# The Role of Dynamic Geometry Software in Teacher–Student Interactions: Stories from Three Chinese Mathematics Teachers

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Dynamic Geometry Software in Mathematics Education

#### 2.2. Framework

#### 2.2.1. Roles of DGS in Mathematics Teaching and Learning: Amplifier and Reorganizer

- DGS is used to facilitate the material aspects of a task without changing it conceptually. For example, a triangle is constructed using the midpoints of its sides and then its medians.
- Alternatively, DGS is used as a visual amplifier in the task of identifying properties. It is easier to observe that three midlines of one triangle always intersect at one point via dragging with DGS than by using a pencil and paper.

- 3.
- DGS can modify the solving strategies of a task because of its powerful functions, which may make the tasks more difficult, similar to creating a parallelogram without using parallel line tools in DGS.
- 4.
- The task itself takes its meaning or its ‘raison d’être’ from Cabri.These two roles show that DGS can be used to generate new tasks in mathematics.

#### 2.2.2. Initial Categories of Focusing Questions

## 3. Research Questions

- What technology-specific actions do mathematics teachers focus on when using DGS in mathematics lessons during teacher–student interactions?
- What roles does DGS play as an amplifier and reorganizer during teacher–student interactions?

## 4. Methodology

#### 4.1. Participants

#### 4.2. Data Collection and Analysis

## 5. Results

#### 5.1. Stories from Mr. ZH’s Class

- What is the maximum length of side c? What is the relationship between sides a, b, and c? What shape is it now?
- What is the minimum length of side c? What is the relationship between sides a, b, and c? What shape is it now?
- Based on the conclusions of questions 1 and 2, what you can find?

#### 5.2. Stories from Madame J’s Class

#### 5.3. Stories from Mrs. Y’s Class

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. An Example of Analyzing Interview Data

- Step 1: We divided this transcription into the following sentences:

- Step 2: For each sentence, we determined the keywords and their meanings. From the transcript, the first and second sentence had the same meanings. Mrs. J thought that DGS could help her lessons a lot. The third sentence implies that Mrs. J was familiar with DGS. The fourth one shows that Mrs. J was confident when using DGS in the lessons. In addition, the last sentence means that for Mrs. J, DGS was more like a supporter.
- Step 3: Afterwards, we discussed each sentence to find out which of them could help us explain how and why the teachers used DGS and interacted with students. For example, in this transcription, because Mrs. J saw DGS as a supporter, she did not always let her students operate DGS, and her questions or feedback always focused on mathematics.

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**Figure 2.**Mr. ZH operated DGS during teacher–student interactions. In this figure, the Chinese terms mean: question 1, please find what is the smallest value of a + b; question 2, please find what is the largest value of a + b; and question 3, what conclusion can you make?

Focus | Description | Example Questions | Coding Diagram |
---|---|---|---|

Focus on technology. | Teacher’s question or feedback is only about the technology. | Try to drag and click the circle button and draw a circle on your screen. | |

Focus on technology to notice mathematics. | Teacher’s question or feedback is related to a technological action, which is related to a mathematical idea. | If you select the center of the circle you have drawn and move it, what do you see? | |

Focus on mathematics with the use of technology. | Teacher’s question or feedback is about mathematics, and they assume that students need to use technology to find answers. | Can you make a circle that circumscribes the triangle given on your screen? | |

Focus on mathematics. | Teacher’s question or feedback are only about mathematics. Additionally, students can answer them without the help of technology. | Try to conclude ways to prove the drawn triangles. |

Task | |
---|---|

Mr. ZH designed one diagram of a triangle whose side lengths AC and BC were determined (AC = 4, BC = 6). Point A could be moved, while points B and C were stable. From the diagram, the trace of point A is similar to a circle. The main question: please find the relationship between the three sides of the triangle. (Here the lowercase letters a, b, and c mean side BC, side AC, and side AB) |

Task | |
---|---|

In the picture, there is a right triangle ABC, ∠ACB = 90°, point D is the midpoint of AB, point E is on segment BC, AE = BE, point M is the midpoint of AE, point G is on segment CM, and point N is on segment BC, which makes ∠GDN = ∠AEB Question 1: as in picture 1, if point G coincides with point M, this proves that quadrilateral DMEN is a diamond. Question 2: as in picture 2, when point G does not coincide with point M and point C, prove GD = DN | |

Figure for question 1 | |

Figure for question 2 |

Task | |
---|---|

Task 1: DGS is used to construct a new triangle, which is congruent with the given triangle ABC, in which AB = 7, AC = 4, BC = 6. In this triangle, point A and B can be dragged, while point C is stable. When point A is dragged, the whole triangle moves without changing its type and size. When point B is dragged, point A does not move. The trajectory of point B is a circle whose center is point A and radius is the length of AB. |

Questions and Feedback | Focus |
---|---|

T: “OK, the first one, what is the maximum of side c?” | Only focused on mathematics. |

St 1: “The maximum is 10” T: “OK, maximum is 10. I have said in the video, so the maximum is 10” | The teacher’s feedback just repeated what the student answered. The answer only focused on mathematics. |

T: “Now what is the relation?” | Only focused on mathematics. |

St 1:” They are all on the same line.” T: “Good, he said on the same line.” | The teacher’s feedback just repeated what the student answered. The answer only focused on mathematics |

T: “Let us see, this is position. OK, now c is 10 and they are all on the same line.” | The teacher moved the point on the screen at that time. The feedback focused on technology to notice mathematics. |

T: “But what is the mathematical relation between the three sides?” | Only focused on mathematics. |

St 1: “b + c = a” T: “b + c = a, good, sit down. b + c = a, we can also say a − b = c.” | The teacher’s feedback just repeated what the student answered. The answer only focused on mathematics. |

Questions and Feedback | Focus |
---|---|

T: “You need to justify what type is the quadrilateral.” | Focused on mathematics. |

St1: “It is diamond.” T: “Diamond. How can you know it?” | The teacher’s feedback repeated what this student answered. Additionally, the question let this student justify why they believed that quadrilateral DMEN was a diamond. Therefore, the feedback point and question focused on mathematics. |

St1: “First point G and M are coincided.” T: “Yes” | The teacher gave positive feedback. They did not focus on mathematics or technology. |

St1: “Then, DM, point D is the midpoint of AB, point M is the midpoint of AE, so DM the median line of this triangle.” T: “DM is the median line of triangle ABE. You know it is median line, then” | The teacher repeated the conclusion this student made. It still focused on mathematics. |

St1:” Then DM is parallel to BE T: DM is parallel to BE” | The teacher repeated the conclusion this student made. It still focused on mathematics. |

St1: “So angle MDN is add angle END is 180” T: “It is this angle, angle MDN and angle END, angle MDN plus angle END is 180” | The teacher wrote down some marks through DGS on the screen. The feedback focused on technology to notice mathematics |

St1: “Then, it said, angle GEN is equal to angle MDN and AEB” T: “Angle MDN is equal to this angle, angle GEN. It means this angle is equal to angle GEN.” | The teacher wrote down some marks through DGS on the screen. The feedback focused on technology to teach mathematics |

St1: “So angle END plus angle GEN is 180, then DN is parallel to ME” T: “Why you justify DN is parallel to ME? St1: So NDME is parallelogram” | This question only focused on mathematics. |

Questions and Feedback | Focus |
---|---|

St2: “In the conditions, AE is equal to BE T: AE is equal to BE, what it means?” | The teachers’ question focused on mathematics. |

St2: “So we know MD is the median line. Then, MD is 1/2 BE” T: “Ok, the median line can help us know not only this one. However, also DM is equal to 1/2 BE.” | The teacher explained why this student arrived at their conclusion. The feedback focused on mathematics. |

St2: “Because point M and N, M is the midpoint of AE, then, so, it is also useful, it means AM is equal to ME and 1/2AE T: Equal to 1/2AE” | The feedback focused on mathematics |

St2: “Then, AE is equal to BE, so MD is equal to ME” T: “MD is equal to “ St2: “ME” T: “So now, when AE is equal to BE, this condition is also useful, AE is equal to BE, another one is ME is equal to, we need another condition, it is ME is equal to 1/2 AE. Ok, from these two conditions, we can find ME is equal to DM, then it can be a diamond” | After this student finished their answer, the teacher made a conclusion that focused on mathematics. |

Questions and Feedback | Focus |
---|---|

T:”Let’s draw another triangle whose sides are also 4, 7, and 6. Then, we can overlap the two triangles and see whether the two triangles coincided, if they coincided, then the two triangles are, congruent. How to draw it?” | The teacher wanted the students to draw a triangle with the help of DGS. Her question and statement focused on mathematics with the use of technology. |

St 1: “First draw one base side” T: “First draw one base side. [teacher operated the computer, choose the tool “segment”] We can find the tool “segments with given length” in the tool “lines”. Then, here, we draw a segment, 7, right? Here we tap 7. [teacher clicked the tool, and tapped “7”] Now you can see a segment and its length is 7. Ok, then, how to draw this point on side AC, that is side DF?” | The teacher used DGS to show the students the whole drawing process. Her question focused on mathematics with use of technology. |

St 2: “Use compass” T: “Use compass, it means the radius of this circle is?” | The teacher’s feedback just repeated what the student answered. It focused on mathematics. |

All St: “4” T: “4, which one is its center?” | Focused on mathematics. |

St 3: “D” T: “D, ok, now we can find circle. [teacher operated the computer, find the tool “circle”] Then, you can click it, it showed “one center and one point”. We need to find the circle and radius, it means the radius is determined, click it. [teacher clicked the tool “circle with center and radius”, then looked at the students] Ok, now point D is the center, click it. What is the radius?” | The teacher used DGS to show the students the whole drawing process. Her statements focused on mathematics with use of technology, while one question focused on mathematics. |

All St: “4” T: “Then, here tap “4”, then you can see a circle. [teacher tapped “4”, the screen showed one circle whose center is D and radius is 4] Ok, then is this side, its length is 6. Ok, its center is point E, what is the radius? 6. [teacher clicked “circle with center and radius”, tapped 6] Ok, now you can find the intersect point of the two circle?” | The teacher used DGS to show the students the whole drawing process. Her statements focused on mathematics with use of technology, while the question focused on mathematics. |

St 4: “Two” T: “Yes, two, it’s the same with the two points up or down” | Focused on mathematics. |

St 5: “Yes, it is same” T: “Ok, same, here I just draw one, this point, which is the intersect point?” | Focused on mathematics. |

St 6:” F” T: “F, it is F here, then, we can draw this triangle. [teacher clicked the intersect point] Link these three points, DEF. [teacher operated the mouse, linked point D, E, F] This triangle, ok, these lines are supportive, we can delete them.” | The teacher used DGS to show the students the whole drawing process. Her statements focused on mathematics with the use of technology. |

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## Share and Cite

**MDPI and ACS Style**

Zhu, F.; Xu, B.
The Role of Dynamic Geometry Software in Teacher–Student Interactions: Stories from Three Chinese Mathematics Teachers. *Sustainability* **2023**, *15*, 7660.
https://doi.org/10.3390/su15097660

**AMA Style**

Zhu F, Xu B.
The Role of Dynamic Geometry Software in Teacher–Student Interactions: Stories from Three Chinese Mathematics Teachers. *Sustainability*. 2023; 15(9):7660.
https://doi.org/10.3390/su15097660

**Chicago/Turabian Style**

Zhu, Fangchun, and Binyan Xu.
2023. "The Role of Dynamic Geometry Software in Teacher–Student Interactions: Stories from Three Chinese Mathematics Teachers" *Sustainability* 15, no. 9: 7660.
https://doi.org/10.3390/su15097660