Classroom observations

In this section, the classroom observation of Tom teaching is reported. Tom’s class conversation for 6 minutes offers a good sense of his teaching style.

A Geometry Lesson

Grade 9 Duration: 3:09 – 3:15 pm Date: November 18, 2002

Mathematical Content: Geometry – the beginning of unit 2-2. Method: Direct teaching, questioning and problem Solving Questions:

Diagram 1

The teacher’s writing on the blackboard was:

2-2 A central angle, an angle in circular segment, an angle of a circular segment

: 1 The length 2 The degrees 3 The location

The length of an arc＝2Πr ×

360 deg rees

The area of a sector of circle＝ 2Πr2 ×

360 deg rees

(1) a central angle: the top of an angle located at the centre of a circle

The degrees The degrees of an arc subtended by a central angle (Tvh1118p1,2t)

This example shows how he lectured on the concept of a central angle and two formulas. He spent six minutes on this.

Tom: You all look at the blackboard, I will tell you all the important points first. There are five kinds of angles: a central angle, an angle in circular segment, an angle in a chord and tangent, an angle located inside of a circle, an angle located outside of a circle. Regarding the latter two kinds of angles, the textbook does not discuss these two

kinds of angles, but there are some mathematical problems dealing with these two kinds of angles. So, you still need to pay attention to these two. After I have lectured and given you these important points, you can organize them and I might test you on these important points. Before I mention these five angles, please look at this arc of AB which

represents three meanings. Firstly, what is this? (He pointed at this arc of AB on the blackboard.) This is the length of an arc. The first one means the length. What does the second mean? How many degrees are there in a circle?

At least six students responded at low volume: Three hundred and sixty degrees.

Tom: Three hundred and sixty degrees. Can this arc represent a small part of the degrees? Secondly, it represents degrees. Today what we will use most are these concepts about degrees. Thirdly, what does it mean in this place? It means location. So, the sign of represents three meanings: the length, degrees and location. How about the length? The formula of the length of an arc! The formula of the length of an arc has been mentioned in our textbook. There were two formulas, one is about the length of an arc and the other is about the area of a sector of a circle. The formulas of the length of an arc and the area of a sector in a circle! Here we can review the formulas of the length of an arc and the area of a sector of circle. What is the formula of the length of an arc?

Students: …

Tom: Louder! I cannot hear you. Students: 2Π

r

360 deg rees

Tom: OK! 2Π

r

360 deg rees

. The area of a sector of circle (pointing to the blackboard) will equal to that 2Π

r

360 deg rees

. One student: degrees.

Tom: We should generally know this concept. Now we will talk about five kinds of angles. The first angle is a central angle. Why do we call it a central angle? The reason is that the top of the angle is located in the centre of a circle, O. [He pointed at O on the blackboard.] This point O means the centre of a circle. The top of an angle is located at the centre of a circle; so, we call it “a central angle”. So, we define a central angle as one where the top of an angle is located at the centre of a circle (Tvh1118p1). [Tom always pointed at the blackboard since here.] Then, how many degrees are there in a central angle? This is an important point. The first important point of today is that the degrees of a central angle equal the degrees of an arc subtended by a central angle. You need to pay special attention to this. If the degrees of an

arc are 40 degrees, pay attention for the degrees of a central angle are 400

. [He wrote X＝400

on the blackboard.] When you consider the degrees of five kinds of angles, you need to pay special attention to the definitions of five kinds of angles. The degrees of a central angle equal the degrees of an arc subtended by a central angle. The degrees of a central angle equal the degrees of an arc subtended by a central angle. [Tom watched and waited for students to finish their writing.] Have you finished writing? The first angle is a central angle. The second angle is an angle in circular segment. … (Tvh1118p2t)

The example above shows one of his typical ways of teaching through directly delivering and explaining his lessons. This example also indicates his fast teaching speed through direct instruction, then moving to the next mathematics concept. For example, he clearly explained the key point of the definition of an arc through several short questions but mostly he answered his own questions when questioning the class. Next, he asked two mathematics formulas without explaining, and then he shifted to directly explain the definition a central angle. In the later part of this lesson, he used lots of mathematics formulas to explain problem solving in the textbook. Moreover, Tom’s authoritative attitude also could be sensed from his demands on students by directly giving orders without asking students’ opinions and from the pressure of a test.

Another example showed how Tom used questioning skills to teach a concept and solved the problem with students’ responses. For instance:

Tom: Keep writing and keep listen to me. … If the length of an arc is longer, does it means that a central angle is bigger?

One student: Yes [in a quiet voice]. Tom: Are my statements right or wrong? One other student: It is not necessarily like this.

Tom: If it is not necessarily so, that means it is wrong. If the length of an arc is longer, then a central angle is bigger. Is this right or wrong? If the length of an arc is longer, then a central angle is bigger. Is this right or wrong?

Some students: Wrong! Wrong! [Students answered at different times]. The teacher: If the length of an arc is longer, then a central angle is bigger.

It is wrong. Tell me where it is wrong. Is it right or wrong? Tell me where is wrong? How do you judge this?

One student: Check the radius.

Tom: Please tell me. Is bigger than ? Some students: Yes!

Tom: IS bigger than or not? Some students: Yes!

Tom: is bigger than , right?

Please tell me, is AOB bigger than COD? Some students: No!

Tom: Is AOB bigger than COD, or not? Students: No!

Tom: It is not. The two angles are equal to each other, right? The longer length of an arc does not mean a bigger central angle. What is the key to judge this? Radius, right? The longer or shorter radius decides the length of an arc. So, you need to pay special attention to this. The longer the length of an arc does not mean the bigger a central angle. Please don’t be cheated by this! This is a key point if it is located in a same circle. Before when I mentioned this question, I did not mention the same circle. So, I did not give you this condition that the two circles are the same circle. I only said that the longer the length of an arc means the bigger a central angle. This statement is wrong. Please pay attention about this! (Tvh11186e).

This example indicates another type of teaching instead of directly asking formulas to solve problems. This showed how Tom challenged students’ thinking in a big class. Through several times of questioning and waiting, students gradually formatted the correct answers and gave short responses. Then Tom concluded the main mathematical ideas and explained the reasons himself. These teaching strategies might echo Tom’s emphasis on teaching students’ understanding. Tom gave chances for students to think and adjust their ideas and later used teacher’s explanations to develop their understanding.

Tom emphasized the importance of using formulas. (1) He felt that some parts of the textbook used too many steps to solve a problem (Tvh1118p4e,5t) and reminded students to avoid those methods in the textbook (Tvh1118p5e). He

recommended students to use a formula e.g. 2Π

r

360 deg rees

to speed up the time (Tvh1118p4e). (2) He gave some short words to help students to remember the relation between some mathematics concepts (Tvh1118p4t, 7t).

Tom emphasized the importance of students’ concentration in his classes. He encouraged students that if they concentrated in classes, they would learn very quickly. Even if they did not do the practice in the textbook; they could easily understand it (Tvh1118p5t).

Student being engaged

Students mostly appeared to concentrate in Tom’s classes during my class observations. That indicated students were either listening or writing notes. For example, on November 18, 2002:

 the first 10 minutes

All students were either listening or copying from the blackboard (Tvh1118p2e).  the next 23 minutes

All students were listening and at least seven students were both listening and writing (Tvh1118p3b).

 the next 35 minutes

All students were looking at Tom and listening to his sharing (Tvh1118p5b).