Does a transformation approach improve students’ ability in constructing auxiliary lines for solving geometric problems? An interventionbased study with two Chinese classrooms
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Abstract
We conducted an interventionbased study in secondary classrooms to explore whether the use of geometric transformations can help improve students’ ability in constructing auxiliary lines to solve geometric proof problems, especially highlevel cognitive problems. A pre and posttest quasiexperimental design was employed. The participants were 130 eighthgrade students in two classes with a comparable background that were taught by the same teacher. A twoweek intervention was implemented in the experimental class aiming to help students learn how to use geometric transformations to draw auxiliary lines in solving geometric problems. The data were collected from a teacher interview, videorecordings of the intervention, and pre and posttests. The results revealed that there was a positive impact of using geometric transformations on the experimental students’ ability in solving highlevel cognitive problems by adding auxiliary lines, though the impact on the students’ ability in solving general geometric problems as measured using the overall average scores was not statistically significant. Recommendations for future research are provided at the end of the article.
Keywords
Teaching of geometry Geometric proof Geometric transformation Auxiliary lines Interventionbased study1 Background and rationale
Geometry has long occupied an important place in the mathematics curriculum in many countries. However, it has also proven to be a very difficult area for both teaching and learning, and has attracted increasing attention from mathematics education researchers, curriculum reformers and developers, as well as practitioners internationally over the recent decades. In China, the provision of geometry in the recent reformed curriculum has shifted the focus from the Euclidean axiomatic system to the use of three thematic approaches: “the nature of geometric figures”, “transformation of geometric figures”, and “geometric figures and coordinates” to present all the contents of geometry, through which it is expected that students will improve their learning of geometry (China Ministry of Education, 2012).
As widely recognized, proof is one of the most challenging parts in students’ learning of geometry (Mariotti, 2006; McCrone & Martin, 2004; Senk, 1985; Weber, 2001). Different researchers have analyzed various possible reasons and studied possible ways to improve students’ understanding of geometric proof (e.g., see Harel, 1999; Hodds, Alcock, & Inglis, 2015; Inglis & Alcock, 2012) and enhance their proof abilities (Golzy, 2008; Hoyles & Jones, 1998; Marrades, 2000), though overall, probably due to its difficulty, it is still largely underresearched, let alone resolved. Researchers are still looking for more solid solutions, especially from classroombased studies, which is also evident in this special issue of Educational Studies in Mathematics.
In geometric proof, adding auxiliary lines is often helpful and in many cases necessary, especially in solving challenging or highlevel cognitive geometric problems. In fact, the word “auxiliary” itself means offering additional help or support. However, to add auxiliary lines is also very difficult for many students (Herbst & Brach, 2006; Senk, 1985; You, 2009). As Chou, Gao, and Zhang (1994) argued, “adding auxiliary lines is one of the most difficult and tricky steps in the proofs of geometry theorems” (p. 2). In this study, we aimed to address students’ difficulty in constructing auxiliary lines by introducing a transformation approach as an intervention in the geometric classroom, and hence to improve their ability in solving geometric problems and, in particular, in solving challenging or highlevel cognitive geometric problems, as understandably students would have more difficulty in solving challenging or highlevel cognitive problems.
Mathematically, it is well known that the concept of transformation has played a central role in modern geometry. In Klein’s landmark writings about the Erlangen Program, geometry was viewed as “the study of the properties of a space that are invariant under a given group of transformation” (Bonotto, 2007; Jones, 2000). Klein’s Erlangen Program provides a fundamental view and a new approach, not only in modern advanced geometry but also in school geometry, particularly in geometric proof. For example, as researchers have pointed out, any proof by congruence in the Euclidean tradition can be done by congruent transformations, such as rotation, translation and reflection, that preserve everything (e.g., as betweenness of points, midpoints, segments, angles and perpendicularity) between the preimage and the image (Barbeau, 1988; Jones, 2000; Nissen, 2000; Ng & Tan, 1984; Willson, 1977).
In China, the contemporary school curriculum has introduced geometric transformation since 2001 when the government issued its new reformed national curriculum standards (China Ministry of Education, 2001). The new curriculum put more emphasis on three types of congruent transformations: translation, rotation and reflection. Following the national mathematics curriculum, Chinese school textbooks have also introduced geometric transformation as a standard topic of geometry, and, in some cases, used it to help students understand geometric proof.
The introduction of geometric transformation in school mathematics is also reflected in the school curricula in other countries, for example, England and the US. In England, the national curriculum requires that secondary school students be taught in geometry to “identify properties of, and describe the results of, translations, rotations and reflections applied to given figures” at Key Stage 3 (Year 7–9) (Department for Education, 2013) and furthermore to “describe the changes and invariance achieved by combinations of geometrical transformation” at Key Stage 4 (Year 10–11) (Department for Education, 2014). In the US, a most influential curriculum document, NCTM’s Principles and Standards for School Mathematics, emphasized that in the teaching of geometry, “instructional programs from prekindergarten through grade 12 should enable each and every student to … apply transformations and use symmetry to analyze mathematical situations” (National Council of Teachers of Mathematics , 2000, p. 41). The inclusion of transformation in the school curriculum is also endorsed in the US widelyadopted Common Core State Standards for Mathematics, which suggests that students should be taught to “verify experimentally the properties of rotations, reflections and translations” in Grade 8 (Common Core State Standards Initiative, 2009, p. 55).
Having briefly described the mathematics and curriculum background, we point out that, from the perspective of educational research, there are mainly two reasons for us to use a transformation approach to facilitate students in adding auxiliary lines in solving geometric problems.
The first reason concerns the practical value of the study. As aforementioned, the reformed national mathematics curriculum in China has placed more emphasis on the transformation approach in presenting the contents of geometry to students. Because of the influence of the national mathematics curriculum on school textbooks and classroom teaching and learning, it is not only more feasible but also more meaningful and relevant than before from the curricular perspective for us to conduct the study exploring relevant issues concerning the use of a transformation approach in the teaching and learning of geometry.
It should also be noted that, along with the latest national curriculum development and reform, both researchers and school practitioners in China have shown increasing awareness about the possible positive impact of introducing geometric transformations on students’ ability in constructing proofs. Accordingly, researchers have also discussed how transformations can be used to help geometry proofs, and, in particular, in adding auxiliary lines (Gao, 2010; Yang & Pan, 1996; Yao, 2010). For example, it has been argued that using a proper transformation can often rearrange a geometric diagram to form figures that students are familiar with (e.g., rightangled triangles, as shown in Fig. 1) while maintaining the properties concerned (e.g., the length of segments or the size of angles), thus helping students better understand where and how to add the auxiliary lines. Similarly, in a recent analysis about exercise problems in school textbooks, Wang (2010) argued that geometric transformations can provide ideas for students to look at all the known conditions and hence to add auxiliary lines in geometric proofs. However, there is a lack of research evidence from classroombased studies, in China or elsewhere, about the effectiveness of using geometric transformation on students’ ability in adding auxiliary lines, thus a reason for this study.
The second reason concerns the theoretical value of the study at a more general level beyond the Chinese educational setting. As we understand, since at least the early 1960s, researchers internationally have shown interest and offered various reasons for using a transformation approach in the teaching and learning of geometry. Researchers have argued that introducing a transformation approach can allow students to solve problems that might otherwise be more difficult (Hollebrands, 2003; Usiskin, 2014). In particular, some earlier studies conducted in the US have provided evidence suggesting that the use of transformation encouraged students to construct geometric proofs flexibly and creatively (Coxford, 1973; Usiskin, 1972; Usiskin & Coxford, 1972). However, we found that there have been virtually no studies, let alone interventionbased ones, on how transformation can be used to improve students’ ability in constructing auxiliary lines (e.g., see Harel & Sowder, 2007). This fact is surprising as well as another motivating factor for us to conduct this study.
Given the relevant curricular and research background outlined above, we decided to conduct an interventionbased empirical study to explore the impact of introducing a transformation approach in the teaching and learning of geometry on students’ ability in adding auxiliary lines for solving geometric problems. We were particularly interested to investigate the impact of the intervention on students’ ability to solve challenging or highlevel cognitive problems. The reason for this is twofold. First, solving highlevel cognitive problems usually requires the use of more than one area of knowledge, including connections between different domains and an understanding of the terms of transformation (Brändström, 2005; Hiebert et al., 2003; Stein & Smith, 1998). Second, through transformation, students can move different parts of geometric figures together, find or visualize their relationships, and hence add auxiliary lines in solving geometric problems (Wang, 2010). In comparison, it appears clear that solving simple or lowlevel cognitive geometric problems is generally less problematic and adding auxiliary lines is usually not required, but, even when it is required, finding how to add auxiliary lines tends to be straightforward and accordingly does not need a transformation approach.
2 Research design and procedure
The study was carried out in the Chinese educational setting. A quasiexperimental design was used, consisting mainly of a classroombased intervention, a pretest and a posttest. In addition, interviews and video recordings were used in data collection.
2.1 Participants
The intervention took place in a western city in China. The selected institution was a junior secondary school covering grades 7, 8 and 9 (aged 12–15). There were ten classes for each of grades 7 and 9, and eight classes for grade 8, with 1560 students and 14 mathematics teachers in total. In many ways, it was a typical mediumsize and averageperforming secondary school in western China. It should be pointed out that the school had been actively involved in the reform of “Mathematics Classroom in Secondary Schools Based on the New Curriculum” and “Learning Plan Guidance”, which implies that they had some practical experiences about implementing the new curriculum, an important reason for our study to take place in the school.
Average scores of two recent tests of the two classes
Recent formative (monthly) test  Recent summative (midterm) test  

Class A (experimental; n = 65)  91.8  98.5 
Class B (control; n = 65)  91.3  98.4 
The participating teacher, who as we noted was the mathematics teacher of both classes, has taught mathematics for 13 years using the textbook series published by Beijing Normal University Press (Ma, 2014b, a). She was an experienced teacher and had won awards including the title of “Teaching Master” at the provincial level. Before implementing the intervention, the researchers explained to the teacher the purpose and procedure of the study and the intervention. In particular, she was told to teach the two classes in the same way and as usual, except for introducing the intervention for the experimental class.
2.2 Pretest and posttest
To compare students’ achievements before and after the intervention, both pretest and posttest were designed and administered to the classes. The design of questions was based on the school curriculum (including the textbook used) as well as the consultation with the participating teacher, so they can be not only better integrated into the ongoing school curriculum but also better fit students’ background of learning.
Distribution of the questions in the pretest and posttest
PreQ1  PreQ2  PreQ3  PreQ4  PreQ5  PreQ6  PreQ7  PreQ8  PreQ9  PreQ10  
Pretest  Use of geometric transf.^{a}  N  N  Y  O  Y  Y  Y  Y  Y  Y 
    Ro  Tr  Ro  Tr  Ro  Ro  Re  Tr  
Adding aux. lines^{b}  Y  Y  O  O  O  N  N  Y  Y  Y  
PosQ1  PosQ2  PosQ3  PosQ4  PosQ5  PosQ6  PosQ7  PosQ8  PosQ9  PosQ10  
Posttest  Use of geometric transf.^{a}  N  N  N  N  O  O  O  Y  Y  Y 
        Re  Ro  Tr  Re  Re  Ro  
Adding aux. lines^{b}  N  N  Y  Y  Y  Y  Y  Y  Y  Y 
As shown in Table 2, adding auxiliary lines was necessary for solving five questions in the prerest (PreQ1, PreQ2, PreQ8, PreQ9 and PreQ10), not necessary for solving two questions (PreQ6 and PreQ7), and optional for the remaining three questions (PreQ3, PreQ4 and PreQ5). In addition, four questions, i.e., PreQ3, PreQ5, PreQ7 and PreQ8, involved rotation: three (PreQ4, PreQ6 and PreQ10) were related to translation, one (PreQ9) was about reflection, and the remaining two (PreQ1 and PreQ2) did not involve any geometric transformation. The main reason for the test to include a small number of nontransformation questions and questions that do not require adding auxiliary lines to solve is for the test to be better integrated with the school curriculum and students’ learning progress.
As also shown in Table 2, for comparison purposes, two types of questions designed in the pretest were also provided in the posttest, including questions (PosQ8, PosQ9 and PosQ10) that involved geometric transformation and could not be solved without auxiliary lines, and questions (PosQ3 and PosQ4) that were not related to geometric transformation but could not be solved without auxiliary lines. In addition, questions PosQ6 and PosQ10 were related to rotation, PosQ5, PosQ8 and PosQ9 to reflection, and PosQ7 to translation. PosQ1 to PosQ 4 were not related to geometric transformation, and they were designed to explore the influence of the twoweek intervention on students solving general geometry problems without the use of transformation and to better fit the school regular curriculum. In addition, the test included three challenging geometric questions, i.e., PosQ6, PosQ8 and PosQ10, which require highlevel cognition to solve, as one of the study aims was to detect the influence of using geometric transformation on students’ ability in adding auxiliary lines to solve highlevel cognitive problems.
The total score of the posttest was 56 marks, with 12 full marks for PosQ6, 8 full marks for each of PosQ7 and PosQ9, and 4 full marks for each of the remaining questions. Like in the pretest, students could receive any mark from 0 to the full marks based on their solutions. The full sets of all the posttest questions with the scoring schemes can be found in Online Resource 2. In addition, examples of students’ solutions, together with the marks given, can be also found in Online Resource 1 for the pretest (e.g., PreQ2 and PreQ6) and Online Resource 2 for the posttest (e.g., PosQ5 and PosQ8).
2.3 Intervention
The instructional intervention used ten proof questions, which were designed specifically for this study. The participating teacher enacted those questions in the experimental group, i.e., Class A, and introduced how to use geometric transformation to find and construct auxiliary lines that could help solve these proof questions. The intervention was carried out in four lessons with each lasting 45 min, spreading over two consecutive weeks in June 2015.
Questions used in instructional intervention
IntQ1  IntQ2  IntQ3  IntQ4  IntQ5  IntQ6  IntQ7  IntQ8  IntQ9  IntQ10  

Use of geometric transformation^{a}  O  Y  O  O  O  Y  Y  Y  Y  O 
Ro  Re  Ro  Ro  Re  Tr  Ro  Re  Ro  Re  
Adding auxiliary lines^{b}  O  N  O  O  O  Y  Y  Y  N  N 
Used in lessons  Lesson 1  Lesson 2  Lesson 3  Lesson 4 
In lesson 1, the teacher used IntQ1 and IntQ2 to introduce the basic idea of geometric transformation and guide students to compare the figure before and after the transformation. In lesson 2, she used IntQ3, 4 and 5 to help students realize that the use of the three transformations could be helpful in constructing auxiliary lines and hence solving the geometric problems. In lesson 3, the teacher linked IntQ6, 7 and 8 to the previous five questions and led students to find the differences and similarities in the use of transformation and auxiliary lines. Finally, in lesson 4, she summarized the instructions of the whole intervention and emphasized the flexibility of geometric transformation in constructing auxiliary lines. Some classroom invention episodes as videorecorded are given in Section 3.2 below.
The four lessons took place during students’ free periods in the afternoons with the consent of the school and the students. The learning environment was the same as in the control class in order to maintain the integrity of the study. The intervention instruction combined selfstudy, group discussion and teacher’s demonstration, at about 20%, 40%, and 40%, respectively, in terms of class time. The first 10 min in every lesson was selfstudy time that was set to make students familiar with the selected questions. Then, students discussed the questions in groups. Finally, the teacher explained the solutions of those questions and explained to students the idea of geometric transformation. As mentioned earlier, all the intervention lessons were videorecorded and the teacher was also interviewed after the intervention was carried out.
2.4 Limitations
It should be stressed that the intervention was carried out in a relatively short duration, which presents both feasibilities and limitations. As researchers have argued, intervention of a short duration can help better control compounding variables and make incorporating the classroombased intervention into existing curriculum structures more practical (Stylianides & Stylianides, 2013). In fact, when we communicated with the participating teacher and school, they agreed that a twoweek duration was the “best fit” in terms of curriculum and school context (e.g., without affecting students’ preparation for the forthcoming endofyear examination).
On the other hand, we must remind the readers of the limitations the short duration also inevitably means that, in particular, the intervention only covered a limited scope of geometric topics from the school curriculum, and moreover it was not intended to detect the longterm effects on students’ learning. In addition, in relation to the exploratory nature of the study, readers should also note that the sample of the study was from a particular school in China, and hence the results of the study should not be generalized to other students and school settings with different curricular, social and cultural backgrounds.
3 Findings and discussion
Quantitative methods including statistical analysis were chiefly used to analyze the numerical scores of students in the pre and posttests. Qualitative methods were used to analyze the test papers collected from the pre and posttests to obtain indepth information about how students actually solved the geometric problems and to analyze the video data and interview transcripts.
3.1 Pre and posttests
Students’ average score on questions in pretest
PreQ1  PreQ5  PreQ3  PreQ4  PreQ10  PreQ6  PreQ2  PreQ8  PreQ9  

Class A  4  4  3.92  3.92  3.68  3.64  3.12  2.96  2.80 
Class B  3.96  3.8  3.92  3.92  3.32  3.60  2.80  2.96  2.64 
Mann–Whitney test results on pretest scores
n  Mean Rank  Rank Sum  U  Z  P  

Class A  64  70.02  4481.28  1759  −1.534  0.125 
Class B  65  60.06  3903.90 
Further examining the pretest questions, as mentioned earlier, adding an auxiliary line was optional to solve PreQ3 and PreQ4. However, all the students in both classes constructed their proofs based on the properties of parallelograms, and no one considered their proof from the perspective of geometric transformations, which we think was largely due to students’ unfamiliarity with transformations and their use in solving such problems. It implies that students need to be explicitly taught in geometric transformation in order for them to use and apply such an approach in solving geometric problems.
Solving PreQ6 does not need any auxiliary line but it requires students to use the thinking of transformation to understand its reasoning process. For PreQ8, there were many potential relationships in its conclusion but most students only gave one possible answer, which resulted in a low score. Reflection was also an optional method to solve PreQ8, which was not found in students’ answer sheets. The result again revealed that students rarely used transformation in geometric proof, which suggests that the instructional intervention provided in our study was meaningful.
Students’ average score on questions in posttest
PosQ1  PosQ2  PosQ5  PosQ4  PosQ8  PosQ10  PosQ3  PosQ9  PosQ6  PosQ7  

Class A  1  1  1  0.99  0.97  0.97  0.90  0.87  0.79  0.75 
Class B  0.98  0.98  0.96  0.96  0.94  0.93  0.94  0.93  0.76  0.89 
Mann–Whitney test results on posttest scores
n  Mean Rank  Rank Sum  U  Z  P  

Class A  58  58.45  3390.10  2033  0.920  0.357 
Class B  64  64.27  4113.28 
As indicated earlier, we were particularly interested to know if the intervention would help students develop their ability in solving challenging or highcognitive level geometric problems. For this purpose, we further classified all the questions in the posttest into two groups based on the cognitive levels required for solving the questions: one includes all the questions of ordinary cognitive level or general questions, and the other includes all the highlevel cognitive questions or challenging questions. As a result, three questions in the posttest, i.e., PosQ6, 8 and 10, were classified as highlevel cognitive questions. To solve these questions, students were required to use a mixture of different kinds of geometric knowledge, such as congruent triangles, the basic properties of quadrilaterals, the application of special angles and deduction of quantitative relationships. In addition, students were expected to identify relevant conditions in these questions and to link them to a transformation approach to effectively find the solutions.

PosQ6: In rhombus ABCD, ∠ADC = 120^{°} and E is a point on the diagonal of AC; connecting point D with point E, there is ∠DEC = 50^{°}. Rotate BC 50 degrees around point B anticlockwise and extend it to intersect the extension of ED at point G.
 (1)
Complete the figure (Fig. 2) according to the instruction above.
 (2)
Prove: EG = BC.
 (3)
Establish an equation to express the quantitative relationship between segments AE, EG and BG: _________________________________________.

PosQ8: The figure (see Fig. 4) shows quadrilateral ABCD with AB = AD, ∠BAD = 120^{°}, and ∠B = ∠ ADC = 90^{°}. Point E is on line BC and point F is on line CD. ∠EAF = 60^{°}. Explain the relationship between segments BE, EF and FD.

PosQ10: The figure (see Fig. 7) shows a//b//c. The distance between lines a and b is 3 while the distance between lines b and c is 1. The distance from point A to line a is 2 and the distance from point B to line c is 3;\( AB=2\sqrt{30} \). Please find a point M on line a and a point N on line c, so that MN ⊥ a and the value of AM + MN + NB is the minimum. In this case, the value of AM + NB is _________.
This question was challenging as it did not offer any clue for students to use transformation, and they would need to reinterpret some information in terms of transformation; for example, “(the length) is the minimum” implies “the use of reflection”. Once students constructed a parallelogram and applied the transformation of reflection to find point M and point N, they could easily solve the problem. Solving this question required students to use their knowledge about the properties of parallelograms, congruent triangles, and the facts that “the sum of any two sides of a triangle is greater than the third one” and “the shortest distance between two points is the length of the segment joining the two points”. Accordingly, the most difficult part was to realize why and how auxiliary lines should be constructed to help solve the question, which required a high level of cognition or understanding about parallel lines and their properties. Although the average scores of the two classes were close in the posttest, an important difference was detected in their ability to add auxiliary lines: 69% of the students in the experimental class added the auxiliary lines correctly, while the corresponding percentage in the control class was only 42%.
A further look at the data collected revealed that, although 13.8% of the students in the experimental class added auxiliary lines incorrectly, in the answers given most of them were trying to construct parallelograms. In contrast, in the control group, 40.1% of the students added auxiliary lines incorrectly and, moreover, most of them just added vertical lines and then connected the given points, and there was no clear sign of using translation or constructing a parallelogram. In conclusion, the students in the experimental class showed considerably better awareness of applying geometric transformation to add auxiliary lines, suggesting that this kind of awareness and ability should be better explicitly taught and developed through teachers’ teaching in classroom.
3.2 Interview with the teacher
Researcher: How did you introduce to students the idea of geometric transformation in your lesson?
Teacher: The basic principle was to let students experience the process of transformation in solving geometric questions instead of demonstrating the transformation approach myself. However, students should know the basic characteristics of transformation before using them. So I first clarified and introduced the basic ideas and properties of the three transformations (translation, rotation and reflection). Using examples in instructions was necessary. The comparison of similar examples was necessary as well, which was also a kind of reinforcement in students’ learning.
Researcher: How did you introduce the use of geometric transformation in constructing auxiliary lines in solving proof problems?
Teacher: Firstly, I led students to explore the impact of translation, rotation and reflection on basic graphs. For example, let students tell the changes after a translation of a triangle: the corresponding sides and angles are unchanged; only the position of the triangle changes. Secondly, I led students to think about the advantages of a transformation in solving proof problems. For instance, students should know that using translation can sometimes make questions easier. Thirdly, for adding auxiliary lines, I guided the students to analyze the existing conditions and observe the characteristics of a figure to reason which transformation can be helpful to the question then to add the corresponding auxiliary lines. In this way, students can gain experiences of adding auxiliary lines from the perspective of geometric transformation.
Researcher: What did you do when students encountered problems in solving the questions?
Teacher: I let them talk to their peers first and then discuss in groups to get different ideas from each other. Finally I summarized different solutions of their discussion and emphasized the solution from a transformation perspective, in which way students can understand the advantages of geometric transformation in solving geometric questions.
Researcher: After the intervention, did you assign exercises for consolidating?
Teacher: Yes. I assigned two similar questions after every lesson.
 IntQ6 In square ABCD, M, N, P and Q are the points on sides AB, BC, CD and DA, respectively. MP = NQ. Prove MP⊥NQ (Fig. 8).
Student A: (Connecting MN and QP), are △MON and △POQ congruent? Can △MON be flipped over to get △POQ? No, it is not possible. So reflection does not help.
Student B: It appears that one of the quadrilaterals AMPD and CPMB can be obtained by rotation of the other. But no, they are not congruent. So it does not help.
Teacher: Apart from reflection and rotation, which cannot really help here, we can consider the third transformation or translation using the given condition MP=NQ to construct rightangled triangles.
Student C: [Now I know] we can construct auxiliary lines through points M and Q, so they are parallel to AD and BC, and then prove that two triangles obtained are congruent.
Teacher: So we used the translation of transformation, translating AD to get ME and translating AB to get NF (Fig. 9). In general, we can use translation to construct parallel lines, and obtain equal angles.
According to the teacher, in solving this problem, a critical step was the use of translation to construct two congruent triangles and find the relationships between angles formed with parallel lines. Hence, a good understanding of geometric transformation is very helpful in solving such problems, and a teacher can play an important role in guiding students develop such an understanding, as indicated earlier in the teacher’s interview.
 IntQ7: In square ABCD shown below, E is a point on the side of CD and F is a point on the side of AD. FB is the bisector of ∠ABE. Prove: BE = AF + CE (Fig. 10).
Teacher: The segment AF and CE are not on the same line. So what should we do?
Student D: Link E and F.
Teacher: But they are still on two different lines. Here, geometric transformations can be helpful to construct auxiliary lines. Will a reflection make the two segments onto the same line?
Student D: No, here [we] should use a rotation.
Teacher: Yes, when you rotate, be careful about the angle and direction of the rotation.
Student E: Rotate △BAF 90 degrees anticlockwise around point B.
Teacher: So is △BAF still inside the square?
Student E: No. (after drawing the image of △BAF, △BCG, see Fig. 11) A goes to C, F goes to G, so △BAF becomes △BCG.
Teacher: Brilliant! The rotation moves all the related sides into one triangle, which is the most important step in solving this question. Adding auxiliary lines (BG and CG in this case) is just an expression of the geometric transformation [So now we just need to prove BE = EG or ∠EBG=∠BGE]
Researcher: What did students feel about the use of geometric transformation? Are there any changes in them?
Teacher: Yes. … Firstly, after the intervention, [my recent experience is] 80% of the students in the experimental class first chose the approach of geometric transformation when facing proof problems, while there is only about 30% in the control class. Secondly, about 65% of the students in the experimental class can use the terminology “translation”, “reflection” and “rotation” accurately in oral communication, while that was only 8% in the control class. Thirdly, about 70% of the students in the experimental class reported to me that the use of geometric transformation saved the time of constructing auxiliary lines. Fourthly, about 60% of the students in the experimental class felt that the use of geometric transformation is helpful, making solving proof problems easier and faster.
From the teacher’s interview, it appears that the intervention made a difference in students’ approach to solving geometric problems and had a positive influence on students’ use of geometric transformation. It appears further that most students also had a positive view about the use of geometric transformation in solving proof problems.
4 Summary and conclusion
This paper reports an interventionbased study to explore whether a transformation approach in teaching of geometry can improve students’ ability in constructing auxiliary lines and hence enhance their learning of solving geometric problems with focus on proof problems and, in particular, highlevel cognitive problems. A classroombased intervention was carried out with a quasiexperimental design in two Chinese secondary classrooms in a twoweek duration.
The results of the study based on the data collected from the pre and posttests as well as the teacher interview showed a neutral to positive impact of the intervention. On the one hand, there appears no statistically significant difference overall in the impact of using geometry transformation on students’ ability in solving general geometric problems, which could be due to the fact that the intervention had a short duration and covered a limited set of geometric topics. On the other hand, encouraging evidence was found to support the use of geometric transformation in solving geometric problems by adding auxiliary lines and hence in enhancing students’ learning of geometry. This is particularly evident in students’ solutions of challenging geometric questions and in the teacher’s observation as reported from the interview data. As described earlier, in solving highlevel cognitive geometric questions, the use of transformation helped students realize more clearly why and how auxiliary lines should be added in order to effectively solve the problem (Yang & Pan, 1996; Wang, 2010), and hence enhance their ability in solving these challenging geometric problems.
The study is a first step in our effort to address the difficulty in teaching and learning of geometric proof with a focus on classroom pedagogy (see also Fan, Mailizar, Alafaleq, & Wang, 2016); it was intended to be an exploratory study and hence has some limitations as explained earlier. In future, we think research in two directions is worth undertaking. The first is research of a more confirmatory nature, especially a study with a larger sample size and different groups of students (e.g., in different countries or in different school settings), a wider or different coverage of geometric contents and a longer duration of intervention, though such a study would be also more challenging to implement. The second is research with a focus on high cognition level or challenging geometric proof problems, as one result of this study is that a transformation approach appears to support students’ ability in adding auxiliary lines to solve highlevel cognitive geometric problems including proof problems and hence enhance their learning of geometry.
Notes
Acknowledgements
The authors wish to thank Mr. Xianfeng Shi and Ms. Hong Li for their assistance in conducting this study and the anonymous reviewers for their helpful comments. The study was supported in part by a research grant from Beijing Advanced Innovation Centre for Future Education (Project No. BJAICFE2016SR008).
Supplementary material
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