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Figure 1: Faction multiplication with table.

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I have learnt the concepts of instrumental and relational understanding on mathematics. Instrumental understanding needs fixed rules and plans for different types of tasks. It is easier for students to understand and get the right answers quickly because they just need to know how to apply the rules to the corresponding questions. However, they may not have a deep understanding of the topic through learning instrumentally but depend on outside guidance for learning each new task. Thus, I prefer relational understanding to instrumental one. Not only mathematical rules, but relational mathematics also requires students to know why the theorems work. Although both of these understanding bring learners to get the same correct answer but relational understanding is way more extensive and impressive. Students with relational understanding are more adaptable to new tasks or challenging questions. This approach can also act as a goal and an agent of them on growth.

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.

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I got inspired by the GeoGebra enrich tasks that foster geometrical thinking. To create Dynamic Geometry Environments (DGEs), teacher should encourage dynamic and narrative modes of thinking, support visualization and learning of motion graphs, mediate variance and invariance, mediates functional dependency, or foster multi-representational thinking. Past BMED courses taught me how to design mathematic meaningful tasks with physical tools, while today's lesson taught me how to deal with online tools such as GeoGebra and Desmo Classroom.

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.

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I have learnt a lot about flipped classrooms from Mr. Chan today. Since I have not experienced flipping learning, I thought it was an ideal but not effective way to teach mathematics, compared to a traditional classroom. I changed my mind after listening to Mr. Chan’s sharing.

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.

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I was interested in the discussion of Hong Kong STEAM education. There is a growing need for STEAM education in the city today. STEAM integration aims to apply skills and knowledge from two or more disciplines to real-world problems to enrich students’ problem-solving skills, creativity, and abilities of collaboration and communication. However, the majority of STEAM activities currently take place outside of the classroom or are even outsourced to private companies. I strongly agree that the Hong Kong government should further develop a completed STEAM education with professional teachers but not private businesses.

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.

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Chau King Hei Boris's micro-teaching (Circumcenter)

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Chau King Hei Boris's micro-teaching (Circumcenter)

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Boris’s lesson about circumcenter impressed me very much. He did a great job on designing Geometry rich tasks. At the beginning of the lesson, he introduced a question “where should the school be located so that the distances between each student and the school keep the same?”. It was a real-life issue that students care about. The interesting opening task fosters students thinking and learning motivation. Rather than reveal the solution directly, Boris guided us with the concept of circumcenter to the correct answer step by step.

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.

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Yuen Ngai Ha Christy's micro-teaching (Coordinate Geometry: Distance between Two Points)

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Yuen Ngai Ha Christy's micro-teaching (Coordinate Geometry: Distance between Two Points)

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The lesson activity designed by Christy acted as scaffolding. Christy introduced a real-life example: building a road with the shortest distance between the entrance and the exit of IKEA. The situational question encouraged students to think and engage in the activity. The first task of IKEA provided a triangle with numerical data so students can apply Pythagoras’s Theorem to solve easily. It is suggested to add a checkbox to hide the triangle so it can be more challenging to arouse students' thinking.

The second GeoGebra task was dynamic and effective. All the points can be dragged freely. Students were allowed to form different right-angled triangles to calculate the distance of AB. It was interactive to conduct two sets of worksheets to solve the question in two different ways. Christy separated two groups of students to try the two methods. It was a good idea to show the results keep the same even applying different methods. After trying several numerical examples, Christy guided students to construct the formula with algebraic symbols step by step. This approach of constructivism allowed learners to acquire new knowledge actively through their discovery of GeoGebra. It was impressive to students. Yet, a minor improvement I would like to suggest is that the second GeoGebra task should include IKEA scenarios in order to make the lesson coherent. Furthermore, since absolute value is not on the syllabus and not covered in textbooks, it should be handled carefully. It is recommended to spend some time introducing the definition of absolute value once we apply it to junior mathematics.

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Figure 1: GeoGebra task of standard score

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Lai Chak Hei Kobe's micro-teaching (Standard Score)

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Lai Chak Hei Kobe's micro-teaching (Standard Score)

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I appreciate the teaching performance of Kobe. The teaching content was fluently presented and rich. I believe he practiced a lot that I should learn from him. At the beginning of the lesson, he introduced a real-world case: Nobi Nobita, a character of Doraemon, got the results of three subjects. Students are asked to determine which subject he performed the best. It was a controversial problem because Nobi got a higher score in subject A than that in subject B but his ranking was lower than that in subject B. It stimulated students' critical thinking and sense of judgment. It also was an interesting opening to introduce the concept of standard score. The transition from one discussion topic to another was very smooth.

Apart from the teaching flow, his task design fostered learners’ statistical thinking. The GeoGebra task visualized the usage of standard score (Fig. 1). He explained that a standard score of zero acts as the average of the class. For those who got lower than the mean, their standard score would be negative. Conversely, the standard score would be positive for those who got higher than the mean. The horizontal distance between a dot (someone’s standard score) and the vertical line on zero means how much he/she performed below/above the average. The definition of standard score was explained clearly with the dot graph. However, the second GeoGebra task was just a whiteboard for students to write down the working of the calculations. It would be more effective if he replaced it with paper because the GeoGebra whiteboard is not easy to use.

Moreover, the hidden message he brought out was meaningful and thoughtful. In the lesson, he did not just teach the mathematical meaning of standard score but also told students that score or ranking might be not as important as we thought. In the case of Nobi, he actually performed well in both subjects. A high score means the level of how much he/she reaches perfection while a higher standard score means how excellent he/she performs compared to the average. The lesson was educational and impressive.

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Leung Sze Ching Celia's micro-teaching (Arithmetic Sequence)

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Leung Sze Ching Celia's micro-teaching (Arithmetic Sequence)

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Celia’s lesson activity was attractive. I got impressed by her magic tricks. At the beginning of the lesson, Celia performed a magic show of playing cards related to arithmetic sequence (A.S.). She asked one of the audience to take a card and she can tell the card correctly without seeing. This opening aroused all students’ curiosity and attention. They were motivated to study the following topic and engage in the class activity. Then, Celia introduced the concept of A.S. and explained the myth of magic was an application of A.S. The playing cards were arranged with a common difference. Once the magician knows the former or the latter card of the taken one, he/she can tell the answer by calculating the common difference. She guided students to take several numbers as common differences to form different sets of playing cards on the lesson worksheet. However, it was a pity that we did not have sufficient time to try the magic on our own. The cards she provided to each group were not enough to form a completed set. It is suggested to give much time for learners to experience the hands-on activity. Overall, the task design was inspiring.

Moreover, the GeoGebra task was effective. It visualized the formula of the sum of A.S. She explained the proof of the formula relationally instead of just telling students how to apply it instrumentally. Yet, the transition from the A.S. formula to the sum of A.S. was quite sudden. The GeoGebra was not related to the magic we discussed at the beginning. The lesson was so tight to learn all the topics so I believe it could be better to separate AS and the sum of AS into two lessons. The follow-up GeoGebra task is advised to deeper the concept of AS, such as visualization of the first term, common difference, and general term. It is suggested to allow students to discover how to form the formula of the A.S. general term with the aid of GeoGebra.