Mathematically gifted and talented students

Mathematically
Gifted and Talented
Students
A Resource Booklet
By Karen Chow
Contents
Introduction…………………………………………………………………………………....1
Identifying Giftedness………………………………………………………………………....2
Issues for Mathematically Gifted Students…………………………………………................3
Catering for the Gifted………………………………………………………………………...4
Differentiated Learning………………………………………………………………………..6
Differentiation in Practice…………………………….………………………………….........8
Online Resource Links……………………………………………………………………….14
Conclusion……………………………………………………………………………………15
References……………………………………………………………………………………16
Appendix I: Gardner’s Multiple Intelligences………………………………………………..17
Appendix II: Presenting the Booklet…………………………………………………………23
Introduction
This booklet is used to raise awareness of mathematically gifted and talented students and
provides some points to consider when identifying and catering for them. Teachers from
other curriculum areas may also find the ideas presented in this resource helpful in their own
classroom and discover techniques that will engage mathematically gifted students in their
classes.
Karen Chow
Mathematically Gifted and Talented Students
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Identifying Giftedness
Indicators of mathematical giftedness include
Unusual curiosity about numbers and mathematical information
Ability to understand and apply mathematical concepts quickly
High ability to identify patterns and think abstractly
Flexible and creative in their strategies to problems
Able to transfer mathematical concepts to an unfamiliar situation
Persistent in solving challenging problems
(Stepanek, 1999)
In addition to some of the traits described above, gifted students typically display
Enthusiasm for knowledge
High reading abilities from an early age
High concentration levels and self-sufficiency
Curiosity in objects and occurrences
(Holton & Daniel, 1996)
However, these indicators should not be used as rules for qualifying students as being
mathematically gifted. Not every mathematically gifted student will display all these
characteristics, or they may emerge at different times depending on the student’s
development. Much of identifying gifted students relies on ongoing assessments and teacher
observations.
Mathematical giftedness can also manifest in three ways (Krutetski, 1976 as cited in Bicknell
& Holton, 2009):
1. Analytical. Analytically gifted maths students tend to think abstractedly with ease. They
solve problems using logic and reasoning.
2. Geometric. Geometrically gifted maths students prefer using diagrams and visual aids to
solve problems.
3. Harmonic. Harmonically gifted maths students can use both geometric and analytic
methods of thinking with ease.
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Issues for Mathematically Gifted Students
Holton and Daniel (1996) state that mathematically gifted students tend to be identified at an
early age; however they may not get the appropriate support. Research has shown that
teachers of mathematically gifted students can become intimidated by them and avoid
interacting with the students to ensure that the power balance does not alter (Holton &
Daniel, 1996). Parental assistance can contribute to the success of gifted students, but the
majority of parents with mathematically gifted children only have primary school level
mathematics ability.
The resulting isolation that mathematically gifted students suffer from, alongside the
stereotypes associated with being gifted in mathematics can cause students to reject their
abilities to fit in with their peers. In addition to this, mathematically gifted students encounter
the same challenges as other gifted learners:
Mental laziness. Gifted students can become complacent in their work and no longer strive
for personal excellence.
Perception that grades are more important than learning. This implies students are no
longer intrinsically motivated; instead they use grades to gauge their successes.
Perfectionism. Unrealistic expectations from teachers, parents and peers may cause a
gifted student to avoid experiences where they may risk failure.
Failure to achieve self-efficacy. This occurs when a gifted student is not sufficiently
challenged, leading students to wonder when they will be exposed for being a “fraud” in
what they do.
Underachievement. Gifted students’ abilities are ignored or not appropriately catered to,
leading to apathy towards mathematics and thus underachievement.
(Ministry of Education, 2000)
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Catering for the Gifted
Mathematically gifted students require a lot of support, both academically and socially. They
need to be challenged in their work and feel as if they are part of a supportive learning
community rather than a burden or oddity. Methods that mathematics teachers can use in the
classroom to support their gifted students include:
1. Flexible grouping
Students should be able to work individually or in small groups. The teacher can use
different types of ability grouping depending on the skills, unit of work or other learning
opportunities available to students. Homogeneous ability groups can prevent the teacher
from having to explain the same concept multiple times to individuals, while
heterogeneous groups encourage student-to-student teaching. By altering the groups as
needed, students work with peers they may not usually interact with; this can broaden their
mathematical perspectives.
2. Having clear expectations of the students
Students should be aware of what is expected of them in terms of mathematical
understanding and their expression of this in written form. For mathematically gifted
students who are able to omit steps in their calculations, this can be frustrating; however
mathematical evidence is necessary to track their progress and uncover any
misconceptions they may have. The expectations may be written or verbal, but they should
always be explicitly stated.
3. Allowing students to create goals that are challenging but achievable
To allow gifted students a sense of self-efficacy, they should be given opportunities to
reflect on their learning and as a result form goals that they believe are worthwhile. Giving
them a task that they may struggle on but eventually succeed in will provide more
satisfaction than completing an easy activity that they could do without thought.
4. Circulating the room and providing one-on-one assistance to students
In a 2007 workshop by Robin Averill and Megan Clark, teacher caring was an aspect
highlighted by students. An expression of this was engaging students mathematically and
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Mathematically Gifted and Talented Students
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taking an interest in them outside of the classroom. This also allows the teacher to view
students from a different perspective and incorporate topics of interest into the lesson. It
can break down the teacher’s preconceived notions of students that could hinder their
learning.
5. Providing students feedback
Feedback shows that the teacher is aware of students’ progress and can give students
guidance in their next learning steps. However, feedback should
Focus on student progress in the topic as opposed to a grade. By eschewing grades, the
attention to social comparisons is reduced and gifted students do not feel singled out by
a high grade.
Note areas of success and areas requiring improvement with specific commentary on
how students could progress. According to Black, Harrison, Lee, Marshall and Willam
(2003), assessment only leads to learning gains when specific guidance is provided for
students. Generic observations such as “Well done,” or one-word comments such as
“Equation?” are not beneficial, and can confuse students.
Be given in a timeframe where students can usefully apply it. Feedback is often
provided to students after it is useful, or students are not given sufficient time to act
upon the suggestions provided (McInerney & McInerney, 2006).
Provision for gifted students in New Zealand usually involves enrichment or acceleration
(MacLeod, 1996).
i. Enrichment. Students discover concepts in more depth either within class or through
programs outside the classroom including Mathematics Olympiads, Mathswell, problem
solving competitions or one-off withdrawal programs (see Online Resource Links section
for more details).
ii. Acceleration. This practice of providing more advanced materials to gifted students
occurs primarily in male-only schools. Students usually skip a year and do NCEA Level
One in year 10. However, acceleration should focus on developing conceptual knowledge
rather than moving students through the same content at a faster pace, as gifted students
find mechanical skills easy to pick up (VanTassel-Baska, 2004).
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Differentiated Learning
Although the strategies outlined above are useful in managing a differentiated classroom,
there should also be techniques that cater to gifted students. Differentiation in the
mathematics classroom in each lesson can enhance the learning of all students, and it can be
adapted to other curriculum areas.
Differentiation can take different forms. The three main areas differentiation can occur are:
1. Content.
What is taught or learnt varies depending on the abilities of the students. Whether this is in
the form of accelerated learning or enrichment for gifted students should be a decision
made on a case-by-case basis. It is essential that gifted students have a solid mathematical
base, before they are moved onto extension exercises; a preassessment may be necessary
to gauge their understanding prior to the teaching. Curriculum compacting as developed
by Joe Renzulli (Tomlinson, 1995) is where students are assessed and taught only the
content that they are missing, before being given the opportunity to work on a project they
have chosen with the teacher’s assistance. However, great care is required in mathematics
when accelerating or enriching students. Ensure that students will not be repeating the
same work the following year and that the project is beneficial to the student’s
mathematical progress and contains an element of reflection.
2. Process.
Gifted students will require specialist instruction in regards to their work and this aspect
focuses on how students are taught or how they learn. This type of differentiation usually
relies on using the upper echelons of Bloom’s taxonomy for gifted students (See diagram).
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Creating
Evaluating
Analysing
Creating
Evaluating
Analysing
Applying
Applying
Understanding
Understan
ding
Remem
bering
Remembering
Diagram 1: The left hand pyramid shows Bloom’s taxonomy for mainstream students, where the emphasis
is on remembering, understanding and applying. For gifted students, the right hand pyramid highlights
creating, evaluating and analysing as areas of focus (Adapted from Ministry of Education, 2000).
Other thinking models that can be used include deBono’s Six Hats, Thinkers Keys or
Krathwohl’s Taxonomy; however it is a good idea to choose one and allow students
become familiar with it rather than introducing numerous thinking models simultaneously.
3. Product.
Real world problems are presented to gifted students, where the context and mathematical
content are entwined and require students to create interdisciplinary connections (Goos,
Stillman and Vale, 2007). Students should be self-evaluating their work and considering
ways to improve their solutions. This aspect should appeal to the different learning styles
the students favour. For more information on learning styles, see Gardner’s multiple
intelligence model in appendix I.
While differentiating all three aspects is optimal, this is not always possible, and for someone
who is starting to implement differentiation it can become an unsustainable practice.
Tomlinson (1995) provides some advice for teachers wishing to differentiate in a classroom:
Have a strong rationale for differentiating in regards to student learning
Begin differentiating at a comfortable pace
Time differentiated activities for student success
Create and deliver instructions carefully
Give students as much responsibility of their learning as possible
Engage students in talking about classroom procedures and group processes
Be flexible – flexibility is the hallmark of a well-differentiated class
Use a variety of resources for students rather than relying on one textbook
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Differentiation in Practice
Assessment of students’ giftedness should be ongoing and it should occur in an inclusive
classroom. Even students who are not identified as gifted but show potential should be given
the opportunity to work at the level gifted students work at. A differentiated curriculum
where students are provided with choices means learning is student-directed, but other than
providing them with multiple resources to work from, how else can teachers differentiate in
the classroom?
Assessing Students
Students need to be assessed so that the teacher is able to gauge what students already know
and pitch content appropriately. However, the preassessment should only be used as a
guideline.
1. Five Most Difficult: Students are given the five most difficult questions prior to the start
of the topic. Those who are able to respond with well-reasoned arguments warrant
extension tasks.
2. Mindmap: Have students design a mindmap of a topic where they visually share
information they know about a topic and the interrelations that grasp prior to being taught
the unit of study.
3. KW Chart: Students are provided with a chart where they record what they wish to know
and what they want to find out. The aspects students want to know are analysed in terms
of learning objectives, while what they want to find out is a basis for creating questions
they may find intriguing and challenging.
4. Exit Cards: Students are required to fill
out a card for their teacher that reflects on
what they have learnt and how they think
they are progressing. This is used to assist
the teacher in planning future lessons.
Karen Chow
Exit Pass
How confident are you in mathematics today?
Give an example of a problem you could not
do before this lesson. Make sure you give the
solution too!
Mathematically Gifted and Talented Students
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Presenting Activities for Differentiation
1. Choice Board
Students are provided with compulsory activities they must complete along with some
choices in what they would like to do as the “main course.”
Mathematics Menu
Entrée (Compulsory)
Skills Questions
Main Meal (Must have one)
True or false factorising quadratic worksheet. One question, two
answers. Are either of these answers correct? Why/why not?
Textbook: Mixed factorising quadratics
Create a game for factorising quadratics
Rewrite the chapter on factorising in your textbook. Include examples.
Dessert(Voluntary)
Write an A5 cheat sheet for factorising quadratics
If you were the teacher, how would you have taught the class
factorising quadratics? Write a paragraph with supporting evidence to
back your claims!
2. KUDs
KUDs is an acronym for Know, Understand and Be Able to Do. It incorporates the choice
board and Bloom’s Taxonomy; however it currently caters for mainstream or struggling
students. This can be adapted for gifted students by focusing on the higher order thinking
aspect in Bloom’s Taxonomy. The sample KUD has been provided from The
Differentiation Toolbox (2009):
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Application I: Integration – Three Approaches
Audience
This lesson is intended for high school Calculus students as they near the completion of their
work with integrals
KUDs
Know
 Five integration
approaches an when to
use them
o U-substitution
o Trig-substitution
o Integration by parts
o Separation of
variables
o Partial fractions
 Integrals can be
approximated using
Riemann sums
Understand
Be Able to Do
 Integration usually
 Solve integrals using the
involves rearranging
five integration
problems in order to use
approaches
a known approach
 Conduct estimations
 Integration is used in
using Riemann sums
the real world to
calculate quantities such
as volume and area
 Real world integration
involves error
Rationale/Description
This application is a version of a learning contract in which students are granted a degree of
flexibility in their assignment, but agree to use those freedoms appropriately to design and
complete quality of work. Throughout the year, students will have been looking at calculus
through three lenses or frameworks – algebraic, numerical and graphical. This assignment
requires the students to look at integration through each of those lenses. It assesses their
mastery of various skills and understanding about integration, but gives them choice in terms
of how that mastery is demonstrated. For each category, students have three choices which
are differentiated according to student interest. Some activities are hands-one and require
students to design and build objects while others are more linear and descriptive. But all
activities focus on the same essential skills and understandings.
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Mathematically Gifted and Talented Students
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Integration – Three Approaches
Algebraic
You’ve probably noticed that our textbook is
not exactly a stellar work of prose. Rewrite
Chapter 5.2 (the section summarizing five
integration approaches) to make it more
approachable to today’s average high school
student. Feel free to be as weird or as funny
as you want, just don’t lose the important
definitions, diagrams or examples
Create a visual to be presented to the class
summarising the five integration approaches
discussed in class. For each approach,
include one worked-out example. For at
least three approaches, include thorough,
step-by-step instructions for using the
approach
We just don’t to calculus for fun; it’s helpful
in real life. Find real life phenomena that
can be calculated using integration. Find one
phenomena for each of the five approaches
discussed in class. For each phenomena,
work out an integration and include a brief
explanation of your work.
Numerical
Choose a container, like a box or jar. Then
fill the container with regular solids (balls,
cubes, etc.). Estimate the volume of your
container based on the calculated volume of
your solids. Take picture of your experiment
and produce a creative visual display (book,
storyboard, poster, etc). Include a discussion
of your procedures and possible sources of
error
Choose any third order polynomial and
lower and upper values for limits of
integration. Then approximate the area
under the curve using two different methods.
The division is up to you, but use at least
five divisions. Include a discussion of
your procedures and possible sources
of error
Research the origin of Riemann Sums. Write
a brief paper explaining their history,
variations, and uses. Include diagrams, a
discussion of error, and a symbolic
definition of Riemann sums. Also, include at
least one worked out problem
involving Riemann sums from
Chapter 5 of your textbook.
Graphical
Complete one block in each category, Graphical, Numerical and Algebraic.
Remember to make your work thoughtful, original, insightful, neat and elegant in expression
Construct a three-dimensional model of a
shape that can be divided either into disks,
washers, or regular prisms. Estimate the
volume by actually dividing the shape into
those small divisions. Then, find the exact
volume of the shape using waterdisplacement. Submit a careful description
of your work
Write a users manual to help students
complete a graphical integration on their TI83’s or TI-89’s. Include step-by-step
instructions with rationale for each step.
Also, include pictures explaining how the
calculator does its math.
Use your graphical calculator to estimate the
integral of a couple different functions using
5 then 10 divisions. Submit a brief paper
discussing the differences in error. Select
functions that will give you significant error.
What about the functions created estimates
that were far off?
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3. Tiered Activities
Students have the same learning objectives in this method; however the level of challenge
differs in the range of tasks provided.
Learning Objective: To factorise quadratic equations
Challenge: The quadratic expressions have been partially factorised. Fill in the gaps.
Challenge Plus: Factorise these quadratic expressions where the coefficient of x2 is 1.
Mega Challenge: Factorise these quadratic expressions where the coefficient of x2 may not be 1.
4. Learning Stations
A range of different learning activities are set up around the room, with students having to
complete some or all of the activities. This method supports the diverse interests students
may have, and gifted students will have the opportunity to work cooperatively with their
peers. Remember to make sure that there are tasks available for fast finishers.
Learning Objective: To factorise quadratic equations
Station One: Solve the quadratic maze using what you know about factorising
Station Two: Write two questions that would give (x – 3) (x + 4) as a solution. Check your
answers with a partner and decide which the best answer is. Why do you think that?
Station Three: Write a song or poem about how to factorise
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Specific Strategies That Support Differentiated Classrooms
These strategies are implemented to optimise the teacher’s time and students’ learning, so
their purpose indirectly benefits gifted students.
1. The Doctor Is In
This is where students need to sign in for an appointment stating what their needs are on
the whiteboard when the teacher is engaged with another student or group. This allows the
teacher to optimise their time spent with students rather than repeatedly responding to the
same query from various students.
2. Mini – Lessons
Mini-lessons are used to facilitate gifted students’ thinking, especially when they are
struggling with the extension activities they are provided.
3. Three Before Me
Students are encouraged to ask three other students for help before resorting to the teacher.
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Online Resource Links
The material in the resource links are for mathematics teachers with gifted students and can
be used to extend and enrich students’ understanding of mathematics.
International Competition and Assessments for Schools
Created by the University of New South Wales, this website provides problem sets and
resources for teachers of gifted students. The webpage given is linked to the mathematics
resource page; however ICAS also caters for other curriculum areas.
http://www.eaa.unsw.edu.au/eaa/mathematics_resources
International Maths Olympiad
This is an international competition that is based in America. Students enter in teams to solve
monthly problems that are used to encourage students to deepen their mathematical
understanding. Sample questions and monthly problems are available for free on the website.
http://www.moems.org/
The National Bank Junior Mathematics Competition
This competition caters for years 9 – 11 where students have one hour to answer up to five
problem solving questions. The emphasis is on students’ ability to display strong
mathematical reasoning. A sample of past competition papers is available on the website.
http://www.maths.otago.ac.nz/nbjmc/JMChome.php
The New Zealand Association of Mathematics Teachers
This webpage contains a variety of problem sets for students.
http://www.nzamt.org.nz/sites/cms/index.php?option=com_content&task=view&id=140&Ite
mid=142
NRich Site
This website is managed by Cambridge University and contains a variety of problem solving
questions at different levels of difficulties. Articles for teachers are also available here.
http://nrich.maths.org/public/
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Mathematically Gifted and Talented Students
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Conclusion
Outside of the mathematics classroom, how can we support these mathematically exceptional
students? In terms of other curriculum areas, Gardner’s multiple intelligence model can be
put to use. Clearly, mathematically gifted students fall under the logical-mathematical
category, so their learning can be enhanced using activities geared towards this intelligence.
These ideas can be used to scaffold a mathematically gifted student in another subject they
may dislike or be struggling in by giving them a different perspective that they can relate to.
It can serve as a way to engage them in a subject that might not usually be associated with
mathematics, so that they can become motivated students in any classroom. The important
part for teachers is communication on students’ progress and methods that can be used to
enhance student learning.
While this resource booklet focuses on issues mathematically gifted students face and how
mathematics teachers can minimise or nullify these, many of the differentiation ideas can be
adapted to different curriculum areas. Differentiation and student development should not
only be for gifted students however – all students should strive to achieve beyond their
potential.
“What is good for the gifted is good for all learners.”
(Delisle, 2000, p1 as cited in Riley, 2004)
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References
Averill, R. & Clark, M. (2007) If They Don’t Care, I Won’t: Showing Caring for Mathematics in
Progress. Retrieved May 25, 2009 from
http://www.nzamt.org.nz/sites/cms/index.php?option=com_content&task=view&id=91&Itemid=1
13
Bicknell, B, & Holton, D. (2009) Gifted and Talented Mathematics Students. In R. Averill & R.
Harvey (Eds.) Teaching Secondary School Mathematics and Statistics: Evidence-Based Practice,
Volume One (pp. 173 – 186). Wellington: NZCER Press
Black, P., Harrison, C., Lee, C., Marshall, B. & Willam, D. (2003). Putting the Ideas into Practice. In
Assessment for Learning: Putting Into Practice (pp. 30 – 57). UK: Open University Press
Easter, A. (2002) Multiple Intelligences. New Zealand: The University of Waikato
Goos, M., Stillman, G. & Vale, C. (2007) Teaching Secondary School Mathematics: Research and
Practise for the 21st Century. Australia: Allen & Unwin
Holton, D. & Daniel, C. (1996) Mathematics. In D. McAlpine and R. Moltzen (Eds.) Gifted and
Talented: New Zealand Perspectives. pp. 201 – 218. Palmerston North, New Zealand: ERDC
Press
McInerney, P. M., & McInerney, V. (2002). Educational psychology: Constructing learning (3rd ed.)
Australia: Prentice Hall.
MacLeod, R. (1996) Educational Provisions: Secondary Schools. In D. McAlpine and R. Moltzen
(Eds.) Gifted and Talented: New Zealand Perspectives. pp. 171 – 184. Palmerston North, New
Zealand: ERDC Press
Ministry of Education (2000) Gifted and Talented Students: Meeting Their Needs in New Zealand
Schools. Wellington: Learning Media
Riley, T.L. (2004) Qualitative Differentiation for Gifted and Talented Students. In D. McAlpine and
R. Moltzen (Eds.) Gifted and Talented: New Zealand Perspectives. Palmerston North, New
Zealand: ERDC Press
Stepanak, J. (1999) Meeting the Needs of Gifted Students: Differentiating Mathematics and Science
Instruction. United States of America: Northwest Regional Educational Laboratory
The Differentiation Toolbox (2009) KUDs. Retrieved 17 September, 2009 from
http://people.virginia.edu/~mws6u/diff/index.htm
Tomlinson, C. A. (1995) How to Differentiate Instruction in Mixed-Ability Classrooms. United States
of America: ASCD
VanTassel-Baska, J. (2004) Effective Curriculum Instruction Models for Talented Students. In J.
VanTassel-Baska (Ed) Curriculum for Gifted and Talented Students. pp. 1 – 12. USA: Corwin
Press
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Appendix I: Gardner’s Multiple Intelligence Model
The material on the following pages relates to Gardner's model of multiple intelligences and
may be used to enhance the learning of all students. Special thanks to Ann Easter for
allowing the reproduction of this material in this booklet.
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Multiple Intelligences
Howard Gardner
Gardner (1983), one of the leading proponents of the Theory of
Multiple Intelligences (MI), defines intelligence as “... sets of
abilities, talents or mental skills which enable individuals to
solve problems or fashion products in such a way as to be
useful in one or more cultural settings.”
According to Gardner (1995), there are at least eight relatively
independent intelligences or areas of special ability:
Verbal Linguistic
Musical Rhythmical
Logical Mathematical
Naturalist
Bodily Kinesthetic
Interpersonal
Visual Spatial
Intrapersonal
Gardner (1998) maintains that while all human beings possess
all of these abilities to some degree, each individual has different
profiles of intelligences. He argues that more students can be
taught effectively if educators take into account their preferred
learning styles and ‘ways of knowing’.
Collated by: Easter, A. (2002). GATE Advisor, School Support Services, The University of Waikato
MULTIPLE INTELLIGENCES
LINGUISTIC INTELLIGENCE is the ability to think in words and to use language
to express and appreciate complex meanings. It is the most widely-shared human
competence and allows us to understand the order and meaning of words and to apply
metalinguistic skills to reflect on our use of language.
Commonly found in: Novelists, poets, scriptwriters, comedians, journalists, editors,
publicists, political leaders, lawyers, and effective public speakers.
Examples of prominent people: Margaret Mahy, Witi Ihimaera, Sam Hunt, Billy T.
James, Kim Hill, Sir Winston Churchill, David Lange, Paul Holmes.
________________________________________________________________________________
LOGICAL-MATHEMATICAL INTELLIGENCE is the ability to calculate,
quantify, consider propositions and hypotheses, and carry out complex mathematical
operations. It enables us to perceive relationships and connections, to use abstract,
symbolic thought, sequential reasoning skills, and inductive and deductive thinking
processes.
Commonly found in: Mathematicians, scientists, engineers, accountants, bankers,
investigators, and computer programmers.
Examples of prominent people: Ernest Rutherford, Albert Einstein, Thomas Edison,
Dr Don Brash, Bill Birch, Dr Jim Sprott, Bill Gates.
________________________________________________________________________________
BODILY-KINESTHETIC INTELLIGENCE is the ability to manipulate objects and
use a variety of physical skills. It also involves a highly-developed sense of timing, and
the perfection of skills and reflexes through the coordination of mind and body.
Commonly found in: Sporting achievers, athletes, gymnasts, dancers, actors,
magicians, surgeons, mechanics, racing car drivers, inventors, and craftspeople.
Examples of prominent people: Susan Devoy, Jonah Lomu, Rob Waddell, Danyon
Loader, Jon Trimmer, Michael Jackson, Harry Houdini, David Copperfield, Marcel
Marceau, Rowan Atkinson, Sir Brian Barret-Boyes, Possum Bourne, John Britten.
________________________________________________________________________________
VISUAL-SPATIAL INTELLIGENCE is the ability to think in three dimensions.
Core capacities of this intelligence include mental imagery, spatial reasoning,
manipulation of images, graphic and artistic skills, and an active imagination.
Commonly found in: Navigators, pilots, sculptors, painters, illustrators, architects,
designers, chess players, strategists, and theoretical physicists.
Examples of prominent people: Sir Peter Blake, Pablo Picasso, Claude Monet, Dick
Frizzle, Robyn Kahukiwa, Sir Michael Fowler, Ian Athfield, Bobby Fischer.
Collated by: Easter, A. (2002), GATE Advisor, School Support Services, The University of Waikato
MULTIPLE INTELLIGENCES
MUSICAL INTELLIGENCE is the ability to discern pitch, rhythm, timbre, and
tone. It enables one to recognise, create, reproduce, and reflect on music. Interestingly,
there is often an affective connection between music and the emotions, and
mathematical and musical intelligences may share common thinking processes.
Commonly found in: Composers, conductors, musicians, vocalists, musical
audiences, recording engineers, piano-tuners, and makers of musical instruments.
Examples of prominent people: Amadeus Mozart, Elton John, Neil Finn, Ron
Goodwin, Dame Kiri Te Kanawa, Sir Howard Morrison, Hirini Melbourne.
________________________________________________________________________________
NATURALIST INTELLIGENCE is the ability to observe, understand and organise
patterns in the natural environment. A naturalist is someone who shows expertise in
the recognition and classification of plants and animals. These same skills of
observing, collecting, and categorising may also be applied to the “human”
environment.
Commonly found in: Vets, zoo-keepers, biologists, botanists, meterologists,
conservationists, outdoor adventurers, and forensic scientists.
Examples of prominent people: Charles Darwin, David Attenborough, David
Bellamy, Eion Scarrow, Diane Goodall, Guy Salmon, Sir Edmund Hillary.
________________________________________________________________________________
INTERPERSONAL INTELLIGENCE is the the ability to understand and interact
effectively with others. It involves effective verbal and non-verbal communication, the
ability to note distinctions among others, a sensitivity to the moods and temperaments
of others, and the ability to entertain multiple perspectives.
Commonly found in: Teachers, counsellors, facilitators, industrial mediators, actors,
politicians, talk show hosts, sales people, public relations officers, and business
managers.
Examples of prominent people: Stephen Covey, Bill Rogers, Winston Peters,
Oprah Winfrey, Jim Hickey, Maggie Barry, Suzanne Paul, Kevin Roberts.
________________________________________________________________________________
INTRAPERSONAL INTELLIGENCE is the capacity to understand oneself one’s thoughts and feelings and to use such knowledge in planning and directing
one’s life. Intrapersonal intelligence involves not only an appreciation of the self, but
also of the human condition.
Commonly found in: Psychologists, spiritual leaders, psychics, mystics, gurus, wise
elders, and philosophers.
Examples of prominent people: Miriam Saphira, Mahatma Ghandi, Mother
Theresa, Sir Hepi Te Heuheu, James K. Baxter, Plato, Socrates.
Collated by: Easter, A. (2002), GATE Advisor, School Support Services, The University of Waikato
Intrapersonal
Naturalist
Interpersonal
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Intuitive/perceptive
Willing to appear different
Self-motivated/independent
Well-developed sense of self
Deeply aware of own feelings
Strong personal values and ideals
Highly-developed sense of purpose
Recognises own strengths/weaknesses
Enjoys collecting things
Interest in natural phenomena
Adapts well to the environment
Enjoys growing/cultivating plants
Empathy/concern for living things
Sorts/labels/classifies natural objects
Keeps detailed records of observations
Likes tramping/field trips/being outdoors
Has many friends
Enjoys being with people
Prefers group/team activities
Sensitive to feelings of others
‘Reads’ social situations well
Co-operates readily with others
Communicates ideas effectively
Effective at mediating disputes
Bodily-Kinesthetic
Visual-Spatial
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Learns best by doing
Skilled at handicrafts
Mechanically-minded
Exceptional control of body
Good timing/reflexes/responses
Likes to find out how things work
Keen to engage in physical activities
Enjoys acting/drama/mime/role-play
Musical
Linguistic
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Possible Traits for Identifying
the Multiple Intelligences
Spells easily
Enjoys reading
Writes fluently
Good listening skills
Expresses ideas precisely
Has a good memory for facts
Likes crosswords, word games
Well-developed sense of humour
May be a confident public speaker
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May be deeply spiritual
Enjoys composing music
Can pitch notes accurately
Skilled at musical performance
Natural sense of rhythm/timing
Able to play a range of instruments
Sensitive to emotional power of music
Responsive to sounds in the environment
Appreciates complex organisation of music
Thinks in pictures
Creates mental images
Advanced spatial awareness
Likes to see the ‘big picture’
Flair for colour/style/design
Enjoys cartoons/illustrations
Likes drawing/painting/sculpting
Easily reads maps/charts/diagrams
Logical-Mathematical
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Thinks abstractly
Is very organised
Keeps orderly notes
Uses logical structure
Enjoys manipulating numbers
Likes working with computers
Approaches tasks systematically
Good at problem-solving activities
Likes experimenting in a logical way
Collated by: Easter, A. (2002). GATE Advisor, School Support Services, The University of Waikato
Learning Activities to Strengthen the Multiple Intelligences
Linguistic
LogicalMathematical
BodilyKinesthetic
Visual-Spatial
• Tell jokes
• Retell stories
• Discuss ideas
• Ask questions
• Play word games
• Speeches/debates
• Create a mnemonic
• Record ideas on tape
• Use similes/metaphors
• Read a variety of texts
• Research topics of interest
• Write poems/letters/essays
• Flow charts
• Brain teasers
• Sequential lists
• Strategy games
• Number patterns
• Statistical analysis
• Develop prototypes
• Create new systems
• Collate information
• Evaluate ideas logically
• Computer programmes
• Compare/contrast/measure
• Gymnastics
• Team games
• Moving around
• Individual sports
• Physical activities
• Body language/mime
• Manipulative materials
• Making/building things
• Hand-eye coordination
• Large/small muscle skills
• Drama/acting things out
• Dance/bodily movement
• Create mindmaps
• Graphic organisers
• Watch a video or film
• Mazes/jigsaw puzzles
• Maps/charts/diagrams
• Photograph/draw/paint
• Demonstrations/models
• Relate to the ‘big picture”
• Colour/highlight/underline
• Imagine/pretend/visualise
• Define/clarify an end result
• Posters/charts/illustrations
Musical
Naturalist
Interpersonal
Intrapersonal
• Raps/jingles
• Writing songs
• Musical scores
• Choral singing
• Listen to music
• Chants/rhythms
• Remember tunes
• Record audiotapes
• Respond to sounds
• Instrumental music
• Melodies/harmonies
• Musical performance
• Collect data
• Make collections
• Nature hikes/field trips
• Observe changes in the
natural environment
• Sketch/photograph natural
objects
• Grow plants/care for pets
• Wildlife protection projects
• Sort/label/classify objects
• Visit zoos/botanical gardens/
natural history museums
• Peer tutoring
• Buddy systems
• Reciprocal teaching
• Brainstorm in groups
• Give/receive feedback
• Negotiation/concensus
• Organise social occasions
• Work in co-operative teams
• Plan/revise/discuss work with
a partner
• Encourage networking and/or
mentor relationships
• Use affirmations
• Keep a diary/journal
• Self-directed learning
• Specify personal goals
• Reflect on past events
• Self-management skills
• Visualise the end result
• Allow time for reflection
• Use wall charts to enhance
peripheral learning
• Identify barriers to personal
learning
Collated by: Easter, A. (2002). GATE Advisor, School Support Services, The University of Waikato
Appendix II: Presenting the Booklet
A presentation was given in conjunction to this booklet, where an example of differentiating in a
mathematics classroom was provided. The following are the associated PowerPoint slides and
notes used in the presentation.
Mathematically gifted students. Intimidating, right?
The idea of a kid in a mathematics class, spouting
numbers like a pro is scary. There’s a likelihood
that they are working well beyond their peers, and
it feels like they aren’t learning anything in class.
And what about beyond mathematics? What
happens to them then? Does their English teacher
pop them in a cupboard at the back of the room?
Okay, so being placed in a cupboard is probably not an issue mathematically gifted students
will face, but there are problems unique to mathematically gifted students every teacher
should consider.
(The resource booklet is an attached document that
provides practical ideas that can be used in the
classroom. This presentation is used as an overview
to this resource with an example of a class where
some of the ideas were implemented. )
According to Holton and Daniel (1996), mathematics
is unique in that children who are mathematically
gifted are usually discovered early. They tend to
have a fascination with numbers and patterns and are
very inquisitive.
Karen Chow
Mathematically Gifted and Talented Students
Page | 23
Although students may be identified early as gifted
in mathematics, they might not be getting the access
or support that they need. They end up hiding the
fact that they’re gifted to conform to the norm. Even
students who get the support that they need to
progress might prefer to appear “normal,” because of
the stereotypes attached to being mathematically
gifted. Nobody wants to appear being different or a
“freak” – and being a freak and being freaky are two different things. Students don’t mind
being freaky, and so the challenge is to make being mathematically gifted freaky. Part of this
is in making the classroom an inclusive one, so that all students feel confident in their
abilities and able to express their opinions without fear of ridicule. Every teacher has a
different method for this, but in general, teachers should avoid making judgements on any
student before they enter the class. It’s easy to look at their previous performance or judge
them on their behaviour and then stick a label on them, but students aren’t cows to be
branded. By labelling them, you can be hindering their holistic development and ignoring
students who could potentially be gifted. That’s why ongoing assessment is important, so that
teaching and learning can be appropriately adjusted for students. But how can we cater for
gifted students in mathematics?
In an observed year 12 class, there were a fair number
of strugglers and some gifted students. The calculus
topic was being taught at the time and although a lot of
the skills used in this topic are found in algebra,
students tended to struggle with putting these ideas
into practice in calculus. For the gifted students, this
wasn’t an issue, so as their peers struggled through the
basics of calculus, they were speeding on ahead. This
isn’t to say that the gifted students were anything alike. There was Bronwyn, a studious young
lady who was motivated to excel. She usually worked independently ahead of the class, and
would rarely ask for any help from the teacher. Nick on the other hand, was a distracted and
distracting young man who liked clicking his joints, high-fiving his friends and rapping in class.
Karen Chow
Mathematically Gifted and Talented Students
Page | 24
The question here was what to do? One word: Differentiate. This refers to educational
differentiation that can be applied across all curriculum areas…not to be confused with the topic
differentiation in calculus. With differentiated teaching, it’s important to start at a pace that
you’re comfortable with, whether it be differentiating content, process or product. This can look
different for different people. In this instance the whole class was taught a concept, and then
given a choice as to what activities they would do for the lesson. If it meant working through
ideas from previous lessons, there were resources for that; otherwise students would work on
concepts presented during the whole-class teaching. For those who had moved beyond this, they
were provided with an extension activity that required more analytical, evaluative thinking. There
were opportunities for all students to extend their learning, and it was student-driven rather than
teacher-directed. This way, students felt that they’d achieved something in class rather than
blindly struggled through an activity or simply performed an exercise they’d done or felt like
they’d done twenty times before.
It does take a lot of time to find resources for this
type of differentiation when students are at
completely different stages of learning. However, if
students are forced to be at a level that they’re not
ready for or have moved past, they’ll lose interest,
and the next thing you know they’re getting into all
sorts of trouble…like clicking joints, high-fiving
friends and rapping in class. Adjusting to the students
in this way is what Tomlinson (1995) calls the “readiness” factor. For gifted students, this might
mean accelerating or enriching what they happen to be working on. Although there is the
acceleration vs. enrichment argument, a range of strategies should be used and this should depend
on where the student is heading next year and whether the activity is beneficial to their
development. If they’re just being drilled in a skill they’ll learn again next year, by giving them
the material this year, you’re just causing problems for next year’s teacher. If they’re being given
a one-off activity that is not evaluated or considered any further, students see it as a useless “gapfiller.”
Differentiating content goes a long way and presumably so does differentiating processes and
products, but other ideas that can be used to help gifted students progress at a pace they’re
comfortable with that works in any class include:
Karen Chow
Mathematically Gifted and Talented Students
Page | 25
Flexible grouping strategies. This can mean having
students work by themselves, in small groups and the
types of grouping you use can change. You might
want the struggling students in a group and gifted
students in another, or you might want mixed ability
groups.
It’s
good
for
students
to work
in
homogeneous and heterogeneous groups to get
different perspectives but it can take a lot of work for
this to be effective.
Having clear expectations of the students. Although they might be operating at different levels,
ensure that students understand what is expected of them either verbally or written on the board.
This can be a point that’s easily overlooked, because it seems so obvious. Somehow the
instructions need to be given out, right? It’s surprising how easy it is to skip over key details
though. Of course, part of having these expectations is making sure that students are given
activities that they’ll learn something from, rather than drilling them with the same old exercises
that they can do in their sleep.
In short, they have goals that are challenging but achievable. It’s the “Why bother?” factor.
“Why should I finish this quickly when all that will happen is that I’ll get more of the same?”
After a while, gifted students will stop trying, and they lose their respect for you – if you’re not
making an effort, why should they?
This is why circulating and one-on-one time with students is important. Talk to them about
their work using various questioning techniques, find out what they might want to do in the next
lesson. However, important point, don’t let them sidetrack you with random talk. Circulating also
doubles as a time when you can give students informal feedback.
Outside of the mathematics classroom, how can we
support these mathematically exceptional students?
Putting aside any extracurricular activities students
can join through the school, in terms of other
curriculum areas, Gardner’s multiple intelligence
model can be put to use. Clearly, mathematically
gifted students fall under the logical-mathematical
category, so their learning can be enhanced using
activities geared towards this intelligence. This isn’t to say that students shouldn’t be encouraged
to develop holistically, but these ideas can be used to scaffold a mathematically gifted student in
Karen Chow
Mathematically Gifted and Talented Students
Page | 26
another subject they may dislike or be struggling in by giving them a different perspective that
they can relate to. It can serve as a way to hook them into a subject that might not usually be
associated with mathematics, so that they can become engaged and motivated students in any
classroom. The important part for teachers is communication on students’ progress and methods
that can be used to enhance student learning.
Mathematics can bring teachers together, and it can
be used in other subjects to engage mathematically
gifted students…so don’t be intimidated if there’s a
mathematically gifted student in your class, or stuff
them into cupboards. Work with their differences
both in and outside mathematics, and you never
know – something very beautiful might come out of
it.
A big thank you to the VUW Gifted and Talented
2009 class and Judy Lymbery for the support
provided in coming up with this presentation. Thanks
also to the year 12 class from TE2 who experienced a
student
teacher
attempting
to
(educationally)
differentiate content for the first time, particularly
Bronwyn and Nick (These aren’t their real names, but
you know who you are!). You’re all some of the
freakiest people around.
Karen Chow
Mathematically Gifted and Talented Students
Page | 27