Concluding Remarks

All the findings echoed three special effects of integrating history in mathematics, which are brought to our attention by Jahnke (2000): the first is replacement – it allows mathematics to be seen as an intellectual activity, rather than as just a corpus of knowledge or a set of techniques; the second is reorientation – it reminds us that mathematical concepts were invented and that this did not happen all by itself; the third is cultural understanding – it invites us to place the development of mathematics in the scientific and technological context in a particular time of certain societies.

Returning to Veal and MaKinster’s PCK model, we believe that the two scholars proposed a valuable model but did not explain how those attributes of PCK influenced on or were connected to one another. Our study indicated that, by reading mathematical texts, the in-service teachers came to understand the genesis of Heron’s Formula and the evolution of its proofs, realising the respective virtues and difficulties of geometrical and algebraic representations. In addition, the teachers’

reflections revealed that the teachers became more knowledgeable of difficulties that might arise when teaching Heron’s Formula and related concepts. As for the vertical connections between their mathematics content knowledge and the top-level attributes of their PCK, we can see that in several occasions in their reflections, they demonstrated that their content knowledge on Heron’s Formula and its proofs connected with the attributes of context, assessment, pedagogy, curricula, and socioculturalism. Besides, we also see that the horizontal links among the five attributes were strengthened. In other words, an in-service teacher’s mathematical content knowledge and PCK might both be enhanced with an HPM approach. This finding is consistent with the results of Smestad (2011) and Clark (2011) regarding pre-service teachers. We believe that this case study serves as a beginning of a direction in research exploring more methods for enhancing the professional knowledge of in-service teachers by using an HPM approach.

A final remark is about the correlation between a teacher’s PCK and her maturity of the knowledge related to HPM. Although Veal and MaKinster mentioned that for the attributes of their PCK model, the usefulness, impact, and understanding would not be fully realised or integrated until a teacher has acquired several years of classroom experience (Veal & MaKinster, 1999), teaching experiences and seniority do not hold a salient position in the model. However, in this study we did observe that a teacher’s PCK and her maturity of the knowledge related to HPM may be positively correlated. The four high school teachers have at least nine years of experience; thus, none of them were novices, and they easily understood the mathematical knowledge in the proofs. However, their reflections differed. Teacher Gamma, who had the longest experience in teaching and self-learning with an HPM approach, generated more ideas on how Heron’s Formula is placed in the mathematical structure of the concept of the area and in the curricular structure in current high school textbooks than other teachers did. She also elaborated more on how it could be used in her own teaching, showing more relevance in PCK. By contrast, the other three teachers, who had less experience in self-learning and applying historical ideas in mathematics teaching, did not have as many opinions on how historical proofs could be utilised in teaching or curricular designs. It would seem that a teacher’s PCK and his or her maturity in HPM (i.e., experience in learning and using historical texts and concepts in mathematics teaching and learning) are positively correlated. This study provides evidence of enhancements of PCK through an HPM approach, so a teacher with more experience in teaching and self-learning with historical material may have improved his or her PCK in the same time. Further studies can be conducted on this topic.

Acknowledgement

The authors would like to thank the two anonymous reviewers for their constructive comments.

However, the authors take full responsibility for the opinions expressed and the accuracy of the contents in the paper.

Reference

Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3(1), 1-29. doi:

10.1111/j.2044-835X.1985.tb00951.x

Clark, K. (2011). Reflections and revision: Evolving conceptions of a using history course. In V. Katz

& C. Tzanakis (Eds.), Recent developments on introducing a historical dimension in mathematics education (pp. 211-220). Washington, DC: Mathematical Association of America. doi:

10.5948/UPO9781614443001.020

Fan, L. (2003). A study on the development of teachers’ pedagogical knowledge. Shanghai, China: East China Normal University Press. (In Chinese)

Fauvel, J., & Gray, J. (Eds.). (1987). The history of mathematics: A reader. London, UK: Macmillan Education in association with the Open University.

Fauvel, J., & van Maanen, J. (Eds.). (2000). History in mathematics education: The ICMI study.

Dordrecht, The Netherlands: Kluwer Academic Publishers.

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: D. Reidel.

Jahnke, H. N. (2000). The use of original sources in the mathematics classroom. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education: The ICMI study. (pp. 291-328). Dordrecht, The Netherlands: Kluwer Academic Publishers. doi: 10.1007/0-306-47220-1_9

Jorgensen, D. L. (1989). Participant observation: A methodology for human studies. Newbury Park, CA: Sage. doi: 10.4135/9781412985376.n1

Katz, V., & Tzanakis, C. (Eds.). (2011), Recent developments on introducing a historical dimension in mathematics education. Washington, DC: Mathematical Association of America. doi:

10.5948/UPO9781614443001

Li, S. (2002). 天算或問 [Some questions about astronomy and mathematics]. In 續修四庫全書編輯 委 員 會 [Continued emperor's four treasuries editorial committee] (Ed.), 續 修 四 庫 全 書 [Continued emperor's four treasuries]. Shanghai, China: 上海古籍[Shanghai Ancient Texts].

(Original work published in 1867)

Li, Z. (2005). A concise history of mathematical education in the late Qing dynasty. Jinan, China:

Shandong Education Press. (In Chinese)

Liu, P. H. (2009). History as a platform for developing college students’ epistemological beliefs of mathematics. International Journal of Science and Mathematics Education, 7(3), 473-499. doi:

10.1007/s10763-008-9127-x

Loomis, E. (1859). Elements of plane and spherical trigonometry, with their applications to mensuration, surveying, and navigation. New York, NY: Harper & Brothers.

Mei, W. (1993). 平三角舉要 [Elements of planar trigonometry]. In 郭書春 [Guo Shuchun] (Ed.), 中國科學技術典籍通彙：數學卷 [Compendium of Chinese texts on science and technology:

mathematics section]. Zhengzhou, China: 河 南 教 育 [Henan Educational]. (Original work published nd)

Rho, G. (2000). 測量全義 [Complete explanation of measurements]. In 故宮博物院 [Palace Museum] (Ed.), 故宮珍本叢刊 [Collected publication of rare books in the palace museum].

Haikou, China: 海南[Hainan]. (Original work published in 1631)

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1-22.

Smestad, B. (2011). History of mathematics for primary school teacher education or: Can you do something even if you can’t do much? In V. Katz & C. Tzanakis (Eds.), Recent developments on introducing a historical dimension in mathematics education. (pp. 201-210). Washington, DC:

Mathematical Association of America. doi: 10.5948/UPO9781614443001.019

Thomaidis, Y. (2005). Two questions on historical conceptions on teaching and learning mathematics.

HPM Newsletter, 60, 10-12.

Tzanakis, C., & Arcavi, A. (2000). Integrating history of mathematics in the classroom: An analytic survey. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education: The ICMI study.

(pp. 201-240). Dordrecht, The Netherlands: Kluwer Academic Publishers. doi:

10.1007/0-306-47220-1_7

Veal, W. R., & MaKinster, J. G. (1999). Pedagogical content knowledge taxonomies [Electronic version]. Electronic Journal of Science Education, 3(4). Retrieved October 1, 2005, from http://unr.edu/homepage/crowther/ejse/vealmak.html.

Yang, K. L. (2004). Constructing a model for high school students’ reading comprehension of geometrical proofs (Unpublished doctoral dissertation). National Taiwan Normal University, Taipei. (In Chinese)