#### Effectively use technologies to explore geometry and develop understanding of geometric relationships.

**SLO 6 Summary**

Geometry courses need to utilize technology to help develop geometric reasoning and deepen students’ understanding of geometric concepts and relationships. Two types of technologies that are beneficial for teaching geometry are dynamic geometry environments (DGEs) and digital proof tools (DPTs). DGEs support student understanding by allowing students to explore properties of geometric figures dynamically, which provides advantages over using paper and pencil. These explorations help students generate their own conjectures, test their conjectures, and provide justification and understanding for theorems. DGEs have been an important tool for teaching geometry since the 1990s. DPTs are an emerging technology that provide students with interactive figures to manipulate and opportunities to practice writing proofs with immediate feedback.

**SLO 6 Narrative**

Technology can help develop students’ understanding of geometry across grade levels and through a variety of aspects. Therefore, the GeT course should utilize technology so students can explore, conjecture, and develop an understanding of geometric relationships. Dynamic geometry software has been shown to help students develop geometric reasoning (Hadas, Hershkowitz, & Schwarz, 2000; Hoyles & Jones, 1998; Jones, 2000).

In addition to using technology to learn geometry, GeT students who are preservice teachers need to understand best practices in using technology in a geometry classroom. Based on a systematic analysis of a set of frameworks related to learning or teaching mathematics with technology, McCulloch et al. (2021) identified four categories of frameworks (Figure 1) that can “inform the framing of how teachers learn to use technology for instruction” (p. 331). Preservice teachers need varied experiences with all four categories, but it is essential that GeT courses have students utilize technology for the learning of mathematics (i.e., first branch of Figure 1) which is the foundation of all the categories. Depending on the audience in the GeT course, GeT instructors may include the other three categories somewhere in their preparation program to prepare PSTs to be able to utilize technology for the teaching of mathematics.

All GeT students need experience as learners with technology. Preservice teachers also need to experience reflection regarding the design of tasks that utilize technology. GeT courses are an ideal place for preservice teachers to experience authentic and meaningful use of technology. The main category of technology that has been widely used by both the GeT and secondary instructors is dynamic geometry environments (DGEs). Emerging technology such as digital proof tools (DPTs) is also introduced because of its potential to be beneficial for teaching geometry as its prominence continues to evolve.

**Dynamic Geometry Environments**

Dynamic geometry environments (DGEs) refer to geometry software that supports the “continuous real-time transformation often called ‘dragging’” (Goldenberg & Cuoco, 1998, p. 351). The dragging feature allows the user to change certain elements (e.g., a point) in a constructed geometric figure and observe the change of the corresponding geometric relationships in the figure. The constructed figures are referred to as “draggable” or “moving” figures, which can provide the user with opportunities to experience “motion dependency” and further explore “logical consequence between properties within the geometrical context” (Mariotti, 2014, p. 159). Geometer’s Sketchpad, GeoGebra, and Desmos are the most commonly used DGEs (Table 1).

The development of DGEs in the 1990s opened up significant new possibilities for teaching and learning geometry. They offer ways to experiment with lots of possible figures at once rather than just one static figure, to make exact measurements easily, and to incorporate geometric exploration and making conjectures. For example, DGEs can be used to explore properties of various types of quadrilaterals so students understand how quadrilaterals are related to each other. DGEs also provide an easy way to demonstrate transformational geometry, with built-in tools for translating, rotating, reflecting, and dilating objects. In order for students to solidify their understanding of geometry, DGEs should be included in a GeT course.

**Emerging Technology – Digital Proof Tools**

Proof and reasoning lie at the heart of any geometry course, and GeT courses are no exception. Many students struggle with proof and recently developed digital tools have been created to help students with the proof process. As an emerging category of technology, DPTs allow the instructors to create and edit geometry proof problems and provide students with feedback or hints to facilitate their proof-writing process. Two-column proofs are usually supported by DPTs because of their neat organization and clear logic flow. Drawing features can help students identify the given conditions or add auxiliary lines in a diagram. CanFigureIt Geometry®, FullProof, and Proof Companion are some of these DPTs (Table 2).

An advantage of DPTs is that they provide opportunities for students to practice two-column proofs and get instant feedback. It is becoming a common practice to use online homework in mathematics courses at the secondary level, and students like it because of the instant feedback. Geometry proof problems are not included in typical online homework systems so DPTs provide a solution to this issue.

**Table 1: Some Commonly Used Dynamic Geometry Environments**

Software | Description of Tool |

Desmos | Basic geometry construction and transformation tools. |

GeoGebra | More powerful geometry construction and transformation tools (compared to Desmos). Visualizing 3D objects and creating 2D nets that correspond to 3D objects. It includes activities with non-Euclidean geometries. |

Geometer’s Sketchpad | The first well-known DGE that was created by Key Curriculum Press. It has comparable capabilities as GeoGebra but requires purchasing a license to access all features. The last commercial version of Sketchpad has been discontinued, but a new free version is in development. |

Spherical Easel | This software is currently being updated and the link will be added later. Allows the users to construct shapes, rotate, and make measurements on a sphere. |

WebSketchpad | Similar to Geometer’s Sketchpad, but allows teachers to select which tools from the Tool Library can be used for a specific activity. |

**Table 2: Some Emerging Digital Proof Tools **

Software | Description of Tool |

CanFigureIt Geometry® | Teachers can select and assign geometry proof tasks through a virtual class platform. Gives students continuous feedback and guidance during the proving process. |

Full Proof | Personalized proof-writing tool using interactive diagrams and an equation editor. Provides detailed feedback, hints, and fine-grained evaluation. |

Proof Companion | Teachers can create or edit a geometry proof to share with students and track their progress. Students simply drag and drop the statements and reasons to their proper position to have their work instantly graded. |

As technology continues to improve, there will likely be new tools to help with teaching and understanding geometry so this list is not exhaustive. Thus, GeT courses will need to evolve so that we can continue to prepare effective secondary geometry teachers.

**References**^{1}

Goldenberg, E. P., & Cuoco, A. A. (1998). What is dynamic geometry? In R. Lehrer & D. Chazan (Eds.), *Designing learning environments for developing understanding of geometry and space* (pp. 351–367). Erlbaum.

Hadas, N., Hershkowitz, R., & Schwarz, B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. *Educational Studies in Mathematics, 44*(1), 127-150.

Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry. In D. Mammana & V. Villani (Eds.), *Perspectives on the teaching of geometry for the 21st century* (pp. 121-128). Kluwer.

Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using dynamic geometry software and their evolving mathematical explanations. *Educational Studies in Mathematics, 44*(1), 55-85.

Mariotti, M. A. (2014). Transforming images in a DGS: The semiotic potential of the dragging tool for introducing the notion of conditional statement.

In S. Rezat, M. Hattermann, & A. Peter-Koop (Eds.), *Transformation – A fundamental idea of mathematics education* (pp. 155–172). Springer.

McCulloch, A., Leatham, K., Bailey, N., Cayton, C., Fye, K., & Lovett, J. (2021). Theoretically framing the pedagogy of learning to teach mathematics with technology. *Contemporary Issues in Technology and Teacher Education, 21*(2), 325-359.

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^{1}An extensive bibliography about DGEs can be found at https://www.dynamicgeometry.com/General_Resources/Bibliography.html.