Developing Math Talent:
A Guide for Educating Gifted and Advanced Learners in Math
by Susan Assouline and Ann Lupkowski-Shoplik
Selected quotes from
Susan Assouline and Ann Lupkowski-Shoplik.
Developing Math Talent :
A Guide for Educating Gifted and Advanced Learners in Math.
Waco, TX, USA: Prufrock Press, 2005; ISBN 1-59363-159-6.
See this book at Amazon.com]
See this book at Google Books]
compiled by Tom Verhoeff
in August 2010.
Bold face emphasis is mine.
Reading this material is no substitute for reading the book.
I hope that it will make you more interested in the material.
NOTE: This book is not a book that provides material
for direct use in teaching math to talented students.
It is about all imaginable issues surrounding the
teaching of math to talented students.
The book contains extensive references, also to specific resources;
a glossary, and an index (the latter could have been even better).
If you just want a summary,
skip directly to Lessons Learned
in the last chapter.
Ch.1: Myths About Mathematically Talented Students
two often-stated developmental myths are "He's too young to start algebra"
and "Students who skip a grade will have social problems when they are
A commonly heard programmatic myth is "But, if you push her ahead now,
she won't have any math left to study in high school."
... the myths about mathematically talented students are either fiction or,
at best, half-truths;
nevertheless, they have acquired mythical power and are often used by
teachers, administrators, and [...] parents as roadblocks to
developing appropriate interventions for mathematically talented students.
- Myth 1:
Only students identified for a gifted program are mathematically talented.
- gifted programs ...
tend to emphasize the all-around gifted student.
[T]his ... effectively eliminates students who have a special talent
in just one area and average ... abilities in other areas. ...
However, there is no guarantee that the mathematically talented
student will actually receive, through the [general] gifted program,
advanced curricular opportunities that correspond to mathematical talent.
- Myth 2:
Results from standardized, grade-level testing are sufficient
for identifying mathematically talented students.
- above-level test helps to measure the students' abilities
[G]rade-level testing ... does not give information that is
precise enough to [differentiate good, talented, and
exceptionally talented students].
- Myth 3:
Gifted students respond equally well to the same curriculum.
Our research has shown that gifted students are a varied group
with respect to their interests ... and abilities ...
[W]e don't recommend just one curriculum for all gifted students.
- Myth 6:
Mathematically talented students are computation whizzes.
Many ... mathematically talented students ... have excellent
conceptual skills, but their skills at computation are less developed.
[T]eachers are often tempted to hold students back from learning
advanced concepts until their "basic skills" catch up.
This ... may actually be detrimental to his or her mathematical development.
- Myth 7:
Mathematically talented students cannot be identified until high school.
[I]t is necessary to identify mathematically talented students well before
high school so that adjustments can be made to their educational programs.
- Using above-level tests, we have successfully identified students
as young as third grade for challenging programs in mathematics ...
- Myth 9:
The best option for mathematically talented elementary school students
- Acceleration should not be dismissed for talented students
automatically because of their young age.
Although enrichment is appropriate and necessary for mathematically
it is not the only option and it might not be the best option
for any particular student.
[I]t is a frequent practice in [the USA] for gifted students
to participate in pull-out programs where the topics ... are ...
unrelated to the regular curriculum. ...
[T]hese ... do not advance the student's understanding of mathematics.
- Myth 11:
If mathematically able students study mathematics at an accelerated pace,
they will run out of math curriculum before they reach high school.
[T]here is always more mathematics to study
(ask any college mathematics professor). ...
Students, educators, and parents might need to be flexible and creative
to ensure that students are receiving the appropriate mathematics course(s).
- Parents are their children's primary advocates. ...
- Effective advocates maintain a positive attitude of cooperation
with the school throughout the process. ...
- Objective information is critical for effective advocacy. ...
- Change is neither easy nor immediate...
We advise parents to concentrate on adapting the current situation
so that their child's needs are met in a timely manner,
rather than trying to overhaul the school system.
What legal options are available when parents perceive
that school policy does not match their child's academic needs?
... [T]here is no [US] federal government mandate requiring school districts
to provide special programs for gifted students.
[G]ifted children are not a [US] constitutionally protected group
[S]ome states [in the USA] have a mandate for the provision of gifted education,
whereas others only "permit" gifted education.
However, ... informal and quasi-formal resolutions are preferable
to those that involve the courts [of law].
The responsibility of finding appropriate programs with adequate challenges
for mathematically talented students ultimately falls on the parents.
Ch.3: Educational Assessment
- Thorough and objective information obtained through educational assessment
is essential in advocating, identifying, and programming
for mathematically talented students.
- Testing is one of four components of an educational assessment.
The other three components include interviews,
observations, and informal assessment.
- [T]here are thousands of tests and well-articulated theories
underlying the development of these tests. ...
Testing should be driven by a question. ...
Tests that measure ability, aptitude,
and achievement have many similarties,
but also some important differences. ...
Linn and Gronlund ... have provided some informative distinctions
by the extent to which the content items are dependent
on specific learning experiences. ...
[A]chievement tests are very much based upon the student's
familiarity with specific subject-matter content;
aptitude tests ... measure problem solving in specific content areas
taught in school;
and [ability tests or] general problem-solving tests
... are unrelated to school learning...
Ch.4: The Diagnostic Testing --> Prescriptive Instruction Model
- The Diagnostic Testing --> Prescriptive Instruction (DT-->PI) model
is a five-step procedure designed to tailor instruction
to a student's learning needs. ...
- Using tests developed for older students (above-level testing)
is an important component of the DT-->PI process.
For these students, the grade-level test does not adequately measure
because answering all of the items correctly informs us that they perform well
compared to other students in their grade.
However, we don't know the extent of their talents
due to the ceiling effect of the test.
In the DT-->PI model, the talented students simply take a test
that was designed for older students,
which has been shown to be extremely effective with talented students.
A more challenging test allows students to demonstrate what they know
and what they don't know in mathematics.
Above-level tests spread out the scores of able students,
helping us to differentiate between talented students and
exceptionally talented students.
This information is extremely helpful for good educational planning...
When [gifted] students take a standardized test that was designed
for their grade level, their responses to all or most of the items are correct,
thus reaching what psychologists call the
Step 1: Determine Aptitude
[E]lementary school students who have scored at the 95th percentile
or above on a nationally standardized [grade-level] test ... are eligible
to participate in above-level testing [i.e., Step 1].
- at least two grade levels
higher than the student's current grade placement ...
We strongly recommend the use of a standardized, nationally-normed
[aptitude] test in [Step 1 itself].
[I]t is important to select [a test] that is
The point of administering an above-level assessment is to determine
what a student doesn't know
so that instruction can be based upon filling in their knowledge gaps.
Standardized tests are especially effective for this purpose
because they provide corresponding instructional objectives.
When Step 1 is completed, the scores will determine whether or not
the student will go to Step 2.
Scores earned by talented students should be compared to scores for
students in the above-level grade for which the test was designed. ...
[W]e recommend that students who earn scores at the 50th percentile
on the [above-level] test administered in Step 1 move on
to Step 2 of the DT-->PI. ...
We do not recommend that talented students who score below the 50th percentile
on the above-level test continue with the DT-->PI process.
Instead, we encourage curricular adjustments such as enrichment
or problem solving activities ...
- Step 2:
In Step 2,
we administer achievement tests, starting with the grade level
that approximates the above grade level of the aptitude test
used in Step 1.
There are three possibilities for students in this step of the model,
and only the second results in moving on to Step 3:
- They do very well on the diagnostic pretest (85th percentile or above).
These students should take a more difficult version of the test.
- They do moderately well
(earning scores between the 50th and 85th percentiles).
These students should move on to Step 3...
- They do not do well (the scores are below the 50th percentile).
These students should exit from the model and
receive enrichment instruction in specialized mathematics topics.
- Step 3:
Readminister and Evaluate Missed Items
The purpose of Step 3 is to gain a more complete understanding
of what topics the student does and does not know
so that instruction can focus on new material.
The first activity of Step 3 is to return the test booklet
to the student and have him or her rework missed or omitted items
or items marked with a question mark.
- [... further details omitted ... ]
The final activity of Step 3 is identifying
who will best serve as the instructor for Step 4.
- Step 4:
- excellent background in mathematics.
This person is referred to as a mentor.
The prescriptive instruction of Step 4 is based on a thorough
analysis of the testing results obtained in Steps 1-3.
Prescriptive instruction may or may not be provided by the classroom teacher,
but it is critical that whoever does provide it
- The mentor designs an instructional program
based upon the diagnostic testing. ...
- In this important phase of the model,
the mentor works with the student on the concepts--not the items--
he or she does not understand.
Very little time is spent on the topics
about which the student has demonstrated mastery.
- Step 5:
On the posttest,
students who score above the 85th percentile for the appropriate
above-level norms are considered to have mastered the material.
[The others] require additional instruction and practice with the material.
After the student has demonstrated mastery and
the mentor is satisfied that he or she has adequately filled in
any knowledge gaps on that particular topic,
the student reenters the process at Step 2,
using achievement tests and materials for the next level or topic.
Thus, the student studies the mathematics topics in a systematic,
sequential fashion, demonstrating mastery before moving on.
The final step ... is posttesting to determine
if students have mastered the content.
After completing the prescriptive instruction [i.e., Step 4],
students take a parallel form
(equivalent in difficulty, but with different items)
of the same [achievement] test used for pretesting.
- What Is the Role of the Mentor?
Stanley ... described the mentor's role as follows:
For the "prescriptive instruction" on needs a skilled mentor.
He or she should be intellectually able, fast-minded, and
well-versed in mathematics
considerably beyond the subjects to be learned by the "mentee(s)."
This mentor must not function didactically as an instructor,
predigesting the course material for the mentee.
Instead, he or she must be a pacer, stimulator, clarifier, and extender.
It is not necessary for the mentor to be a trained mathematics teacher.
... [E]ngineers, college professors, undergraduate math majors, and
graduate students in mathematics have been successful mentors.
It is critical that the mentor have a good understanding of both
the mathematics the student is currently studying and
the mathematics he or she will study in the near future.
- parents mentor their children,
even if they hae the appropriate background in mathematics.
We do not recommend that
- Sometimes, it is also suggested that a high school student
might be a mentor.
Our experience has been that high school students ...
would have difficulty managing the responsibilities of planning
for mentoring sessions and meeting consistently on a weekly basis.
- The mentor-to-student ratio is often 1:1.
However, a skilled mentor can work successfully with up to five students ...
- How are the Mentoring Sessions Conducted?
[D]aily instruction is not necessary.
We recommend that the mentor and student meet for a total of 2 hours
For students in fifth grade and older,
it may be best to meet once a week for 2 hours.
Because younger students may be more easily fatigued,
we recommend shorter, more frequent sessions
(twice a week for an hour each time).
- spiral approach
of most mathematics textbooks,
where students are not expected to master a concept
because they will be exposed to it the following year.
During their meetings,
the mentor works with the student on the principles
(not the specific test items) the student didn't understand.
Students must demonstrate mastery on one topic
before moving on to the next,
but they are not required to work through every page of a text.
This process is in sharp contrast to
- What Happens Between Meetings?
- Homework completed on a daily basis (4-5 days per week)
is essential in a mentoring situation
because the student does not meet with the mentor every day.
However, the homework should not take more than 20 minutes
to complete each day.
Students participating in the mentor-paced program should be
self-motivated and self-disciplined.
Parents remain actively involved by
supervising the completion of homework.
[I]deally, the student will find participation in the DT-->PI process
more challenging and more fun than regular math classes in school.
- Many Teachers Worry That Removing
Talented Students From Their Classes Will Remove an Important Role Model
for Average Students
- [E]ffective role models must be somewhat close in ability
to those who would benefit from exposure to the models.
Social comparison theory tells us that average students would not look upon
exceptionally talented students as academic role models.
[T]his remains a common myth.
- [ ... other arguments omitted ... ]
- How Are Student Evaluation Issues
For work that is completed,
students should receive both a grade and credit.
- How Do Educators Respond to the
Mentor-Paced Student's Need for Acceleration?
By definition, mathematically talented students
who participate in a mentor-paced program will be accelerated in mathematics.
Research over the last 50 years has shown that acceleration
for mathematically talented students is an appropriate and useful option ...
- How Effective is the DT-->PI Model?
Many students [participating in the DT-->PI]
have mastered an entire year's worth of material
with just 75 hours of instruction.
Olszeweski-Kubilius ... noted that,
"Talent search summer programs have shown that some students
can learn at a much faster pace than heretofore believed,
without sacrificing the level of subject mastery or preparation
for future courses, all with higher student satisfaction" ...
Ch.5: Talent Searches for Elementary Students
Research findings on the EXPLORE test have been consistent each year.
The most important finding is ... [that] the scores of those students
who were already in the top 5% of their age group
(and, therefore, at or near the ceiling of the grade-level test)
were spread out when they took the EXPLORE as an above-level test ...
[T]heir test results are spread out into a new normal curve ...
- The goal of an academic talent search is to discover,
via above-level testing,
students who are ready for advanced academic challenges
that are not typically offered as part of the regular school curriculum.
- Talent search participants,
although much younger than the students for whom the tests were intended,
typically earn higher scores on the tests
than the students for whom the test was developed.
Thus, initial concerns that students who participated in the talent search
would be unnecessarily frustrated are unfounded.
- There are many educational options for mathematically talented students,
ranging from enrichment within the regular classroom to
No one option is "right" for all students.
- Acceleration is appropriate and necessary for many
mathematically talented students.
Acceleration as a successful program option has been strongly researched
for several decades.
- Enrichment and Acceleration
- acceleration versus enrichment
is decades old.
Enrichment proponents are concerned that accelerating students
means that they will learn information in a superficial manner,
while advocates of acceleration are frustrated by
the repetitive tasks students are asked to do
even if they have already demonstrated mastery of a topic.
The debate about
- [A]cceleration as an educational strategy for challenging gifted children
is clearly supported by research,
but has received only minimal support from educators in the field.
The four forms of enrichment are:
More of the same [material that was already mastered]...
- Irrelevant academic enrichment:
... challenge the academic experiences ...;
however, ... has very little to do with the talent area ...
- Cultural enrichment:
... not relevant to the specific
talent area, [but] ... has cultural merits ...
- Relevant academic enrichment:
... exposure to special topics in the specific talent area...
- Comments Concerning Acceleration.
[M]any school officials are categorically opposed to students
accelerating in school ...
This opposition seems to stem from either one bad experience with
acceleration or a major concern with students' social development.
Accelerated students may have fewer opportunities to interact
with same-age peers.
However, we do not feel that this is a sufficient reaon
to hold students back.
[F]ive important points concerning acceleration:
- Healthy social development means learning to get along
with people of all ages and skills.
- There are many opportunities outside of school
to interact with age peers.
- Having a large number of same-age friends isn't as critical
to a child's social development as having a few close friends
with whom to share ideas.
- An indicator of social readiness for acceleration is
a preference for playing with more mature children.
- Finally, long- and short-term planning is essential. ...
We also noted:
Acceleration accomplishes much that is useful. ...
If a student is not permitted to accelerate,
he or she may be ... denied the opportunity to study
a more advanced subject later.
[W]aiting to study a subject eliminates the chance to take new,
more challenging courses.
The mathematically talented student
who craves higher level mathematics and related subjects
would be wise to pursue those interests.
- combination of acceleration and enrichment makes the most sense.
By definition, relevant enrichment in mathematics must have some
acceleration and approprirate acceleration will have an enrichment component.
Schiever and Maker ... carefully explored the various forms of
enrichment and acceleration and concluded, ... that a
- What Are the Program Options
for Mathematically Talented Students?
- Instructional Options
Within the Regular Classroom
- Appropriate Options
- Breadth/Depth approach: The same curriculum, greater depth
- Enrichment topics: Extend or enrich the regular curriculum
- Math-related independent study projects:
Investigate a math topic
- Curriculum compacting: Eliminate some curriculum
to allow more time for other activities
- Telescoping: Spend less time in a course of study
(complete 3 years of high school in 2 years)
- Subject-matter acceleration in mathematics:
Move up a grade for mathematics
- Ability grouping: Groups of advanced students
study math together
- Less-Appropriate Options
- Tutor other children: ...
This is not a good substitute for learning new material.
- Isolated, self-paced instruction: Student works ahead
in the textbook at his/her own pace.
May result in feelings of isolation; student probably won't learn
- Subject-Matter Acceleration.
Moving up a grade just for the mathematics class can be a good option
for students who are exceptionally talented in only one content area...
One disadvantage of this acceleration is that the pace of
the class a grade higher might still be too slow
for these quick learners
and there may be very little content that is substantially new ...
- Instructional Options
Outside of the Regular Classroom
- Mathematics competitions and clubs
- Summer programs for gifted students
- Weekend programs for gifted students
- Individually paced programs: The DT-->PI model
- Magnet schools
- Distance learning and correspondence courses
- Issues in Planning Programs
for Mathematically Talented Students
- A "one-size-fits-all" program doesn't fit all.
- Students may be gifted in math, but not in other subjects.
- The gifted program might not address the needs of the
matheatically talented students.
- "Acceleration vs. enrichment" is a false dichotomy.
(... are not mutually exclusive )
- Acceleration doesn't necessarily produce gaps.
- Students extremely talented in mathematics
may make computation mistakes.
- Special programs need to be integrated into district-wide objectives
so they can survive changes in personnel.
We have developed a "To Do" list to help institutionalize
the program so that it becomes a part of the school and
not dependent upon one or two individuals:
- Create ownership for the constituents (parents, students,
principals, gifted coordinators, and regular classroom teachers).
- Document what has happened.
Make a handbook for the new people who will be joining the team.
- Devise a written plan. ...
- Gather data and conduct evaluations that demonstrate
the program works. ...
- Listen for the success stories. ...
- Make sure there is articulation within the curriculum.
People who will be involved at the high school level
need to be aware of this program so they can begin planning
for these students when they get to higher grade levels.
- Evaluating Programs
for Mathematically Talented Youth
Belcastro ... outlined a set of principles that gifted programs
for mathematically talented elementary students should meeet and
proposed a series of excellent questions
that program personnel should ask in reviewing their program:
- Is the program for mathematically talented elementary students
part of the regular curriculum?
It is recommended that delivery of services occur through the
subject area (mathematics),
not through a general gifted program
where mathematics would be only a part of the activities. ...
- Is there a rigorous identification procedure? ...
- Is the program in effect every day? ...
- Does the program provide placement and
interaction with peers who are mathematically talented? ...
- Is faster pacing of the mathematics group facilitated? ...
- Are students challenged at their own level
using advanced strategies?
- Are teachers selected who are trained in the education
of mathematically gifted?
Ch.7: Curricula and Materials
- It is important to develop differentiated curricula
for mathematically talented students.
- Teachers of mathematically talented students should go
beyond the textbook to differentiate the curriculum by using
manipulatives, math games, and computer programs.
- Key Elements of
a Curriculum for Mathematically Talented Students
Johnson ... has recommended six key elements of a curriculum for
mathematically talented students:
- The scope of content must be as broad as possible.
- The content must be presented at a greater depth and
with higher level of complexity, combined with abstraction of concepts.
- The curriculum must be presented with a discovery orientation
that allows for exploration of concepts.
- Instruction should continue to focus on problem solving.
- Teachers should use a metacognitive approach to
solving problems, that is, teachers should take advantage of systems
such as Polya's, in which students actively think about their
problem-solving process as they
(a) understand the problem,
(b) make a plan to solve it,
(c) carry out the plan, and
(d) look back to evaluate the process and the solution.
- Mathematics should be connected to other disciplines. ...
- Essential Topics
for Mathematically Gifted Elementary Students
- Problem Solving
- Geometry and Measurement
- Math Facts and Computation Skills
- Arithmetic and Algebraic Concepts
- Computer Programming
- Estimation and Mental Math
- Structure and Properties of the Real Number System
- Probability and Statistitcs
- Spatial Visualization
- Ratio, Proportion, and Percent
- Computer Programming
in the primary grades.
Logo is useful for developing logic skills, solving geometry problems,
and exploring concepts.
Rather than using computers in math class strictly for drill and practice,
mathematically talented students benefit from using them
for developing their computer programming skills.
Students can begin studying simple programming such as
- Spatial Visualization
- most highly correlated with
mathematics achievement and least addressed in the mathematics curriculum" ...
Spatial visualization means the ability to visualize spatial transformations
(e.g., mentally rotating a cube).
Wheatley ... cited a number of research studies that document the importance
of recognizing spatial visualization as an ability in mathematics.
Wheatley claimed that "of all intellectual factors,
spatial visualization is the
Ch.8: Teaching Mathematically Talented Students
- Along with parents, teachers play a critical role
in the academic and social/emotional development of students.
- The Preschool Years and Impact of
Parents and Teachers on the Mathematically Gifted
- magical years,
during which time parents rediscover the joy of learning something new
through their children's first experiences.
Parents of mathematically precocious children are astounded with
their child's fascination with numbers.
This fascination typically precedes any interest in numbers
displayed by a child's agemates, and often will exceed that of the
child's peers throughout the school years. ...
The years before a child enters kindergarten are
- preschool teachers
are more sensitive than their elementary and middle school colleagues
to their students' need for individualized and developmentally
It is almost as if a child enters an elementary school, and bam!,
the door to curiosity, exploration, and individualization slams shut.
Combined, our work with mathematically talented students
spans nearly 30 years and, to date, we find that
- The Middle School Years:
How the Teacher and Classroom Culture Make or Break the Student
- Sparking the Joy of Learning
Within the Current Culture of Schools
Ch.9: Case Studies
- Mathematically talented students can be identified at a young age,
and their parents are often the first to recognize their talents.
- Objective information is essential in helping parents to be
effective advocates for their children.
- The Diagnostic Testing --> Prescriptive Instruction model
helps ensure that exceptionally mathematically talented students
study mathematics at a steady rate.
- Long term planning is essential so that students are always studying
an appropriately challenging level of mathematics.
- Mathematically talented students benefit from finding
an intellectual peer group;
this may be especially important for mathematically talented girls.
- Lessons Learned
As a result of working with ... many [talented] students,
we have learned numereous lessons.
- Parents know their kids.
They are essential advocates for their children.
- Mathematical ability
can be recognized at a very young age.
- be persistent in their advocacy efforts.
Because school personnel often take a "wait and see" attitude or
"just say no,"
parents need to
- above-level testing.
School-based assessments are typically grade-level in nature, and
many educators are not aware of the need or procedure involved
- Children don't need to be tested
until they are ready for systematic programming.
Although it is not essential to have
standardized testing at a very young age,
it is when students enter the school system.
- driven by a question.
All assessment (testing) needs to be
- The Diagnostic Testing --> Prescriptive Instruction model is useful
for helping mathematically talented youth study mathematics
at the appropriate level and pace.
- cognitive structures that characterize mathematical maturity.
Even extremely talented students need time to develop the
- Mathematically talented students should study mathematics
at a steady rate.
- students will run out of math"
before they graduate from high school.
Choices made when students are in elementary school may affect
their high school mathematics program.
Educators often fear that "
- Students may need to look outside of their school system
for appropriate mathematics.
- Mathematically talented girls may have special needs.
among studying mathematics at an accelerated pace,
studying other academic subjects, and
participating in extracurricular activities.
Students should strive to achieve
- Talented students benefit from finding an
intellecutal peer group.
- not "pushing" their children.
Most parents of talented students we have met are
Feedback on this page is welcome.