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.
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[ 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. In fact, 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

p.1 ... 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 teenagers." 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."

p.2 ... 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. Not true: p.2... Many 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. ...; ...; p.4 ... 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. Not true: p.5... [G]rade-level testing ... does not give information that is precise enough to [differentiate good, talented, and exceptionally talented students]. ... An above-level test helps to measure the students' abilities more accurately.

Myth 3: Gifted students respond equally well to the same curriculum. Not true: p.5 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. Not true: p.9 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. Not true: p.10 ... [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 is enrichtment. Not true: p.12 Although enrichment is appropriate and necessary for mathematically talented students, it is not the only option and it might not be the best option for any particular student. Acceleration should not be dismissed for talented students automatically because of their young age. ...; ... [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. Not true: p.15 ... [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).

Ch.2: Advocacy

p.19

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?
p.25 ... [T]here is no [US] federal government mandate requiring school districts to provide special programs for gifted students. ...
p.26 ... [G]ifted children are not a [US] constitutionally protected group of individuals.
... [S]ome states [in the USA] have a mandate for the provision of gifted education, whereas others only "permit" gifted education. ...
p.27 ... However, ... informal and quasi-formal resolutions are preferable to those that involve the courts [of law]. And even if you would be able, through court actions, to force a school to adapt the curriculum for a talented student, the court will not (help) determine an appropriate curriculum. The school could try to get away with an ineffective change.
p.28 The responsibility of finding appropriate programs with adequate challenges for mathematically talented students ultimately falls on the parents.

Ch.3: Educational Assessment

p.63

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. ...

p.67 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

p.105

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.

p.107-108 ... 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 test's "ceiling." For these students, the grade-level test does not adequately measure their talents 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...

Step 1: Determine Aptitude: p.108 ... [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]. So, this is an entry condition, to be ascertained prior to Step 1.; ...; p.111 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 at least two grade levels higher than the student's current grade placement ...; ... 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.; ...; p.112 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. ...; p.113 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: Diagnostic Pretest

p.113 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.

...

p.114 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: p.116 ... 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.; p.117 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 ... ]; p.118 The final activity of Step 3 is identifying who will best serve as the instructor for Step 4.

Step 4: Prescriptive Instruction: p.118 ... 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 have an excellent background in mathematics. This person is referred to as a mentor.; 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: Posttesting: p.119 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.; p.119-120 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.

What Is the Role of the Mentor?: p.122 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. ...; p.123 We do not recommend that parents mentor their children, even if they hae the appropriate background in mathematics. ...; 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?: p.123 ... [D]aily instruction is not necessary. We recommend that the mentor and student meet for a total of 2 hours per week. 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).; p.124 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 the 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.

What Happens Between Meetings?: p.124 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.; p.125 ... [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: p.131 ... [T]his remains a common myth. ... [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. ...; [ ... other arguments omitted ... ]

How Are Student Evaluation Issues Determined?: p.131 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?: p.132 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?: p.137 ... 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

p.141

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.

p.149-150 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 ... thus, differentiating between good, talented, and exceptionally talented students.

Ch.6: Programming

p.173

There are many educational options for mathematically talented students, ranging from enrichment within the regular classroom to radical acceleration. 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

p.178-179 The debate about 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. ...

[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:

Busywork: 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.

p.182 ... [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. ...

...

p.183 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. ...

p.184 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. ...

...

p.185 ... Schiever and Maker ... carefully explored the various forms of enrichment and acceleration and concluded, ... that a 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.

What Are the Program Options for Mathematically Talented Students?

Instructional Options Within the Regular Classroom

p.186 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 material well.

...

p.190 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 ... cf. the spiral approach

Instructional Options Outside of the Regular Classroom

p.195

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

p.210

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 p.211)
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.
p.213-214 ... 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:
1. Create ownership for the constituents (parents, students, principals, gifted coordinators, and regular classroom teachers). ...
2. Document what has happened. ... Make a handbook for the new people who will be joining the team. ...
3. Devise a written plan. ...
4. Gather data and conduct evaluations that demonstrate the program works. ...
5. Listen for the success stories. ...
6. 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

p.218-220 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

p.223

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

p.226 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

p.228

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: p.232 ... 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 Logo in the primary grades. Logo is useful for developing logic skills, solving geometry problems, and exploring concepts.

Spatial Visualization: p.235 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 most highly correlated with mathematics achievement and least addressed in the mathematics curriculum" ...

Ch.8: Teaching Mathematically Talented Students

p.259

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: p.261 The years before a child enters kindergarten are 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. ...; p.262 ... Combined, our work with mathematically talented students spans nearly 30 years and, to date, we find that preschool teachers are more sensitive than their elementary and middle school colleagues to their students' need for individualized and developmentally appropriate curricula. ... It is almost as if a child enters an elementary school, and bam!, the door to curiosity, exploration, and individualization slams shut.

The Middle School Years: How the Teacher and Classroom Culture Make or Break the Student: p.266 ...

Sparking the Joy of Learning Within the Current Culture of Schools: p.272 ...

Ch.9: Case Studies

p.287

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

p.319 ... 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.
p.320 Because school personnel often take a "wait and see" attitude or "just say no," parents need to be persistent in their advocacy efforts.
p.321 School-based assessments are typically grade-level in nature, and many educators are not aware of the need or procedure involved in above-level testing.
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.
p.322 All assessment (testing) needs to be driven by a question.
The Diagnostic Testing --> Prescriptive Instruction model is useful for helping mathematically talented youth study mathematics at the appropriate level and pace.
p.323 Even extremely talented students need time to develop the cognitive structures that characterize mathematical maturity.
Mathematically talented students should study mathematics at a steady rate.
p.325 Choices made when students are in elementary school may affect their high school mathematics program. Educators often fear that "students will run out of math" before they graduate from high school.
Students may need to look outside of their school system for appropriate mathematics.
Mathematically talented girls may have special needs.
p.325 Students should strive to achieve balance among studying mathematics at an accelerated pace, studying other academic subjects, and participating in extracurricular activities.
Talented students benefit from finding an intellecutal peer group.
p.326 Most parents of talented students we have met are not "pushing" their children.

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