Relationship between university teachers’ beliefs about teaching mathematics and their instructional practices

Abstract

This study delves into the correlation between the beliefs of five university mathematics teachers regarding the teaching of mathematics and their instructional practices. Adopting a case study methodology, this research aimed to thoroughly investigate this relationship. The collection of data was carried out through questionnaires, and the analysis was performed using IBM SPSS Statistics 20. The results indicate that the beliefs held by teachers significantly influence their instructional practices. Teachers with a belief in the transmission model of teaching mathematics are more inclined towards teacher-centered methodologies, whereas those who subscribe to a discovery or connectionist perspective favor student-centered approaches. Moreover, a clear relationship was found between teachers’ beliefs concerning the nature of mathematics, its teaching methodology, and the learning process, and their instructional practices. Recognizing the importance of teachers’ beliefs about the teaching and learning of mathematics is essential, as these beliefs profoundly affect instructional practices. It is suggested that teacher education and ongoing professional development programs should focus on modifying beliefs regarding mathematics and its learning processes to enhance instructional practices. Continuous professional development interventions aimed at refining these beliefs could lead to improved outcomes for students.

Keywords:

• Effective mathematics instruction
• learning process
• instructional practice
• transmission view
• connectionist or discovery view

REVIEWING EDITOR:

• Stephen Darwin, Universidad Alberto Hurtado, Chile

SUBJECTS:

• Teachers & Teacher Education
• Higher Education
• Philosophy of Education

Introduction

The concept of beliefs has garnered significant attention in research, yet consensus on its definition remains elusive. It falls upon researchers to demystify the notion of beliefs, which are often deduced from the alignment between individuals’ actions and verbal expressions as suggested by Pajares (1992). The stronger the alignment, the more deeply ingrained the belief is presumed to be. However, individuals may harbor beliefs that are either unspoken or subconscious, yet these beliefs still exert influence over their thoughts and behaviors. Unearthing these hidden beliefs offers a window into an individual’s underlying motivations and thought processes. Moreover, delving into the origins and impacts of beliefs sheds light on their formation and potential avenues for alteration or refinement. This paper focuses in on the interplay between university mathematics teachers’ beliefs about effective mathematics instruction and their instructional practices.

Xie and Cai (2018) highlight how a teacher’s perception of mathematics is shaped by their beliefs about the discipline. Ernest (1989) elaborates that a teacher’s comprehension of mathematics encompasses both their knowledge and beliefs regarding teaching and learning. While knowledge is pivotal, it is the teacher’s beliefs that predominantly dictate their pedagogical approach, interaction with students, and assessment methods. These beliefs mold instructional strategies and student engagement; for instance, a teacher who perceives mathematics as a domain for creativity and collaboration might emphasize group activities and problem-solving in their lessons, contrary to a teacher with a more rigid perception of mathematics, who may lean towards rote learning and individual tasks.

Teachers’ belief systems regarding mathematics comprise three interconnected facets: their understanding of mathematics itself, their teaching methodology, and their views on the learning process. As Ernest posited in 1989, a teacher’s conceptualization of mathematics profoundly influences their teaching and learning beliefs, making the exploration of these perspectives vital for discerning effective mathematics instruction. Ernest’s framework has been widely adopted and applied in educational research, underpinning the necessity of grasping a teacher’s mathematical outlook for effective teaching and learning (Beswick, 2012; Bryan et al., 2007; Muhtarom et al., 2019; Yang et al., 2020).

Belief systems are essentially the organization of beliefs around a specific concept or entity, consisting of primary or derivative beliefs that can be either central or peripheral. Notably, teachers’ beliefs are not solitary entities but are interconnected (Philipp, 2007). These systems are dynamic, capable of evolution as individuals reassess their beliefs in light of new experiences (Thompson, 1992). The interconnected nature of these beliefs, their theoretical consistency, and their spectrum from instrumentalist to problem-solving views offer a comprehensive understanding of teachers’ pedagogical orientations. Table 1 illustrates how these beliefs are connected, with the beliefs in the same row being theoretically consistent and those in the same column forming a continuum.

Table 1. Teachers’ beliefs systems (adapted from Beswick, 2012 and Ernest, 1989).

Beliefs about the nature of mathematics	Teacher’s role	Beliefs about mathematics teaching	Beliefs about mathematics learning
Instrumentalist	Instructor	Content-focused with an emphasis on performance	Skill mastery, passive reception of knowledge
Platonist	Explainer	Content-focused with an emphasis on understanding	Active construction of understanding
Problem-solving	Facilitator	Learner focused	Autonomous exploration of own interest

Despite extensive research on teachers’ beliefs across different educational levels, there remains a paucity of studies by university teachers examining their own beliefs about mathematics teaching and its impact on higher education learning outcomes (Speer, 2008). This gap in research, especially in the context of Ethiopian higher education, hampers efforts to enhance the quality of university-level mathematics education. Addressing this gap, our study aims to elucidate the beliefs and practices of University Mathematics Teachers (UMTs), offering insights into their pedagogical approaches.

The nexus between teachers’ perceptions of effective mathematics instruction and their pedagogical methods is intricate and has been the focal point of extensive research. This body of work underscores a significant correlation between teachers’ beliefs about mathematics and their instructional practices (e.g. Beswick, 2012; Muhtarom et al., 2019; Yang et al., 2020). However, the translation of beliefs into practice is mediated by several factors, including teachers’ mathematical knowledge, experience, and teaching context. Despite these complexities, understanding this relationship is crucial for advancing mathematics education. By comprehending how teachers’ beliefs influence their teaching, we can devise targeted interventions to elevate teacher quality and student learning outcomes.

Despite the significant advancements in the field of mathematics education, a gap persists between teachers’ views of effective mathematics teaching and their instructional practices. The nature of this relationship is not always straightforward. Teachers’ beliefs are not always directly translated into their instructional practices. There are a number of factors that can mediate this relationship, such as teachers’ knowledge of mathematics, their experience, and the context in which they teach (e.g. Bryan et al., 2007; Cai & Wang, 2010). Despite these challenges, understanding the relationship between teachers’ beliefs and their instructional practices is important for improving mathematics education. By understanding how teachers’ beliefs influence their teaching, we can develop more effective interventions for improving teacher quality and student learning.

This study presents a detailed examination of the connection between university mathematics teachers’ perceptions of effective teaching and their applied instructional strategies. Central to our inquiry was an in-depth analysis of the beliefs of five UMTs at Assosa University regarding what constitutes effective mathematics instruction. The guiding question for this research was: How do teachers’ perceptions of effective mathematics teaching shape their instructional approaches? This investigation entailed a comprehensive exploration of their views on the essence of mathematics, the characteristics of adept mathematics instruction, and the learning process, aiming to understand how these beliefs correlate with their reported teaching practices.

By gaining insight into the beliefs held by mathematics teachers at Assosa University, we can thoroughly analyze the factors that shape their teaching methods and pedagogical approaches. This understanding also offers valuable perspectives on the effectiveness of current teaching approaches and curriculum design at the university. The exploration of these beliefs aims to enhance the existing knowledge and support advancements in mathematics education at Assosa University, taking into account its distinct cultural and educational environment. Understanding the specific beliefs of these teachers and their application in the classroom is essential for comprehending their influence on student learning.

Literature review

Teachers’ beliefs about mathematics

Xie and Cai (2018) emphasized that teachers’ perceptions of mathematics are significantly influenced by their beliefs regarding the discipline. Ernest (1989) argued that a mathematics teacher’s comprehension of the subject is an amalgamation of their knowledge and beliefs about teaching and learning mathematics. Although knowledge plays a crucial role, it alone cannot explain the diversity in teaching approaches among teachers. Pajares (1992) and Philipp (2007) highlighted that teachers with comparable levels of mathematical knowledge might adopt different instructional strategies, such as problem-solving or traditional didactic methods. Furthermore, a teacher’s beliefs about teaching and learning mathematics are often shaped by their personal experiences as both students and educators, in addition to cultural and societal norms. These beliefs profoundly affect instructional design, student interactions, and the evaluation of learning outcomes.

The relationship between teachers’ beliefs about mathematics, effective teaching practices, and their instructional approaches has been well-documented (Fennema et al., 1996), including their propensity towards student-centered methodologies (Heck et al., 2008; McGee et al., 2013). The discourse on the composition of teachers’ mathematical beliefs has evolved over time, encompassing their views on the nature of mathematics, mathematics teaching, and learning (Ernest, 1989). Askew et al. (1997) delineated teachers’ orientations towards these aspects into three categories: transmission, discovery, and connectionist, noting that teachers may integrate elements from these orientations, despite potential contradictions.

Teachers’ perceptions of mathematics include a broad spectrum of conscious and unconscious thoughts, beliefs, mental representations, and preferences that underlie their educational philosophy towards mathematics (Cai, 2007; Cai & Wang, 2010). Distinguished philosophies on the nature of mathematics among teachers include instrumentalist, Platonist, and problem-solving perspectives. The instructional approach adopted by a teacher is heavily influenced by their interpretation of teaching roles, classroom activities, and interactions (Beswick, 2012; Ernest, 1989). Various models have been proposed to describe teachers’ beliefs about teaching, including the instructor, explainer, and facilitator teacher models (Xie & Cai, 2018). These perceptions are intricately linked to their beliefs regarding the process of learning mathematics, encompassing their understanding of how mathematics is learned and the activities deemed effective for learning.

Mathematics teachers’ instructional practice

In the realm of mathematics education research, instructional practices have been classified through various lenses. Qualitative research offers detailed insights via case studies or ethnographic methods, while some studies adopt a multi-method approach, quantifying observational data. Surveys provide self-reported data from teachers about their instructional practices, often augmented with classroom observations to enhance validity (e.g. Heck et al., 2008). These methodologies yield valuable information on the effectiveness of instructional practices, contributing to the development of professional development programs.

Efforts to categorize teachers’ instructional practices, such as teacher- or student-centered approaches, have been extensive (Heck et al., 2008; Swan, 2006; Tarr et al., 2008). Findings indicate that instructional methods can vary based on the subject matter and curriculum resources. Although there may be slight variations in teaching methods, the congruence between teachers’ self-reported and observed practices suggests a consistency in instructional execution (Swan, 2006). This consistency underscores the influence of subject matter and resources on instructional strategies and highlights the alignment between teachers’ beliefs and their actual teaching practices.

Materials and methods

Methodology

This study utilized the case study approach to explore the relationship between teachers’ perspectives on effective mathematics teaching—including their views on the nature of mathematics, its teaching, and learning—and their instructional practices. The research employed a questionnaire adapted from Swan (2006, 2007) to assess teachers’ beliefs and practices in the context of mathematics education. The survey was conducted at the outset of the semester to capture teachers’ beliefs and practices at a specific point in time.

Sample

The study sample comprised five experienced mathematics teachers from Assosa University, who voluntarily participated during the 2020/2021 academic year. These individuals were teaching ‘Applied Mathematics I’ to pre-engineering students and were selected based on their expertise, experience, and willingness to participate. The selection process excluded two of the seven teachers responsible for the course: one due to insufficient teaching experience and the other, a female teacher, due to maternity leave. This strategic selection aimed to ensure that the study benefited from the insights of teachers deeply familiar with applied mathematics, thereby enhancing the investigation of teaching methodologies and belief systems in engineering mathematics instruction. All participants were male lecturers with backgrounds in Mathematics Education or Applied Mathematics, and all held Master’s degrees in Applied Mathematics with specializations in areas like Numerical Analysis and Modelling. Their teaching experience varied widely, enriching the study with diverse perspectives.

Research instruments

The study utilized questionnaires for both teachers’ beliefs and practices, derived from Swan’s work (2006, 2007) (Appendix A & B). The beliefs questionnaire probed teachers’ views on effective mathematics teaching across three dimensions—nature of mathematics, mathematics teaching, and learning—asking participants to allocate percentages to transmission, discovery, and connectionist views, summing to 100% per dimension. Based on Swan’s (2007) findings, teachers responses were categorized into ‘transmission’ and ‘discovery/connectionist’ groups, with the latter label applied to those indicating a 50% or higher alignment with discovery or connectionist views.

The instructional practices questionnaire sought self-reported data on teaching approaches, distinguishing between teacher-centered and student-centered practices. Participants rated the frequency of twenty-five classroom behaviors on a five-point Likert scale, enabling an assessment of their instructional tendencies without direct classroom observation. This approach aligns with literature suggesting a general congruence between self-reported and observed instructional practices (Polly et al., 2013; Swan, 2006). Thirteen of these behaviors may be categorized as teacher-centered and twelve as student-centered. For each item, participants rated themselves on a 5-point Likert scale, with 1 representing ‘almost never’ and 5 representing ‘almost always.’ Scores ranged from 25 to 125, with higher scores indicating more student-centered behaviors.

Data analysis

Descriptive statistics, including mean and standard deviation, were employed to explore potential relationships between teaching practices and beliefs in transmission versus discovery/connectionist orientations. Data coding and analysis utilized IBM SPSS Statistics 20, achieving Cronbach’s alpha coefficients of 0.86 and 0.84 for the beliefs and practices questionnaires, respectively. These coefficients indicate a high degree of internal consistency among respondents, affirming the reliability of the survey results and their suitability for drawing substantive conclusions about teachers’ beliefs and instructional practices. The consistency and reliability of these findings underscore the clarity of the survey questions and their effectiveness in measuring the intended constructs.

Results

This study establishes a coherent relationship between University Mathematics Teachers’ (UMTs’) beliefs about the nature of mathematics, the methodologies and learning processes involved, and their teaching practices. This coherence is further supported by their self-reported pedagogical practices in the classroom, highlighting the critical importance of understanding UMTs’ beliefs about mathematics teaching and learning. These beliefs significantly influence their instructional practices, underscoring the vital role they play in shaping educational outcomes.

The discovery of such coherence suggests that UMTs possess a deep understanding of the principles of effective mathematics teaching and are capable of integrating these principles into their classroom practices. This integration has the potential to improve learning outcomes and elevate the quality of mathematics education within Higher Education Institutions (HEIs).

Analysis of responses from five UMTs regarding their views on effective mathematics teaching across three distinct categories (transmission, discovery, and connectionist views) reveals a preference for a ‘transmission’ approach, with lesser emphasis on discovery or connectionist methods. However, a closer examination indicates a slight preference for a ‘connectionist’ approach, emphasizing collaboration over individual discovery. These findings have significant implications for enhancing teacher education curricula and developing professional development paths for UMTs, encouraging the adoption of more effective teaching strategies.

The detailed responses from the five UMTs about their conceptions of efficacious mathematics teaching across three distinct categories are summarized in Table 2. The Table delineates the means and standard deviations of transmission, discovery, and connectionist views as perceived by the five UMTs, achieved by calculating the mean of individual statements within each category. The findings suggest that a predominant number of UMTs subscribe to a ‘transmission’ view of teaching mathematics, with minimal inclination towards discovery or connectionist views. A nuanced comparison between discovery and connectionist perspectives reveals a marginal inclination among UMTs towards a collaborative ‘connectionist’ approach over an individualistic ‘discovery’ one. This reflects UMTs also recognize the importance of practical engagement and application beyond mere memorization, indicating a preference for collaborative, connectionist approaches that emphasize social interactions and dialogue in learning.

Table 2. Results from the beliefs questionnaire.

Mathematics is:	Statement	Mean (%)	SD
MT	A specific body of knowledge and standard procedures exists, along with a set of universal truths and rules that must be imparted to students.	53	9.75
MD	A creative subject requires the teacher to assume a facilitating role, enabling students to develop their own concepts and methods.	24	9.62
MC	A collaborative process between the teacher and the student, where they jointly develop a network of interconnected ideas through meaningful discussions.	23	4.47
Learning is:
LT	An individual activity based on watching, listening and imitating until fluency is achieved.	46	6.52
LD	An activity that involves practical exploration and reflection by an individual.	28	8.37
LC	An interactive activity where students engage in discussions to challenge their thinking and reach a deeper understanding.	26	8.22
Teaching is:
TT	Structuring a linear curriculum for students involves providing verbal explanations and ensuring understanding through practice questions. It is also important to address any misunderstandings that may arise when students struggle to grasp the material being taught.	45	7.07
TD	Assessing a student’s readiness to learn, creating a stimulating environment to encourage exploration, and preventing misunderstandings through carefully sequenced experiences.	32	10.95
TC	A non-linear dialogue between teachers and students that explores meanings and connections through verbal communication. This dialogue explicitly addresses and resolves any misunderstandings that may arise.	23	9.75
Transmission: mean of MT, LT, TT Discovery: mean of MD, LD, TD Connectionist: mean of MC, LC, TC		144	14.75
		84	23.29
		72	14.40

Note: In the first column; the first letter represents Mathematics (M), Learning (L) or Teaching (T). The second letter refers to Transmission (T), Discovery (D) or Connectionist (C) beliefs.

These insights have profound implications for mathematics education, advocating for a shift towards collaborative learning and the integration of mathematical concepts. Such a shift could enhance understanding and application, fostering environments where students actively construct knowledge through exploration and discussion. This approach not only deepens understanding of mathematical concepts but also has the potential to spark a passion for the subject, leading to future innovations.

Table 3 summarizes the UMTs’ self-reported instructional practices, which helped to relate their views to mathematics, its teaching, and learning as depicted in Table 2. Accordingly, the findings of the study illustrate that UMTs who align with discovery or connectionist ideologies about the essence of mathematics and learning are inclined to employ more student-centered instructional practices. Conversely, those who harbor transmission beliefs predominantly use more teacher-centered practices. This underscores the pivotal role teachers’ beliefs regarding mathematics and learning hold in molding their instructional practices. The study indicates that teacher beliefs strongly influence their teaching methods, with those who support student-centered ideologies using more interactive techniques. Reflecting on one’s own beliefs about mathematics education can lead to more effective teaching strategies in the classroom.

Table 3. UMTs’ self-reported instructional practices.

No	Types of practice	Practice	M	SD
1	T	Students learn through doing exercises	4.0	1.00
2	T	Students work on their own, consulting a neighbor from time to time	3.4	.55
3	T	Students use only the methods I teach them	3.0	1.41
4	T	Students start with easy questions and work up to harder questions	4.6	.55
5	S	Students choose which questions they tackle	3.6	.55
6	S	I encourage students to work more slowly	3.2	.84
7	S	Students compare different methods for doing questions	4.6	.55
8	T	I teach each topic from the beginning, assuming they know nothing	3.6	1.52
9	T	I teach the whole class at once	3.0	1.58
10	T	I try to cover everything in a topic	4.2	1.79
11	S	I draw links between topics and move back and forth between topics	4.4	.55
12	S	I am surprised by the ideas that come up in a lesson	4.0	1.22
13	T	I avoid students making mistakes by explaining things carefully first	3.4	.55
14	T	I tend to follow the textbook or worksheets closely	4.4	.55
15	S	Students learn through discussing their ideas	2.6	.55
16	S	Students work collaboratively in pairs or small groups	3.4	.55
17	S	Students invent their own methods	3.2	1.64
18	T	I tell students which questions to tackle	3.4	.55
19	S	I find out which parts students already understand and do not teach those parts	4.0	1.73
20	S	I teach each student differently according to individual needs	2.8	.84
21	T	I only go through one method for doing each question	1.6	.89
22	T	I tend to teach each topic separately	4.4	.55
23	T	I know exactly what mathematics the lesson will contain	4.4	.55
24	S	I encourage students to make and discuss mistakes	3.8	.45
25	S	I jump between topics as the need arises	3.0	1.22

Note: In the second column; T represents teacher-centered statements, S represents student-centered statements.

These results underscore the importance of addressing such beliefs in teacher education and professional development programs to promote effective teaching practices. By confronting and reconstructing negative beliefs and attitudes towards mathematics, we can foster a positive belief system that reflects in instructional practices and, consequently, in student experiences and outcomes. Addressing these beliefs early in teacher training and continuing professional development is crucial for developing practices that better support student learning and engagement, thereby enhancing the quality of mathematics education and student academic achievements.

Discussion

The primary objective of this research was to explore University Mathematics Teachers’ (UMTs’) beliefs about effective mathematics teaching and how these beliefs influence their instructional practices, particularly within the context of Higher Education. Notably, there is a significant gap in the literature regarding UMTs’ perspectives in this area. Understanding these beliefs is essential for driving meaningful and lasting changes in teaching methodologies, ultimately leading to improved policies and practices that enhance the quality of higher education in mathematics.

The findings reveal that UMTs are able to clearly articulate their beliefs about effective mathematics teaching, and these beliefs significantly shape their instructional practices. This is supported by data from two distinct instruments used in the study. In the field of mathematics education, research increasingly focuses on the complex interplay between teachers’ beliefs about the nature of mathematics, their pedagogical preferences, their chosen teaching models, and the actual instructional practices they implement. Several studies have shown a positive relationship between specific instructional practices and student learning outcomes, yet the detailed mechanisms through which instructional methodologies impact student learning are not fully understood (Carpenter et al., 2000; Hamilton et al., 2021; Polly et al., 2013).

Our in-depth analysis indicates congruence between UMTs’ belief systems and their self-reported instructional methodologies, emphasizing the critical role of these beliefs in determining teaching approaches. Therefore, professional development programs for teachers should be designed to address and refine these foundational beliefs to enhance pedagogical effectiveness.

The categorization presented in Table 1, echoes the widespread academic consensus on the beliefs of mathematics teachers regarding effective teaching. Beliefs about the inherent nature of mathematics significantly influence the development of related teaching and learning beliefs (Cai, 2007; Conner et al., 2011; Cross, 2009; Ernest, 1989; Xie & Cai, 2021). It is crucial for teacher education programs to focus on strengthening UMTs’ understanding and beliefs about the core nature of mathematics, as this foundation can positively impact their overall teaching approach and student learning experiences.

Despite ample evidences that teacher’ beliefs directly influence their classroom practices, questions remain about the consistency of this reflection in instructional practices (Hamilton et al., 2021). The academic community is divided, with some researchers noting discrepancies between professed beliefs and observed practices (Raymond, 1997), while others report a harmonious alignment (Ernest, 1989; Polly et al., 2013; Swan, 2006). Our findings support the latter, suggesting that teachers are often keenly aware of their beliefs and endeavor to align them with their teaching methods, promising more effective instructional outcomes.

Moreover, there is a clear link between UMTs’ beliefs and their chosen instructional practices. UMTs with a ‘transmission’ view of mathematics tend to use teacher-centered strategies, whereas those with a ‘discovery/connectionist’ perspective favor student-centered methodology (Philipp, 2007; Polly et al., 2013; Swan, 2006, 2007). Emphasizing student-centered strategies is crucial for maximizing learning gains, as they encourage active engagement and develop critical thinking and problem-solving skills. Thus, it is vital to support UMTs in adopting student-centric instructional practices in mathematics education.

Conclusion and implications

This study mainly focused on teachers’ perspectives on the efficacy of mathematics teaching, noting both commonalities and slight differences in beliefs among participants. Further research is needed to validate these findings and to refine teacher education programs.

Teachers’ beliefs about mathematics teaching significantly influence their instructional methods. For example, those who perceive mathematics as an exploratory subject may integrate hands-on activities, while those holding traditional views may prefer repetitive drills. These beliefs also shape teachers’ expectations for their students and the tasks they assign.

A teacher’s beliefs about mathematics profoundly impact their approach to teaching. Acknowledging these beliefs is critical for designing impactful professional development programs. Our research indicates that university mathematics teachers have a strong understanding of mathematical interrelations but could benefit from more structured content knowledge. Comprehensive understanding encompasses not only depth in mathematical concepts but also an appreciation for their interconnectedness.

In conclusion, there is a discernible relationship between teachers’ beliefs about mathematics and their classroom practices. These beliefs guide instructional decisions, classroom interactions, and assessment methods. Therefore, understanding teachers’ beliefs is key to creating professional development programs that align with these beliefs and promote effective classroom practices.

The study underscores the need for further research to evaluate the effectiveness of UMTs’ teaching methods, compare them against global standards, and identify best practices for teacher training.

Author contributions

Second and third authors were supervising first author (PhD candidate). Therefore, first author is the main author and second and third authors were contributing authors.

Acknowledgments

This study is part of first author’s PhD dissertation. We especially wish to thank all five mathematics teachers and students who participated in this study.

Disclosure statement

No conflict of interest is declared by author.

Additional information

Funding

During his PhD studies, the first author of this research received financial support from the Addis Ababa University College of Education and Behavioral Studies in Ethiopia.

Notes on contributors

Yosef Kasa

Yosef Kasa, a PhD candidate in the department of Science and Mathematics Education at Addis Ababa University in Ethiopia. He holds a Bachelor's degree in Mathematics Education From Bahir Dar University and a Master's in Mathematics Education from Addis Ababa University. With over 12 years of experience in research and community services, he has taught various mathematics courses at the university level. His main research interests include higher education, teacher knowledge domains, PCK, and beliefs. He has published articles and presented at national conferences on these topics.

Solomon Areaya

Solomon Areaya holds a Bachelor's degree in Mathematics, a Master of Arts in Curriculum & Instruction, and a PhD in Education. Presently, he serves as a full-time lecturer at Addis Ababa University, where he has attained the rank of Associate Professor in the Department of Science and Mathematics Education. His research interests encompass Mathematics Education, Curriculum Implementation, and Teacher Education.

Mulugeta Woldemichael

Mulugeta Woldemichael, is an assistant professor in the Science and Mathematics Education Department at Addis Ababa University in Ethiopia. His research areas are Statistics & Mathematics Education, STEM Education, Teacher Education, and Instructional Technology. He teaches mathematics from early grades to post-graduate education levels in Ethiopia. In particular, Mulugeta has more than 10 year of teaching experiences in undergraduate and graduate programs in Mathematics Education. He has been a reviewer for mathematics curriculum elaborations and text books at Ministry of Education, Ethiopia.

Appendix A.

Teachers’ beliefs questionnaire

Teacher code: ___________________

This questionnaire is designed to collect data for a study about Relationship between university teachers’ beliefs about teaching Mathematics and their instructional practices. Therefore, we kindly request you to respond to each item and provide the correct response.

Thank you for your cooperation!!

Instruction: Indicate the degree to which you agree with each statement below by giving each statement a percentage, so that the sum of the three percentages in each section is 100.

Table

Component/ characteristic	Statement	Percent
Mathematics is:
MT	A given body of knowledge and standard procedures; A set of universal truths and rules which need to be conveyed to students
MD	A creative subject in which the teacher should take a facilitating role, allowing students to create their own concepts and methods
MC	An interconnected body of ideas which the teacher and the student create together through discussion
Learning is:
LT	An individual activity based on watching, listening and imitating until fluency is attained
LD	An individual activity based on watching, listening and imitating until fluency is attained
LC	An interpersonal activity in which students are challenged and arrive at understanding through discussion
Teaching is:
TT	Structuring a linear curriculum for the students; giving verbal explanations and checking that these have been understood through practice questions; correcting misunderstandings when students fail to ‘grasp’ what is taught
TD	Assessing when a student is ready to learn; providing a stimulating environment to facilitate exploration; avoiding misunderstandings by the careful sequencing of experiences
TC	A non-linear dialogue between teacher and students in which meanings and connections are explored verbally. Misunderstandings are made explicit and worked on

Note. In the first column; the first letter represents Mathematics (M), Learning (L) or Teaching (T). The second letter refers to Transmission (T), Discovery (D) or Connectionist (C) beliefs.

Appendix B.

Teacher practices questionnaire

Teacher code: ________

This questionnaire is designed to collect data for a study about Relationship between university teachers’ beliefs about teaching Mathematics and their instructional practices. Therefore, we kindly request you to respond to each item and provide the correct response.

Thank you for your cooperation!!

Instruction: Please indicate the frequency with which you utilized each of the following practices in your teaching by placing a ‘✓‘ mark in the appropriate column. Teachers were asked to rate each behavior on a scale: 5 almost always, 4 most of the time, 3 half the time, 2 occasionally, 1 almost never.

Table

#	Types of practice	Practice	1	2	3	4	5
1	T	Students learn through doing exercises
2	T	Students work on their own, consulting a neighbor from time to time
3	T	Students use only the methods I teach them
4	T	Students start with easy questions and work up to harder questions
5	S	Students choose which questions they tackle
6	S	I encourage students to work more slowly
7	S	Students compare different methods for doing questions
8	T	I teach each topic from the beginning, assuming they know nothing
9	T	I teach the whole class at once
10	T	I try to cover everything in a topic
11	S	I draw links between topics and move back and forth between topics
12	S	I am surprised by the ideas that come up in a lesson
13	T	I avoid students making mistakes by explaining things carefully first
14	T	I tend to follow the textbook or worksheets closely
15	S	Students learn through discussing their ideas
16	S	Students work collaboratively in pairs or small groups
17	S	Students invent their own methods
18	T	I tell students which questions to tackle
19	S	I find out which parts students already understand and do not teach those parts
20	S	I teach each student differently according to individual needs
21	T	I only go through one method for doing each question
22	T	I tend to teach each topic separately
23	T	I know exactly what mathematics the lesson will contain
24	S	I encourage students to make and discuss mistakes
25	S	I jump between topics as the need arises