To build relational understanding, we explored 4 kinds of tasks in the class, including searching for patterns, analyzing a situation, generalizing relationships, and experimenting and explaining. We conducted a learning activity about multiplication and division of fractions. We tried to calculate 2/3 × 1/5 and 2/3 ÷ 1/5 with tables. For multiplication of the fractions, we first drew a table with 3 columns and 5 rows which represented dividends of 3 and 5 respectively. We then shaded two of the 3 columns and one of the 5 rows. We then can see 2 out of 15 blocks are colored with overlap and so the answer is 2/15. The table with shaded blocks are shown in Fig. 1.

The activity visualized the concepts of faction multiplication and division. It also encouraged students thinking relationally. After analyzing the situation with learners, teacher should guide them to generalize the relationships and explain them. In this activity, we concluded the relationship of fraction multiplications was a/b × c/d = ad/bc. The activity inspired me to design teaching tasks with relational understanding.

 

To make an effective GeoGebra task, we should support the affordances of DGEs mentioned above. For example, it should allow students to drag some points or lines to show the variance and invariance and visualize the motion graphs. For editing skills, we should use colors well to highlight lines, angles, or regions. We also can use buttons and checkboxes to hide and show some hints instead of showing answers directly. A good GeoGebra task should encourage students to think and solve the problem on their own with teacher’s guidance. 

 

Moreover, shearing is a good way to visualize the concepts of area. It is a geometrical continuous case of dissecting for conserving area. During the lesson, we discussed an interesting task with shearing, Eda & Asuza’s Problem (Lesson 2 – GeoGebra). At the beginning of class, teacher asked students to straighten the border without changing the area of Eda and Azusa’s land. It was an attractive introduction which increases students’ learning motivation for the topic. Without revealing the solution directly, teacher guided students to solve the problem with the following tasks about dissecting and shearing a triangle step by step. The tasks allowed them to drag a vertex of the triangle along parallel lines so that they can try to figure out that the area of the triangle remains unchanged if its height and base keep the same. The lesson design was excellent and student-centered. I will take it as a reference in my future task design.

 

Mr. Chan recorded what he teaches as videos and uploaded them to YouTube. In his class, students need to watch the tutoring videos before the lesson so that they can have further discussions during school time. After learning basic knowledge through videos at home, students can digest the new chapter and raise effective questions in the lesson. Then, teacher can create a more interactive learning environment such as holding online responsive quizzes, or hands-on activities. Also, flipping videos allow students to learn at their own pace so the classroom differentiation problem can be greatly solved. Students with lower abilities can pause the video and watch it repeatedly until they understand while students with higher abilities can try on some challenging questions or help tutor other classmates during lesson time. Moreover, lecture recording videos help learner review for exams. If students forgot what they learn before, they can replay the lecture videos to have revision anytime. It also reduces teachers’ workload by repeating the same thing. 

 

However, recording flipping videos is not easy. Mr. Chan pointed out some points we need to notice when shooting. First of all, we should be careful about the length of the video. Based on the study done by Ball State University (2011), the majority of students preferred flipped classroom videos with a length of fewer than 7 minutes. Videos with long duration may lower the willingness to watch so we should not include too many examples in the videos. In addition, our presentation should be clear. To avoid background noise, he advised us to use an external microphone. He also taught us how to make the effect of flashing the highlighted part. It did not require as many video editing skills as I thought. He built my confidence to create flipping videos in the future.

 

Last summer, I was a STEAM development intern at a tech startup in Hong Kong Science Park. I designed teaching materials and implemented STEAM education events. Since most of the staff members had degrees in engineering or computer science, the company did not have sufficient professional tutors. Their lesson design was teacher-centered and lack of interaction. Technical knowledge is indeed worth students’ study but it should be taught with educational skills. Thus, I suggest schoolteachers should attend STEAM training and design their own STEM projects so students can have a better STEAM learning experience. 

 

Some projects introduced in class were really interesting and inspiring. I appreciate the STEAM design of the “3D Keychain” Project the most. We explored the functions of 3D CAD on Tinkercad. It visualizes figures in 3 dimensions efficiently and helps users build their own design 3D models. The project asked students to create keychains and consider the possibilities and costs of 3D printing. They are required to find out the volume of their products. The task design integrated the disciplines of art, mathematics, engineering, and technology. This kind of inquiry-based learning also stimulated students’ creativity and problem-solving skills. STEAM learning helps students apply mathematical knowledge to daily-life problems so it makes mathematics meaningful and encourages their learning motivation.

 

There were in total 3 GeoGebra tasks. All of them are dynamic and related to each other. In the first task, student A and student B on the graph act as two distinct points. We were asked to find the best location of the school in order to keep the distances between each point and the school the same. We were allowed to move the point of the school and know the distances. To focus on the track of the school we moved to, Boris brought us a message: all the fair locations fell on the perpendicular bisector of the line joining two students. Not only provide a single example, but Boris also guided us to prove the perpendicular bisector in a general case. It helped students develop their skills in geometrical proof. Then, the second task provided three students on the graph acting as three distinct points. Dealing with the problem, in the same way, Boris led learners to the definition of circumcenter. The activity was applied with constructivism so that students can build the knowledge of circumcenter through their experience on GeoGebra tasks, instead of understanding passively. 

 

At the end of the lesson, Boris took an obtuse triangle as a counterexample in the third task. He drew a conclusion that the circumcenter is not always the best location of the school among three students. When the three students’ locations form an obtuse triangle, the distances are fair but too far for all of them. The task design revealed the gap between the ideal of mathematics and real-world problem. It was really meaningful and inspiring.