Mathematically gifted and talented students - A resource book.pdf

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


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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.

Karen Chow


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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 Karen Chow


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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 Karen Chow


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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!


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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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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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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

• • • • • • • •

• • • • • • • •

• • • • • • • •

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

• • • • • • • •

• • • • • • • •

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 • • • • • • • • •

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

• • • • • • • • •

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 • • • • • • • • •

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.

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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.

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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:

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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

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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.

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