Opportunities to learn mathematics pedagogy and learning to teach mathematics in Swedish mathematics teacher education: A survey of student experiences

ABSTRACT A research-based teacher education rests on research on the practices of teacher education. We undertook a survey of the opportunities to learn mathematics pedagogy and to teach mathematics perceived by students in their last year of teacher education. Questionnaires were distributed to 753 students graduating in June 2020 from 13 Swedish universities, and the response rate was 11.2%. Overall, new graduates perceived good opportunities to learn mathematics pedagogy, but there were substantial variations in opportunities to learn specific competencies of teaching, such as analysing learners’ answers or leading a mathematical discussion. A similar unevenness in perceived opportunities to learn from the practicum experience was found. These results point to less-than-optimal opportunities to learn core practices of mathematics teaching. We recommend that institutions take a more systematic approach to programme coverage, and to create opportunities to use theory/research to deconstruct practice and to inform rehearsing ‘approximations of practice’ in campus-based activities.


Introduction
Reinforcement and conditioning guarantee behavior, and training produces predictable outcomes; knowledge guarantees only freedom, only the flexibility to judge, to weigh alternatives, to reason about both ends and means, and then to act while reflecting upon one's actions.Knowledge guarantees only grounded unpredictability, the exercise of reasoned judgment rather than the display of correct behavior.(Shulman 1986, 13) Shulman argued that knowledge is more important than training.However, engaging knowledge in making reasoned judgment is not straightforward, it requires deliberate opportunities to practice.Hence, mathematics teacher education faces the challenging task of ensuring the development of student teachers' knowledge (theoretical, research-based, and practice-derived) as well as their ability to utilise this knowledge strategically in their future practice.Increasingly, research grapples with understanding the interplay between these components.This is reflected in the shifts from mathematical knowledge for teaching to teachers' mathematical work (Ball 2017), in the call for reframing teacher knowledge around dimensions of classroom practice (Schoenfeld 2020), and in the more situated approach of TEDS-FU (Kaiser et al. 2015).It is also a strong factor in pedagogies of practice, which Grossman et al. suggest can be described using the concepts of representations, decomposition, and approximations of practice (2009).In their study of teaching of future practitioners of theology, psychology, and education, they found that prospective teachers had fewer opportunities in their university-based coursework to engage in approximations of practice.
Based on these findings, Grossman et al. saw the need for further research on what constitutes 'defensible decompositions of practice ' (2009, 2093), while Krainer and Llinares (2010) called especially for more research 'looking at the outcomes of different types of teacher education ' (p. 705).Following this line of thinking, we have undertaken to explore the learning opportunities of mathematics student teachers during their teacher education, and to follow them into their first years in the profession to understand the ways in which their education has prepared them for practice (the TRACE project).In the current paper, we focus on learning opportunities during teacher education, in particular opportunities to learn a range of components of mathematics pedagogy and opportunities to learn to teach mathematics from practice.
As with learning mathematics, opportunities to learn are crucial to learning to teach (Klemenz, König, and Schaper 2019).To identify opportunities to learn in mathematics teacher education, we analyse materials used in the programmes (work in progress), and students' perceptions of their learning opportunities (current paper).We are interested in which components of professional practice are represented in teacher education in both course work and teaching practice.Previously, TEDS-M (Teacher Education and Development Study in Mathematics, Tatto et al. 2012) has explored student teachers' experienced opportunities to learn mathematics pedagogy and to learn from their practicum experiences in a range of countries, but not including Sweden.Using TEDS-M as a starting point but wanting to embrace the interplay between cognitive and situated knowledge, we surveyed students' perceived opportunities to learn principles, to practise, and to get feedback on more specific aspects of teachers' mathematical work.
The research questions guiding the study are: (1) What opportunities to learn components of mathematics pedagogy and practice these do Swedish mathematics student teachers perceive?(2) What opportunities to learn from their practicum do Swedish mathematics student teachers perceive?
Based on the answers to these questions, we discuss the extent to which the respondents' programmes appear to offer a reasonable range of representations of practice, and sufficient opportunities for students to make connections between theory/research and practice.

Opportunities to learn -From OTL for learners to TEDS-M
Opportunities to learn (OTL) is a concept connected to the perpetual question about equal opportunity for school-going learners.The assumption is that if learners have equivalent conditions, they will have the same opportunities for (higher levels of) learning.Early on, OTL was understood as 'content coverage' (Stevens 1996) or as time for learning (Suter 2017).However, the measures of OTL changed over time, from contact time and whether content had been covered, to considering the cognitive level of content, 'curricular coherence', and 'curriculum pacing' as well (Reeves 2005).Ways to determine OTL have included classroom observations and looking at learners' workbooks.Teachers and later also learners have been asked to indicate if content had been taught prior to the testing (Pelgrum 1989;Suter 2017).Over time, measures of OTL were used to explain correlation between countries' strong mathematics test performances and their inclination to teach more advanced mathematics (Suter 2017).
Research on learner performance has drawn attention to the importance of teachers' knowledge (Krainer et al. 2015) and competencies (Kaiser et al. 2015).Correspondingly, international comparative studies in mathematics education have increased the focus on teachers and teacher education.Most significant in this respect has been TEDS-M.Seventeen countries participated in this first international large-scale study about (initial primary and secondary) mathematics teacher education.The goal was not least to inform policy (Krainer et al. 2015).Teacher education programmes within and across countries were compared, as OTL was viewed as being influenced both by national policies and by teacher education institutions (Blömeke et al. 2012;Schmidt et al. 2008).
As with the studies on learners, TEDS-M relied on self-reporting on OTL by student teachers, but also analysed the prescribed curricula.TEDS-M assumed that prospective teachers need to learn mathematical content knowledge (MCK), general pedagogical knowledge, and subject-specific pedagogical knowledge (MPCK).Combining this with a vision of what teachers should be able to do, they identified several areas which it was felt teacher education should address, and conducted tests as well as surveys of perceived OTL (Krainer et al. 2015).They asked student teachers about their opportunities to learn in a coherent programme, to learn from practicum, and to learn mathematics pedagogy.Here, we focus on the two latter of these.
TEDS-M operated with an opportunity to learn mathematics pedagogy index, which was based on responses to eight items addressing 1. Foundations of mathematics; 2. Context of mathematics education; 3. Development of mathematics ability and thinking; 4. Mathematics instruction; 5. Development of teaching plans; 6. Mathematics teaching; 7. Mathematics standards and curriculum; and 8. Affective issues in mathematics.(Tatto et al. 2012, 183) It is a very crude measure of OTL to focus simply on whether a content area has been addressed, and not on what level, to what extent, or how.However, this was partially covered by other components of the TEDS-M study.
To assess the opportunities to learn mathematics instruction from their practicum experience, the future teachers participating in TEDS-M were asked to indicate how often (often, occasionally, sometimes, never) they had been required to engage in eight types of activities, such as observing 'models of the teaching strategies they were learning in their respective courses' (Tatto et al. 2012, 190).

Context: Swedish teacher education
Swedish teacher education has many similarities to teacher education in other countries (Gilroy 2014).There are separate teacher education programmes for grades R-3, 4-6, 7-9, and 10-12.Learning outcomes, length of programme, and distribution of study time are regulated nationally, but each institution has the freedom to decide on the specific implementation.The length of the R-3 and 4-6 programmes is four years (240 ECTS), while the secondary programmes take 4-5½ years (240-330 ECTS), depending on subject specialisation.The programmes contain 60 ECTS for educational core topics and 30 ECTS teaching practice, whilst the remainder are content and content education (i.e.'didactics', PCK, or methods) courses.The programmes are capped by individual student research theses (30 ECTS).(For further detail on Swedish mathematics teacher education see Christiansen et al. 2021.)According to the national law for higher education (SFS 1992(SFS :1434)), content and organisation of teacher education must be based on research.It is expected that students read some research, that they engage in research-like activity, and that lecturers are active researchers.However, Alvunger and Wahlström (2018) found that course literature in Swedish teacher education was mostly based on secondary sources, with only about 10% being research articles.There is limited research on Swedish mathematics teacher education.

Previous research
Internationally, there is a substantial number of research studies on student teachers' learning from their education, yet few studies which have focused on the opportunities to learn offered by the programmes.TEDS-M was the first large-scale international study of its kind, and only a limited number of studies have followed.The studies generally point to opportunities to learn but also to areas of improvement.
TEDS-M found that university-based opportunities to learn together with beliefs were factors in students' content knowledge and PCK (Kaiser et al. 2015).They constructed cohorts of countries based on the balance between MCK and MPCK in the programmes.A secondary analysis of the TEDS-M data for primary teachers from Switzerland, Germany, and Norway (considered representative of a Didaktik tradition), and the USA (considered representative of a curriculum tradition), suggests that MCK is foregrounded in the Didaktik tradition, and MPCK in the curriculum tradition (Werler and Tahirsylaj 2020).
Two additional studies have surveyed preservice teachers' perceived OTL, one of language student teachers in Germany (König et al. 2017), and one of early childhood or primary student teachers in Spain (Fuentes-Abeledo et al. 2020).German students who perceived the relationship between their university and their practicum school(s) as more coherent felt they drew more learning from their practicum with respect to lesson planning (König et al. 2017).This is in line with other work suggesting the importance of the university-school partnership -and mentoring -to opportunities for student teacher to learn from their practicum experiences (Lillejord and Børte 2016;Lloyd, Rice, and McCloskey 2020).
The Spanish students had opportunities to engage in practices related to classroom instruction, but fewer opportunities to practise educational guidance or participate in school activities (Fuentes-Abeledo et al. 2020).Interviews with Swedish students have indicated some of the shortcomings they experienced in their education around the possibility of building a 'teaching stance' and around connecting theory and practice (Emsheimer and Silva 2011).Swedish teacher education has also been criticised for modelling but not making the didactical organisation explicit, suggesting that the teacher education classes limit students' OTL (Asami-Johansson, Attorps, and Winsløw 2020).
The idea that OTL can be measured by self-reported experiences has been challenged.For instance, greater variation in reported OLT was found within than between programmes in the USA, suggesting that the reported OTL has as much to do with the individual respondent as with the programme (Cohen and Berlin 2020).
Several studies have taken a different perspective on what student teachers have the opportunity to learn.As a result, the TEDS-M instruments have been supplemented with video-based tasks to investigate noticing and deliberate practice of German TEDS-M participants after graduation (Kaiser et al. 2017).They suggest that teacher experience is better described 'with a two-dimensional model distinguishing between contentrelated knowledge (MCK, MPCK, and speed in mathematics error recognition) and performance-related competencies (GPK [general pedagogic knowledge], noticing, and reasoning)' (p.96).The study makes a strong argument for not reducing mathematics pedagogy when researching OTL.A scenario-based instrument was also used in studying the impact of opportunities to learn in Flemish preschool teacher education (Torbeyns, Verbruggen, and Depaepe 2020).As second-and third-year students outperformed firstyear students, their result indicates the presence of OTL in the programme, particularly in the theoretical courses on PCK.
A broader perspective is also taken in a study of the mathematics courses in secondary teacher education and their coherence with the education courses (Player-Koro 2011, Beach, 2011).Practices were criticised as inappropriate for teacher education: . . . the mathematics instruction that is shaped in the context of teacher education is, like the traditional school mathematics education, built around a ritualised practice based on the ability to solve exercises based on mathematical concept that is conveyed through an examined-textbook-based content.[sic] (Player-Koro 2011, 338) An alternative to self-reporting of OTL is to analyse content coverage of course literature.

Development of questionnaire
The questionnaire development was guided by the TEDS-M questions on opportunities to learn mathematics pedagogy and to learn from practicum experiences.To get more detailed data on perceived OTL of a broader range of mathematics pedagogy, and thereby a better basis for determining the extent to which a reasonable range of representations of practice were included in the programmes, the number of questions was expanded.The final version contained 143 multiple-choice questions -71 in part A and 72 in part B, and 46 open questions -29 in part A and 17 in part B. The open questions were formulated as encouragements to describe, exemplify, motivate, comment, narrate, or mention.

General opportunity to learn mathematics pedagogy
We used the eight categories from TEDS-M (listed earlier) and their question formulations albeit translated, with the addition of a ninth, assessment.Rather than a yes-no option, we gave the students the options: Has been given too much space in the programme; Has been treated very well and thoroughly; Has been treated to some extent; Has not been treated or treated to a very limited extent; Don't know.

Opportunity to learn specific aspects of mathematics pedagogy
We used two categories of Mathematical Knowledge for Teaching (Ball, Thames, and Phelphs 2008), namely Knowledge of Content and Students (KCS) and Knowledge of Content and Teaching (KCT), to select specific aspects of mathematics pedagogy.For instance, for KCS, we included questions about interpreting learners' reasoning, both theoretically and practically.For KCT, we asked questions about the opportunities to learn planning, teaching methods, inclusive teaching, and assessment.The questions about teaching methods were further detailed, informed by previous research.For instance, we asked whether students had learned how to use comparing and contrasting examples in teaching, a key component of facilitating learning from examples (Kullberg, Runesson, Kempe, and Marton 2017).We aimed to cover core components of practice without making the questionnaire too overwhelming.

Opportunities to learn from practicum
We used the eight types of activities from TEDS-M as our starting point.Rather than asking how often respondents had been required to engage each activity, we asked how often they had worked with the activity.We also changed TEDS-M formulations that seemed to imply a view of theories as directives for practice, since the term 'theory' may be used differently in Swedish teacher education.Instead, and in line with the idea of knowledgeinformed reasoned judgment as desired, we asked about using theory and research results.Based on the input from colleagues, we included additional questions on specific opportunities to learn from practicum.We were aware of the role of mentoring and school-university partnerships to the opportunities to learn from practicum (Christiansen and Österling under review;Ellis, Alonzo, and Nguyen 2020;Lloyd, Rice, and McCloskey 2020;Mena et al. 2016), yet chose not to include questions about these factors.The development of the questionnaire went through several iterations.It was piloted with third-year students in two programmes.They were asked to comment on the coverage of (their) mathematics teacher education, clarity of formulations, and construction of answer options.This process led to only minor adjustments, wherefore no further testing or development was undertaken.

Sample and representativity
Twenty-four of the Swedish higher education institutions have mathematics teacher education programmes.Thirteen agreed to participate in the survey and assisted in distributing our questionnaire to 816 students registered for their last term.Delivery succeeded for 753 students.We received 85 completed responses to part Aa response rate of 11.2%, varying from 3.2% to 32.0% from different institutions.The distribution of respondents on programme specialisation is shown in Table 1.Fifty-five respondents completed part B. The response rate is disappointing but not unexpected; web-surveys have notoriously low response rates (Fan and Yan 2010).
Compared to the national distribution of graduating teachers for the different grade levels from previous years (https://www.uka.se/fakta-om-hogskolan/om-lararutbildning.html), future grade 7-12 teachers are overrepresented in our sample.This may have to do with the fact that they are more likely to identify themselves as mathematics teachers, whereas teachers for grades R-6 have mathematics as one of several teaching subjects.Compared to the national data, women are overrepresented amongst the teachers for grades R-3 and 4-6, and under-represented for the grades 10-12.

Data collection
The electronic questionnaire was sent out to the potential respondents in the spring term of 2020, followed by three reminders.We followed the recommendations based on previous research (Fan and Yan 2010), such as informing about the questionnaire before sending out the link, indicating the importance of the response, etc. Ethical guidelines were strictly followed.

Analysis
The data were downloaded from the on-line questionnaire.Since few questions contained an option not to answer, limited cleaning of the data was required.Responses to each multiple-choice question were summarised numerically, and all responses to open questions were extracted.For the present paper, questions concerning the same or related aspects of KCS or KCT were grouped, as were questions on practice.Results were summarised within these groupings of questions, using descriptive statistics.This choice was informed by the aim of obtaining a first overview of perceived OTL, but also in part a result of the limited number of respondents.
In several questions, we used a Likert scale from 0 to 10.In the result section, a score in the interval [7;10] is taken to represent perceived good or very good opportunities, and a score in the interval [0;3] to represent perceived poor or very poor opportunities to learn.The responses to all Likert scale questions covered all possible scores from 0 to 10, except a few cases where no one had given the score zero.The higher scores generally dominated, with modes for all but two questions with dual models being five or above.
As the sample from each institution was small, no comparisons between institutions were undertaken.As the gender distribution varies between programme specialisations nationally, and 80% of the respondents were women, gender was not used as an independent variable.

Validity issues
The main threat to validity in surveys is the response rate and the representativity of the sample.As is common for on-line surveys, not least for lengthy ones, our response rate was low.Assuming a margin of error of 10% and a fairly homogenous population, we could be fairly confident the sample would be representative; however, the population spans several institutions with different histories and different locations, and four programmes.It is therefore possible that the selection does not represent the populationfor instance, that sample respondents are more or less satisfied with their education than the average student.This must be taken into account when engaging the results.
The large number of questions, guided by existing categories of mathematics teacher knowledge, has helped to increase coverage, yet it is clear that our instrument cannot cover all facets of mathematics pedagogy nor learning from practicum.The difficulties in 'measuring' PCK and teaching quality are well documented; asking respondents about opportunities means that answers will depend on interpretations of both 'OTL' and various aspects of PCK (such as 'distinguish different types of mathematical reasoning and solutions').Clearly, the lack of a uniform specialised language in the field may also lead to a range of different interpretations of the questions.
The questions from TEDS-M were translated into Swedish, keeping in mind that concepts do not always correspond across cultures (Andrews and Diego-Mantecón 2015).For instance, we did adjustments to the TEDS-M questions to allow for different notions of 'theory'.We also asked respondents how often they had worked with an activity instead of how often they had been required to engage with an activity; this does not address the OTLs built into the programme as much as perceived real OTLs.
A concern in a study of this nature is the issue of self-reporting.Indeed, we found cases of students within the same programme who perceived their opportunities to learn rather differently.As Cohen and Berlin (2020) suggested, the perceived OTL may have as much to do with the individual respondent as with the programme.We plan to repeat the study to investigate this issue further.
The variation in responses to the more specific questions on OTL mathematics pedagogy indicates that broad measures of perceived OTL may disguise differences between programmes.As with analysing teaching, there is a need to continue systematic development of instruments for researching OTL in teacher education.
For the present study, we were interested in the students' perceived OTL, hence we did not combine the study with the scenario-based approaches used to determine the effect of teacher education (by Kaiser et al. 2017;Torbeyns, Verbruggen, and Depaepe 2020).This would, however, be a worthwhile follow-up study.Similarly, it would be worthwhile to further investigate coherence between mathematics and mathematics education courses as was done by Player-Koro (2011).

Results
In the following three sections, we present what the respondents suggested about their opportunities to learn.First, we present the general opportunities to learn mathematics pedagogy experienced by the respondents, based on the questions from TEDS-M.Second, we go into aspects of opportunities to learn mathematics pedagogy specific to our study, namely learning about: learner thinking, planning, assessment, and teaching mathematics.Third, we report on the respondents' perceived opportunities to learn from their practicum experiences.We state clearly when the results are from part B of the questionnaire; if nothing else is stated, the results are from part A.

Experienced opportunities to learn broad topics in mathematics pedagogy
To enable contrasting with the TEDS-M results, we treated the Likert scale answers on coverage 'too extensively', 'very well and thoroughly', and 'to some extent' as 'covered', and the answers 'has not been treated or treated to a very limited extent' and 'don't know' as 'not covered'.Thirty-nine respondents (46%) answered that all nine topics had been covered, an additional 27 (32%) that seven or eight topics had been covered.The mean was 7.5 ± 2.0 or a coverage of 0.83 ± 0.2.The outlier is a respondent who answered 'don't know' for all nine topics.The result compares favourably to the TEDS-M results, although our low response rate does not allow us to generalise to the greater population.
The topics which most respondents thought had been covered were development of mathematical proficiency and thinking (80 respondents or 94%), summative and formative assessment (79 or 93%), curricula and examined outcomes (77 or 91%), teaching mathematics, for instance, using representations or building from learners' thinking (73 or 86%), and planning (72 or 85%).Assessment was the only topic that more than half of the respondents thought had been treated thoroughly.These summative results disguise the fact that 11 respondents (13%) felt that their programme only to a limited extent addressed planning, and that nine (11%) felt the same regarding mathematics teaching.
The two topics that most respondents felt had been treated very little or not at all were affect (feelings, attitudes, motivation, mathematics anxiety) (26 or 31%), and the context of mathematics education (role in society, gender, class, and ethnicity) (22 or 26%).
While this result indicates good overall coverage in the programmes across institutions, responses to questions on more specific competencies showed greater dispersion in perceived opportunities to learn and thereby in range of representations of professional practice.

Perceived opportunities to learn specific components of mathematics pedagogy
We asked about opportunities to learn about learner thinking, planning, assessment, and a range of aspects related to instruction.

Learning about learner thinking
Almost 80% felt they had had some or good opportunities to learn about learner thinking practically and/or theoretically, but there were substantial variations in the extent to which they felt they had had the opportunity to learn about specific aspects of learner thinking.
In part A, we asked the respondents to what extent they felt they had had opportunity to learn theoretically about learner thinking and learning (scale from 0 to 10); to what extent they had worked with learner thinking and learning in specific teaching situations (scale from 0 to 10); and to what extent they felt they had been given feedback on their work with learner thinking and learning (4 options).Part B of the questionnaire contained eight additional questions (4 options for each).
Many students felt they had had good opportunity to learn about learner thinking.However, ten (12%) felt they had had little opportunity to learn about learner thinking in any respect (theoretically, practically, and through feedback).
Focusing on theoretical learning about learner thinking, 37 (44%) felt that they had been given good or very good opportunities to learn, with 15 (18%) giving a score on the Likert scale of 9 or 10.Yet, as many as 19 (22%) felt that they had had very limited opportunity.On average, the prospective secondary teachers had experienced more opportunities for this aspect, which is not surprising given that they will become subject specialists.However, the spread in responses was also higher for this group.
Good opportunities to work with learner thinking in practice were indicated by 41 respondents (48%).Opportunity to collect and analyse data about learners' learning was felt to have occurred often or sometimes by 54 respondents (64%), rarely by 20 (24%) and never by eight (9%).The opportunity to actively adapt research results about learners' common difficulties occurred often or sometimes according to 35 respondents (41%), rarely for 30 (35%), and never for 17 (20%).
Twenty-two respondents commented on opportunities to work with learner thinking in practice.Some mentioned their practicum as a less rewarding setting for working with learner thinking in practice.Some mentioned positive experiences about construction of learner tasks or looking at learners' solutions in their campus courses.
Table 2 shows the questions from part B and the frequencies of responses from the 55 future teachers who answered this part.For many specific competencies or knowledge areas, a large share of the students (ranging from 33% to 60%) felt they had had only little or no opportunity to learn.The least perceived opportunities were learning to identify 'gifted' learners, followed by learning theories about developing learners' mathematical thinking and learning, and learning about research results.
Many respondents listed topics where they felt they had particularly good opportunities to learn.These included learning to interpret the level of learners' reasoning, threshold concepts, explanatory models or theories, and analysing learner answers.Many felt that they had learned a lot but would have liked even more.Together, these results indicate that almost all respondents perceive reasonable opportunities to learn at least some aspects of learner thinking, but that some aspects appear to get more attention than others.We note in particular, that less than half of the respondents felt they had had good opportunities to engage with and apply research results on learner thinking.

Learning (about) planning
While many students experienced good opportunity to learn about planning, 16 (19%) felt they had had very limited opportunities across all the dimensions (learn principles for, practise, and get feedback).When it came to specific competencies, more than half of the respondents on part B indicated limited opportunity to learn some of these, such as choosing teaching materials, adjusting tasks, or planning for formative assessment during lessons.
The 55 respondents who answered part B most frequently indicated good or sufficient opportunities to formulate learning goals for a lesson and identify key ideas in a mathematical topic.Worst were the perceived opportunity to learn about choosing, designing, and reformulating tasks, choosing pedagogical resources, and planning thematic or interdisciplinary teaching.Less than half of the 55 respondents perceived sufficient opportunities to learn how to prepare for making adjustments to lessons in response to learners or to adjust tasks -two key aspects of reasoned judgment.

Learning about assessment
Some opportunities to learn about principles for assessment was noted by 85% of the respondents, with 25 respondents giving a score on the Likert scale of 9 or 19, but 28% felt they had had limited opportunity to practise and get feedback.The respondents on part B helped us to get a sense of where the focus had been.Most indicated that they had had great or sufficient opportunity to learn about different forms of assessment, formative assessment specific to mathematics (both 41 or 75%), or summative assessment specific to mathematics (42 or 76%).Good or sufficient opportunity to learn about marking specific to mathematics and communicating with learners or caregivers about learners' exhibited proficiency and knowledge were only reported by 22 (40%) and 16 (29%), respectively.
Some respondents commented that the assessment they had learned about in their courses was not implementable.Practicum, workshops, seminars, field studies, and 'group days', were where the respondents had opportunities to work practically with assessment, suggesting that this component of teaching is well represented and with opportunities for engaging in approximations of practice during course work.Some had marked national exams and class tests, and used exit tickets.Some mentioned working with collecting evidence and working with learners' progression, while others felt they had seen examples of summative assessment only.

Learning about teaching mathematics
When it came to students' opportunities to engage with aspects of teaching mathematics, we asked about opportunities to learn different instructional approaches, methods for engaging a whole class of learners, ways of using examples, ways of utilising mathematical communication, and working inclusively.We go into some detail with the responses to these questions, as they concern some of the core practices of teaching mathematics.
Between a quarter and a third of the respondents indicated that they had had limited opportunities to learn about and practise different teaching approaches.The results were similar for whole-class teaching.Forty percent felt they had had limited opportunity to practise inclusive teaching.The respondents of part B answered questions about using examples, and of these, only half felt they had had the opportunity to learn about comparing and contrasting examples.
Good opportunities to learn principles for different teaching approaches were experienced by 38 respondents (45%), poor opportunities by 22 (26%).In the comments, 'too much' was mentioned by a respondent who would have preferred more content knowledge focus, while 'too little' was mentioned by someone who wanted to learn more about teaching learners who dislike or struggle with mathematics.
Opportunity to practise different teaching approaches was perceived as good and poor, respectively, by around a third each of the respondents (38% and 33%).They generally referred to their practicum experiences.Several pointed to the limitations of working in someone else's class over time: 'One has been able to experiment during practicum, but it is still the mentor who decides'; 'It is difficult during practicum to go and "do something new" in a different way from what learners are used to'.Others had felt supported by their mentor to test any approach they liked.Some respondents indicated that it is up to the students to take initiative to practise different teaching approaches.
Just a little over half of the respondents felt they had received regular feedback on their teaching approaches (45, or 53%), while five (6%) said never.
Good opportunities to learn principles for teaching whole class or larger groups of learners were perceived by 28 respondents (33%), some opportunities by 33 (39%), and limited opportunities by 24 (28%).Several indicated that they were uncertain whether they had met this, e.g.'If we had this, then it was for homogenous groups where all learners are "perfect", that is motivated, capable, attentive and all on the same level.Not particularly realistic!'Others felt that it had been touched on, for instance, in connection to learners' varied competency levels and interactions, or in relation to sociocultural perspectives on learning.
Good opportunities to practise whole-class teaching were noted by 39 respondents (46%), while 23 (27%) said they had had limited opportunities.From the comments, it appeared that respondents had mostly practised this during practicum periods, though field study and work experience were also mentioned.
The respondents to part B were asked about specific aspects of teaching and indicated greater opportunities here than on some of the previously discussed questions.However, only 53% felt that they had had opportunities to learn theoretical knowledge about teaching approaches, as well as how to consciously choose an approach.Sixty percent or more of these respondents indicated that they had learned theories/principles for: presenting to a large group or a class, explaining mathematical procedures to learners, using examples in presentations, and explaining or representing mathematical concepts to learners.Only 26 (47%) replied that they had learned theories or principles for explaining mathematical connections, laws, or theorems to learners.We cannot say if another approach was more frequently addressed, but we note that 34 (62%) said they had learned about how to introduce new content through learner activity.Opportunities to learn theories or principles for choosing examples was more commonly perceived (by 34 of 55, or 62%), compared to learning theories or principles for comparing and contrasting examples (28, or 51%).
We included four questions about mathematical communication in part B. Good or some opportunity to learn about the use of mathematical language/discourse in teaching was felt by nearly half (49%), while fewer had learned about types of constructive questions and principles for asking these, or about leading a discussion amongst learners (36% and 34%, respectively).
When it came to opportunities to learn inclusive teaching approaches, 36 respondents to part A (42%) said they had had good opportunities to learn principles for this, 21 (25%) that they had had poor opportunities.Some said they had had many lectures on this, others that it had only been addressed in relation to 'weak' learners, or that it had been abstract or focused on theory, and one mentioned having learned from their mentor about inclusion.
Only 19 (22%) felt they had had good opportunities for practising inclusive teaching, the majority indicating poor opportunity (40%).Some felt they had not had the opportunity to practise inclusive teaching because their practica had been in fairly homogenous groups.One mentioned that their education had focused on inclusion generically, had not dealt with different groupings of learners, and that it had been quite general, such as getting advice about giving clear instructions.There was a strong correlation between the extent to which students felt they had had the opportunity to practise inclusive teaching approaches and getting feedback on it (0.71).
The respondents on part B felt they had had better opportunities to learn about learners who have difficulties in mathematics (30, or 55% answered good or sufficient), and language development (26 or 47%), compared to learning about learners with strong emotional responses to mathematics (12, or 22%) and learners from different socioeconomic backgrounds (14, or 25%).
Forty-one of the respondents to part B provided additional comments.One mentioned the disconnect between the practicum and the campus courses when it came to teaching.Several mentioned what they had worked with, such as asking open and challenging questions, leading learners forwards through discussions and group tasks, daring to let something take time, working with comparing and contrasting examples, the characteristics of quality problems or tasks, how to explain or represent mathematical concepts, and how to give a presentation.Some felt they had not learned enough about teaching in a way that favours all learners, leading a discussion, asking questions to lead learners on, teaching whole class, how to choose and use examples, explaining mathematical connections and explaining mathematical reasonings in a way appropriate to the group of learners, and how to work with language ('Mathematics is a communicative subject').
These results suggest limitations in our current teacher education practices.We will return to this in the discussion.

Opportunities to learn principles across topics
After looking at the responses to the individual questions, one concern remained: To what extent is the perception of limited opportunities clustered within the sample of respondents?To answer this, we collated the responses to the seven main questions about the opportunities to learn principles (planning, engaging learner thinking, working with whole class, teaching for 'mathematical abilities', 1 different instructional approaches, inclusive teaching, and assessment).We found that 14 (16%) perceived overall poor opportunities to learn principles, while 30 (35%) perceived they had had good opportunities to learn principles.A small group of students felt that their opportunities had been very uneven across the seven questions.The prospective secondary teachers generally felt they had had better opportunities to learn principles, which again is not surprising given that they become subject specialists.The data set is too limited to make claims about differences between programmes in this respect, but it suggests that institutions could benefit from considering their coverage of principles for these core practices.

Perceived opportunities to learn to teach mathematics from practicum
We asked the respondents seven questions about their learning from practicum, as per Table 3.The responses indicate reasonable opportunities to learn from practica, with more than 60% answering 'often or sometimes' to five of the seven questions.The responses to the remaining two questions suggest some limitations in the opportunities to connect content from campus courses or research to practice.
Eleven respondents provided additional comments.Some mentioned the nature of the mentoring, that theory and practice were rarely or never connected, that working with learners' challenges was not connected to mathematics, that the mathematics teaching in the practicum schools was of low quality, and that learning from practicum had been on the students' own initiative.One student felt that they were not treated as a student but as a resource.As previously mentioned, it is well known that the quality of the mentoring and of the university-school partnership is crucial to the opportunities for student teacher to learn from their practicum experiences, and hence it is not surprising that some respondents indicated reduced OTLs as a result of less than optimal mentoring or partnerships.
In relation to learning about specific instructional practices, we asked respondents to indicate, on a scale from 0 to 10, the extent to which they had done the following (Table 4): Eighteen respondents (21%) indicated poor opportunities to practice across the programme, while 25 (29%) indicated good opportunities.Generally, there is room for improvement so that more students recognise more frequent opportunities to learn, particularly through rehearsing inclusive teaching, adapting research results, and observing a range of instructional approaches.

Discussion and conclusion
The results give us an overall indication of the opportunities to learn perceived by Swedish mathematics student teachers.Across the areas, students appeared to have better opportunities to learn theories and principles than to practise or get feedback on the various domains of learning, which adds important knowledge to our understanding of the links between theory and practice in mathematics teacher education, while also raising new questions.Still, some respondents perceived poor opportunities for learning principles and for practice across their programmes.
A little over a third of the respondents felt that they had rarely or never received feedback on their practise of various components of teaching, which may indicate issues related to mentoring.The issue of feedback was not covered in previous studies, despite it generally being considered vital to learning, and this suggests that this is deserving of increased attention in teacher education programmes and research.
The students reported good opportunities to learn about assessment, curriculum, planning, building on learners' thinking, analysing learners' answers, and developing mathematical proficiency and thinking.Perceived to be less well covered were affect, context, choosing resources and tasks, planning thematic teaching, theories about developing learning thinking and about teaching approaches, identifying 'mathematically gifted' learners, and marking.Our respondents perceived more opportunities to practise planning, working with learner thinking and whole-class teaching than to practise inclusive teaching and applying research.
Teaching mathematics in an increasingly multicultural society in a way which provides the best possible opportunities to learn for all learners rests on balancing a number of considerations in making informed decisions.We designed the questionnaire to cover what we considered core practices in this respect.And while many respondents indicated good opportunities for learning to navigate different instructional approaches, engaging a class of learners, choosing and deploying examples, involving learners in mathematical communication, and working with learners of different competence levels or backgrounds, between a third and half perceived limited opportunities to do so.Furthermore, those that commented on these questions generally indicated that they had had better opportunity to learn these aspects during their practica than during course work.This alludes that there is scope for improving the school-university partnerships (see also Lillejord and Børte 2016).
As a whole, these results suggest that Swedish mathematics teacher education needs to give more attention to ensuring that core practices are indeed represented in the programmes; that research-informed principles underlying such practices are addressed in ways explicitly linked to rehearsing in contexts other than in the full-blown teaching during practica; and that these clearly involve engaging in reasoned judgment.Despite all students being required to engage research during their education, the opportunities to relate research to practice also appear to need strengthening.Previous research suggests that these concerns are not unique to Sweden (see, for instance, Christiansen and Österling under review).
Whole-class teaching is prevalent in mathematics lessons.It may not be the desired practice in teacher education (see discussion in Christiansen, Österling, and Skog 2019), but it is a practice the student teachers are likely to end up using and hence it seems reasonable that they be prepared for this.Yet many students felt they had had limited opportunities to learn about, practise, and get feedback on this.
The use of examples is crucial to teaching (Kullberg, Runesson Kempe, and Marton 2017;Mhlolo 2013).Whether teaching through exposition or through engaging learners in exploration activities, the comparison across examples with carefully picked variation is the basis for conceptual development (op.cit.).However, our results exhibited great variation in the extent to which respondents thought they had learned about the use of examples in teaching.This invites further consideration by the institutions offering these programmes.
A majority of students reported having had opportunity to learn about how to introduce new mathematical content through learner activity, but more than a third claimed not to have learned about principles for use of different types of questions, nor about leading a mathematical discussion.If this reflects actual teacher education practices, it means that an oft-desired approach to teaching is only partially covered, as also suggested by Asami-Johansson, Attorps, and Winsløw (2020).
Finally, after the substantial influx of refugees to Sweden and the increased difference in learner performance relative to socio-economic background, together with well documented discrimination issues in grading (Hinnerich, Höglin, and Johannesson 2015), it is surprising that inclusive teaching is not given more consistent attention in the programmes.
We are unable to say, based on students' reports of their perceived OTL, whether these results reflect the weighting of content in actual programmes, but we strongly feel that they invite teacher education institutions to consider whether this is the case, and if so, if this is intended or a result of the history of the programmes.To us, the results suggest a need for a more systematic approach to programme coverage in our practices, and a clearer and more coherent grounding in relevant research.

Note
1.The Swedish school curriculum operates with so-called 'mathematical abilities': reasoning, problem-solving, conceptual understanding, communication, and procedural ability.Upper secondary school has two additional 'abilities': modelling and relevance.

Table 1 .
Distribution of respondents on programme specialisation.

Table 2 .
Frequency of responses to questions from part B concerning OTL about learner thinking.n = 55.

Table 3 .
Frequencies of responses to questions from part A concerning OTL from practica.n = 85.

Table 4 .
Mean score of responses to questions about opportunities to practise.n = 85.During your teacher education, to what extent did you have the opportunity to: Mean score . . . in actual instruction, work with learners' learning and thinking?6.1 ± 2.97 . . .practise working with learners in a large group or the whole class?5.9 ± 3.12 . . . in specific teaching situations, work with the learners' development of the mathematical abilities?5.5 ± 2.89 . . .try out different teaching methods in mathematics instruction?5.3 ± 2.93 . . .try out inclusive approaches with different groups of learners?4.3 ± 2.86