There are something that inspires me in the first lesson 

 

 

The video that we watched on the lecture is thought provoking. I can still remember the example on the textbooks that the talker stated, which rely heavily on calculation and input numbers into the formula. The talker also said we should provide more space for student to think, so that they can know what information they need to solve a question. I totally agreed on the point, Mathematics is a subject that build on the thinking, rather than boring computation, the message that he brought is impressive.

 

The second thing that impress me is the demonstration of using GeoGebra to calculate the slope of tangent. The use of GeoGebra does not stop on the concept introduction, but also applied on the examples to find the slope of function at a point. I found this part impressive because I have thought using Dynamic Geometry Software in the classroom, but I always used it only in part of lesson, instead of incorporating it throughout the lesson. I think this approach is impressive, and the student will learn that they can really use the software to help them finish the exercise, check the answer, not just understand the concepts.

 

 

Not only the technological tool impresses me, but also the concrete teaching tool. 

Using the coin to comprehend the addition and subtraction of negative numbers can help students to comprehend the operation of negative number. When I was a form 1 student, the teacher did not pay too much emphasis on explaining the concept “Why minus a negative number will be adding the opposite number?”. However, using this teaching tool, the idea is represented clearly. Therefore, I look forward to using other kind of teaching tools, no matter concrete or virtual, to present the concept clearly to student, that is what I thought about the first lesson.

 

 

To achieve this goal, one direction deliver of knowledge is not feasible. Instead, classroom discourse would be a powerful tool for this. For the first thing, classroom discourse can encourage student participate in the lesson, rather than a spectator in the maths lesson. For another, classroom discourse can encourage the exchange of idea between different students, which can further bring out the collision between different ideas, this is a crucial step to let student think mathematically.

 

Among the skills of classroom discourse, I think the “Wait Time” is the most powerful one. I will provide some personal usage of it in here, but before that, let me introduce my view on the wait time first. I think the biggest usage of wait time is buying time for students to think, to organize the words said by the teacher. I am not saying that using “wait time” in any moment of lesson is powerful, we must use it when concept or thinking process itself is abstract and difficult. 

 

In this way, the wait time skill would provide students a good chance to stop and think. I think wait time can combine with the speaking skills. When teacher would like to explain a difficult proof in the lesson, he or she can point out the key part for students, but not the whole part (in Chinese, we call 將細節留白), and then use wait time for students to fill in the place where explanation is missing. For example, if we are teaching the sum of interior angle of polygons, we can offer examples when n=4, the polygon can divide into 2 triangles. When n=5, the polygon can divide into 3 triangles. When n=6, the polygon can divide into ____ triangles. Then teachers can use wait time to foster students’ mathematical thinking (finding pattern from observation), which is a good approach to achieve the goal I have mentioned above.

 

Pros: Help teaching reduce a lot of time, since many teaching process can be replaced by student watching videos at home, then during lesson in schools, teachers can have more flexibility (in terms of time) on teaching, like assigning exercise for students to do.

 

Cons: once if students are not able to finish the video at home (no matter due to any reason, like sick or too busy or not feeling interested), their progress will be affected, and it is impossible for teachers to help those students to catch up during the lesson time. (As mentioned in the lesson) As such, the effectiveness of flipped classroom is concerned.

 

Moreover, the mode of learning, in my opinion, is quite monotonous and boring, as students are required to watch the video at home and then finish the exercise on the lesson and repeat the same process for every day. If I am a student of this class, I would think Math is about doing exercise all the days. As such, I am afraid the implementation of flipped classroom is not feasible, or at least we have to modify it to cater for need of different students.

 

Therefore, it comes to the discussion of balancing between the degree of interesting and time efficiency of the approach of flipped classroom. I suggest we can incorporate some interesting stories or games of maths to attract students’ eyeball, which encourage student to watch the video at home, and then the in class activity can be an extension of flipped classroom video to arouse their motivation of participating into the class.

Introducing Concurrent-An approach to let student explore Mathematics

The topic of Anson is introducing the four centers, with focus on circumcenter and incentre. To explain the concept of concurrent, Anson had designed an activity to let students try to make 3 lines concurrent by hand, which is very difficult. This process and let students notice that it is impossible to make 3 lines intersect at a point if we draw it randomly or do adjustment by hand. As such, if student see 3 lines are concurrent, they will have an impression in mind that this is not a coincidence. 

 

I think Anson has done a great job on the part of introducing the concurrency of three lines. This part can arouse their imagination on property of concurrency. Anson can therefore guide students to think, “if three lines are really concurrent in some cases of construction, is there any reason to make them concurrent?” I think this question can further bring student to explore the concept of concurrent and take the first step to let students to explore the world of Mathematics, I think this attitude is very crucial when we study Mathematics, because this is one of the important ways to make student keep interest on learning Mathematics. 

 

The part impressed me most in the Anson’s lesson is that he did aware this part and tried hard to let student to let students to explore, let students to do experiment in the lesson. 

 

As for the suggestion of the lesson, I think after student finding that the perpendicular bisectors and angle bisectors concurrent, Anson can encourage them to think about why they are concurrent. It is because with the support of introduction activity, they will be curious on the concurrent situation. As such, If we can guide student to explore step by step, it will be very helpful to build up the mind of exploration of students, also the lesson will be more consistent. I think apart from learning Mathematical knowledge, learning how to study Mathematics is also important.

 

Furthermore, in terms of the content of activity, it is also very interesting that we can drag the position of typhoon to see whether we have no.8 signal or not, and when will the typhoon being the closest to Hong Kong is also some question interesting to discover and investigate. 

 

I think the activity also integrate with the element of modelling, we simplify the path of typhoon to be straight line and fixing other variables (like the variation of strength of typhoon), to make it become a 2D trigonometry modelling problem. What we need to do is just approximate the distance between line and a point. I think to combination of modelling and daily life example is a good try in the class activity. In fact, I have discussed this activity with Prof. Law, he also said that the using of modelling into activity is a good try and a chance to let students to know some basic idea of modelling. If I have chance, I will also consider adding the element of modelling into the activity, and try to make it more interesting. 

The topic of microteaching of Chloe is doing the point of division. The main idea of the activity is using the similar triangle and the ratio to determine the horizontal distance and vertical distance of each interval and then determine the coordinates of the point of division.

 

The idea was very clear, but I am afraid that students may not be able to observe the two pairs of triangles are similar. For this, I have a few suggestions to make the activity more intuitive.

1.     Using amination of dilation and diminishment

The nature of similar triangle is basically enlargement and contraction. Therefore, if we can enlarge the blue and red triangles in the figure, student can see that the red triangle will complete overlap with blue triangle. This is the first way to see it is similar 

2.     Using slope to observe 

Any two points on the straight line has the same slope as the whole straight line. The same slope implies the inclination of red and blue line are the same, together with the right angle, we see that the three pairs of angles with red triangle and blue triangle are the same, and so they are similar. 

I hope these two ways can help students to comprehend why two triangles must be similar, which is a good chance for them to bulid up the sense of Geometry. (Observing similar triangles) 

 

But I have a question to think here. When we derive the midpoint formula, we did not use the similar triangle, we just use the ratio 1:1 there. Can we do that in similar ways without using similar triangle? Here, I provide another way of not using similar triangle, which is a ratio argument. We consider the simple case when A=(0,0), B(6,3) first. If C is a point on AB such that AC:CB=1:2. Then we can use the ratio to see that since the whole length is of the “amount” 1+2=3, the horizontal length of AB is 6, and so the horizontal length of AC is 6*1/3. Similar to y coordinate, so the coordinate of C will be (6*1/3, 3*1/3). 

Next, if the coordinate of A is not (0,0), which is (3,5). We can reduce to the previous case by translating to the point A to the origin and then we use the argument in the first case. Of course, just like the attached screen capture, we may deliver it via GeoGebra, so that the whole idea becomes more intuitive. Hope these different kind of approach can help student learn better in this concept.

 

1.     Points outside the mirror should be reflected to points inside the mirror, and vice versa.

2.     Points on the mirror should be reflected to the points itself.

3.     The closer the point to the mirror, the closer the refection point to the mirror, and similar thing holds when the point is distant from the mirror.

To do circle inversion, the word “mirror” should be replaced by the circle. In this case, the first point is clear. 

For point 2, it will be “points on the circle will be reflected to itself” 

For point 3, it will be “The closer the point to the circle, the closer the point of reflection to the circle and vice versa”

Under these properties, we can comprehend why inversion is defined like this way, because the equation OR times OR*=r^2 (where O is centre R* is reflection point of R) , which is a constant, this implies the property 1,2,3 is automatically satisfied and so it can be regard as a circle inversion. I think this approach can transform the concept into simpler language. 

 

I think the most difficult part for students to learn this topic is “Why do we have to define the inversion of circle like this?”, which may be unclear when we just look at Simon’s lesson. However, I have some ideas on improving this. The main point is also “Capsulizing the idea of reflection”, from point 3 as above, we see that the description is like the distance of point and its reflection point from center is at inverse proportion. Therefore, if we set OR time OR*=constant, and use point 2, we can see that the constant is r*r=r^2, which is the definition of inversion Simon introduced in the lesson. I think it is more important to let student know how does the definition come from. If student just use GeoGebra to test the property of inversion, although we still can get the definition, student might find it confusing that how do we get the applet for inversion for the first place, how does everything come from? I hope that this approach can make the difficult topic a bit easier to learn.

Teacher to student: Question after activity 1: Whether we can solve the problem by only using the result of activity 1? Alex, Ryan and Wilson have expressed their view on the question and I try to guide them that we did not yet arrive the conclusion.  However, I think I should add more that: If we use the property of slope of -1, then it is still not clear that the line will be perpendicular because we only have the statement in another direction)

For the word choice, I did aware to avoid the word “if and only if”, “converse statement”

 

I think most of the problem lies in the details in designing the activities, including how I present it, when do I need to show the lines or figures, which function should be exclusive to teacher to avoid confusion, whether I pay emphasis on what we are going to learn. I think these things can be improved when I try to bring this activity into actual lessons. Although there are some unexpectable incidents like some student may need more time to open the GeoGebra applet, I tried my best to solve these sudden events in a clam and quick manner.