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Exploring the educative power of an experienced mathematics teacher educator-researcher

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Abstract

This paper aims to explore the educative power of an experienced mathematics teacher educator-researcher (MTE-R) who displayed his insights and strategies in teacher professional development (TPD) programs. To this end, we propose a framework by first conceptualizing educative power based on three constructs—communication, reasoning, and connection—and then we extend the conceptualization with another two dimensions: the reciprocal facilitator-learner relationships involving educators, teachers, and students, as well as a bridge between research and practice. Based on both self-study and case-study approaches, we further elaborate features specific to the MTE-R’s educative power which includes communication using an approach of creating educative phenomenology, reasoning by mapping teachers’ ideas onto emergent models to solve problems in educative challenges, and connection between research and practice by coordination. In particular, the core of the educative power that supported the MTE-R to initiate at-the-moment actions was his insights into the essence of mathematics, and the learning of students and teachers. We believe that the conceptual framework in this study offers a powerful tool that could guide the analyses of educative power, especially for those studies related to the initiation of at-the-moment actions and the implementation of TPD programs.

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Notes

  1. All the teachers’ names presented in the paper are pseudonyms.

  2. RME is the abbreviation of realistic mathematics education.

  3. The Book of Rites (Chinese: 禮記) is a collection of texts describing the social forms, administration, and ceremonial rites of the Zhou Dynasty (c. 1046–256 bc) in China. Some sections contain details of the life and teachings of Confucius (see http://en.wikipedia.org/wiki/Book_of_Rites).

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Acknowledgments

The authors would like to thank all of the participating teachers for their engagement in the workshop. This paper is part of a research project partially funded by the National Science Council of Taiwan (NSC 100-2511-S-003-036-MY3). The views and opinions expressed in this paper are those of the authors and not necessarily those of the NSC.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Taiwan Normal University, Room M202, No. 88, Sec. 4, Ting-Chou Rd., Taipei, 11677, Taiwan

    Kai-Lin Yang & Fou-Lai Lin

  2. National Hsinchu University of Education, Hsinchu, Taiwan

    Hui-Yu Hsu

  3. Ming Chi University of Technology, Taipei, Taiwan

    Jian-Cheng Chen

  4. University of Taipei, Taipei, Taiwan

    Ying-Hao Cheng

Authors
  1. Kai-Lin Yang
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  2. Hui-Yu Hsu
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  3. Fou-Lai Lin
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  5. Ying-Hao Cheng
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Corresponding author

Correspondence to Fou-Lai Lin.

Appendices

Appendices

1.1 Appendix A: Adam’s design for exploring the essence of ratio concept

Please make comparisons between the following items:

  1. A.1.

    Da-Ming and Hsiao-Ming are measuring their heights. Da-Ming is 165 cm and Xia-Ming is 172 cm. Who is taller? Please explain how you judge which one is taller.

  2. A.2.

    A-Ming bought two bags of oranges. The bag with golden oranges cost NT 300; the other bag with organic oranges cost NT200. Please explain how you judge which one is more expensive.

  3. A.3.

    A-Ming made two glasses of sweetened water. He added 3 sugar cubes to the first glass of water and 5 sugar cubes to the other one. Which one is sweeter? Please explain how you judge which one is sweeter.

  4. A.4.

    Da-ming and Hsiao-ming ran on the playground after school. Da-ming ran 400 m, and Hsiao-ming ran 300 m. Who runs faster? Please explain how you make the comparison.

  5. A.5.

    Da-ming and Hsiao-ming played basketball on the basketball court. All the afternoon, Da-ming shot 10 goals, and Hsiao-ming shot 15 goals. Who shoots more accurately? Please explain how you make the comparison.

  6. A.6.

    Different activities were held in two activity centers in the school. The former gathered 200 people, and the latter had 400 people. Which one is more crowded? Please explain how you make the comparison.

  7. A.7.

    There are two slides in the park. The height of the former is 4 m, and the latter is 5 m. Which one is steeper? Please explain how you make the comparison.

1.2 Appendix B: Zhuang’s design for learning \( \sqrt{2}+\sqrt{2}=\sqrt{8} \)

  1. B.1.

    As shown below, a square with side length 2 and area 4 (the diagram on the left side) is folded by moving four vertices to the center and then creating another square with area 2 (see the diagram on the right side).

figure a

Question: What is the side length of the folded square?

  1. B.2.

    As shown below, two squares with side length 2 are placed side by side.

figure b

Question: What is the above diagram?

Question: What is the length of the longer side of the diagram? Why? Please explain the reasons.

  1. B.3.

    Following question 2, four squares with area 2 are juxtaposed and form a bigger square.

figure c

Question: What is the area of the juxtaposed square?

Question: What is the side length of the juxtaposed square? Why? Please explain the reasons.

  1. B.4.

    As show on the above diagram.

Question: \( \sqrt{2}+\sqrt{2}=? \) Why?

Question: \( \sqrt{2}+\sqrt{2}=\sqrt{2+2}? \) True or false? Why?

Question: \( \sqrt{2}+\sqrt{2}=2\times \sqrt{2} \); \( 2\times \sqrt{2}=\sqrt{2\times 2}? \) True or false? Why?

1.3 Appendix C: Adam’s design for calculating the sum of infinite series

  1. C.1.

    There is the infinite series: \( 1+\frac{1}{3}+{\left(\frac{1}{3}\right)}^2+{\left(\frac{1}{3}\right)}^3+\cdots +{\left(\frac{1}{3}\right)}^n+\cdots \) Please answer the following questions:

  2. C.1.1.

    What is the answer of the infinite series?

(A) \( \frac{1}{3} \) (B) \( \frac{1}{2} \) (C) \( \frac{2}{3} \) (D) The answer can not be obtained.

  1. C.1.2

    Please design a word problem and make the answer equaling to the following series \( \left\lceil 1+\frac{1}{3}+{\left(\frac{1}{3}\right)}^2+{\left(\frac{1}{3}\right)}^3+\cdots +{\left(\frac{1}{3}\right)}^n+\cdots \right\rfloor \circ \)

  2. C.1.3

    Is there any other solution for solving the word problem you created? If not, please design another word problem equaling to the infinite series. (Reminder: do not mention \( \frac{1}{3} \) in your designed problem)

  3. C.2.

    A word problem: “A and B are playing paper-scissors-stone game. The game will end till someone wins”.

  4. C.2.1.

    Please answer the probability that A will win.

  5. C.2.2

    Please list two solutions to the above question and combine them into an equation.

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Yang, KL., Hsu, HY., Lin, FL. et al. Exploring the educative power of an experienced mathematics teacher educator-researcher. Educ Stud Math 89, 19–39 (2015). https://doi.org/10.1007/s10649-014-9589-4

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