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Questions or Reflections for Mathematically Gifted Students raised by the author


(a) Definition 4.3.2 Gifted children are those who show exceptional achievement or potential in one or more of the followings

1 - (i) a high level of measured intelligence;

(ii) specific academic aptitude in a subject area;

(iii) creative thinking - high ability to invent novel, elaborate and numerous ideas;

(iv) superior talent in visual and performing arts such as painting, drama, dance, music etc;

(v) natural leadership of peers - high ability to move others to achieve common goals; and

(vi) psychomotor ability - outstanding performance or ingenuity in athletics, mechanical skills or other areas requiring gross or fine motor coordination (Adapted from Education Commission Report No.4 (1990), p.47)

Besides, the following points are also some of the common

1 This definition is based on that used in Marland's report to the Congress of the United States in 1972 on "Education for the Gifted and Talented".

characteristics of gifted students.

 Have excellent memory

 Remember a large amount of vocabularies

 Enjoy solving problem especially numbers

 Ask probing questions

 Be highly sensitive

 Have longer attention span

 Be strong sense of justice and idealism

 Have high curiosity

 Put ideas and things together that not in a traditional way (Adapted from Webb, Gore, Amend, and DeVries, 2007) Many educators also advocate that gifted students are not defined only with high IQ scores they obtained. Each gifted student should have his or her own strengths in different

if their strengths would be nurtured more fully.

2. Characteristics and Learning Needs of Mathematically Gifted Students

In point 2, the author will just focus on discussing the characteristics and learning needs of mathematically gifted students.

Mathematically gifted students can interpret relationships among different topics, concepts and ideas even without their teacher’s instruction and guiding given before (Heid, 1983).

Because of those gifted students’ intuitive understanding of mathematical functions and processes, they may skip over steps to find out answers but can’t explain how to find out the answers (Greenes, 1981).

Besides, they would additionally demonstrate some special features. They could also demonstrate in solving different math questions in a greater depth and breadth way (Sheffield, 1994).

Mathematically gifted students would always want to know that “how” and “why” the mathematical ideas come up with rather than how to compute the respective mathematical problems only (Sheffield). Therefore, let us see what mathematically gifted students could exclusively demonstrate while learning mathematics as follows:

Highly able mathematics students should demonstrate to

 display mathematical thinking and have a keen awareness for quantitative information in the world around them.

 think logically and symbolically about quantitative, spatial, and abstract relationships.

 perceive, visualize, and generalize numeric and non-numeric patterns and relationships.

 reason analytically, deductively, and inductively.

 reverse reasoning processes and switch methods in a flexible yet systematic manner.

 work, communicate, and justify mathematical concepts in creative and intuitive ways, both verbally and in writing.

 transfer learning to novel situations.

 formulate probing mathematical questions that extend or apply concepts.

 persist in their search for solutions to complex, "messy,"

 organize information and data in a variety of ways and to disregard irrelevant data.

 grasp mathematical concepts and strategies quickly, with good retention, and to relate mathematical concepts within and across content areas and real-life situations.

 solve problems with multiple and/or alternative solutions.

 use mathematics with self-assurance.

 take risks with mathematical concepts and strategies.

 apply a more extensive and in-depth knowledge of a variety of major mathematical topics.

 apply estimation and mental computation strategies.

(Adapted from Sheffield, 1999)

From the above mentioned, we also found that it is quite fantastic for mathematically gifted students could learn and solve mathematics problems so brightly.

Most mathematics teachers would also select some harder problems from textbooks or other resources to let their mathematically gifted students get higher sense of satisfactory of learning mathematics. By the way, how can mathematics teachers let those mathematically gifted students learn or solve mathematics problems more fully?

That is why some educators advocate that students, who are highly gifted in mathematics, need a separate or supplementary mathematics programme, tailored to their needs and abilities (Gavin, et. al., 2009; Gavin & Sheffield, 2010).

The implementation modes for gifted education in Hong Kong would be introduced and discussed in points 3 & 4 below and how the author implements the gifted courses. It is hoped that readers would understand more how to nurture mathematically gifted students more fully and divergently.

3. Implementation of Gifted Education in Hong Kong

The policy on gifted education for schools in Hong Kong was first recommended by the Education Commission Report No.4 (1990) and it also suggested guiding principles for the implementation of gifted education in Hong Kong.

In accordance with the guiding principles, the three-tier gifted education was adopted in 2000 as shown below.

Figure 3.1: The Three-tier Implementation of Gifted Education in Hong Kong

(a) Level 1 refers to using pedagogies that could tap the potential of students in creativity, critical thinking, problem solving or leadership in the regular classroom;

(b) Level 2 refers to offering pull-out programmes in disciplinary or interdisciplinary areas for the more able students within the school setting; and

(c) Level 3 refers to provision of learning opportunities for the exceptionally gifted students in the form of specialist training outside the school setting.

(Figure 3.1 and 3-level Descriptions are adapted from EDB’s


About top 2% of qualified students in each school may be nominated by their school to join the off-school support programs of level 3.

Therefore, if gifted students are qualified to join the off-school support programs, school teachers should encourage them to enroll the programs.

4. Design and Implementation of Summer Gifted Courses for Mathematically Gifted Students

a) Program Framework

The courses held by the author including different topics such as “Math used in the Workplace”, “Math Wonders”,

“Beautiful Geometry” and so on are specially designed for mathematically students between 8 and 12 years old who are studying in Primary 3 to 6 in Hong Kong Government Subsidy Schools, Direct Subsidy Scheme Schools, International School. Because of preventing from interest conflict, the author who is a secondary mathematics teacher only designs and holds gifted courses for primary mathematically gifted students.

mainly be based on a model of academically talented youth (Stanley, 1991).

 situations related to mathematics in context

 a need to apply knowledge and skills to solve unique problems

 opportunities to be a creative and critical learner

 experiences to foster appreciation of the beauty and utility of mathematics

 exploration of mathematics as it applies to daily life

 opportunities for gifted learners to increase their understanding of the role of mathematics in the workplace

Besides, problem-solving tasks, collaborative group work and in-depth discussions among course participants would have to be specially emphasized during different learning activities in all courses so that the course participants’

communication skills, intrapersonal skills, problem solving skills as well as higher order thinking skills would be enriched.

Conceptually-oriented themes and open-ended exploration in each session of each course are also deliberately embedded.

b) Selection of Course Participants

For those who are interested in joining the summer courses, they may be nominated by their class teacher, subject teacher, mentor, social worker or parents and then their nominators must fill out a nomination form to be downloaded from the center’s website. In the nomination form, their nominator must simply describe what talents the applicant equips with. On the other hand, it is also welcoming to provide some additional information by a nominator how well the applicant is suitable for taking course(s) they enroll. At last, the center would consider case by case what course(s) an applicant would be offered.

c) Topic Selections for Summer Courses

Before designing summer gifted courses, the author would also take so many references of local and other countries’

professional organizations for gifted education what mathematically gifted courses they offer or do not offer.

Besides, the course participants and their parents would be also asked upon the completion of the courses what courses they expect that CAISE would hold next year.

that three clusters of concepts and skills that included as follows:

 Fluency with whole numbers,

 Fluency with fractions, and

 Particular aspects of geometry and measurement.

Wheatley (1983, 1988), Johnson (1994) and Sheffield (1994) also suggested essential topics for mathematically gifted elementary students as shown below.

 Arithmetic and algebraic concepts

 Computer programming and robotics

 Estimation and mental math

 Geometry and measurement

 Math facts and computation skills

 Probability and statistics

 Problem solving

 Ratio, proportion, and percent

 Spatial visualization

 Structure and properties of the real number system In 1994, Sheffield also suggested many topics for mathematically gifted students and some of the topics are shown below.

 Fractals and chaos

 The Pythagorean theorem and Pythagorean triples

 Fibonacci numbers

 Finite differences

 Pascal’s triangle

 Figurate numbers

Johnson and others at the Center for Gifted Education at the College of William and Mary (2004) have developed criteria for curriculum and resources for mathematically gifted

 Learning Materials must contain a high level of sophistication of ideas.

 The design of the learning materials have to be very challenging to the mathematically gifted students.

 The selection of learning materials must contain different levels, interests and backgrounds and the level of learning materials should have no upper boundary.

 Higher order thinking skills and problem-solving skills must be involved in projects.

 Learning materials must have less emphasis on basis skills.

 Opportunities for student exploration must be based on students’ interest

Based on the topics and the criteria suggested by different scholars for mathematically gifted education, the author has been designing his summer courses in that way.

d) Learning Activities of Each Lesson during Courses In general, there will be also a main theme for each session of a course. Each session’s theme must be closely related to the course. Course participants would not only grasp

knowledge in the course they seldom learn them in school normal curriculum, but also appreciate the beauty of the topic of mathematics. It is expected that the course would inspire the participants that learning mathematics is more meaningful to their lives and would motivate some of them to choose Mathematics as their undergraduate study in future in university.

Besides, each session of any courses at least involves three main elements called 3C and 3C stands for “Critical Thinking Skill”, “Communication Skill” and “Creativity”.

3C would make courses more challenging and interesting to course participants. Furthermore, the design and selection of learning tasks would provide several opportunities for course participants to enrich their higher order thinking skills.

The design and selection of learning activities during each session is also to expect that the course participants would be highly motivated to think about how to solve the questions first with their prior knowledge. The tutor expects that the course participants may think about how and why to solve the mathematical problems in divergent ways.

during the courses over the years. In Figure 4.1, the course participants would be asked to put off their shoes and then put their shoes like a shape “L” on the floor. Then, they would be asked to guess that how many additional shoes were necessary to put on the floor to form a triangle with the 2 lanes of shoes. Of course, the course participants generally felt free to give their answers, by the way, the main purpose of the activity was to arouse course participants’ interests in learning the topic first.

In addition, the course participants would be guided by the tutor that how and why they could find the correct answer at last. In most learning activities throughout different courses, the course participants would have to be guided how to solve the learning tasks creatively and critically on their own first.

Figure 4.1: Course participants suggested different approaches to estimate how many shoes that were needed to form a right-angled triangle.

In Figure 4.2, course participants would be asked how to use geometrical approach to solve some algebraical identities by using different sizes of paper.

In Figure 4.3, course participants tried to find a diagonal of a cuboid after they learned Pythagoras’ theorem. Even some course participants had learned the theorem on their own before, they also could not solve the problem in a short period of time because they only learned and applied the theorem in a plane before, but not in a solid instead. Therefore, a same similar problem would be more challenging if it was applied in another scenario.

Figure 4.3 A group of course participants worked together to find out the length of diagonal of the cuboid by using Pythagoras’


The author learned how to use origami for learning mathematics especially geometry in 2012 from Mr. TAM who is famous in Hong Kong because origami lets students learn geometry in a more interesting and effective way. Figure 4.4 was an example that course participants attending a course of

“Math Wonders in Daily Life” tried how to find out some specific angles by using folding a piece of paper.

Then, the tutor would also explain why and how they could find some specific angles with folding a piece of paper. During the activity, most course participants were very concentrated on the tasks.

Figure 4.4: A participant tried to fold a piece of paper to find some specific angles the tutor asked.

In a course on “Math Wonder in Daily Life”, course participants were first guided to learn fractal geometry such as Sierpinski triangle and perimeter of korch snowflake. Then, all groups would make a model of Sierpinski tetrahedra shown in Figure 4.5 to know the three equations for Siepinski tetrahedra guided by the tutor.

Let be the number of tetrahedra, the length of a side and

the fractional volume of tetrahedra after the nth iteration.

Then, the three equations for Siepinski tetrahedra are as follows:

Most course participants were very concentrated on making the model and never wanted to stop the activity even the lesson had finished for long time.

Figure 4.5: A group of course participants displayed their completed model with a very high sense of satisfaction.

5. Course Participants’ General Feedbacks obtained upon the Completion of a Program on “Math in the workplace” in 2012 for Mathematically Gifted Primary Students

Below are 4 tables to list out students’ degree of satisfaction with the program, which lesson(s) they like most, which

lesson(s) they like least and students’ suggestions for the program.

Table 1: Students’ degree of satisfaction with the program (Adapted from Kwan, A.C.K. & Yuen, M, 2013)

Which lesson(s) did you like most, and why?

Lesson 1 Typical comments:


[3 students rated this lesson as the best.]

Lesson 3 Typical comments:

I can assemble “Math Cubes”.

The topic on “3 side views” is very interesting.

I love Mathematics and Medicine.

[4 students rated this as the best lesson]

Lesson 4 Typical comments:

Since my Dad is a surveyor, I now have more understanding about his job

[1 student rated this lesson as the best].

Lesson 5 Typical comments:

I have learned about investment before, and found it very interesting to learn more.

[2 students rated this as the best lesson]

Lesson 6 Typical comments:

I think the task is very interesting and challenging.

I love learning and applying formula.

It like this session because a big gift is given!

It is because the application of mathematics on that day is very interesting.

[4 students rated this as the best lesson]

Other comments

Students also expressed their preferences for different categories of concepts or content. Geometry was most popular, with statistics rating second.

Additionally, 2 students reflected that they liked all lessons because they were able to learn new mathematics knowledge, and because the course tutor performed well in teaching.

Table 2: Students’ perceptions of the lesson they appreciated

most (Adapted from Kwan, A.C.K. & Yuen, M, 2013)

Which lesson (s) you like least, and why?

Overall Typical comments:

I loved all lessons.

Mr. Kwan performs very well.

Each lesson is very good.

Every lesson is so sound.

Each lesson was very interactive and well taught.

[8 students reflected that there was no lesson they disliked]

Lesson 1 Typical comments:

It is because I need to handle many statistics graphs.

It is boring (4 students)

I do not have much interest in banking.

Lesson 3 Typical comments:

I do not understand much of the material.

Lesson 4 Typical comments:

I did not understand some things and questions (2 students).

It is because the challenging task on that day is very difficult.

Lesson 5 Typical comments:

I found it boring.

No guest speaker (1 student):

It is boring.

Medical treatment facilities I don’t understand.

Interest (1 student)

Table 3: Students’ perceptions of the lesson they liked least (Adapted from Kwan, A.C.K. & Yuen, M, 2013)

What are your suggestions for improving this programme?

No comment from 3 students

Sample comments from individuals:

Hope for more guest speakers and each lesson to last one hour.

Shorten duration of each lesson.

Add more gifts (rewards), and each of us also has gifts. (A view expressed by three students).

The venue should not be too far away. (A view expressed by several students).

More encouragement to motivate our thinking.

Conduct this programme in summer vacation Don’t be so boring.

Change the time to 1.5 hour.

May increase number of lessons and duration for each lesson.

Increase duration for each lesson so as to have more discussion.

Contents of this programme should be made very difficult


Add more elements in this programme.

Longer duration for each lesson.

Table 4: Students’ suggestions for improving this programme (Adapted from Kwan, A.C.K. & Yuen, M, 2013)

General speaking, most course participants were satisfactory with the course held by the author in 2013. Then, the author keeps on designing and conducting the gifted program for primary mathematically gifted students to enroll after 2013.

Besides, he also takes a reference of the data summarized in the 4 tables as shown above to modify the following courses he holds. Some course participants have enrolled the author’s courses for the consecutive three years. The author also thinks that mathematics teachers could also take a reference of the data as shown above when they design some learning materials for their mathematically gifted students.

Updated Development of Gifted Education in the World

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