The Development of Pre-Service Teachers’ Understanding of the Knowledge Necessary to Teach Mathematics: A Case Study in Malawi
Keywords:
mathematics, MalawiSynopsis
Globally, progress in education has been made in recent years to promote learning opportunities for all. However, significant gaps remain in low- income countries. In Malawi, a developing country where the educational system is under development, primary education is a major concern. Several initiatives have been made to bridge this gap through improving the quality of teacher education, but research has shown that there is still a need to further understand the learning process of pre- service teachers during the teacher education program.
The present study aimed to gain knowledge about the primary pre- service teachers’ process of learning, particularly, the development of their understanding of the knowledge necessary to teach mathematics. The current study addressed the question: How do pre-service teachers develop their understanding of the knowledge necessary to teach mathematics throughout teacher education? In the study, the knowledge necessary for teaching mathematics refers to the knowledge that teachers need to carry out the tasks of teaching mathematics.
To further examine this matter, the overall question was divided into three sub-research questions: (1) What understanding do pre-service teachers have of the knowledge needed to carry out the tasks of mathematics teaching at the beginning of their teacher education? (2) To what extent does the pre-service teacher’s understanding of the knowledge needed to carry out teaching tasks evolve through the discussion of practical experiences in college? (3) How do pre-service teachers develop their understanding of the knowledge necessary to carry out the tasks of teaching throughout teacher education?
These questions were compiled into a qualitative case study with six pre- service teachers in a two-year primary teacher education program at a college in Malawi. Each of these pre-service teachers represented a different profile with teaching experience, subject preferences in high school, and a favorite subject to teach during college. The research occurred over three different moments in a two-year teacher education program in which the pre-service teachers were enrolled: an initial moment at the beginning of the program consisted of a questionnaire survey and individual interviews; a second moment during teaching practice that involved mathematics lessons observations and post-lesson interviews; and a third moment at the end of the program that included a focus group discussion.
The data gathered were transcribed and analyzed using a thematic analysis approach. The themes of analysis were designed based on the six domains of mathematical knowledge for teaching theory. Findings show that pre-service teachers develop different paths of development of their understanding of the knowledge needed for teaching mathematics during teacher education and that such development has influences in how they acknowledge effective teaching in Malawi.
The current thesis includes four articles that present the main data and results of the study. The first two articles present findings from an analysis of the pre-service teachers’ understanding of the subject matter knowledge and pedagogical content knowledge, and the third article presents an analysis of the pre-service teachers’ understanding when discussing teaching practice. The fourth article explores the understanding pre-service teachers developed throughout the teacher education program.
The contribution of this thesis is to not only offer new empirical and theoretical insights to teacher education but also to suggest a path for further research in teaching knowledge.
References
Aguinis, H., & Solarino, A. M. (2019). Transparency and replicability in qualitative research: The case of interviews with elite informants. Strategic Management Journal, 40, 1-25.
https://doi.org/10.1002/smj.3015
Akinsola, M. K., & Awofaba, A. O. A. (2009). Effect of personalization of instruction on students' achievement and self-efficacy in mathematics word problems. International Journal of Mathematical Education in Science and Technology, 40(3), 389-404.
https://doi.org/10.1080/00207390802643169
Allmark, P., Boote, J., Chambers, E., Clarke, A., McDonnell, A., Thompson, A., & Tod, A. M. (2009). Ethical issues in the use of in- depth interviews: Literature review and discussion. Research Ethics Review, 5(2), 48-54.
https://doi.org/10.1177/174701610900500203
Attride-Stirling, J. (2001). Thematic networks: An analytic tool for qualitative research. Qualitative Research, 1, 385-405.
https://doi.org/10.1177/146879410100100307
Axelrold, J. (1973). The university teacher as artist. San Francisco, CA: Jossey Bass.
Ball, D. L. (1990). Prospective elementary and secondary teachers' understandings of division. Journal for Research in Mathematics Education, 21(2), 132-144.
https://doi.org/10.5951/jresematheduc.21.2.0132
https://doi.org/10.2307/749140
Ball. D. L. (2017). Uncovering the special mathematical work of teaching. In G. Kaiser (Ed.), Proceedings from the 13th International Congress of Mathematics Education (pp. 11-34). New York, NY: Springer.
https://doi.org/10.1007/978-3-319-62597-3_2
Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J.
Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 83-104). Westport, CT: Ablex Publishing.
Ball, D. L., & Forzani, F. M. (2009). The work of teaching and the challenge for teacher education. Journal of Teacher Education, 60(5), 497-511.
https://doi.org/10.1177/0022487109348479
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407.
https://doi.org/10.1177/0022487108324554
Banda, K. N. (1982). A brief history of education in Malawi. Blantyre, Malawi: Dzuka Publishing Co.
Barkatsas, A., & Malone, J. (2005). A typology of mathematics teachers' beliefs about teaching and learning mathematics and instructional practices. Mathematics Education Research Journal, 17, 69-90.
https://doi.org/10.1007/BF03217416
Bassarear, T. (1997). Mathematics for elementary school teachers. Boston, MA: Houghton Mifflin.
Bauer, M. W. (2000). Classical content analysis: a review. In M. W. Bauer and G.Gaskell (Eds.), Qualitative researching with text, image and sound (pp.131-151). London, UK: Sage.
https://doi.org/10.4135/9781849209731
Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., &. . .Tsai, Y. M. (2010). Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47, 133-180.
https://doi.org/10.3102/0002831209345157
Baxter, P., & Jack, S. (2008). Qualitative case study methodology: Study design and implementation for Novice Researchers. The Qualitative Report, 13(4), 544-559.
Bendixen, L. D., & Feucht, F. C. (Eds.). (2010). Personal epistemology in the classroom: Theory, research, and implications for practices. Cambridge, England: Cambridge University Press.
https://doi.org/10.1017/CBO9780511691904
Ben-Peretz, M. (2011). Teacher knowledge: What is it? How do we uncover it? What are its implications for schooling? Teacher and Teacher Education, 27(1), 3-9.
https://doi.org/10.1016/j.tate.2010.07.015
Bergmark, U., Lundstrom, S. Manderstedt, L., & Palo, A. (2018). Why become a teacher? Student teachers' perceptions of the teaching profession and motives for career choice. European Journal of Teacher Education, 41(3), 266-281.
https://doi.org/10.1080/02619768.2018.1448784
Beswick, K. (2012). Teachers' beliefs about school mathematics and mathematicians' mathematics and their relationship to practice. Educational Studies in Mathematics, 79(1), 127-147.
https://doi.org/10.1007/s10649-011-9333-2
Biesta, G. (2007). Why "what works" won't work: Evidence-based practice and the democratic deficit in educational research. Educational Theory, 57, 1-22.
https://doi.org/10.1111/j.1741-5446.2006.00241.x
Bourdon, J., Frölich, M., & Michaelowa, K. (2010). Teacher shortages, teacher contracts and their effect on education in Africa. Journal of the Royal Statistical Society, 173(1), 93-116.
https://doi.org/10.1111/j.1467-985X.2009.00601.x
Boyatzis, R. E. (1998). Transforming qualitative information: Thematic analysis and code development. Thousand Oaks, CA: Sage.
Brant, J. (2006). Subject knowledge and pedagogic knowledge: ingredients for good teaching? An English perspective. Edukacja, 94(2), 60-77.
Bråten, I. (2010). Personal epistemology in education: Concepts, issues, and implications. In E. Baker, B. McGaw, & P. Peterson (Eds.), International encyclopedia of education (Vol. 5, pp. 211-217). Oxford, UK: Elsevier.
https://doi.org/10.1016/B978-0-08-044894-7.00480-2
Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77-101.
https://doi.org/10.1191/1478088706qp063oa
Brown, S., & McIndyre, D. (1993). Making sense of teaching. Buckingham (UK) and Philadelphia (PA): Open University Press.
Bruckmaier, G., Krauss, S., Blum, W., & Leiss, D. (2016). Measuring mathematics teachers' professional competence by using video clips (COACTIV video). ZDM Mathematics Education, 48, (1-2), 111-124.
https://doi.org/10.1007/s11858-016-0772-1
Cady, J. A., & Rearden, K. (2007). Preservice teachers' beliefs about knowledge, mathematics, and science. School Science and Mathematics, 107(6), 237-245.
https://doi.org/10.1111/j.1949-8594.2007.tb18285.x
Carrillo, J., Climent, N., Contreras, L. C., & Muñoz-Catalán, M. C. (2013). Determining specialised knowledge for mathematics teaching. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the Eighth congress of the European Society for Research in Mathematics Education (pp. 2985-2994). Antalya, Turkey: METH.
Chamoso, J. M., Cáceresa, M. J., & Azcárateb, P. (2012). Reflection on the teaching-learning process in the initial training of teachers. Characterization of the issues on which pre-service Mathematics teachers reflect. Teaching and Teacher Education, 28(2), 154-164.
https://doi.org/10.1016/j.tate.2011.08.003
Chanaiwa, D. (1993). Southern Africa since 1945. In A. Mazril & C. Wondji (Eds.), General history of Africa VIII: Africa since 1935 (pp. 249-281). Berkeley, CA: Heinemann UNESCO.
Charalambous, C. Y., & Litke, E. (2018). Studying instructional quality by using a content-specific lens: The case of the Mathematical Quality of Instruction framework. ZDM Mathematics Education, 50, 445-460.
https://doi.org/10.1007/s11858-018-0913-9
Chevallard, Y. (2000). La recherche en didactique et la formation des professeurs: problématiques, concepts, problems. In M. Bailleul (Ed.), Actes de la x e école d'été de didactique des mathématiques (pp. 98- 112). Caen, France: IUFM.
Chinnapan, M., & White, B. (2015). Specialised content knowledge: Evidence of pre-service teachers' appraisal of students' errors in proportional reasoning. In M. Marshman, V. Gieger & A. Benisson (Eds.), Proceedings of the 38th Annual Conference of the Mathematics Research Group of Australasia (pp. 157-164). Sunshine Coast, Australia: Merga.
Cohen, D. K. (2011). Teaching and its predicaments. Cambridge, MA: Harvard University Press.
https://doi.org/10.4159/harvard.9780674062788
Coimbra, M. N. C. T, & Martins, A. M. O. (2013). Case studying educational research: A way of looking at reality. American Journal of Educational, 1(9), 391-95.
Conley, A. M., Pintrich, P. R., Vekiri, I., & Harrison, D. (2004). Changes in epistemological beliefs in elementary science students. Contemporary Educational Psychology, 29, 186-204.
https://doi.org/10.1016/j.cedpsych.2004.01.004
Costa, C., Breda, Z., Pinho, I., Bakas, F., & Durão, M. (2016). Performing a thematic analysis: An exploratory study about managers' perceptions on gender equality. The Qualitative Report, 21(13), 34-47.
https://doi.org/10.46743/2160-3715/2016.2609
Creswell, J. (2005). Educational Research: Planning, conducting, and evaluating quantitative and qualitative research (2nd Ed.) Upper Saddle River, NJ: Pearson Education.
Creswell, J., & Plano Clark, V. (2018). Designing and conducting mixed methods research, (3rd Ed.). Thousand Oaks, CA: Sage.
Darling-Hammond, L. (2006). Constructing 21st century teacher education. Journal of Teacher Education, 57(3), 300-314.
https://doi.org/10.1177/0022487105285962
Darling-Hammond, L., & Bransford, J. (Eds). (2005). Preparing teachers for a changing world: What teachers should learn and be able to do. San Francisco, CA: Jossey-Bass.
Darling-Hammond, L., & Youngs, P. (2002). Defining "Highly Qualified Teachers": What Does "Scientifically-Based Research" Actually Tell Us? Educational Researcher, 31, 13-25.
https://doi.org/10.3102/0013189X031009013
Davis, B., & Simmt, E. (2006). Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61, 293-319.
https://doi.org/10.1007/s10649-006-2372-4
Davydov, V. V. (1999). What is real learning activity? In M. Hedegaard & J. Lompscher (Eds.), Learning activity and development. (pp. 123−138). Aarhus, Denmark: Aarhus University Press.
Day, C. (2002). Developing teachers: The challenges of lifelong learning. London, UK: Routledge.
https://doi.org/10.4324/9780203021316
De Regt, H. W. (2004). Discussion note: Making sense of understanding. Philosophy of Science, 71, 98-109.
https://doi.org/10.1086/381415
Delaney, S., Ball, D., Hill, H., Schilling, S., & Zopf, D. (2008). "Mathematical knowledge for teaching": Adapting U.S. measures for use in Ireland. Journal of Mathematics Teacher Education, 1(3), 171- 197.
https://doi.org/10.1007/s10857-008-9072-1
DeSantis, L., & Ugarriza, D. N. (2000). The concept of theme as used in qualitative nursing research. Western Journal of Nursing Research, 22(3), 351-372.
https://doi.org/10.1177/019394590002200308
Ding, M. (2016). Developing preservice elementary teachers' specialized content knowledge: The case of associative property. International Journal of STEM Education, 3(9), 1-19.
https://doi.org/10.1186/s40594-016-0041-4
Dudek, C.M., Reddy, L.A., & Lekwa, A. (2019). Measuring teacher practices to inform student achievement in high poverty schools: A predictive validity study. Contemporary School Psychology, 23, 290- 303.
https://doi.org/10.1007/s40688-018-0196-8
Duthilleul, Y. (2005). Lessons learnt in the use of contract teachers. International Institute for Educational Planning: Synthesis Report. Paris, France: UNESCO, IIEP.
Enyedy, N., Goldberg, J., & Welsh, M. (2005). Complex dilemmas of identity and practice. Science Education, 90(1), 68-93.
https://doi.org/10.1002/sce.20096
Fajet, W., Bello, M., Leftwich, S. A., Mesler, J. L., & Shaver, A. N. (2005). Preservice teachers' perceptions in beginning education classes. Teaching and Teacher Education, 21(6), 717-727.
https://doi.org/10.1016/j.tate.2005.05.002
Farrell, J., & Oliveira, J. (1993) (Eds.). Teachers in developing countries: Improving effectiveness and managing costs. Washington, DC: Economic Development Institute.
Fennema, E., & Franke, M. L. (1992). Teachers' knowledge and its impact. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 147-164). New York, NY: Macmillan Publishing Co, Inc.
Ferguson, L. E., & Brownlee, J. L. (2018). An investigation of preservice teachers' beliefs about the certainty of teaching knowledge. Australian Journal of Teacher Education, 43(1), 94-111.
https://doi.org/10.14221/ajte.2018v43n1.6
Fernandez, C. (2014). Knowledge base for teaching and pedagogical content knowledge (PCK): Some useful models and implications for teachers' training. Problems of Education in the 21th Century, 60, 79- 100.
Fives, H., & Buehl, M. M. (2008). What do teachers believe? Developing a framework for examining beliefs about teachers' knowledge and ability. Contemporary Educational Psychology, 33, 134-176.
https://doi.org/10.1016/j.cedpsych.2008.01.001
Fives, H., & Buehl, M. M. (2010). Examining the factor structure of the teachers' sense of efficacy scale. Journal of Experimental Education, 78, 118-134.
https://doi.org/10.1080/00220970903224461
Fives, H., Lacatena, N., & Gerard, C. (2015). Teachers' beliefs about teaching (and learning). In M. G. Gill & H. Fives (Eds.), International handbook of research on teachers' beliefs (pp. 249-265). New York, NY: Routledge.
https://doi.org/10.4324/9780203108437
Freire, P. (1985). Pedagogia do oprimido [Pedagogy of the Oppressed]. Rio de Janeiro, Brazil: Ed. Paz e Terra.
Freire, P. (1996). Pedagogia da autonomia - saberes necessários à prática educativa [Pedagogy of autonomy - the necessary knowledge for educational practice]. São Paulo, Brazil: Editora Paz e Terra.
Frith, H., & Gleeson, K. (2004). Clothing and embodiment: Men managing body image and appearance. Psychology of Men & Masculinity, 5(1), 40-48.
https://doi.org/10.1037/1524-9220.5.1.40
Gage, N. L. (1978). The scientific basis for the art of teaching. New York, NY: Teacher College Press.
Gardner, C., & Williamson, J. (2007). The complexities of learning to teach: "Just what is it that I am doing?" In T. Townsend & R. Bates (Eds.), Handbook of teacher education: Globalization, standards and professionalism in times of change (pp. 691-710). Dordrecht, Netherlands: Springer.
https://doi.org/10.1007/1-4020-4773-8_47
George, A. L., & Bennett, A. (2005). Case studies and theory development in the social sciences. Cambridge, MA: MIT Press.
Gerring, J. (2004). What is a case study and what is it good for? American Political Science Review, 98(2), 341-354.
https://doi.org/10.1017/S0003055404001182
Givvin, K. B., Stipek, D. J., Salmon, J. M., & MacGyvers, V. L. (2001). In the eyes of the beholder: Students' and teachers' judgments of students motivation. Teaching and Teacher Education, 17(3), 321-331.
https://doi.org/10.1016/S0742-051X(00)00060-3
Goos, M. (2013). Knowledge for teaching secondary school mathematics: What counts? International Journal of Mathematical Education in Science and Technology, 44, 972-983.
https://doi.org/10.1080/0020739X.2013.826387
Government of Malawi. (2016). Education sector performance report 2015−16. Lilongwe, Malawi: Ministry of Education, Science, and Technology.
Government of Malawi. (2018). 2018 Malawi population and housing census: Main report. Zomba, Malawi: National Statistical Office.
Grimm, S. R. (2006). Is understanding a species of knowledge?, The British Journal for the Philosophy of Science, 57(3), 515-535.
https://doi.org/10.1093/bjps/axl015
Grossen, B. (Ed.). (1996). What was that project follow through? [Special issue] Effective School Practices, 15(1), 1-85.
Grossman, P. L. (1990). The making of a teacher: teacher knowledge and teacher education. New York, NY: Teachers College Press.
Grossman, P., & McDonald, M. (2008). Back to the future: Directions for research in teaching and teacher education. American Educational Research Journal, 45(1), 184-205.
https://doi.org/10.3102/0002831207312906
Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P. (2009). Teaching practice: A cross-professional perspective. Teachers College Record, 111(9), 2055-2100.
Haddock, A., Millar, A., & Pritchard, D. (Eds.). (2009). Epistemic value. Oxford: Oxford University Press.
https://doi.org/10.1093/acprof:oso/9780199231188.001.0001
Hamilton, L., & Corbett-Whittier, C. (2013). Using case study in education research. Los Angeles, CA: Sage.
https://doi.org/10.4135/9781473913851
Harper, D. (2016). Online etymology dictionary. Retrieved on March 24, 2020, from http://www.etymonline.com
Hatisaru, V. (2018). Teachers' beliefs about knowledge of teaching and their impact on teaching practices. In B. Rott, G. Törner, J. Peters- Dasdemir, A. Möller, Safrudiannur (Eds.), Views and Beliefs in Mathematics Education (pp. 149-159). Cham, Switzerland: Springer.
Hayes, N. (1997). Theory-led thematic analysis: Social identification in small companies. In N. Hayes (Ed.), Doing qualitative analysis in psychology (pp. 93-114). Hove, UK: Psychology Press.
Hiebert, J., & Grouws, D. A. (2007). Effective teaching for the development of skill and conceptual understanding of number: What is most effective? Research Brief for NCTM. Reston, VA: National Council of Teachers of Mathematics.
Hiebert, J., Morris, A. K., Berk, D., & Jansen, A. (2007). Preparing teachers to learn from teaching. Journal of Teacher Education, 58, 47- 61.
https://doi.org/10.1177/0022487106295726
Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. Cognition and Instruction, 26(4), 430-511.
https://doi.org/10.1080/07370000802177235
Hill, H. C., Rowan, B., & Ball, D. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42, 371-406.
https://doi.org/10.3102/00028312042002371
Hill, H. C., Sleep, L., Lewis, J., & Ball, D. L. (2007). Assessing teachers' mathematical knowledge: What knowledge matters. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 111-156). Charlotte, NC: Information Age Publishing.
Hofer, B. K. (2002). Personal epistemology as a psychological and educational construct: An introduction. In B. K. Hofer & P. R. Pintrich (Eds.), Personal epistemology: The psychology of beliefs about knowledge and knowing (pp. 3-14). Mahwah, NJ: Lawrence Erlbaum
https://doi.org/10.4324/9781410604316
Hofer, B. K., & Pintrich, P. R. (1997). The development of epistemological theories: Beliefs about knowledge and knowing and their relation to learning. Review of Educational Research, 67(1), 88- 140.
https://doi.org/10.3102/00346543067001088
Jablon, J. R., Dombro, A., & Dichtelmiller, M. L. (2007). The power of observation (2nd Ed.). Washington, DC: Teaching Strategies.
Jacinto, E. L., & Cedro, W. L. (2006). Semelhança de triângulos: atividades de ensino de geometria para o ensino fundamental. In: III Congresso de Pesquisa, Ensino, e Extensão - CONPEEX, Goiania, Brazil: UFG.
Jacinto, E. L., & Cedro, W. L. (2012). A atividade pedagógica no PROEJA e a aprendizagem dos conceitos matemáticos [The pedagogical activity in the PROEJA and the learning of mathematical concepts]. Perspectivas da Educação Matemática, 5, 68-87.
Jacinto, E. L., Kazima, M., Jakobsen, A., & Bjuland, R. (2020). Mathematics Teacher Education in Malawi. In D. R. Thompson, M. A. Huntley & C. Suurtamm (Eds.), Research issues in mathematics education: International perspectives on mathematics teacher education (in press). Charlotte, NC: Information Age Publishing.
Jakobsen, A., Kazima, M., & Kasoka, D. N. (2018). Assessing prospective teachers' development of MKT through their teacher education: A Malawian case. In E. Norén, H. Palmér, & A. Cooke (Eds.) Papers of NORMA 17, The Eighth Nordic Conference on Mathematics Education, Stockholm, May 30-June 2, 2017 (pp. 219- 227). Göteborg, Sweden: Swedish Society for Research in Mathematics Education.
Jakobsen, A., Thames, M. H., Ribeiro, C. M., & Delaney, S. (2012). Using practice to define and distinguish horizon content knowledge. In International Congress in Mathematics Education (Ed.), Pre- proceedings of the 12th International Congress in Mathematics Education (pp. 4635-4644). Seoul, Korea: International Congress in Mathematics Education.
Jao, L. (2017). Shifting pre-service teachers' beliefs about mathematics teaching: The contextual situation of a mathematics methods course. International Journal of Science and Mathematics Education, 15, 895- 914.
https://doi.org/10.1007/s10763-016-9719-9
Javadi, M., & Zarea, K. (2016). Understanding thematic analysis and its pitfall. Journal of Client Care, 1(1), 34-40.
https://doi.org/10.15412/J.JCC.02010107
Joffe, H., & Yardley, L. (2004). Content and thematic analysis. In D. F. Marks & L. Yardley (Eds.), Research methods for clinical and health psychology (pp. 56-68). London, UK: Sage.
Johnson, T. (2011). Elementary preservice teachers' mathematical knowledge for teaching: using situated case studies and educative experiences to examine and improve the development of MKT in teacher education (Doctoral dissertation). University of North Carolina at Chapel Hill.
Jones, M. G., & Vesilind, E. M. (1996). Putting practice into theory: Changes in the organization of preservice teachers' pedagogical knowledge. American Educational Research Journal, 33(1), 91-117.
https://doi.org/10.3102/00028312033001091
Kasoka, D. N., Jakobsen, A., & Kazima, M. (2017). Preparing teachers for teaching: does initial teacher education improve mathematical knowledge for teaching? In T. Dooley & G. Gueudet, Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME 10) (pp. 3328-3336). Dublin, Irland: European Society for Research in Mathematics Education.
Kayuni, H., & Tambulasi, R. (2007). Teacher turnover in Malawi's Ministry of Education: Realities and challenges. International Education Journal, 8(1), 89-99.
Kazima, M. (2014). Universal basic education and the provision of quality mathematics in Southern Africa. International Journal of Science and Mathematics Education, 12(4), 841-858.
https://doi.org/10.1007/s10763-013-9434-8
Kazima, M., Jakobsen, A., & Kasoka, D. N. (2016). Use of mathematical tasks of teaching and the corresponding LMT measures in the Malawi context, The Mathematics Enthusiast, 13(1), 171-186.
Kennedy, M. M. (2008). Sorting out teacher quality. Phi Delta Kappan, 90(1), 59-63.
https://doi.org/10.1177/003172170809000115
Kilic, H. (2015). Preservice teachers' perceptions about teacher knowledge. Procedia - Social Behaviour Sciences, 191, 1838-1842.
https://doi.org/10.1016/j.sbspro.2015.04.306
Koponen, M., Asikainen, M. A., Viholainen, A., & Hirvonen, P. E. (2016). Teachers and their educators' views on contents and their development needs in mathematics teacher education. The Mathematics Enthusiast, 13(1/2), 149-170.
https://doi.org/10.12973/eurasia.2017.01209a
Krauss, S., Brunner, M., Kunter, M., Baumert, J., Blum, W., Neubrand, M., & Jordan, A. (2008). Pedagogical content knowledge and content knowledge of secondary mathematics teachers. Journal of Educational Psychology, 100(3), 716-725.
https://doi.org/10.1037/0022-0663.100.3.716
Krosnick, J. A. (1999). Survey research. Annual Review of Psychology, 50, 537-567.
https://doi.org/10.1146/annurev.psych.50.1.537
Kunje, D., Lewin, K., & Stuart, J. (2003). Primary teacher education in Malawi: Insights into practice and policy. London, UK: Department for International Development.
Kvanvig, J. (2003). The value of knowledge and the pursuit of understanding. New York, NY: Cambridge University Press.
https://doi.org/10.1017/CBO9780511498909
Lauermann, F., & Karabenick, S. A. (2013). The meaning and measure of teachers' sense of responsibility for educational outcomes. Teacher and Teacher Education, 30, 13-16.
https://doi.org/10.1016/j.tate.2012.10.001
Lavigne, A. L. (2014). Beginning teachers who stay: Beliefs about students. Teaching and Teacher Education, 39, 31-43.
https://doi.org/10.1016/j.tate.2013.12.002
Le Cornu, R., & Ewing, R. (2008). Reconceptualizing professional experiences in pre-service teacher education… reconstructing the past to embrace the future. Teaching and Teacher Education, 24(7), 1799- 1812.
https://doi.org/10.1016/j.tate.2008.02.008
Leikin, R., & Zazkis, R. (2010). The content-dependence of prospective teachers' knowledge: A case of exemplifying definitions. International Journal of Mathematical Education in Science and Technology, 4, 451-466.
https://doi.org/10.1080/00207391003605189
Letwinsky, K. M., & Cavender, M. (2018). Shifting preservice teachers" beliefs and understandings to support pedagogical change in mathematics. International Journal of Research in Education and Science, 4(1), 106-120.
https://doi.org/10.21890/ijres.382939
Levitt, K. (2001). An analysis of elementary teachers' belief regarding the teaching and learning of science. Science Education, 86, 1-22.
https://doi.org/10.1002/sce.1042
Liamputtong, P. (2007). Researching the vulnerable: A guide to sensitive research methods. Thousand Oaks, CA: Sage.
https://doi.org/10.4135/9781849209861
Lindquist, G., & Coleman, S. J. (2008). Introduction: Against belief? Social Analysis, 52(1), 1-18.
https://doi.org/10.3167/sa.2008.520101
Maguire, M., & Delahunt, B. (2017). Doing a thematic analysis: A practical, step-by-step guide for learning and teaching scholars. All Ireland Journal of Teaching and Learning in Higher Education, 8(3), 3351-3364.
Malawian Institute of Education. (2017). Syllabus for initial primary teacher education: Mathematics ministry of education, science and technology. Domasi, Malawi: Malawi Institute of Education.
Mapolelo, D. C., & Akinsola, M. K. (2015). Preparation of mathematics teachers: Lessons from review of literature on teachers' knowledge, beliefs, and teacher education. American Journal of Educational Research, 3(4), 505-513.
Marshman, M., & Porter, G. (2013) Pre-service teachers' pedagogical content knowledge: Implications for teaching. In V. Steinle, L. Ball & C. Bardini (Eds.), Mathematics education: Yesterday, today and tomorrow. Proceedings of the 36th annual conference of the Mathematics Education Research Group of Australasia (pp.474-481). Melbourne, Australia: Merga.
Mazzoni, G., & Kirsch, I. (2002). False autobiographical memories and beliefs: a preliminary metacognitive model. In T. Perfect, & B. Schwartz (Eds.), Applied metacognition (pp. 121-145). Cambridge, UK: Cambridge University Press.
https://doi.org/10.1017/CBO9780511489976.007
Ministry of Education, Science, and Technology. (2016). Malawi government education statistics. Lilongwe, Malawi: Department of Education Planning.
Ministry of Education. (2005). Initial primary teacher education teaching syllabus: Numeracy and mathematics. Domasi, Malawi: Malawian Institute of Education.
Minor, L. C., Onwuegbuzie, A. J., Witcher, A. E., & James, T. L. (2002). Preservice teachers' educational beliefs and their perceptions of characteristics of effective teachers. The Journal of Educational Research, 96(2), 116-127.
https://doi.org/10.1080/00220670209598798
Mitchell, R., Charalambous, C. Y., & Hill, H. C. (2014). Examining the task and knowledge demands needed to teach with representations. Journal of Mathematics Teacher Education, 17, 37-60.
https://doi.org/10.1007/s10857-013-9253-4
Moshman, D. (2016). Epistemic Cognition. In R. Levesque (Ed.), Encyclopedia of adolescence. New York, NY: Springer.
https://doi.org/10.1007/978-3-319-32132-5_17-2
Mosvold, R., & Fauskanger, J. (2013). Teachers' beliefs about mathematical knowledge for teaching definitions. International Electronic Journal of Mathematics Education, 8(2-3), 43-61.
Mosvold, R., & Fauskanger, J. (2014). Teachers' beliefs about mathematical horizon content knowledge. International Journal for Mathematics Teaching and Learning, 9(3), 311-327.
Moura, M. O. (Ed.). (2010). A atividade pedagógica na teoria Histórico- Cultural [The pedagogical activity in the Cultural-hitorical theory]. Brasília, Brasil: Liber livro.
Ndlovu, Z., Amin, N., & Samuel, M. A. (2017). Examining pre-service teachers' subject matter knowledge of school mathematics concepts. Journal of Education, 70, 46-72.
Needham, R. (1972). Beliefs, language, and experience. Oxford, UK: Basil Blackwell.
Neto, V. P. B., & Costa, M. C. (2016). Saberes docentes: Entre concepções e categorizações [Teacher knowledge: Between conceptions and categorizations]. Tópicos Educacionais, 22(2), 76-99.
Ng, D., Mosvold, R., & Fauskanger, J. (2012). Translating and adapting the mathematical knowledge for teaching (MKT) measures: The cases of Indonesia and Norway. The Mathematics Enthusiast, 9(1-2), 149- 178.
Ng, W., Nicholas, H., & Williams, A. (2010). School experience influences on pre-service teachers' evolving beliefs about effective teaching. Teaching and Teacher Education, 26(2), 278-289.
https://doi.org/10.1016/j.tate.2009.03.010
Noble, H., & Smith, J. (2015). Issues of validity and reliability in qualitative research. Evidence Based Nursing, 18(2), 34-35.
https://doi.org/10.1136/eb-2016-102306
https://doi.org/10.1136/eb-2015-102054
Novak, J. D. (1998). Learning, creating, and using knowledge: Concept maps as facilitative tools in schools and corporations. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
https://doi.org/10.4324/9781410601629
Nowell, L., Norris, J., White, D., & Moules, N. (2017). Thematic analysis: Striving to meet the trustworthiness criteria. International Journal of Qualitative Methods, 16, 1-13
https://doi.org/10.1177/1609406917733847
O'Meara, N., Prendergast, M., Cantley, I., Harbison, L., & O'Hara, C. (2019). Teachers' self-perceptions of mathematical knowledge for teaching at the transition between primary and post-primary school. International Journal of Mathematical Education in Science and Technology, 51 (4), 497-519.
https://doi.org/10.1080/0020739X.2019.1589004
O'Neill, S., & Stephenson, J. (2012). Does classroom management coursework influence pre-service teachers' perceived preparedness or confidence? Teaching and Teacher Education, 28, 1131-1143.
https://doi.org/10.1016/j.tate.2012.06.008
Osmanoglu, A. (2016). Prospective teachers' teaching experience: Teacher learning through the use of video. Educational Research, 58(1), 39-55.
https://doi.org/10.1080/00131881.2015.1117321
Pajares, M. (1992). Teachers' beliefs and educational research: Cleaning up a messy construct. Review of Educational Research, 62(3), 307- 332.
https://doi.org/10.3102/00346543062003307
Pape, S. J., & Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical understanding. Theory into Practice, 40(2), 118-127.
https://doi.org/10.1207/s15430421tip4002_6
Parlett, M., & Dearden, G. (Eds.) (1977). Introduction to illuminative evaluation: studies in higher education Cardiff-by-the-Sea. California, USA: Pacific Soundings Press and Guildford, UK: Society for Research into Higher Education.
Patrick, H., & Pintrich, P. R. (2001). Conceptual change in teachers' intuitive conceptions of learning, motivation, and instruction: The role of motivational and epistemological beliefs. In B. Torff & R. J. Sternberg (Eds.), Understanding and teaching the intuitive mind: Student and teacher learning (p. 117-143). Hillsdale, NJ: Lawrence Erlbaum.
Perrenoud, P. (1993). Práticas pedagógicas, profissão docente e formação [Pedagogical practice, teacher profession and training] Perspectivas sociológicas. Lisboa, Portugal: Dom Quixote.
Philipp, R. A. (2007). Mathematics teachers' beliefs and affect. In Lester, F.K. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 257-315). Charlotte, NC: Information Age Publishing.
Pillen, M., Beijaard, D., & Brok, P. D. (2012). Tensions in beginning teachers' professional identity development, accompanying feelings and coping strategies. European Journal of Teacher Education, 36, 240-260.
https://doi.org/10.1080/02619768.2012.696192
Ponte, J. P. (2011). Teachers' knowledge, practice, identity: Essential aspects of teachers' learning. Journal of Mathematics Teacher Education, 14, 413-417.
https://doi.org/10.1007/s10857-011-9195-7
Provost, L. E., (2013). The multifaceted nature of mathematics knowledge for teaching: Understanding the use of teachers' specialized content knowledge and the role of teachers' beliefs from a practice- based perspective. (Doctoral Dissertation). Retrieved from https://scholars.unh.edu/dissertation/741
QSR International Pty Ltd. (2018). NVivo 12. Version 12.1.249 [Computer software]. Retrieved from https://qsrinternational.com/nvivo-qualitative-data-analysis-software
Quick, G., & Siebörger, R. (2005). What matters in practice teaching? The perception of schools and students. South African Journal of Education, 25, 1-4.
Rebmann, K., Schloemer, T., Berding, F., Luttenberger, S., & Paechter, M. (2015). Pre-service teachers' personal epistemic beliefs and the beliefs they assume their pupils to have. European Journal of Teacher Education, 38(3), 284-299.
https://doi.org/10.1080/02619768.2014.994059
Rockoff, J. E., Jacob, B. A., Kane, T. J, & Staiger, D. O. (2011). Can you recognize an effective teacher when you recruit one? Education Finance and Policy, 6(1), 43-74.
https://doi.org/10.1162/EDFP_a_00022
Rosenberg, J. P., & Yates, P. M. (2007). Schematic representation of case study research designs. Journal of Advanced Nursing, 60(4), 447-452.
https://doi.org/10.1111/j.1365-2648.2007.04385.x
Rowland, T. (2013). The knowledge quartet: The genesis and application of a framework for analyzing mathematics teaching and deepening teachers' mathematics knowledge. Sisyphus-Journal of Education, 1(3), 15-43.
Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255-281.
https://doi.org/10.1007/s10857-005-0853-5
Ryan, M. E., Carrington, S. B., Selva, G., & Healy, A. H. (2009). Taking a 'reality' check: Expanding pre-service teachers' views on pedagogy and diversity. Asia-Pacific Journal of Teacher Education, 37(2), 155- 173.
https://doi.org/10.1080/13598660902804303
SACMEQ (2010). Trends in achievement levels of grade 6 learners in Malawi. Policy Brief. Retrieved from www.sacmeq.org
Salagi, E. (2018). Exploring the changing context of initial primary teacher education in Malawi: Towards pre-service teacher retention from a recruitment and mentoring perspective (Doctoral dissertation), University of Liverpool, UK.
Sandoval, W. A. (2005). Understanding students' practical epistemologies and their influence on learning through inquiry. Science Education, 89, 634-656.
https://doi.org/10.1002/sce.20065
Schleicher, A. (2016). Teaching excellence through professional learning and policy reform: Lessons from around the world. International summit on teaching profession. Paris, France: OECD Publishing.
https://doi.org/10.1787/9789264252059-en
Schommer, M. (1990). Effects of beliefs about the nature of knowledge on comprehension. Journal of Educational Psychology, 82, 498-504.
https://doi.org/10.1037/0022-0663.82.3.498
Schommer, M. (1994). Synthesizing epistemological belief research: Tentative understandings and provocative confusions. Educational Psychology Review, 6, 293-319.
https://doi.org/10.1007/BF02213418
Schön, D. A. (1987). Educating the reflective practitioner: Toward a new design for teaching and learning in the professions. San Francisco, CA: Jossey-Bass Higher Education Series.
Selling, S. K., Garcia, N., & Ball, D. L. (2016). What does it take to develop assessments of mathematical knowledge for teaching? Unpacking the mathematical work of teaching. The Mathematics Enthusiast, 13(1/2), 35-51.
Sheridan, L. (2016). Examining changes in pre-service teachers' beliefs of pedagogy. Australian Journal of Teacher Education, 41(3), 1-20.
https://doi.org/10.14221/ajte.2016v41n3.1
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
https://doi.org/10.3102/0013189X015002004
Shulman, L. S. (1987). Paradigms and research programmes in the study of teaching: a contemporary perspective. In M. Wittrock (Ed.), Handbook of research on teaching (pp. 505-526), New York, NY: Macmillan.
Silverman, D. (2005). Doing qualitative research: A practical handbook (2nd Ed.). London, UK: Sage.
Sinatra, G. M., & Kardash, C. A. (2004). Teacher candidates' epistemological beliefs, dispositions, and views on teaching as persuasion. Contemporary Educational Psychology, 29, 483-498.
https://doi.org/10.1016/j.cedpsych.2004.03.001
Sleep, L. (2009). The work of steering instruction toward the mathematical point: A decomposition of teaching practice. American Education Research Journal, 49(5), 935-970.
https://doi.org/10.3102/0002831212448095
Smith, K., & Lev-Ari, L. (2005). The place of the practicum in pre- service teacher education: The voice of the students. Asia-Pacific Journal of Teacher Education, 33(3), 289-302.
https://doi.org/10.1080/13598660500286333
Smith, M. U., & Siegel, H. (2004). Knowing, believing, and understanding: What goals for science education? Science & Education, 13, 553-582.
https://doi.org/10.1023/B:SCED.0000042848.14208.bf
Snyder, D. W. (2010). Reflections of pre-service teachers on their own teaching practices. Faculty Publications - College of Education, 7, 55- 60.
Stake, R. E. (1995). The art of case study research: Perspective in practice. London, UK: Sage.
Stake, R. E. (2008). Qualitative case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), Strategies of qualitative inquiry (pp. 119-149). Thoursand Oaks, CA: Sage.
Star, J. R., & Strickland, S. K. (2008). Learning to observe: Using video to improve preservice mathematics teachers' ability to notice. Journal of Mathematics Teacher Education, 11(2), 107-125.
https://doi.org/10.1007/s10857-007-9063-7
Starkey, L. (2010). Teachers' pedagogical reasoning and action in the digital age. Teachers and Teaching: Theory and Practice, 16(2), 233- 244.
https://doi.org/10.1080/13540600903478433
Starman, A. B. (2013). The case study as a type of qualitative research. Journal of Contemporary Educational Studies. 64(1), 28-43.
Stephenson, J. (2018). A systematic review of the research on the knowledge and skills of Australian preservice teachers. Australian Journal of Teacher Education, 43(4). 121-137.
https://doi.org/10.14221/ajte.2018v43n4.7
Stipek, D., Givvin, K., Salmon, J., & MacGyvers, V. (2001). Teachers' beliefs and practices related to mathematics instruction. Teaching and Teacher Education, 17(2), 213-226.
https://doi.org/10.1016/S0742-051X(00)00052-4
Strauss, A., & Corbin, J. (1990). Basics of qualitative research techniques and procedures for developing grounded theory. London, UK: Sage.
Street, C. T., & Ward, K. W. (2012). Improving validity and reliability in longitudinal case study timelines. European Journal of Information Systems, 21(2), 160-175.
https://doi.org/10.1057/ejis.2011.53
Stringer, E. T. (1996). Action research: A handbook for practitioners. Thousand Oaks, CA: Sage.
Tardif, M. (2002). Saberes docente e formacão professional. [Teacher knowledge and Professional Training]. Petrópolis, Brazil: Editoria Vozes.
Tellis, W. (1997). Introduction to case study. The Qualitative Report, 3(2), 1-14.
https://doi.org/10.46743/2160-3715/1997.2024
The Southern and Eastern Africa Consortium for Monitoring Educational Quality [SACMEQ]. (2010). SACMEQ III project results: Pupil achievement levels in reading and mathematics. Retrieved November 1, 2019, from http://www.sacmeq.org
Thomas, G. (2011). A typology for the case study in social science following a review of definition, discourse, and structure. Qualitative Inquiry, 17(6), 511-521.
https://doi.org/10.1177/1077800411409884
Thompson, A. G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 127-146). New York, NY: Macmillan.
Tripathi, P. N. (2008). Developing mathematical understanding through multiple representations. Mathematics Teaching in Middle School, 13(89), 438-445.
https://doi.org/10.5951/MTMS.13.8.0438
Turner-Bisset, R. (2001). Expert teaching: Knowledge and pedagogy to lead the profession. London, UK: David Fulton Publishers.
Ulusoy, F. (2020). Prospective teachers' skills of attending, interpreting and responding to content-specific characteristics of mathematics instruction in classroom videos. Teaching and Teacher Education, 90, 1-13.
https://doi.org/10.1016/j.tate.2020.103103
United Nations Development Program. (2018). Human development indicators and indices: 2018 Statistical update team. Retrieved November 2, 2019, from http://hdr.undp.org/sites/default/files/2018_human_development_stati stical_update.pdf
United Nations Educational, Scientific and Cultural Organization [UNESCO]. (2015a). Regional overview: Sub-Saharan Africa. Education for all Global Monitoring Report 2015. Retrieved November 4, 2019, from https://en.unesco.org/gem-report/sites/gem- report/files/regional_overview_SSA_en.pdf
United Nations Educational, Scientific and Cultural Organization [UNESCO]. (2015b). Sustainable Development Goal for Education cannot advance without more teachers. UNESCO Institute for Statistics Fact Sheet 33. Retrieved January 3, 2020, from https://unesdoc.unesco.org/ark:/48223/pf0000234710
United Nations Educational, Scientific and Cultural Organization [UNESCO]. (2016a). Unpacking sustainable development goal 4 education 2030 guide. Retrieved November 12, 2019, from http://unesdoc.unesco.org/images/0024/002463/246300E.pdf
United Nations Educational, Scientific, and Cultural Organization [UNESCO]. (2016b). The world needs almost 69 million new teachers to reach the 2030 education goals. Montreal, Canada. Retrieved January 21, 2020, from https://unesdoc.unesco.org/ark:/48223/pf0000246124.
United Nations Educational, Scientific, and Cultural Organization [UNESCO]. (2019). Meeting commitments: Are countries on track to achieve SDG 4? Global Education Monitoring Report - Target 4 Quality of Education. Paris, France. Retrieved March 14, 2020, from https://unesdoc.unesco.org/ark:/48223/pf0000369009
United Nations International Children's Emergency Fund [UNICEF]. (2019). Education Budget Brief: Towards improved education for all in Malawi. Retrieved November 1, 2019, from https://www.unicef.org/esa/sites/unicef.org.esa/files/2019- 04/UNICEF-Malawi-2018-Education-Budget-Brief.pdf
Vaismoradi, M., Bondas. T., & Turunen, H. (2013). Content analysis and thematic analysis: Implications for conducting a qualitative descriptive study. Journal of Nursing & Health Sciences, 15, 398-405.
https://doi.org/10.1111/nhs.12048
Wasserman, N., & Stockton, J. (2013). Horizon content knowledge in the work of teaching: A focus on planning. For the Learning of Mathematics, 33(3), 20-22.
Waters-Adams, S. (2006). The relationship between understanding of the nature of science and practice: The influence of teachers' beliefs about education, teaching and learning. International Journal of Science Education, 28(8). 919-944.
https://doi.org/10.1080/09500690500498351
Watt, H. M. G., & Richardson, P. W. (2007). Motivational factors influencing teaching as a career choice: Development and validation of the FIT-choice scale. The Journal of Experimental Education, 75(3), 167-202.
https://doi.org/10.3200/JEXE.75.3.167-202
Wideen, M., Mayer-Smith, J., & Moon, B. (1998). A critical analysis of the research on learning to teach: Making the case for an ecological perspective on inquiry. Review of Educational Research, 68(2), 130- 178.
https://doi.org/10.3102/00346543068002130
Wilkins, J. (2008). The relationship among elementary teachers' content knowledge, attitudes, beliefs, and practices. Journal of Mathematics Teacher Education, 11(2), 139-164.
https://doi.org/10.1007/s10857-007-9068-2
Wilkinson, S. (2004). Focus group research. In D. Silverman (Ed.), Qualitative research: Theory, method, and practice (pp. 177-199). Thousand Oaks, CA: Sage.
Wilson, P. S., Cooney, T. J., & Stinson, D. W. (2005). What constitutes good mathematics teaching and how it develops: Nine high school teachers' perspectives. Journal for Mathematics Teacher Education, 8(2), 83-111.
https://doi.org/10.1007/s10857-005-4796-7
Wittgenstein, L. (2007). Zettel. Berkeley, CA: University of California Press.
Worden, D. (2015). The development of content knowledge through teaching practice. Ilha do Desterro, 68(1), 105-119.
https://doi.org/10.5007/2175-8026.2015v68n1p105
Yang, X., Kaiser, G., König, J., & Blömeke, S. (2020). Relationship between pre-service teachers' knowledge, beliefs and instructional practices in China. ZDM Mathematics Education, 52(3), 281-294.
https://doi.org/10.1007/s11858-020-01145-x
Yaya, S., Bishwajit, G., &, Shah, V. (2016). Wealth, education and urban-rural inequality and maternal healthcare service usage in Malawi. BMJ Global Health, 1, e000085.
https://doi.org/10.1136/bmjgh-2016-000085
Yin, R. K. (2002). Case study research: Design and methods. Thousand Oaks, CA: Sage.
Zhang, Y., Hawk, S. T., Zhang, X., & Zhao, H. (2016). Chinese preservice teachers' professional identity links with education program performance: The roles of task value belief and learning motivations. Frontiers Psychology, 7(573), 1-12.
