that he have [sic!] a conception of the present state of knowledge in his field, and that he generally be capable of following its further development” (Klein, in Rowe 1985, p. 128, highlighted by the authors). Moreover, he uses the term “standpoint” several times to describe certain beliefs towards mathematics or mathematical preferences (cf. Klein 1932/2016, e.g., p. 4, p. 155, p. 195).

Taking such a higher standpoint is – as described above – necessary but not sufficient to cope with teaching situations. PSTs who have a similar standpoint will not necessarily have the same points of reference at their disposal. Moreover, they will not necessarily be able to use the same perspectives or link the given content to a teaching situation in the same way. Partially due to its philosophical background (Stegmaier, 2019), mathematical orientation is therefore a highly individual concept, and it will be important to focus on all parts of mathematical orientation and not just the standpoint alone. This broad focus does not contradict earlier ideas. Klein (1932/2016, p. 256 f.) himself claims the importance of orientation, observing that “if you lack orientation, if you are not well informed concerning the intuitive elements of mathematics as well as the vital relations with neighboring fields, if, above all, you do not know the historical development, your footing will be very insecure.”

Even today the concept of orientation is used when describing the requirements of teaching (e.g., Schoenfeld 2011; Hannah et al., 2011). Schoenfeld (2011), for example, defines orientation – similarly to Stegmaier (2019) – as something a teacher needs for “in-the-moment decision making” (p. 457). In contrast to Stegmaier, however, in his understanding orientation is restricted to being “a broad category that includes beliefs, values, preferences, and tastes” (Schoenfeld, 2011, p. 460). The original geographical meaning of orientation leads us to follow Stegmaier and argue that orientation also includes aspects of knowledge and abilities, as described above.

Part of orientation is, therefore, as described above, knowledge of pieces of mathematical content and the connections between them. Klein refers to this when he demonstrates the “mutual connection between problems in the various disciplines” (Klein, 1932/2016, p. 2) in his lectures. When describing the problem of “double discontinuity,” Klein (1932/2016) also points to the further connections a teacher must make:

The young university student found himself, at the outset, confronted with problems, which did not suggest, in any particular, the things with which he had been concerned at school. (…) When, after finishing his course of study, he became a teacher, he suddenly found himself expected to teach the traditional elementary mathematics (…); and, since he was scarcely able, unaided, to discern any connection between this task and his university mathematics, he soon fell in with the time honored way of teaching, and his university studies remained only a more or less pleasant memory which had no influence upon his teaching. (Klein, 1932/2016, p. 1, authors’ italics)

In difference-oriented frameworks, the double discontinuity is sometimes interpreted as PSTs’ inability to connect pieces of mathematical content in the school mathematical world to pieces of content in the university mathematical world or, alternatively, as describing the lack of knowledge about these connections. Dreher et al. (2018), for example, distinguish between bottom-up and top-down connections between the two mathematical worlds. Similarly, Zazkis and Mamolo (2011) characterize their understanding of horizon knowledge through connections between school mathematical topics and different pieces of university mathematical content. However, if we read Klein’s definition of “double discontinuity” literally, it is not about connecting different pieces of mathematical content; rather, it concerns the lack of ability to connect the mathematical content presented in university courses to the task of teaching. This understanding is used in the conceptualization presented in this paper, where mathematical orientation describes the ability to link any kind of piece of mathematical content from the reference map – without explicitly differing between school and university mathematics – to specific teaching situations. This conceptualization leads to two distinct types of connections, both of which are part of a mathematical orientation. To distinguish between them, we call connections within the mathematical landscape and marked on the reference map paths, maintaining the metaphorical language. We interpret the connections mentioned by Dreher et al. (2018) and Mamolo and Zazkis (2011) as special paths within the one-world model that cross school mathematical boundaries. Connections between pieces of mathematical content and teaching situations, on the other hand, are referred to as links (cf. Figure 2). Thus, mathematical orientation notably highlights the situational component of teaching and raises the question of how these links can be described and characterized with regard to the different tasks a teacher faces.

2.3 Research questions

Obviously mathematical orientation is a content-specific concept, as it will depend on the reference map that PSTs have at their disposal. Additionally, mathematical orientation is, as pointed out, highly individual. Having introduced the different sub-concepts and their dependencies, we can narrow the focus of our empirical approach in this study to four research questions on mathematical orientation:

(RQ1) Which points of reference and paths are marked in PSTs’ reference maps?

(RQ2) Which mathematical perspectives are included in PSTs’ perspective toolboxes?

(RQ3) How can PSTs’ mathematical attitude towards mathematical content be characterized?

(RQ4) How do PSTs use their reference map, perspective toolbox, and mathematical attitude to link mathematical content to teaching situations?

For the purpose of this article, we will narrow down the questions by only considering one specific topic, decimal expansion. Our aim is to identify possible indications for all four parts of mathematical orientation and show common collective trends by deliberately choosing a sample group from two different international courses.

To answer the research questions, we had PSTs write reflections on selected pieces of mathematical content as part of their coursework. The PSTs were requested to describe three hypothetical teaching situations in which specific mathematical content could be relevant for their later work as teachers. Additionally, they were given further guiding questions for reflection, for example how the presented mathematical content enriched their understanding and professional growth as a teacher; which difficulties arose when dealing with the mathematical content; whether their perspective on the specific mathematical content changed during this course; whether the mathematical content would be covered in school; which questions from students could arise in this context; and where they thought this mathematical content could be useful for their teaching, either directly or indirectly.

3.1 Sample and courses

For this study, two groups of PSTs with different backgrounds and requirements were deliberately chosen from two different countries, namely Switzerland and Norway. Both university mathematics courses have been explicitly developed for PSTs; moreover, both discuss school mathematical topics with a university mathematical focus, as opposed to being courses that focus principally on didactics or pedagogy. In both courses, the focus is on mathematical content that will be useful in teaching situations, but there are significant differences between the two in terms of audience. The PST groups differ in background knowledge, which impacts the presentation of topics within the courses. Furthermore, the groups aim to teach mathematics at different grades in secondary school, so they have different school mathematical boundaries within the mathematical landscape.

One group attends a course at the University of Teacher Education Lucerne designed for PSTs teaching grades 7 to 9. This course integrates mathematical and didactical content on topics from arithmetic and algebra (Hölzl, 2013). The mathematical focus is on clarifying school mathematical content and then relating it to topics in university mathematics. For the PSTs, this course is part of the bachelor’s program and is taken in their fourth semester. Before starting this course, PSTs have already taken two introductory courses, one giving them a mathematical overview and the other being an equivalent course on geometrical topics.

The second group attends a course at the University of Oslo which is designed for PSTs in an integrated five-year master’s program. Most course participants will become teachers, with about half teaching grades 7 to 9 and half teaching grades 10 to 12. The course is taken in their eighth semester. The students have a solid mathematical foundation: They have all taken several courses in university calculus, and a few have even taken a course in abstract algebra. However, the goal of the mathematics courses they have taken at university is to train future mathematicians rather than to give them the kind of understanding that will help them as teachers of mathematics. Thus, much of the material covered in the course considered here consists of topics they have already seen but with a different focus, namely as related to school mathematics.

3.2 Topic selection

Both groups’ reflections on the topic of decimal expansions were analyzed. The two courses covered this topic to a comparable extent, but each had a slightly different focus. A professional requirement for mathematics teachers in this topic is to explain why \(0.\stackrel{-}{9}=1\), why a number is rational if and only if it has a terminating or repeating decimal expansion, and which rational numbers have a terminating decimal expansion. Students in the classroom see early on that \(\frac{1}{2}=0.5\) has a terminating decimal expansion; that \(\frac{1}{3}=0.\stackrel{-}{3}\) has an immediate-repeating decimal expansion, where the repeating block starts right after the decimal point; and that \(\frac{1}{6}=0.1\stackrel{-}{6}\) has a delayed-repeating decimal expansion, where there are one or more digits between the decimal point and the repeating block. What makes the decimal expansions of these numbers so fundamentally different is a natural mathematical question, which is not normally addressed in school but that teachers should understand and that provides an opportunity to see paths between different areas of mathematics. The two basic pieces of mathematical content are (A) to understand that a number with an immediate-repeating decimal expansion can be written as a fraction with a denominator that is a “9-block,” that is, \({10}^{r}-1 (r\in \mathbb{N})\); and (B) to show that if n is relatively prime to \(10\), then n divides a 9-block, so that \(\frac{1}{n}\) can be extended to a fraction with a denominator that is a 9-block.

There is an interesting historical angle to this. In 1677, Leibniz claimed that if n and 10 are relatively prime, then the decimal expansion of \(\frac{1}{n}\) is simply repeating and that the length of the repeating block is a factor of \(n-1\). However, in 1685, Wallis pointed out that this fails for \(\frac{1}{21}=0.\overline{047619}\), since 6 does not divide 20. Lambert claimed in 1758 that Leibniz’s claim was true if n is prime but was only able to prove this in 1769, using Fermat’s Little Theorem, which shows that \({10}^{p-1}-1\) is divisible by p if p is prime and thus that \(\frac{1}{p}\) can be extended to a fraction with a denominator equal to \({10}^{p}-1\). If we use Euler’s Theorem, we know that if n is relatively prime to 10, then n will divide \({10}^{\varphi \left(n\right)}-1\), where \(\varphi \left(n\right)\) is the number of \(1\le k<n\) that are relatively prime to n. It follows that the length of the repeating block is a factor of \(\varphi \left(n\right)\). For example, the length of the repeating block of \(\frac{1}{21}\) was seen above to be 6, which divides \(\varphi \left(21\right)=12\) but does not divide \(21-1=20\) (cf. Bullynck, 2009). This example shows that to gain an understanding of decimal expansion, it is necessary to introduce techniques from number theory and modular arithmetic and that even famous mathematicians struggled with these problems.

In the Norwegian course, the first piece of content (cf. A) is handled by a simple formal argument, while the second piece of content (cf. B) leads to a detailed discussion of number theory leading up a proof of Euler’s Theorem. In the Swiss course, the emphasis is on explaining the first part (cf. A), with comparison to the finite case, while the second piece of content (cf. B) is only illustrated through examples. Additionally, in this course the question “Why does \(0.\stackrel{-}{9}=1\)?” was covered intensively by discussing several different possibilities to explain this fact.

3.3 Data collection and evaluation

We collected and evaluated 85 reflections written by the PSTs on the topic of decimal expansions, 56 from the Swiss course and 29 from the Norwegian course. Evaluation of the reflections was based on qualitative content analysis, according to Mayring (2008). The PSTs’ statements were subjected to a theoretically guided and systematic interpretation process. The reflections were pre-structured by means of categorization (i.e., a class of analysis aspects defined by our theoretical framework on mathematical orientation) and then interpreted according to our research questions (cf. Figure 3).

Qualitative content analysis adapted from Mayring (2008) and Buchholtz and Behrens (2014)

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The procedure began with the written reflections, considering the question of insights into the individual mathematical orientation of PSTs. The Swiss and Norwegian written reflections were translated into English for comparison. In the process of deriving coding categories from the presented framework on mathematical orientation, initial categories were defined and used to subsequently code a sample selection (deductive categorization). Coding entailed assigning individual text elements and statements to the initial categories and developing coding rules. The initial categories were deductively formulated in cases where we could rely on existing research. For example, the different perspectives found in Klein’s (1932/2016) lectures (see Allmendinger 2013) served as initial categories when analyzing the reflections about the perspective toolbox. In a revision phase, we checked for new categories in favor of our conceptualization that emerged from the sample data (inductive categorization) and refined the categories along with the data, with existing categories getting generalized or combined where appropriate. Deductive and inductive categorization complemented each other during coding. Final definitions, coding rules and guidelines were formulated according to the results from a selection of cases and used to evaluate and interpret all the material of both courses, relating the results back to the research questions. We present one of the coding manuals that was used in the appendix. All 85 reflections were coded within this process. To ensure the coding process had sufficient reliability, the two groups were coded separately by two of the authors, and one author coded both groups. We will provide further details of the categories in the immediate proximity of the results.

From the PSTs’ written reflections, we were able to reconstruct several indications of mathematical orientation. We found different points of reference and paths within the personal reference map (see 3.1). Furthermore, we identified different perspectives taken by the PSTs when they analyzed decimal expansions (see 3.2). The reflections also gave insight into the PSTs’ mathematical attitude: They wrote about how university mathematics helped them grow as both mathematicians and teachers (see 3.3). Points of reference, perspectives, and attitude were used to link the content to concrete but hypothetical teaching situations. Here, we were able to identify several types of links (see 3.4).

4.1 Pieces of mathematical content as part of the reference map (RQ1)

Asking the PSTs to imagine concrete teaching situations where they would need knowledge about aspects of decimal expansions enabled us to gain an insight into whether they identify this content as a point of reference on their reference map (cf. RQ1). When coding, we didn’t have any strong initial categories here, but based on inductive coding, we determined which aspects of the reference map became apparent within the reflections.

The reflections gave some insight into where the PSTs locate the given content in the reference map, for example within or outside school mathematical boundaries, as seen in the two contrasting examples below:

“The content 0.999…=1 can be addressed in school. I find both variants [of proofs] equally plausible and comprehensible for students. I would probably even go so far as to carry out both variants shown.” (Swiss PST #4)

“I hardly think any of the questions will come up in high school. These things are not something I can remember ever wondering about or learning anything about. However, it was still very interesting!” (Norwegian PST #3)

In other cases, PSTs described paths to other mathematical content, even though they were not explicitly asked to do so, thus integrating the content into their existing knowledge structure. A semantic network of mathematical content could be created that ensures the long-term retention of the knowledge content. These reflections give us broader insight into the totality of the PSTs’ reference maps in regard to existing points of reference and paths between them:

“We see that periodic decimal expansions involve limit operations, so this task and examples can be used in the introduction to calculus.” (Norwegian PST #20)

“The different [types of] fractions have interesting aspects, which can also be related to other school mathematical topics.” (Swiss PST #11)

Specifically, the PSTs saw paths between decimal numbers and prime numbers:

“Yes [my perspective changed], because I didn’t know there were different types of rational numbers and how they were related to prime factorization.” (Swiss PST #53)

“In addition, this is a great way to link prime numbers to school mathematics.” (Norwegian PST #13)

4.2 Perspectives on pieces of mathematical content (RQ2)

Felix Klein addressed the mathematical needs of PSTs by taking different perspectives (see Allmendinger, 2013). A mathematical perspective concentrates on diverse ways to build a path between a mathematical topic as addressed at university to a mathematical topic as part of school curricula. A didactical perspective concentrates on various aspects of how to present mathematical content in the classroom in comparison to its presentation at university. It also includes discussions of simplification (see Kirsch, 1977) and different structuring and visualizations. Finally, Klein takes a historical perspective, describing how mathematical topics have evolved over time, as knowledge of this process might have consequences for teaching practice as well. Additionally, Klein focused on several different mathematical principles and highlighted these throughout his lectures, such as the principle of mathematical interconnectedness, the genetic principle, the principle of intuition, and the principle of application orientation. When coding, we deduced, our initial categories from these principles and perspectives (see appendix).

The PSTs, for their part, also took different perspectives on mathematical content when linking it to teaching situations (cf. RQ2). We could identify a content-related perspective, which resembles the Kleinian mathematical perspective; a principle-related perspective, where the PSTs focused on underlying mathematical principles, as also seen in Klein’s lectures; and a presentation-related perspective, where links were made by drawing conclusions for their own teaching practice from the way the content was presented in the university course, which is related to the didactical and historical perspective of Klein.

Taking a content-related perspective in the written reflections, PSTs linked the mathematical content to a teaching situation simply by referring to the content. This linkage became apparent when they identified the content as part of the school curriculum and planned to discuss the content itself or found it likely that questions concerning it could be posed:

“I think it would have been useful for the students to calculate 1/2, 1/3, 1/6, and 1/7 by hand and look for what makes the decimals repeat or end.” (Norwegian PST #1)

“I think the question of whether 0.9$ [sic!] is equal to 1 is the question that will come up most often in a teaching context. This is a type of question that I think can be asked by both weak and average students. Questions linked to different types of decimal expansion (finite, repetitive, and delayed repetitive) I think will be asked less often, and by curious students... However, I think this type of question can provoke more frequent questions if the students are challenged to investigate different types of decimal numbers (e.g., by performing the divisions 1:8, 1:3, and 1:7 by hand).” (Norwegian PST #10)

Some reflections, on the other hand, showed that the PSTs focused more on general mathematical principles and practices, as became apparent when they discussed the mathematical content and concentrated on this when linking it to a teaching situation. They acknowledged the relevance of mathematical principles in teaching practice, for example the principle of classification, the principle of abstraction, or the application orientation (as seen in Klein’s lectures as well):

“Therefore, it [the topic of decimal expansions] can be used to incorporate a basic mathematical experience. For example, that there are classifications, to distinguish distinct types of decimal fractions.” (Swiss PST #53)

“Irrational numbers, on the other hand, can only be argued, so they require abstract thinking from students.” (Swiss PST #1)

“In areas with real-life examples, for example in measurements or other descriptions, fractions or decimal numbers are often given. It [the topic of decimal expansions] also helps in practice to get a better overview of different sizes.” (Swiss PST #14)

Another perspective was taken by the PSTs in the written reflections in relation to the presentation of mathematical content in their own lessons. In this perspective, PSTs focus on the way the mathematical content was presented within the university courses. From this, they draw conclusions for their teaching. Some PSTs’ reflections indicate that the presentation chosen in the course was illuminating and could be adapted in the classroom by the PSTs, while others perceived the presentation as unsuitable for teaching.:

“Through the cognitive conflict of the decimal number 0.999…[= 1] the possibility is offered to actively involve the students in finding the approach. In doing so, they question, research, try out, which contributes centrally to the basic understanding afterwards.” (Swiss PST #53)

“To introduce the question of whether \(0,\stackrel{-}{9}=1,\)I would start with this picture to arouse their curiosity and then go deeper into it with the approach that the connection between \(0,\stackrel{-}{9}\) and 1 can be directly established. However, of course, step by step and firmly simplified.” (Swiss PST #7)

On the one hand, therefore, this perspective involves concrete questions of presentation in the literal sense, such as visualizations and the degree of abstraction, as in the previous examples. On the other, it also concerns questions of structuring content in a way that enables students to gain intellectual insights:

“I also think that students will remember things better if they know why something is like that.” (Norwegian PST #26)

“I think that it could be useful to work on many topics through discovery. The students should work out rules, etc. themselves and not simply learn them by heart (e.g., when a periodic decimal fraction decomposition results).” (Swiss PST #12)

4.3 Standpoints towards pieces of mathematical content (RQ3)

We understand a PST’s mathematical attitude as a collection of individual standpoints, e.g., perceptions and opinions of mathematics, especially regarding the presented mathematical content and its relation to teaching practice (cf. RQ3). Some indications in the written reflections enabled us to describe individual PSTs’ mathematical attitude in terms of their self-confidence in teaching decimal expansions and their experiences of learning the topic themselves. When coding, we defined the initial categories in terms of PSTs self-confidence and motivation for learning about decimal expansions.

The indications of how secure the PSTs feel about decimal expansions give insight into how solid their point of reference is or, as Klein says, how secure their footing is. The more secure a PST feels about the mathematical content, the more likely they are to use it in teaching situations in the future. Most PSTs described their attitude as positive, having gained confidence by dealing with the topic:

“Nevertheless, it is important that we as teachers are prepared for the students to ask questions about these topics, because the ability to take the students’ questions seriously can facilitate curiosity and understanding for the students. ... Otherwise, the knowledge brings security to me as a teacher. When I am more confident in my own knowledge, I will become a more confident teacher who is able to meet the students’ questions in a good way. A third arena where this topic is important can be when it comes to preparing for classes.” (Norwegian PST #18)

“The chapters […] have taken away my fear of decimal fractions.” (Swiss PST #7)

Additionally, we found sequences referring to compassion towards the PSTs’ future students. The mathematical content covered gave them insight into problems students might encounter. They could transfer their own learning difficulties to the difficulties they anticipated their students might face:

“Compassion towards the students if they do not understand and want to understand the difference between finite, periodic, mixed periodic.” (Swiss PST #48)

“I think students may ask how to convert fractions to decimal numbers and find that in itself difficult. I think students also find it difficult with numbers that are not exact so that they have to round off. Then, I think they find it difficult to know when to round off. I remember I might have wondered myself why the decimal expansion is so different for different fractions, so it was fun to learn about this!” (Norwegian PST #8)

Finally, PSTs felt motivated by having learned new mathematical content, and they developed enthusiasm for further deepening their mathematical knowledge:

“Tasks, such as \(0,\stackrel{-}{9}=1,\) I find very exciting. I also think that these tasks will be fun for my students in the future because they are thought-provoking and not so easy to grasp.” (Swiss PST #49)

“I was not aware of how the inversion of a pure periodic or a mixed periodic decimal fraction works. Admittedly, this impressed me a bit.” (Swiss PST #6)

“Personally, I had only few questions about decimals in school that are not covered here. The fact that a number is rational even when it is finitely repetitive is fascinating, and it would have been cool to hear about that when I was at school myself.” (Norwegian PST #22)

4.4 Links between pieces of mathematical content and teaching situations (RQ4)

The PSTs were asked to imagine where the given mathematical content would be relevant in their school practice and to link it to a hypothetical teaching situation (cf. RQ4). We notice that all three preconditions – reference map, perspective toolbox, and mathematical attitude – are combined in this task.

“We see that periodic decimal expansions involve limit operations so such tasks and examples can be used in the introduction to calculus and to motivate teaching in calculus. In my own understanding, I feel that the introduction to number theory and how the results from that topic are used to prove the theorems and conclusions [...] is a useful tool for my own teaching by making me more secure in myself. In addition, I assume that this knowledge will also be useful in other parts of school mathematics which I have not thought too much about.” (Norwegian PST #20)

In this quote we recognize several aspects of the PSTs reference map, as he or she draws paths from decimal expansions to calculus. Further, we can determine the perspective used. Here the content-related perspective is the most present, as the path described focuses on the mathematical connection. Finally, he or she describes how the topic influences their attitude, gaining confidence.

When coding, we did not have any initial categories at hand, so we relied more on inductive coding. The nature of these links can be distinguished regarding the question of how the content is implemented in the classroom. For example, some PSTs believe the content to be part of the school curriculum and that they will use the content directly when presenting it to potentially all students:

“So that the students do not blindly trust the calculator, it helps from time to time to solve tasks by means of written division. The students also benefit from this because they develop a better feeling for the numbers and can thus convert fractions into decimal numbers using a simple method.” (Swiss PST #25)

Other reflections indicate that the PSTs want to have this content at hand as an option to challenge able and interested students further with proofs, discovering mathematical laws, or finding additional tasks. The content may therefore sometimes, but not always, be used directly or even as background knowledge:

“The question of whether a number is rational if and only if it has a terminating or repeating decimal expansion I myself have encountered at school from students of different strengths. This was in connection with numbers and being able to decide whether a number is rational or not. The vast majority agreed that π is not a rational number and that 3.14 was a rational number because it could be written as 314/100. When the students had to test their knowledge themselves, it was clear that they had not understood why it is so, but only that certain decimal numbers were rational and that these had to be memorized. Here, a simple proof could advantageously have been carried out to show that it does not only apply to a small number of examples, but generally applies to numbers with terminating or repeating decimal expansion.” (Norwegian PST #14)

In addition, PSTs relate university mathematical content to their professional actions in certain teaching situations. On the one hand, examples include out-of-class activities such as selecting tasks and content when preparing the lesson:

“Personal understanding of fractions simplifies lesson design.” (Swiss PST #27)

“Knowing how to see if a fraction is finite or periodic is certainly helpful in preparing lessons to choose good fraction examples.” (Swiss PST #42)

“The knowledge about the conversion of fractions into decimals helps me to understand textbook tasks faster and to break them down for the students in an understandable way.” (Swiss PST #25)

On the other hand, PSTs see the value of the university mathematical content in interactive situations in the classroom, such as giving explanations to students, valuing students’ answers in a differentiated way, and reacting adequately:

“I can plausibly explain to students how and why fractions can be classified using different prime factors.” (Swiss PST #29)

“I can adapt the teaching by explaining a concept at different levels” (Norwegian PST #7)

“When the student gives a result, the method can be used to briefly consider whether that result is logical at all.” (Swiss PST #3)

“With regard to my own competence development, I think it has been instructive to understand how, based on the denominator of a rational number, you can predict what type of decimal expansion the associated decimal number has and, not least, what possibilities there are for what the length of the repeating block can be (if the decimal expansion is repetitive). This is knowledge that makes me safer in the classroom […] in terms of giving students good answers to questions they ask related to decimal expansion.” (Norwegian PST #10)

4.5 Conclusions

The qualitative content analysis of the written reflections enabled us to identify different categories within the four aspects of mathematical orientation according to our presented framework.

We were able to show where PSTs located the mathematical content (in this case, several aspects of decimal expansions) within their reference map, for example with regard to the school mathematical boundaries, and in relation to other mathematical topics by drawing paths between them. Further, we were able to identify that three different mathematical perspectives were taken, namely content-, principle-, and presentation-related. With regard to the mathematical attitude, we found different standpoints towards the mathematical content presented in the course and identified three different categories: PSTs described having gained confidence, having strengthened their compassion towards the students, and feeling enthusiasm for the specific mathematical content or mathematics in general (cf. Figure 4).

PSTs also drew different links to teaching situations that could be distinguished with regard to the question of how the content was implemented (directly with the whole class, as an option to challenge able students, or indirectly as background knowledge) or with regard to the question of when the link is made (during out-of-class activities while preparing a lesson or during in-class activities).

Findings within the reflections on decimal expansions

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We conclude that the conceptualization of mathematical orientation is a key variable in university teaching to address the mathematical needs of PSTs. The analysis of the written reflections gave insight into all four aspects of orientation – points of reference, perspectives, mathematical standpoints, and links to teaching situations. We were able to identify a wide variety of different categories showed in both groups. The framework provides a terminology that allows us to describe and differentiate the links between the contents of the study and later school activities and identifies possible conditions for making the links. The results give indications that the framework can be used as an analytical tool within the discussion of how to strengthen PSTs’ mathematical knowledge base. However, further studies are needed to demonstrate its effectiveness to answer specific research questions.

5.1 Can the identified categories be transferred to other mathematical topics?

Obviously mathematical orientation is a content-specific concept, as it will depend on the reference map that PSTs have at their disposal. In our study we only presented reflections on one specific topic. With this, we only got insight into parts of the individual reference maps. Further on, the question remains if the different perspectives, aspects of attitude and the manner of the links found for this mathematical topic can be found when regarding other mathematical topics as well or if additional or other categories will occur. Further research will have to examine PSTs reflections with regards to different mathematical topics.

5.2 Is it possible to reconstruct PSTs individual mathematical orientation?

With regard to the theoretical foundation of the framework building on the work of Stegmaier (2019), mathematical orientation is an individual concept. In the presented study, we have so far identified different possible components of all four parts, but we see a limitation of the study with regard to reconstructing mathematical orientation of individual PSTs as a whole.

For example, we assumed that the Norwegian PSTs would possess a more detailed reference map than the Swiss PSTs, due to their broader background and the way the topic was discussed. The reflections did not support this assumption. However, this might be due to the setting of this study: We presented a specific piece of content and did not explicitly ask the PSTs to locate the content within their knowledge structure (i.e., personal reference maps) or to describe possible paths to other pieces of content by themselves. To gain a broader understanding of the PST’s individual reference maps, the guiding questions will have to be adjusted and reflections on more topics will have to be evaluated. Additionally, it might be helpful to present a teaching situation instead and ask the PSTs to name relevant points of reference to which they can relate this situation.

Regarding PSTs perspective toolboxes, we found that although we could identify three different perspectives, most PSTs reflections in both sample groups could only be assigned the content-related perspective. We see several reasons for this. Firstly, PSTs were not forced to show multiple perspectives, so it was possible to name three different situations by using the same perspective. Secondly, being asked about a link between a piece of content and a teaching situation after having attended a mathematical course discussing content issues might subconsciously have led them to embrace a content-related perspective. Finally, the topic in question is closely related to school mathematical content, which might steer the PSTs towards the content-related perspective more than if other pieces of content were considered where links were less obvious. Preliminary findings in reflections on other topics support this assumption. We obtained written reflections from PSTs following other courses, for example addressing complex numbers or differential equations, topics which are not particularly closely related to school mathematical content. These reflections showed a far wider variety of perspectives than those in the example addressed in this article and will be subject to further research. To access all available perspectives, it could help to conduct additional interviews in which guided questions are put to the PSTs.

Similar observations can be stated for the other two parts of orientation. Adjusting and extending the study setting, especially by analyzing reflections of PSTs on various different topics and conducting additional interviews, can help to reconstruct individual mathematical orientations. The focus and aim will be laid on identifying different profiles of mathematical orientation.

5.3 Is it possible to identify the contribution of a mathematical course to mathematical orientation?

The findings within the written reflections enabled us to compare the mathematical orientation of the two sample groups from (a) different university mathematics courses, (b) different countries and school systems and (c) with different course settings and audiences. Therefore, in the two courses the focus, connections to other topics and the level of abstraction were varying. Under these conditions, we were surprised to find that the PSTs showed similar reflections, especially regarding their mathematical attitude, and that they were able to establish similar links to teaching situations.

The results, especially when the two groups are compared, can help identify in what ways university mathematics courses can strengthen the orientation of PSTs. There were several indications in the reflections that orientation changed or was enhanced. Our guiding questions about the new insights gained by the PSTs in the courses or the problems they face highlighted this in particular.

However, to draw explicit and valid conclusions, it would have been necessary to reconstruct mathematical orientation before and after a course to gain a deep insight into how the course itself contributed to this orientation. By enhancing the empirical study in this way, the presented framework can be used to analyze university mathematics courses in a differentiated way with regard to such courses’ relevance for PSTs’ future teaching.

5.4 How can the framework help to develop and refine mathematical courses?

The focus on mathematical orientation can help us systematically design or improve mathematical courses, without substituting creative content designs to convey relevance experiences, but providing guidelines to choose among the variety of examples as well as how to present them.

For this, all four parts of mathematical orientation must be considered. With regard to the reference map the topic selection in the course can be inspired by the question “Which pieces of mathematical content can be useful in teaching situations?” Obviously, topics should be chosen where a path can be drawn to a school mathematical topic. This path should be discussed within the course. With regard to the perspective toolbox, the focus of the course can be set to the question “How can different perspectives be evoked?” With regard to the attitude, it should be taken into account that a course will obviously contribute to or shape the standpoint of a PST to a specific topic in some way. For example, deliberately discussing a topic which the PST has not seen before, such as a new number system, can create an “alienation effect,” which will help the PST see the analogy between what they are learning themselves and what they will be teaching their students in the future. This realization may strengthen their compassion towards the challenges faced by their students. Finally, even in courses which focus strictly on content issues, links to teaching situations should be discussed, at least on an exemplary level. Klein pointed to this in his lectures, aiming.

“to make it easier for you to acquire that ability which I look upon as the real goal of your academic study: the ability to draw […] from the great body of knowledge taught to you here as vivid stimuli for your teaching.” (Klein, 1932/2016, p. 2).