Data Analysis and Findings

With the collected reflections written by those teachers, we can now explain why reading ancient mathematical texts might enhance teachers’ content knowledge and PCK, according to Veal and MaKinster’s taxonomy. The reflections were originally written in Chinese. Quotations were translated into English by the authors from the teachers’ reflections. As mentioned earlier, the teachers’

reflections were analysed by the participating author and the non-participating one to seek trustworthiness through triangulation. Therefore, the excerpts of the reflections quoted in this section objectively represent to a certain extent the changes of the teachers. In what follows we list our findings about the teachers’ enhancements and provide excerpts of the teachers’ reflections as the evidence in our analysis.

1. Enhancements of mathematical knowledge and the connection between content knowledge and socioculturalism

From the previous section, we know that Heron’s Formula was first proposed in an historical scene due to the need for solving real-world problems, and geometrical representations were used in early proofs. Those proofs expanded the teachers’ understanding of the mathematical content knowledge on Heron’s Formula. Teachers Alpha wrote,

[W]hen I first saw Heron’s original proof, on the one hand, I was alerted to the scant knowledge I had; on the other hand, I had a sudden realisation, because it was only natural that Heron’s idea came from geometrical concepts and figures…[H]ardly any textbook version mentions Heron’s original proof.

Veal and MaKinster’s terminology can be applied in describing the change in Teacher Alpha’s professional knowledge. First, the teacher’s mathematical knowledge was improved for he learned how to prove Heron’s formula with Euclidean geometry. Second, the connection between content knowledge and the knowledge of context and socioculturalism were enhanced. The reason we say this is that Teacher Alpha realised he had actually known that, in Hellenistic Alexandria, where Heron worked, there was no symbolic algebra (because it was invented in the eighteenth century), and all geometrical propositions were proved with Euclid’s approach and Aristotelian logic. Teacher Alpha linked the mathematical content and socioculturalism in history.

2. Enhancements of the awareness to different representations and structures of proofs We also observed that all the teachers, except Teacher Beta, mentioned that they noticed the different representations – geometrical and algebraic – used in the texts. The same three teachers also reflected on how they might be able to use the characteristics and differences among the proofs in lessons, thus enriching their teaching repertoire. For instances, Teacher Gamma wrote:

[Heron’s and Mei’s proofs] had the same basic strategy: first they used the inscribed circle to obtain the area of the triangle, and then they used proportions to obtain the relationships between rs and segments s, s-a, s-b, and s-c. Both proofs have only one inconvenience in teaching; namely, because we cannot see the role a ‘height’ plays, teachers cannot link

‘1/2 × base × height’ to Heron’s Formula. For this problem, Li Shanlan’s proof provides a solution.

This excerpt provides evidence that the teachers’ content knowledge on the structures of the proofs was enhanced.

3. Enhancements of the connection between the teachers’ content knowledge and assessment.

Assessment is another attribute that is connected to content knowledge. Teacher Delta stated that:

[U]sing Pythagorean Theorem to prove Heron’s Formula may be able to prevent students from perceiving that Heron’s Formula is merely a procedural knowledge.

Teacher Gamma also wrote in another place about the aspect of assessment:

Knowing the geometrical counterparts of s, s-a, s-b and s-c […] helps in problem-solving, such as in solving the [following] problem [that was used in an exam]: ‘Given △ABC, AB= 4, BC= 5, and CA= 6. Its inscribed circle touches three sides at points D, E, and F. If the area of △AEF and △BDF are x and y, respectively, then x y: = ?’

These reflections, in Veal and MaKinster’s terms, can be considered as improvements of the connection between content knowledge and assessment, because, after reading Heron’s proof, Teacher Gamma could see the help students might receive from those proofs, and she could identify the link between the proofs and assessments of students’ knowledge about a triangle and its inscribed circle.

4. Enhancements of the connection between content knowledge and pedagogy

Another enhancement, the connection between content knowledge and pedagogy, can be observed in Teacher Beta’s reflection on Li Shanlan’s proof:

Li Shanlan seemed to narrate [his proof] from the end, using ‘reverse reasoning’ and

‘transitional objects’…This seems to be different from proving strategies in modern texts. In them, solutions of a problem often begin with the ‘givens’, and then through several

‘transitional objects’ [that are found or created], the problem is solved. But they never explain why those transitional objects are necessary. By contrast, Li Shanlan’s proof began with the

‘result’, and then investigates what ‘transitional objects’ were necessary to achieve this result; therefore, he created and proved them. This is why he arrived at this form of solution strategy, which is actually closer to students’ natural learning curve…[Li Shanlan’s proof]

looks like Confucius’ Analects, in which the questions and answers between the master and the pupils are used to elaborate and solve problems.

As the reader can see, in this excerpt, Teacher Beta mentions the connection between content knowledge and pedagogy, since he compared the teaching strategies among modern textbooks, Li Shanlan’s proof and the Analects.

5. Enhancements of the awareness to students’ possible difficulties and its connection with content knowledge

The excerpt of the reflection in the previous subsection also addresses how students’ problems can be solved after the teacher recognises them. Teacher Delta, too, mentioned how he recognised students’ difficulties after reading those proofs:

[T]he proofs of Heron’s formula…are not so natural, in both the past and the present, because whatever method you use, they are all very tricky. Heron’s original proof and Mei Wending’s proof require using the radius of the inscribed circle, and you even have to construct very specific lines to fit the requirement…Although Li Shanlan used proportions, he still needed the inscribed circle. And for those who do not know proportional line segments that well, it is difficult to understand Li Shanlan’s proof at first glance.

Thus, reading historical proofs also help teachers understand students’ difficulties, because the teachers encountered the same difficulties. Therefore, we can see there is an enhancement of the knowledge of students, and its connection with content knowledge.

6. Enhancements of the connections between content knowledge and curricula

In addition, the implicit connection between geometrical and algebraic representations also helped the teachers consider the knowledge context and different strategies for arranging teaching material. Teacher Delta said,

In the context of senior high school mathematics, it is very natural to verify Heron’s Formula with Laws of Sine and Cosine, and we can conveniently check how well students learned the two Laws. If we leave this context, then it is not easy for students to deduce Heron’s Formula naturally.

He also noticed the limitation of presenting the formula in this way:

It seems to be a conclusion for Laws of Sine and Cosine, with no further room for development.

Clearly the connections between the knowledge of context and curricula were enhanced, since Teacher Delta could compare the historical proofs with those in modern textbooks, and realised not only why the author of textbooks present this formula in the context of trigonometry, but also that the textbook proof has no other development, suggesting that using historical proofs might introduced more relevant mathematical concepts in the curriculum.

7. Enhancements of the connections among content knowledge, pedagogy, and curricula Further understanding of the content knowledge on Heron’s Formula enabled the teachers to discover that it can be used as a bridge between the trigonometry contents in junior and senior high school levels, and Teacher Gamma even developed a feasible design for teaching materials. Teacher

Gamma stated that in Taiwan, the basic area formulae of triangles and rectangles taught in elementary school mathematics, the Euclidean propositions about triangles taught in junior high school mathematics, and trigonometry taught in senior high school, are taught separately with very few connections among them. Geometrical proofs of Heron’s formula, if taught in the senior high school level, could serve as a bridge linking contents in the elementary and junior high school levels (such as 1/2 × base × height, triangle congruence properties, similar shapes, inscribed circles and so on) to senior high school mathematics. Those proofs in turn will help students understand the nature and value of the algebraic Laws of Sine and Cosine in a geometrical manner. Finally, through the learning process, all triangle area formulae could be summarised with trigonometry. This shows the teacher’s professional growth through this HPM approach, as well as the connections among content knowledge, pedagogy and curricula. According to Veal and MaKinster’s model, these connections are vertical connections among attributes, as are several other enhancements described in the previous subsections.

According to the analysis of teachers’ reflections, the reading of historical texts of mathematics can improve teachers’ comprehension of the mathematical content knowledge in Heron’s Formula, complementing current textbooks. Those proofs deepened the teachers’ understanding of triangle area and its formulae, so they could extend their teaching material for geometry. Furthermore, they felt the convenience of algebraic methods compared to geometrical representations, which shows that they became more perceptive to the panorama of the structural context of the mathematics curriculum.