How can teachers and parents make accommodations for elementary school students who are extremely talented in mathematics? One solution is to design an individualized, mentor-paced program. Two such programs have been developed at the University of North Texas and the University of Iowa. The purposes of the present article are to describe how students are selected for mentor-paced instruction, and to explain how the mentoring program is set up.
The Julian C. Stanley Mentor Program (JCSMP) for mathematically talented elementary students began in January, 1990. The JCSMP is modeled after the successful, fast-paced mathematics programs for talented junior high school students developed by Dr. Julian C. Stanley (Bartkovich & Mezynski, 1981; Stanley, 1976; Stanley & Benbow, 1986). The JCSMP, which is administered by the Study of Mathematically Precocious Youth at the University of North Texas, identifies talented youth and matches them with mentors who design individualized programs for them.
In September, 1990 the Investigation of Mathematically Advanced Elementary Students (IMAES) was established. The IMAES is also modeled after Dr. Stanley's fast-paced mathematics programs, and it is administered by the Connie Belin National Center for Gifted Education. The goals of both IMAES and JCSMP are to identify mathematically talented students, give them challenging work, and encourage them to move ahead as fast as their abilities and motivation allow.
The diagnostic testing--prescriptive instruction (DT-PI) model on which the JCSMP and IMAES are based was described in a recent issue of The Gifted Child Today (Lupkowski, Assouline, and Stanley, 1990). The first stage of the DT-PI model is to identify talented students via an out-of-level aptitude test. Then, a student's level of achievement is assessed through subject matter achievement tests, and instruction is prescribed based upon information gathered from that testing.
DT-PI is a mentor-paced model for individualizing mathematics instruction for students who are extremely talented in mathematics. The DT-PI approach to providing instruction can be thought of as an alternative to wholesale grade- or subject-skipping. It avoids inappropriately placing a young student who has advanced cognitive skills into a grade that may be higher, but where mathematics is taught at a pace more appropriate for average students.
Listed here are some concerns about the process as well as explanations of some of the details of developing the programs. These may help parents and teachers understand the DT-PI model as it is used in the IMAES and JCSMP programs.
Who is Eligible for the Mentor-Paced Programs? Students in the upper elementary grades who score at the 95th percentile or above on a mathematics section of a nationally standardized achievement test battery (such as the Iowa Tests of Basic Skills) are good candidates to take an out-of-level aptitude test and to be considered for special programming. The lower level of the Secondary School Admission Test (SSAT), which was developed for students in grades 5-7, serves as the out-of-level aptitude test. Students take the Quantitative portion; those scoring at the 50th percentile or above when compared to students 2 years older on the out-of-level SSAT are recommended for further diagnostic testing of their specific level of achievement. Mentor-paced instruction is based upon the results of that assessment (Lupkowski and Assouline, in review).
Besides scoring high on aptitude and achievement tests, students who participate in the mentor-paced programs should be mature and highly motivated. Participation in the program should be reserved for children who are eager to study more mathematics. The program demands a great deal of time and effort on their part. It is critical for children who participate in such a demanding program to want to be a part of it, not just responding to parent's or teacher's desires.
In addition to student aptitude and motivation, a third consideration for placing students in the mentor program is the level of cooperation of the school. For the most part, schools have been eager to have their students participate in these special programs because they understand that JCSMP and IMAES serve as a supplement to what schools are able to provide. One of the most important ways in which that cooperation is assured is through communication. Letters from IMAES and JCSMP staff explaining the program and meetings with parents, mentors, and school personnel facilitate that communication.
How is the Level at Which Instruction Begins Determined? A student's knowledge of mathematics is assessed using a battery of achievement tests. The battery of out-of-level achievement tests includes the first two levels of the Basic Concepts and Computation tests of the Sequential Tests of Educational Progress (STEP) and tests from the Cooperative Mathematics series (beginning with Structure of the Number System and Arithmetic all the way through Calculus). Students scoring at a high level on these tests may participate in a continuum of activities based on their interests, needs, and abilities. Classroom teachers who are interested in applying this model might consider using chapter tests and final examinations provided by textbook publishers for diagnosing what a student has already mastered or has yet to learn.
It may seem as though a great deal of time is spent in administering standardized tests. However, only a small portion of the total time devoted to the mentor-paced program is allocated to standardized testing and most of that is in the beginning stages of the program. Once instruction begins, the time between standardized testing sessions is usually several months, although mentors continuously test children via chapter tests and teacher-made tests to be certain that they understand the material they are studying.
One of the children who participated in a mentor-paced program was "Matthew," a 10-year-old fifth grader who had been studying fifth-grade mathematics. He qualified for participation in the program by scoring at the 92nd percentile on the Quantitative section of the SSAT when compared -to seventh-grade boys. Clearly, Matthew has exceptional abilities in mathematics. During the spring of his fifth-grade year he took a series of achievement tests; on one test in which he was compared to eighth graders he scored at the 89th percentile on Basic Concepts and at the 85th percentile on Computation. Based on the results of diagnostic testing, Matthew was matched with a mentor who helped him fill in his few gaps in elementary school mathematics. Then he moved into the pre-algebra curriculum (using Merrill's Pre-Algebra text). By the time he entered sixth grade in the fall, Matthew had completed the pre-algebra textbook and was ready to begin Algebra I. A more complete description of Matthew's participation in the mentor-paced program is found in the sidebar.
Who Are the Mentors? We recommend that mentors be adults such as trained mathematics teachers, engineers, college professors, or mathematics education majors. In addition to a strong background in mathematics, the mentor needs to have the maturity to work with younger students and a good rapport with talented youth. Sometimes it is suggested that a high school student might serve as an effective mentor. However, our experience has been that high school students typically are busy with their own course work and extracurricular activities, and thus would have difficulty managing the responsibilities of planning for the mentoring sessions and meeting consistently on a weekly basis.
Graduate students from the mathematics department at the University of North Texas serve as mentors for the students in the JCSMP, and University of Iowa undergraduate education majors serve as mentors for the students in the IMAES. Typically the mentor-to-student ratio is 1:1. However, skilled mentors can work with small groups of students (2-5) who are at about the same level in their understanding of mathematics (Moore & Wood, 1988).
The description of the different mentoring relationships offered by IMAES and JCSMP is meant to indicate that there is not a hard and fast rule concerning who should serve as mentors. Each local district must determine what system will work best for them.
How Are the Mentor-Paced Programs Administered? Staff members of the JCSMP and IMAES regularly receive telephone calls and letters from parents and teachers of mathematically talented students. Their role is to identify the students and assess their levels of mathematical knowledge via aptitude and achievement tests and then match them with appropriate mentors. Staff members of the IMAES and JCSMP provide workshops for teachers in which the model is explained, and they train mentors in the DT-PI process. They regularly consult with mentors concerning students' progress and help mentors find appropriate materials to use with their students. IMAES and JCSMP staff members also oversee the administration of post-tests and make an effort to keep school personnel informed of the progress of the students.
How Often Do Mentors and Mentees Meet? Typically, mentors and students meet once a week for 2 hours throughout the school year. During that time, they go over homework assigned during the previous meeting, clarify questions, and investigate new topics. One 2-hour time period per week is preferable to several shorter time periods because it permits mentor and mentee to study topics in greater depth without interruption. We recognize that this schedule might not fit easily into the school day; it has been necessary to plan after-school and weekend sessions for many of the children in the JCSMP and IMAES programs. However, children often work on their mentor-assigned homework during the daily mathematics period while other children are doing their seat work. One school system that was extremely eager to have a student participate in the mentor-paced program allowed the mathematics mentor to go into the classroom twice a week for an hour to work with a student.
Who Should Administer Student-Progress Tests? It is preferable that tests be administered by the mentor or in collaboration with school personnel. This will insure that the testing is conducted under standardized conditions. Achievement testing is necessary to insure that students receive proper credit for work done, since that work is not part of the regular curriculum.
All testing should include a written report of the results, a description of the child's behavior during the testing, and recommendations for prescriptive instruction based upon the results. The report should be shared with parents as well as appropriate school personnel (e.g., principal, counselor, and other teachers).
Will Students in the Program Have Less Opportunity to Interact with Others about Their Studies? Although students who participate in a mentor-paced program typically spend a great deal of time working alone, the mathematics they study is not learned in a vacuum. Because they are working in a one-on-one or small-group situation, student and mentor can toss around ideas about the material being studied. The mentor will be able to provide more accurate answers to questions than peers can; also, the mentor has the knowledge and ability to extend the student's questions and ask new questions of that student.
Another concern is that students will have less opportunity to interact with their peers and may even be "isolated" from them by participating in the mentor-paced process. Remember, however, that mathematics class is only one place in which students interact with each other. It is strongly recommended that students participate in mathematics clubs and contests, which give them a chance to interact with other students in a mathematics context. They are encouraged to participate in unstructured play as well as more structured activities, such as sports and music. These varied activities will give them an abundance of opportunities to spend time with their agemates.
The time spent interacting with peers should be balanced by adequate time alone. It is not necessary for students to spend every spare moment interacting with peers. Allowing talented students to have plenty of independent thinking time is important; children need time to reflect, plan, and dream. This "alone time" allows talented students the opportunity to think about what they have learned and experienced. The time can be used to formulate new questions and to investigate them. Thus, what is sometimes perceived as "isolation" by adults actually has many positive aspects.
What Should Mentor-Paced Program Participants Do While Classmates Are in Regular Math Class? Before beginning a mentor-paced program, educators, parents, and administrators need to discuss the options available for a student while his or her classmates are in math class with their regular teachers. These options include: reading, working on the computer, going to the library, and working on mentor-assigned homework. Students in a mentor-paced program should be included for classroom games and other group activities. However, it is inappropriate to require students to sit through the regular mathematics class if they are already working with a mentor on more advanced mathematics. Not only would the talented student be bored in the regular class, but he or she may become disruptive.
If the student works on mentor-assigned homework while the other children are in the regular mathematics class, he or she will need a quiet setting with adult supervision. It is important to structure that situation so that the student does not feel that he or she is being ostracized or punished. Not only should parents, mentors, and school personnel discuss this, but also the students should be made fully aware of the reasons for the special arrangements.
Won't the Extraction of Talented Students Remove an Important Role Model for Average Students? This worry is a valid concern for teachers, especially in light of the cooperative learning movement that is gaining momentum in schools throughout the country. This concern is also mirrored in statements that question the validity of any type of ability grouping. By formally identifying elementary students who are talented in mathematics and providing instruction that is appropriately accelerative and challenging, we become part of a larger debate. It would be impossible, however, to discuss the advantages and disadvantages of cooperative learning or ability grouping here. Ability grouping and cooperative learning, as they interface with gifted education, are thoroughly explored in articles by Robert E. Slavin and Ann Robinson in Volume 14, number 1, of the Journal for the Education of the Gifted.
To be effective, role models must be somewhat close in ability to those who would benefit from exposure to the models (Schunk, 1987). Large differences in ability may promote arrogance on the part of the high-ability student. If teachers teach a mixed-ability group at the appropriate pace and depth for those who are extremely talented mathematically, less-able students experience unnecessary pressure from a set-up for failure. Mixed ability grouping in mathematics, with extreme variation in aptitude, increases the management problems of teachers. On the other hand, teachers of students who participate in a mentor-paced program can concentrate on teaching the majority of their students and be satisfied that their highly-able students' needs are being met.
How Are Issues of Grades, Credit and Placement Resolved? Before children begin participating the mentor-paced program, we ask parents to visit with school personnel determine how children will receive grades and credit for the mathematics work that they will be doing. Each school has resolved these issues in it’s own way. For example, some have made special notes on children's report cards indicating what grade-level mathematics students are studying. Mentors continually assess students' progress, so they are frequently asked to turn in grades to teachers. In other cases, mentors give chapter tests to teachers to administer and that is how a grade for a particular marking period is determined. It is important to maintain careful records of students' progress so that, when the time comes to make a placement decision, students will be placed at the proper level in mathematics.
If Students Who Participate in the Program Will Be Accelerated, What Happens Then? The mentor-paced program is based on a process of diagnostic testing and prescriptive instruction designed to move students through the curriculum at a challenging pace. By definition, students will be accelerated in mathematics. Thus, the importance of planning before entering this process cannot be overemphasized. For example, elementary students who have participated in a mentor-paced program may be ready to take a high school mathematics class long before they are high school students. Special arrangements such as transportation from one school to another may be needed for the students. Older students may complete all of the available mathematics courses a year or more before graduating from high school. For these students, arrangements may be made with a local college or university or correspondence school so that the study of mathematics will not be interrupted.
How Long Do Students Participate in the Mentor-Paced Program? Because the mentor-paced programs are individualized, the length of time the students participate is very flexible. The mentor-paced program is usually used as a bridge to prepare students to take the next level of mathematics in their own schools. One student was almost ready to take an Algebra II class; she spent the last 4 weeks of the summer "cleaning up" her knowledge of Algebra I with a mentor in order to be ready to take Algebra II in her school. Other students have participated in the program for a semester or a full school year. One exceptional student who was studying geometry as a fifth grader will probably participate in a mentor-paced program for a number of years while he takes other subjects with his age-mates. If students who have completed the mentor program and are taking a regular class find at some future point that they are again well beyond their classmates in mathematics, they can re-enter the mentor-paced program.
Is There a Cost to the Program? Parents are expected to bear the financial burden of the JCSMP, although a grant from the Texas Association for the Gifted and Talented has provided some partial scholarships. Parents pay mentors at the same hourly rate that is charged by the University of North Texas Mathematics Department tutors who assist college students having difficulty in their classes. Parents also pay a book and materials fee.
Since the tutors for the IMAES are undergraduate students at the University of Iowa, they do not receive payment for their services; however, they receive practicum credit. Undergraduates require more direct supervision than do graduate student mentors.
What Efforts are Made to Cooperate with the Schools? Most of the school personnel who have become aware of the mentor-paced programs have been extremely cooperative. In many cases, teachers have been the first to recognize talent and have encouraged parents to contact SMPY or UNT or the Belin National Center for Gifted Education. In other cases, parents were the first to investigate the program. Parents are always encouraged to provide school personnel with information, including reprints of pertinent articles and test reports, and to discuss educational options for their child. A letter is sent to school personnel describing the JCSMP and inviting their support of the program for children who have qualified.
IMAES engages similar support with cooperating schools. IMAES mentors receive university supervision and practicum credit, and they agree to keep a log of all activities engaged in with the student. The mentor keeps the classroom teacher informed of the student's progress and of materials presented to the student. This may be done by making duplicate copies of lesson plans, and by talking, either in person or on the phone, with the classroom teacher.
Programs such as IMAES and the JCSMP are conducive to a system of "dual mentoring" as described by Clasen and Hanson (1987). In dual mentoring, one mentor (the classroom teacher) attends to the developmental needs of the youngster and the other (the university graduate or undergraduate student) attends to the intellectual needs. Thus, the two mentors work in tandem to foster both the intellectual and the socio-emotional development of the youngster. The expertise of these two mentors is supplemented by university personnel who administer the program and train the mentors.
Clasen and Hanson (1987) describe basic steps and goals in setting up and carrying out a double mentoring relationship. We have adapted these as follows:
Could This Model Be Used in Other Subject Areas? The mentor-paced approach is useful for subjects other than mathematics. For example, students have mastered the topics of high school biology, chemistry, and physics in mentor-paced programs (see Stanley & Stanley, 1986). Other subjects that lend themselves well to this approach include foreign languages, grammar, and writing skills. The same process of out-of-level diagnostic testing followed by prescriptive instruction can be applied.
Matthew: A Parent’s Perspective Matthew entered the mathematics mentor program at age 10, in the early spring of his fifth-grade year. In grades one through three/his school provided accelerated math instruction for mathematically talented students. When he was in the fourth grade, we moved and had to advocate to ensure that Matthew had challenging instruction.
In the middle of Matthew's fifth-grade year, we moved again. In the former school, he had been studying sixth-grade mathematics. The new elementary school principal looked at Matthew's records and our request for math instruction at the current level of acceleration, and flipped the file shut. "Well," he said, "Looks like Matt's going to be ahead of his classmates. We just don't do that sort of thing."
Soon after that meeting, the school's teacher of gifted learned that a local university had a new mentor-paced program for elementary students gifted in math. We contacted the administrator of that program and made an appointment for individual testing. Prior to the test, Matthew and the examiner talked about the testing procedures and the mentor program itself. The examiner spent some time setting him at ease by explaining to Matthew that the tests were to be used as a measuring device and not for a grade. She explained to him that he might encounter math problems he wouldn't be able to answer and that he shouldn't worry about them; those unanswered questions would give her information about what level of mathematics he would be ready to study next.
It was also explained that the decision to enter the mentor program would be primarily Matthew's. We were told that the program's goal is to reach those students who are internally driven by a need to learn mathematics. Care is taken when testing and speaking to families to discern whether the desire to study more math comes from the student or from a parent. Participation in the program would require meetings outside of his regular school day and doing homework.
Matthew did not appear to have a high anxiety level about all this or about the testing situation itself; the examiner relates that he hummed throughout most of the testing.
The mentor assigned to us was "Mr. B.," a doctoral student in mathematics. Our arrangement was for them to meet for 2 hours a week, for an hourly tutoring fee. During this time, they would cover the material to be taught, outline assignments, and work out any problems in understanding the previous materials.
We were assured that Matthew and Mr. B. would meet on a trial basis until each of them determined that they were comfortable with the arrangement. Mr. B. is a genial, gentle man and he and Matthew hit it off right away, although I later learned that both of them had been concerned about the mentoring sessions. Matthew admitted that he had been afraid that he would be asked to work problems he didn't understand, and be embarrassed if he couldn't answer them. Matthew soon learned that he would indeed be asked to work difficult problems, but that Mr. B. would be there to instruct and encourage him. Matthew was given his mentor's home telephone number and told to use it if he ran into difficulties. The first telephone call was agonizing for Matthew. He was literally unable to dial because he just couldn't believe that a teacher actually meant it when he said, "Call for help." He learned that Mr. B. really did want him to call. With a week between sessions, it's important for the student to know that he can reach the mentor with questions, and not remain stuck in one spot.
Mr. B. said later that one of his concerns had been wondering if his student would have questions and comments of his own. With Matthew, this was not a problem! Matthew had lots of questions about the material they were covering, about tangential concerns, and about the dissertation Mr. B. was writing.
Matthew's enthusiasm for math, always high, was now totally set loose. He was being asked to work harder and longer on the subject than he ever had before. On the average, he spent an hour a day on math homework during the week, plus the 2-hour mentoring session. At first, it took some adjustment for Matthew to learn to plan his time efficiently, but after about 3 weeks he fell into a workable schedule.
After they gathered speed, Mr. B. and Matthew covered nearly a chapter a week in the textbook. Although Matthew's time investment into mathematics had increased over that spent even in regular accelerated classroom work, he was not resentful of that fact. His efforts were no longer being spent in endless repetition of problems that he already knew how to do. Once when I asked how many he had to do, Matthew's answer was, "Mr. B. told me to do every third one in this section until I understand it. Then I can go on."
Mastery of the subject had become the main objective, and Matthew had to be able to demonstrate that mastery. He was expected to have his assignments done and also took periodic assessment tests. Sometimes a test would show that a certain concept needed more work to be fully understood, and they would go over it again, probably from a different direction.
This program removes adversarial aspects, together with its resulting anxieties, from the learning process. Even as a student is tested for program entry, the goal is not so much a matter of winning a distinction as seeing if the program is a good fit. Tests given to program participants are used to determine the appropriate time to move on to new materials--not for grades. Teaching is for the purpose of understanding. Concepts are taught until proficiency is demonstrated. The prize is not a grade, but the opportunity to move on. For students like Matthew, who are driven from within to know, knowledge truly is its own reward.
What happens in the school when a child enters a mentor-paced program? There is a lot of variation on this point, and it is best negotiated with each individual school. In our case, the principal thought it logical that, since math was so easy for Matthew, he shouldn't mind repeating the fifth grade curriculum while also doing his mentor assignments. His explanation was that since Matthew was currently in the fifth grade, the state needed a grade for fifth grade math filled in on Matthew's report card. We eventually came to an acceptable compromise, whereby Matthew had 4 days a week to do his daily mentor program assignment, and the fifth day belonged to the classroom teacher to give weekly tests for a school math grade. Matthew was satisfied with this arrangement, and grateful for the time in class.
In the months they spent together, Mr. B. and Matthew covered the transitional math textbook used in the middle school. They worked on a less regular basis throughout the summer. Late in the summer, Matthew was retested and it was proposed that it was time for him to re-enter the school mathematics curriculum. At this point, Matthew had also completed fifth grade and was ready to move from the elementary school building to the middle school.
Matthew was ready to begin Algebra, which in our school system was offered as an eighth-grade honors class. Mr. B. and I went to speak with the middle school administrators. I took full records of Matthew's acceleration in math up to this point and his mentor took documentation of Matthew's abilities, as charted by the pre- and post-tests administered by the mentor-paced program. He also spoke eloquently as an advocate for Matthew.
To our utter amazement and joy, the middle school was totally supportive. Due to scheduling conflicts, they could not put Matthew in a class with eighth graders, so their proposal was for Matthew to meet daily with the Algebra teacher in the resource room. Matthew does the same assignments and takes the same tests as the students in the regular classes she teaches. He enjoys the one-on-one relationship with his teacher, who has been excited and supportive regarding this new arrangement. They are good friends now, too. It is not exactly like his mentorship with Mr. B.: They are not free to explore tangential material quite as much. But there is also no ambiguity about grades; because he is being taught standard materials by a regular classroom teacher, the school will give him whatever grade he earns. We are doubly fortunate because these administrators took the initiative to see that Matthew will be given high school credit for the work he does while in middle school.
We have learned to take Matthew's education one year at a time. Thanks to our involvement in the mentor-paced program, however, we feel that we can look at the future with some confidence that he is not going to be frustrated and discouraged in his pursuit of his math studies. His teacher this year is already talking about the course of study she is planning for his next 2 years in the middle school. When he reaches high school, he will complete the math sequence there. When he runs out of available classes (as he is apt to do about his junior year), we live close to a major college where he can continue his studies.
Our satisfaction with the mentor-paced program lies not in its ability to turn out mathematicians, but in the serendipitous effect that taking care of academic needs has on the whole child. Matthew is productive and contented, secure in knowing that he is not likely to have to repeat any academic material. This had sometimes been a major worry and frustration. His mentorship was especially rewarding -- Mr. B. holds a special place among all the teachers Matthew has ever had.
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The mentor-paced approach is appropriate for very talented students who are highly motivated to move ahead in mathematics. For those students who are well above average when compared to their age group but not extremely talented, other accommodations might be made.
References
Bartkovich, K.G., & Mezynski, K. (1981). Fast paced precalculus mathematics for talented junior high school students: Two recent SMPY programs. Gifted Child Quarterly, 25(2), 73-80.
Clasen, D.R., & Hanson, M. (1987). Double mentoring: A process for facilitating mentor-ships for gifted students. Roeper Review, 10(20), 107-110.
Lupkowski, A.E., & Assouline, S.G. (in review), Extending the Talent Search model to elementary students: The potential of the SSAT for identification.
Lupkowski, A.E., Assouline, S.G., & Stanley, J.C. (1990). Applying a mentor model for young mathematically talented students. The Gifted Child Today, 13(2). 15-19.
Moore, N.D., & Wood, S.S. (1988). Mathematics with a gifted difference. Roeper Review, 10(4), 231-234.
Price, J., Rath, J.N., & Leschensky, W. (1989). Merrill pre-algebra: A problem-solving approach. Columbus, OH: Merrill.
Schunk, D. (1987). Peer models and children's behavioral change. Review of Educational Research, 57(2), 149-174.
Stanley, J.C. (1976). Special fast-mathematics classes taught by college professors to fourth-through twelfth-graders. In D.P. Keating (Ed.), Intellectual talent: Research and development (pp. 132-159). Baltimore: Johns Hopkins University Press.
Stanley, J.C., & Benbow, C.P. (1986). Youths who reason exceptionally well mathematically. In R.J. Sternberg & J.E. Davidson (Eds.), Conceptions of giftedness (pp. 361-387). New York: Cambridge University Press.
Stanley, J.C., Lupkowski, A.E., & Assouline, S.G. (1990). Eight considerations for mathematically talented youth. The Gifted Child Today, 13(2), 2-4.
Stanley, J.C., & Stanley, B.S.K. (1986). High-school biology, chemistry, or physics learned well in three weeks. Journal of Research in Science Teaching. 23(3), 237-250.
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