Research for Better Schools

Using TIMSS Videos to Improve Learning of Mathematics: A Resource Guide

Detailed Description and Commentary: Hong Kong Lesson 1

This lesson focuses on square numbers and square roots. It is the second in a sequence of lessons working towards the Pythagorean theorem and its applications. Though the students’ first language is Chinese, the lesson is in English. The lesson is 34 minutes long. There are 44 students in the class.

Lesson outline

0:00–17:53 (Whole class) The class discusses squaring and square roots.
0:06–3:56

The class discusses the meaning of the square, using the example, “What is the square of 3?” A student says it is equal to 9. The teacher asks students to show her the steps in calculation and outlines them by writing:
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A student answers 9, which the teacher records after the last equal sign, then 3 times 3, which is written after the second equal sign, and then the teacher asks a student to write what the square of 3 is, which the student writes as 32 after the first equal sign.
1:29 The teacher asks, “What is the square of –3?”
2:10 A student comes to board and writes “(–3)².”
2:29 Another student comes to board and writes “(–3) × (–3).”
3:56–5:56 The class discusses finding the positive value of a if a squared equals 9, and use of the radical sign.
5:56–9:19 The class discusses finding the negative value of a if a squared equals 9. The teacher asks for the answer and then how to find the answer.
8:44–9:00 A student comes to the board and, with some prompting from the teacher, writes “a² = 9.”
9:19–10:09 The class discusses radical36 and –radical36. The teacher asks students to work on question 2 on p. 162 (see the Resources section of the CD).
10:12–12:18 Students work individually as the teacher circulates.
12:18–14:56 (Whole class) The teacher calls on students for answers to the different parts of question 2. She summarizes some of her comments by saying that if a² is equal to nine, there are two answers. The values of a may be positive or negative three. If there is no sign before the square root, it is positive (and there are two answers); if there is a negative sign before the square root, then the answer is negative.
17:53–19:46 As the teacher circulates, students work individually to find the value of a if the square of a is 64, and the value of a if the square of a is 81.
19:46–21:57 (Whole class) Students come to the board to write answers.
21:57–27:31 (Whole class) Discussing the square root of -4, squared, and then the square root of -4.
27:31–28:31 (Whole class) The teacher summarizes the lesson.
28:33 (Whole class) The teacher assigns homework: reading pp. 163–164 and questions on p. 164
28:33–34:16 Students work on homework as the teacher circulates.

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Mathematical and pedagogical comments

Discussion of square root and radical sign

U.S. college students sometimes have difficulty with square root and radical sign and are also sometimes uncertain about whether Square roota has one or more values. This lesson gives an example of an effective way to discuss these topics.

Instructional strategy

This lesson follows a consistent pattern: the teacher asks simple numerical questions, in stages, to introduce increasingly complex properties of square root notation and conventions. Students then do similar examples. Although the lesson is only 34 minutes long, it proceeds at a leisurely pace.

Instructional decisions

The teacher commentary provides specific reasons for particular teaching decisions.

Making connections

Researchers classified segment 25:01–27:28 as “making connections.”

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Screen captureTechnical details

File name: NB10301719

Pages from the teacher’s version of textbook are in the Resources section of the CD. Some parts appear to be poorly scanned—a few words and sentences are missing and footnotes may have been cut off.

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Further discussion

Relationship with previous learning

The national research coordinator comments (0:00, 5:14) that students should have learned the concept of square root and associated terminology and symbol in earlier grades. The teacher comments (0:00):

“Students have learned about square and square root in elementary school. But it was focused on practicing the calculations only. They are not very clear on many of the concepts, resulting in future difficulties when learning algebra. Therefore, I concentrated on discussing the conceptual problems with students in this lesson. I wanted to let the students know about their insufficiencies and speculate from their common mistakes. Consequently, they can build an accurate mathematical concept.”

Students need to know:

Relationship with future learning

This material will be useful when learning:

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Detailed mathematical and pedagogical comments

Instructional decisions

(0:37) The teacher comments:

“A student stated that 3² = 6. This is a common mistake made by students. In algebra, they are usually mistaken that a² = a × a = 2a. Therefore, it is necessary to clarify this mistake. I guided the student to write down each necessary step through questioning. This can help students to better understand why 3² = 9 rather than 6.”

At other points in the lesson, the teacher mentions specific reasons for particular teaching decisions; see the teacher commentary in the Resources section on the CD.

(6:00–10:30) When the teacher asks why, she answers herself and gives as the reason that the chosen number has a square equal to 9. She does not discuss the possibility of other solutions.

(22:28–24:18) The national research coordinator says (22:27) that “square root of the square of negative four" is a very demanding question for Hong Kong students". One student gives a very good answer. The teacher then summarizes the answer, helping to solidify the correct answer given by the student.

The next problem she asks is what is the square root of -4? The only possibilities are stated to be a positive number, a negative number, or zero. Each of these is ruled out. As the national research coordinator writes: "[this] is very rigorous reasoning for students at this level".

Making connections

The country researcher says (25:01):

“The teacher poses the following question: Is the square root of negative four equal to negative two, positive two, or does it have no solution? This problem has a ‘using procedures’ problem statement (see comment at 6:01), however as it is implemented the class uses mathematical reasoning to explain why there is no solution.

Problems that were stated as ‘using procedures’ but then solved by ‘making connections’ accounted for, on average, 9% of the problems per lesson in Hong Kong (Video Report, figure 5.10).”

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Uses of the lesson

  1. Focus question: This series of lessons leads to the Pythagorean theorem. What should students know about squares and square roots before starting to study the Pythagorean theorem?

The instructor should ask viewers/participants to write down their responses.

Instructor and participants should read the Chapter Overview and the comments to teachers on the first page of the textbook in the Resources section on the CD.

  1. Focus question: “Are there additions to the list of what students need to know in order to understand the statement of the Pythagorean theorem?”
  1. Focus question: “Here is a video of a teacher preparing students to learn the Pythagorean theorem. What concepts necessary to understanding the statement of the Pythagorean theorem does this clip address?”

The instructor should show the clip (0:06–3:56): discussing the meaning of the square: what is the square of 3? (The teacher commentary above discusses this segment.)

  1. Focus question: “What does this clip address? How is this relevant to learning and using the Pythagorean theorem?”

The instructor should show the clip (3:56–5:56): finding the positive value of a if the square of a is 9.

  1. The use of questions in a mathematics lesson: “How does the teacher use questions differently throughout the lesson?”
  1. (22:46–27:28) Repetition of key mathematics concepts (involving procedures):
  1. See N2 for an incomplete proof of the Pythagorean theorem.

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