the challenge of challenging gifted students cesa #11 workshop october 19, 2009 dr. stephen...

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The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis [email protected] 1

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Page 1: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The Challenge of Challenging Gifted Students

CESA #11 WorkshopOctober 19, 2009

Dr. Stephen [email protected]

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Page 2: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Goals for this morning

• Review basic concepts to develop a common language

• Examine the “emerging definitions” of giftedness

• Promote parent & teacher dialogue• Assess needs for workshops 2 & 3

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Page 3: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

2005 Gifted and Talented Definition

Gifted and talented children and youth are those students with outstanding abilities, identified at preschool, elementary, and secondary levels.

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Page 4: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

2005 Gifted and Talented Definition

These students are capable of high performance when compared to others of similar age, experience, and environment, and represent the diverse populations of our communities.

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Page 5: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

2005 Gifted and Talented Definition

These are students whose potential requires differentiated and challenging educational programs and/or services beyond those provided in the general school program.

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Page 6: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

2005 Gifted and Talented Definition

Students capable of high performanceinclude those with demonstratedachievement or potential ability inany one or more of the followingareas:

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Page 7: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

2005 Gifted and Talented Definition

General intellectualSpecific Academic subjectsCreativityLeadershipVisual and performing artsFrom the MGTDC (MDE) Advisory Committee

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Page 8: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Giftedness is asynchronous development, placing the gifted child “out of phase” with:

SelfAge peersAt level tests, assessments, expectationsMuch of popular culture

(The Columbus Group, 1991)

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Page 9: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Two examples of asychronicityA kindergarten teacher is explaining how

dinosaurs are discovered, and states, “A geologist is a scientist who studies these fossils.”

Jenny, age 4 says, “I don’t mean to be rude Ms. Mays, but it’s a paleontologist that examines the dinosaur bones.”

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Page 10: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

A middle school example

• The majority of the 200,000 middle school students who take the SAT and ACT score as well or better than high school seniors

• The stronger students from that cohort can absorb one year of a high school course in 3 weeks

• The strongest of those candidates can absorb one year of a high school course in 10 days

A Nation Deceived

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Page 11: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Giftedness is “Abnormal”

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Top 3 -10% of population in any given area of ability

Page 12: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Gifted People Are Different

• Neurosystem• Perception• Behavior• Environment

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Page 13: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Neurology

• Larger Frontal Lobes• Faster synapses• More efficient processes

See Sanjay Gupta

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Page 14: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

My Beliefs About This Topic

• Gifted students exist, are an exceptional population, and require accommodations to be challenged in school

• Gifted students are the most underserved population in most schools (i.e., they learn the least)

• Serving them appropriately would benefit them and every other student in a school setting

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Page 15: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The four achievement gaps

• Racial• Gender• Economic• Aptitude (between what is being learned, and

what could be learned with appropriate GT programs and services)

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Page 16: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Gifts vs. TalentsF. Gagne

“Giftedness refers to measures of potential, of untrained natural ability, while talent is reserved specifically for

indices of achievement, of the performance attained as the result of a

systematic program of training and practice.”

(Gagne 1995)16

Page 17: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

A Common G/T Vocabulary

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Gifted Talented

• high aptitude • high achievement• nature • nurture• ability • performance• potential • environment• threshold • accomplishment• endowment • output

(Gagne 1995)

Rigor & Challenge

Differentiation

Page 18: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

How Much Time is Wasted in a typical classroom for GT’s?

• 140 IQ = 50% of their time• 170 IQ = 99% of their time

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Hollingworth (1942), Renzulli, Silverman (1991)

Page 19: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The National Research Center on the Gifted and Talented: Classroom Practices Study

Approximately 40-50% of traditional classroom material could be eliminated for targeted students.

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Page 20: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Gagne’s Metric SystemLevel Label Ratio IQ SD

5 Profoundly 1:100,000 165 + 4.3

4 Exceptionally 1:10,000 155 + 3.7

3 Highly 1:1,000 145 + 3.0

2 Moderately 1:100 135 + 2.3

1 Mildly 1:10 120 + 1.3

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Page 21: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Questions and comments?

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These ideas square with my beliefs. I’d like to

add. . ..

These are the ideas that are going around in my head.

This made me uncomfortable!

Some of the ideas with which I disagreed . . .

Page 22: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Why are GT’s Underserved?

22From Get Off My Brain, by Randy McCutcheon. Illustrations:Pete Wagner

Level 1 2 3 4 5 w/o DI

Page 23: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Coercive Egalitarianism

Forced regression toward the mean through indifference or neglect

Stephen Schroeder-Davis

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Page 24: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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Gifted Children

$0.03

Drug Abuse

Prevention $2

Reading First $3

No Child Left Behind $64Children with

Disabilities $32

Federal Education Budget

Page 25: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Training, preparation, programming

• No states have comprehensive policies in gifted education in all areas.

• only 77 of 3500 HEIs offer GT courses

• 18 states offer no teacher preparation

• Only 11 mandate GT fundingNAGC National Conference Report, 2007

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Page 26: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

NCLB and GT’s

• Remedial, deficit-based• Teach what is tested (narrowing curriculum, which

“homogenizes” talent)• One-size-fits all education to a HIGHLY diverse

population• Curriculum reduced to basic skills emphasis• Goals are statistically impossible to meet• “Adequate yearly progress” does not apply to, refer to,

or even acknowledge needs of- GTs• No incentive to challenge high ability studentsSee M. Gentry

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Page 27: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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High-Achieving Students in the High-Achieving Students in the Era of NCLBEra of NCLB

Thomas B. Fordham Institute

Page 28: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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BackgroundBackground• First two studies of a multifaceted research

investigation of the state of high-achieving students in the No Child Left Behind (NCLB) era.

• Part I: An Analysis of NAEP Data, by Tom Loveless: achievement trends for high-achieving students since the early 1990s and, in more detail, 2000.

• Part II: Results from a National Teacher Survey, by Steve Farkas and Ann Duffett: reports on teachers’ own views of how schools are serving high-achieving pupils in the NCLB era.

Page 29: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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High-achieving students made minimal High-achieving students made minimal progress since 2000progress since 2000

• While the nation’s lowest-achieving youngsters made rapid gains from 2000 to 2007, the performance of top students was languid.

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Page 30: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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Page 31: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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Page 32: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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Struggling students command attention…Struggling students command attention…

• Teachers are much more likely to indicate that struggling students, not advanced students, are their top priority.

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Page 33: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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• Low-achieving students receive dramatically more attention from teachers.

Page 34: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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but that doesn’t reflect teachers’ own viewsbut that doesn’t reflect teachers’ own views• Teachers believe that all students deserve their fair

share of attention.

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Page 35: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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ImplicationsImplications

• Languid growth of high-achieving students is associated with the introduction of NCLB (and, earlier, with state accountability systems).

• Most teachers, at this point in our nation’s history, feel pressure to focus on their lowest-achieving students.

Page 36: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Teachers: does the Fordham study reflect your reality?

1. Share agreements! 1. Share disagreements!

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Yes, in these ways No, because

Page 37: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The nature of the curriculum

• Mass produced, typically at “grade-level”• “Spirals,” with frequent repetitions• Paced (at best) for the average learner• Requires a high degree of differentiation, flexibility

and accelerative options to work for advanced learners

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Page 38: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Grouping Definitions: Tracking

• Tracking: sorting students, usually once a year, by ability level and then scheduling all of their classes together:

• Uni-dimensional• Inflexible• Permanent (at least for that year)• Placement criteria may be invalid or irrelevant

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Page 39: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

What would happen if GT’s were challenged appropriately?

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Learners begin here

Challenging, differentiatedCurriculum for all

Gifted learner’s faster learning pace

Achievement “gap” increases due to appropriate growth for all students

Page 40: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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Page 41: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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Page 42: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

What Zone Am I In?

• Too Easy• I get it right away…• I already know how…• This is a cinch…• I’m sure to make an

A…• I’m coasting…• I feel relaxed…• I’m bored…• No big effort

necessary…

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On Target I know some

things…

I have to think…

I have to work…

I have to persist…

I hit some walls…

I’m on my toes…

I have to re-group…

I feel challenged…

Effort leads to success…

Too Hard I don’t know where

to start…

I can’t figure it out…

I’m spinning my wheels…

I’m missing key skills…

I feel frustrated…

I feel angry This makes no

sense…

Effort doesn’t pay off…

THIS is the place to be… THIS is the achievement zone…

Page 43: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The “emerging view” of talent development focuses on:

• Environment• Effort• Coaching• “luck”• The “10,000” hour rule• Could be misconstrued to discount aptitude

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Page 44: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The four sources of the emerging view on an aptitude continuum

• “Talent is Overrated” (Colvin, 2008) essentially denies the validity of heritable (intellectual) traits focusing on (deliberate) practice, hard work and passion

• “The Talent Code” (Coyle, 2009) reluctantly acknowledges, but heavily discounts, heredity (aptitude), focusing on ”deep practice,” ignition, master coaching, and myelin (!)

• “Mindset” (Dweck, 2006) overtly acknowledges aptitude, but focuses on effort, persistence, and risk-taking (a “growth” mindset)

• “Outliers” (Gladwell, 2008) overtly and consistently acknowledges aptitude,but focuses on environment, practice, mentors, and “luck”

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Page 45: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Gifts Talents

10,000 hour ruleDaniel Levitin, Michael Howe, Malcolm

Gladwell, and Many, many others

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Page 46: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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Developmental Process K-12

Intrapersonal CatalystsMotivation•values•interests•efforts•persistence•work habits

TemperamentPersonality

Physical attributes

Giftedness(potential)•Intellectual

•Creative •Socio-Affective

•Sensori-Motor

Talents(Skills)

Academic•Language•ScienceArts•Visual•Drama•Social action•Chess•Video gamesSportsLeisure

Milieu: physical, cultural, social, familial

Persons: parents, teachers, mentors, peers

Provisions: programs, activities, services

Events: encounters, awards, accidents

Gagne’s Talent Development Model

Page 47: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Another interpretation: “Talent is Overrated” (Colvin, 2008)

“Deliberate practice is difficult. It hurts.”

“Deliberate practice” is focused, intense, specific practice designed to increase performance (+ hard work + passion) = talent

How are gifted students to engage in deliberate practice and hard work, let alone develop passion, without challenging school experiences?

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Page 48: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Another interpretation: “Mindset” Dweck, 2006)

“Those with the growth mindset found setbacks motivating. They’re a wake-up call.”

How are gifted students to experience setbacks and mistakes without challenging school experiences?

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Page 49: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Another interpretation: “The Talent Code” (Coyle, 2009)

“Deep practice . . . takes events we would normally strive to avoid-namely, mistakes-and turns them into skills.”

How are gifted students to experience “deep practice” without challenging school experiences?

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Page 50: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Another interpretation:”Outliers” (Gladwell, 2008)

What does the “Hamburg Crucible” and the 10,000 hour rule imply for gifted students and talent development?

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Page 51: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The Beatles’ “Hamburg Crucible”

From 1960 - 1962 the Beatles played in Hamburg, Germany:

• Five trips• 270 nights• 8 hours per night, 7 nights a week• 1,200 live performances in 18 months

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Page 52: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The Beatles’ “Hamburg Crucible”2

Does this mean that any four musicians playing 1,200 live performances, could equal the Beatles’ legacy?

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What do you think?

Page 53: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The Beatles’ “Hamburg Crucible”3

Lennon: “We had to try even harder, put our heart and souls into it . . . we had to play for 8 hours and so we really had to find a new way of playing.”

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Page 54: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The Beatles’ “Hamburg Crucible”4

Biographer Philip Norman, “They learned not only stamina. They had to learn an enormous amount of numbers-cover versions of everything you can think of-not just rock and roll, a bit of jazz too. When they came back, they sounded like no one else. It was the making of them.”

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Page 55: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

The Beatles’ “Hamburg Crucible”5

Author Gladwell: “The Beatles are undeniably talented. Lennon and McCartney had a musical gift of the sort that comes along once in a generation.”

Have you ever heard of “Rory and the Hurricanes”? They too were in Hamburg for a long time - but didn’t become the Beatles.

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Page 56: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

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Developmental Process K-12

Intrapersonal Catalysts

Work ethic

Passion and

Persistence

Degree of Giftedness(potential)Can’t be fulfilled without appropriatecurriculum challenges and GTprogramsand services!

(Skills)Academic•Language•ScienceArts•Visual•Drama•Social action•Chess•Video gamesSportsLeisure

Milieu: physical, cultural, social, familial

Persons: parents, teachers, mentors, peers

Opportunities: programs, activities, services

Events: encounters, awards, accidents

Luck

10,000 hours

GT’s should “succeed” by learning, not by exceeding an arbitrary standard

Page 57: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Gagne’s Formula from 1995 (!)

Aptitude + Catalysts + Practice = Achievement

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High Aptitude/Catalysts/Practice

Less Aptitude/Catalysts/Practice

Virtually everyone can improve significantly in virtually any endeavor, but that does not mean everyone is gifted. It does mean that all students needan appropriately challenging education to thrive!

Page 58: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

Related questions

If you were to practice with the same intensity and for the same duration, could you eventually equal:

• Tiger Woods?• Maya Angelou?• Steven Spielberg?• Could you become an Olympic swimmer?

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Page 59: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

My Conclusions & Recommendations

• Heritable differences are real, and need to be accommodated from grade K - College graduation

• Gifted students are an exceptional population, requiring specialized programs and services if they are to optimize their development

• Appropriate teacher training and classroom differentiation work, but need to be applied to all students

• Teachers need to understand and practice “high-end” differentiation if schools are to work for GTs (our afternoon session)

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Page 60: The Challenge of Challenging Gifted Students CESA #11 Workshop October 19, 2009 Dr. Stephen Schroeder-Davis stephen.schroeder-davis@elkiver.k12.mn.us 1

On to the NAGC proposal

This fall, a select committee of GT experts submitted a new, expanded definition of giftedness to the National Association for Gifted Children (NAGC) in the hopes of gaining more federal funding and expanded support for gifted children. That (unpublished) definition follows.

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