The following article has been excerpted from Math Education for Gifted Students, one of six exciting books in the Gifted Child Today Reader Series. This series brings together the best articles published in Gifted Child Today, the nation's most popular gifted education journal. Each book in the series is filled with exciting and practical classroom ideas, useful summaries of research findings, and discussions of identification and classroom management, and informed opionion about educating gifted children.

Chapter 9

One summer several years ago, I taught at the Governor’s School of North Carolina, a 6-week residential program for academically or intellectually gifted rising high school seniors in North Carolina. Student eligibility for this program included multiple criteria: aptitude test scores in the 92nd–99th percentile; achievement test scores in the 92nd–99th percentile; scholastic performance records, such as transcripts and class rank; and personal data, such as school and community awards, honors, and activities (for more information, see http://www.ncgovschool.org). The intent of the academic program at the Governor’s School was to challenge and stimulate these gifted students by emphasizing theory over memorization of fact, particularly contemporary and progressive theories that stimulate innovative thought in a rapidly changing culture.

To meet this goal for the Governor’s School program, the curriculum had to be appropriate. Gallagher and Gallagher (1994) suggested four ways that a curriculum could be modified for gifted students: acceleration, enrichment, sophistication, and novelty. Coleman (2001) asserted that sophistication is what gifted students thrive on, but, of the four aforementioned modifications, it is the one that most often eludes teachers who attempt to modify curricula for gifted students. Because the unique environment of the Governor’s School allowed the classroom teacher unusual flexibility in devising the curriculum, I concentrated on sophistication as I developed the curriculum for a 3-week session on mathematics problem solving.

A number of approaches to teaching mathematical problem solving are possible (e.g., Posamentier & Wolfgang, 1996). Kroll and Miller (1993) found that student beliefs about the processes of problem solving, ways to approach problem solving, and beliefs about their own ability to engage in problem solving may be important factors between successful and unsuccessful problem solvers. I developed a curriculum that integrated the five process standards espoused by the National Council of Teachers of Mathematics (NCTM, 2000): problem solving, reasoning and proof, communication, connections, and representations. My goal was to create a 3-week mathematics experience focusing on sophisticated mathematics that would engage gifted students in all of the process standards highlighted by the NCTM. My hope was that this experience would have a lasting impact on student beliefs about mathematics problem solving.

The structure for the problem-solving course was to meet for 1 hour and 15 minutes once a day for 15 days, spread over 3 weeks. One class of 15 students met in the morning, and a second class of 16 students met in the afternoon. In keeping with the overall goals of Governor’s School, I wanted students to be able to take advantage of the numerous out-of-class activities, so I did not assign heavy workloads to be done outside the classroom. I designed a mathematics problem-solving curriculum emphasizing the following concepts:

• Understanding underlying mathematics is much more powerful for developing problem-solving capabilities than merely applying algorithms.
• Similar mathematical tools apply across a broad range of problems, even problems that may seem unrelated to each other.
• Mathematical analysis leads to useful insight in a wide variety of situations.

Many of the specific problems and situations I used to accomplish these goals are well-known to experienced mathematics teachers, but the manner of using them to highlight connections and complexity may not have been considered extensively. Along with a few administrative tasks related to students’ mathematics journals and distribution of the first set of eight problems, the first day began with mathematics-oriented pictorial puzzles designed to stimulate creative thinking and to emphasize the value of teamwork. For example, a picture of a wavy line (a sine function) bisected horizontally by a dashed line represented “sine on the dotted line,” and a picture of Santa Claus with “2L-2L” above his head represented “No L (Noel).”

As students pondered these pictorial puzzles, I encouraged them to say aloud any ideas they may have. Typically, a student would say something that wasn’t completely correct, but that would stimulate another student to follow up on the idea that eventually led to a correct solution. I used this activity to emphasize the importance and usefulness of teamwork; without group contributions, it would usually be much more difficult and time-consuming to solve these puzzles. I strongly encouraged discussion and collaboration throughout the course, and this activity set the appropriate tone. Many of the students seemed to view teamwork and collaboration as an impediment to accomplishing mathematics tasks quickly, but perhaps they hadn’t yet experienced enough of the kind of challenging mathematics tasks where the value of teamwork becomes evident.

The rest of the first day was devoted to students in teams of two solving a problem that didn’t have any readily apparent mathematical equation or formula leading directly to a solution. (Figure 9.1 summarizes this problem.) As students worked on this task, they began to appreciate the value of developing their own notation systems and approaching the problem in an iterative sense. I challenged students who came up with a solution if they could do better. If not, I asked, how could they justify their solution as optimal?

The next day, students continued with this problem, presenting solutions and ideas to the class. Typically, there were a number of different approaches and notation systems, and students began to appreciate different ways of thinking about the same situation. I then challenged the class to justify why a given solution was the optimal one. Usually, one or more groups had already made strides in this direction. With help, students were usually able to convince each other of an optimal solution of 533 bananas to market. Along the way, some algebraic equations were often generated, and students could see that sometimes a problem doesn’t require the solver to search for an appropriate formula, but rather to create one of his or her own. I then asked students to generalize the solution to “beginning with x bananas, with the market y miles away, and Cori eating z bananas per mile traveled.” Typically, I recommended they generalize one parameter at a time, work out a solution to that, and then add in another parameter. I used this to highlight the point that problem solving takes time, and sometimes it is helpful to begin with the specific and move toward the general case. The value of teamwork was again emphasized by this problem, with students benefiting from ideas presented by others.

The Cori the Camel activity demonstrated the value of understanding why a certain algorithm works to lead toward a solution, which led to the next part of the course. The next portion of the course, approximately 4–5 days long, was spent having students analyze why some very basic algorithms work. I had students explain the reasons behind subtraction techniques, including the little nines and ones put on top when borrowing. They also described the reasons behind the algorithms for adding and multiplying fractions, as well as the algorithm for multiplication and division of multidigit numbers. In some cases, no student was able to explain an algorithm fully, providing me with an opportunity to do so. All these students could certainly do these problems effortlessly, but a number of them admitted to never having thought about why a given algorithm works.

I then taught students an algorithm to extract by hand the square root of a number to any desired degree of precision and asked them to explain why that procedure works. (See Figure 9.2 for this algorithm, and see http://MathCentral.uregina.ca/ RR/database/RR.09.95/grzesina1.html for a geometric view why this algorithm works, or see http://jwilson.coe.uga.edu/ EMT668/EMAT6680.F99/Challen/squareroot/sqrt.html for an algebraic perspective.) This square root algorithm was chosen in order to have students think about something that most have never seen before, in contrast to explaining procedures they already knew very well. With a little practice, students could easily apply the algorithm, but they had a difficult time explaining why it worked. With extensive guidance from me, students eventually grasped the mathematical foundations behind the algorithm. Many of them told me that, although they had been good at doing these sorts of computations since elementary school, they had never thought about why they work and were surprised at the depth of understanding needed in order to understand the algorithms fully.

To show that algorithms can be taught and learned at mathematical levels higher than elementary arithmetic, I then taught students a beginning calculus example: how to take the derivative of a polynomial function, a topic presented in any first-year calculus text. Although some had already seen this in their previous math courses, others had not. This algorithm is much simpler than the square root algorithm, and, within a few minutes, all the students could easily apply it. However, they couldn’t explain why the procedure produced the derivative of the function.

In order for them to understand this procedure fully, I first had to detour into teaching (or reviewing, depending on the student’s background) about permutations, combinations, and the binomial theorem. I highlighted the strong connection between combinations and the binomial theorem, pointing out how the coefficients from the binomial theorem can be thought of as coming from computations of combinations (see Figure 9.3 for an example).

I then used a definition of a derivative to be a limit of the slope of a function and demonstrated to the students how the list of simple rules given earlier, when applied to a polynomial, will produce the derivative. This definition included a binomial term, and the mathematics simplified to the point where the second number in the given row of Pascal’s Triangle turned out to play an important role (see Figure 9.3). The link between Pascal’s Triangle, combinatorics, and derivatives was used to develop the simple rules that took only a few minutes to master 2 days earlier. I pointed out to the students that I could teach them how to take a derivative in minutes, but that to truly understand it took 2 days of preliminary background.

While we were examining these various algorithms over the course of a week, students were working on the first problem set given out on the first day. As they brought questions to class, we would address them, which allowed students to have input into the topics considered day to day. The problems on this problem set were designed to utilize the mathematics discussed during the algorithm problems, as well as expose students to potentially new areas of mathematics such as modular arithmetic (see Figure 9.4 for a few sample problems from this set).

Sample problem 1 in Figure 9.4 stimulated thinking related to modular arithmetic, and, once students had a chance to discuss this problem in class and see approaches taken by others, they were usually able to solve it. Part (b) of this problem asked them to generalize their solution strategy, which required them to understand more deeply what was driving their solution. As always, I stressed the mathematical thinking, not the specific answer. The second sample problem afforded multiple solution paths, which provided for interesting classroom discussion. Students might simply look for a pattern, which is all that is actually asked for. They could then write their solution as a sum of terms. They could also recognize that the triangular numbers, which incidentally show up in Pascal’s Triangle in the second diagonal (1, 3, 6, 10, . . .), are in the pattern (see Figure 9.3). Use of the triangular numbers combined with formulas developed earlier for combinations led to a closed form solution for sample problem 2 in set 1. Students usually ended up at different points on these solution spectra, and classroom discussions of the thinking that went into various approaches often proved insightful for all students. The value of previous knowledge applied to a problem that seemed completely different from the initial context of exposure to Pascal’s Triangle intrigued many students. After about two class periods were spent on this problem set, students began to appreciate the value of experience in a variety of mathematical thought because very diverse problems could often be attacked with similar thinking. Although it took some time to follow the mathematical detour for understanding the algorithm for taking derivatives of polynomials, the knowledge gained by this detour turned out to be useful in other contexts.

The second half of the course was spent showing how mathematics could be used to analyze a variety of situations. Students were given a second problem set of nine problems in order to establish a common base for future class discussions (Figure 9.5 shows a few sample problems from this set). Many of these problems were again carefully chosen to use prior topics we had discussed, as well as stretch students’ mathematical experiences in new directions. Part (b) of the first problem leads to a result called Gomory’s Theorem, but is something that offers multiple opportunities to seek a solution. The last three sample problems all can be solved with techniques that relate back to Pascal’s Triangle. The seeming disparity in those problems hides some underlying commonalities. As with the first problem set, students worked on these problems throughout the second half of the course, bringing questions and ideas to class to bounce off of each other as needed.

The second portion of the course analyzed the well-known Tower of Hanoi problem (see Figure 9.6). Students were given manipulatives to use if needed, and they suggested starting with a small number and building up a pattern. Students worked with recursive formulas to solve this problem and then translated that recursive formula into a closed form solution. With classmates, students were generally able to do this, and so I then challenged them to develop closed form solutions for more general linear recursive formulas. This led to much more complexity and special conditions that needed to be applied in some cases. This activity not only introduced students to thinking recursively, but also to appreciate again the value of teamwork, the value of starting simple and building up to more complex situations, and the ubiquity of some basic mathematical processes useful for approaching a variety of problem situations.

During this portion of the course, students also explored computational science techniques as an approach to problem solving. The Web site http://www.shodor.org/interactivate/activities/index.html provides a number of applications appropriate for experimentation with ideas related to fractals, chaos theory, and probability, among others. These activities exposed students to the idea that not all of mathematics leads to closed form solutions or explicit equations, but mathematical ideas can still be useful for solving a wide variety of problems.

In the second half of the course, students also analyzed a game strategy for a number of different games such as Sprouts (see Figure 9.7). Both recursive thinking and portions of Pascal’s triangle are relevant in the analysis of strategy for winning this game, helping students to see that mathematical knowledge applied in one context can reappear in seemingly unrelated areas. As time permitted, various games were mathematically analyzed for strategy. Rudimentary graph theory problems were also introduced. The final few days were again spent discussing any of the questions from problem set 2 that had not yet been resolved.

In general, throughout the course, not only were students exposed to mathematical thinking in a variety of situations, but they also got to see how some of the same mathematical ideas used earlier were useful in new situations. I chose to use Pascal’s Triangle as one of my unifying themes because of its versatility, but other choices are also possible. As the reader may notice, the number of different problems attempted was relatively small for such a mathematically capable group of students. However, I attempted to include a representative sample of problems from different mathematical branches so that students could gain some appreciation of the breadth of mathematical thinking, as well as the depth. The goal was to balance breadth and depth of mathematical topics within a short 3-week session without sacrificing too much of either one.

As current reform efforts strive to make curricula more student-centered (NCTM, 2000; National Research Council, 1996), one group that may nevertheless be overlooked is student input into the curriculum and its implementation. Student voices are sometimes absent from discussions of school policies and practices. Nieto (1994) highlighted this deficiency and argued for the importance of students having input into schooling processes, particularly students from backgrounds typically considered out of the mainstream. Gifted students, while not specifically mentioned by Nieto, may also fall into a category out of the mainstream, not having their voices heard on curricular matters of importance to them. Hatchman and Rolland (2001) recommended that educators and policy makers listen to student voices in order to implement school reform effectively. They argued that such input is extremely helpful in creating schools where students thrive academically. Kordalewski (1999) summarizes specific teaching strategies that may help encourage this input from students, reflecting a growing movement to consider what students have to say as being important.

There is a continuing need to highlight the importance of student voices in research. “Virtually no research has been done that places student experience at the center of attention. . . . Rarely is the perspective of the student herself explored” (Erickson & Schultz, 1992, p. 467). This article attempts to address this void, providing a forum for gifted students’ voices to be heard concerning their reactions to the mathematics problem-solving course they experienced at the Governor’s School. Students themselves may be the best judges of the effectiveness of a curriculum and its implementation, and they may provide guidance on aspects helpful to consider for future courses.

In order to evaluate the success of this endeavor to create a course with more mathematical sophistication than often seen in regular high school courses, I asked students to respond in writing to several questions (see Figure 9.8). These prompts were provided to aid in developing introductions to the mathematics journals the students had been keeping throughout the course. Students were encouraged to limit this introduction to one page in an attempt to distill their main thoughts down to the most important ideas.

A content analysis of the students’ responses to these prompts and their own unsolicited comments about the course revealed some interesting themes that provide insight into what these gifted students value in an educational setting for the learning of mathematics. A total of 28 students responded to these prompts, but not all students responded to all of them. Here is what they said.

Prompt 1: What abilities are important for someone to become an adept mathematics problem solver in situations that may arise in genuine contexts?

In response to prompt 1, the most common theme that emerged (12 students) was to emphasize the importance of understanding why, instead of merely how. A. H. expressed this emphasis by writing, “Understanding math is much more important than being able to complete 60 math problems correctly. I have learned that the thought process is very important and that now more than ever I must become responsible for my learning.” Some students were surprised to learn that they really didn’t understand very rudimentary mathematics on a deep level. For example, M. L. wrote, “Basic, fundamental principles are examined in new lights to my amazement when I realized that I can do and understand differential equations, but not arithmetic. Thank you for expanding my mind.” The course emphasis on the importance of the process that leads to understanding, as opposed to the product that results from mindlessly applying an algorithm, is reflected in L. E.’s comment, “The thinking is more important than the product.”

A second common theme emerging from this prompt (7 students) was the need to be able to think creatively, to not be artificially restricted in strategies for solving problems. R. Y.’s comment, “[This course gave me] strength to realize that it’s OK to be wrong; being wrong on one thing can bring you a right answer on another thing,” highlighted the value of making mistakes and going down dead-end pathways on the way to improving mathematical problem-solving skills. The creative aspect of mathematics came to the fore in K. W.’s assertion, “To be a good problem solver, one must think ‘outside the box’—think deeper than usual.”

Prompt 2: Has this course helped you think about mathematics differently than a typical high school course might have, and what are your opinions about any differences?

In response to prompt 2, student comments clustered into three themes: most high school courses and textbooks don’t go deep enough (8 students), problem solving takes time that is not usually made available in most high school courses (5 students), and most high school courses are concerned with preparing you for the test, not for understanding (4 students).

Criticism of the lack of depth in high school courses centered on both the activities implemented by the teacher and the textbooks and other resources used. “In high school, [the teachers] rarely say why [mathematical algorithms] work. Now I know to either question why, or to figure it out myself.” This response from C. M. indicated a realization that he could take some of his learning into his own hands if he were unsatisfied with what a course offered—a powerful recognition of the ability to control one’s own learning. Not only are classroom activities criticized for not going deep enough, but textbooks received similar criticism. According to L. S., “Textbook questions are sometimes rather basic and don’t require deep thinking.”

The second theme emerging from this prompt concerned the lack of time in most courses. R. F. recognized the value and need for adequate time, writing, “Problem solving takes time. It isn’t always simple questions you can answer off the top of your head.” M. J. contrasted her Governor’s School experience of mathematical problem solving with her experiences in her high school, also focusing on the valuable resource of time: “This is definitely not a high school course, for the simple reason that [in high school] there’s not enough time or not enough teachers willing to go through with [the extensive development of a mathematical idea].”

The third theme arising from this prompt revealed some interesting insights about how testing can detract from a course’s value. T. M.’s view that “high school courses are too vague. They don’t go deep, just try to quickly prepare you for the test” attributed the lack of depth of high school mathematics courses at least partly to testing pressure. A. H. expressed a similar view: “In a typical high school math class, there is such a narrow focus that you are learning math simply for the test, not for later applications.” J. L. equated the absence of testing at the Governor’s School with educational value: “My high school teachers do not explore why. Most importantly, I will remember the testless, fully educational, non-pressure-giving courses this summer.” Student comments falling under this theme expressed a common sentiment that much of testing is detrimental to the educational value of their coursework in their home high schools.

Prompt 3: Would you recommend a similar course to other high school seniors?

In response to prompt 3, students strongly recommended (16 students) a similar course for all students, regardless of their mathematical ability. C. R. believed such an approach would even help students who may not have a mathematical inclination: “I would recommend this class to anyone willing to learn more math to help them with any type of problem, not just math-oriented people, but anybody.” These gifted students seemed to find the approach taken in this course to be beneficial to them, and many believed that other students would likewise benefit from such an approach. B. L. highlighted the emphasis on thinking as particularly beneficial: “I think [this course] was very beneficial to me and everyone should be exposed to this kind of thinking.” Similar sentiments expressed by many students underlined the potential for an appropriate emphasis in mathematics courses to impact positively the learning of a wide spectrum of students.

Prompt 4: What would be the most important idea(s) that a reader should get from reading your journal?

Many of the most important ideas listed in the responses to prompt 4 included themes already discussed. In addition, a number of students mentioned that they really enjoyed the challenge (7 students) and that they better understood the rich tapestry of mathematics as it relates to applications in the real world (7 students). Many students discussed the exposure to really thinking about underlying mathematics and connections between seemingly different applications as part of the most important and enjoyable aspect of this course. T. H. expressed this by writing, “Problem-solving class helped me to think this year. It was one of the best things at Governor’s School.” L. E. focused on the challenging aspect of the course: “This course offered me a great challenge and has done as much for me in 3 weeks as a typical high school course would.” M. A. expressed her enjoyment: “It is really fun thinking through things.”

Comments pertaining to the connections between mathematics and applications, as well as connections within the field of mathematics itself, were also seen as important aspects of the course. K. T. revealed a surprising gap in her mathematical experience considering that she had had formal schooling in mathematics for at least 11 years: “I’ve never had to think about math as it relates to everyday life.” The use of mathematics to analyze applications different from most test questions may have prompted C. K. to write, “Problem-solving is much more difficult in real life than it is on tests.” The opportunity for student input into the direction of classroom discussions and input into time spent on portions of mathematics that interested them was appreciated by students such as K. T., who wrote, “I was glad that [the teacher] would get into math tangents that interested him and us.” This approach allowed the rich interconnections between mathematical topics to be explored as class interest dictated, which helped to illuminate the existence of connections of which students may not have previously been aware.

The students in this program were successful problem solvers in order to be eligible for the Governor’s School program. In spite of this success, many of them expressed a common faulty impression that all mathematics problems could be solved quickly and directly. For example, A. F. wrote in his mathematics journal, “Not all problems can be solved, and the person who tries the most and takes the most time is usually the one who gets the answer. I used to think that, if I took longer than about 15 minutes on a problem, I must have been doing it wrong and I felt stupid. Now I realize that there are some problems that require time to solve.”

Students also tended to believe that there was one right way to solve a given problem, a common misconception highlighted by Kroll and Miller (1993). After the experience of this problem-solving course, H. R. described a shift away from this misconception: “The questions in real life are so much more complex than those encountered in most courses.” These types of student comments suggest that, in spite of these students being among the most mathematically able high school seniors in North Carolina, they had not truly experienced much sophisticated mathematics that would challenge their misconceptions. If, as Halmos (1980) suggested, problem solving is the heart of mathematics, then even these mathematically gifted students, on the cusp of their last year in high school, may not have yet developed this core skill to their full potential.

Coleman’s (2001) assertion that bringing more complexity and abstraction to a subject (sophistication) is needed to meet the desires of gifted students seems to be supported by these student comments. Some models of curricular modification for gifted students focus on content acceleration and fast-paced instruction. For example, McCarthy (1998) researched a model whereby middle school students complete 4 years of high school math in 2 years during their regular school day. While such models certainly have a valuable role to play in educating gifted students, student comments in the previous section reinforce the idea that modification via sophistication, rather than acceleration, may be an effective way to motivate gifted students. None of the students complained, either in journal writing or informally in conversation, about the pace of the class or the amount of content coverage. In fact, many of them indicated that they particularly enjoyed exploring fewer topics in more depth.

As I read the students’ journals, I was struck by the overwhelming sense of excitement and energy as they wrote about the abstraction and depth provided throughout the course; this aspect is what seemed to capture their interest. A theme that emerged was the desire to have more time to study topics in depth, which supports Kaplan’s (2001) call for educators to become advocates for creating the time gifted students need for optimal learning. Student comments about time and the focus of classroom instruction due to tests also have implications for the trend of ever-increasing standardized testing in our educational system. These comments highlight the danger that ill-considered testing may rob our most capable students of an appropriate curricular approach. Strong advocates in this arena are needed to ensure that gifted students (as well as other students) do not lose out on appropriate instruction.

Another aspect of this experience that may have contributed to the students’ positive responses was the collaborative nature of the course. The scoring procedures of standardized tests such as the SAT compare students to one another, which implies an inherently competitive framework. All the students at the Governor’s School score well on standardized tests in order to be eligible for the program, which may indicate a propensity to consider schooling as an academically competitive activity. In contrast to competition, I encouraged cooperation and collaboration throughout the course.

Diezmann and Watters (2001) found that gifted students preferred collaboration to independent work only when the task was sufficiently challenging. As in their study, student collaboration in this course was self-initiated, as opposed to teacher-directed. Although formal cooperative learning groups were not assigned, students were encouraged to work together by discussing approaches and thoughts related to the mathematics being studied. Garduno (2001) likewise found that gifted students were more motivated when they were able to advance at their own pace, were grouped with others of similar ability, and were engaged with tasks involving complex processes, as opposed to merely computational procedures. The structure of the Governor’s School program, wherein gifted students are all housed together in a dormitory setting 7 days a week throughout the course, seemed to be conducive to collaboration, as evidenced by student comments about working together on problems. A number of student comments pointed out the insight gained by seeing other students’ thought processes and expressed appreciation for the opportunity to do so. Students were able to establish the pace of their work to some extent because they had significant input into the daily topics and particular problems addressed in class. The admissions requirements for the Governor’s School ensured that all students in the class were of similar mathematical ability, and the course intentionally focused on complex mathematical processes, as opposed to merely computational procedures. This confluence of conditions satisfied those highlighted by Garduno, and students’ comments imply a high degree of motivation on their part.

All of the students quoted in this article had already completed precalculus in their home high school before enrolling in the Governor’s School, and some had completed as much as 2 years of calculus along with additional advanced math courses offered through local universities. Although such advanced high school math courses are intended to serve academically gifted students, student comments indicated that educators may need to consider modifications to such courses to stimulate our academically gifted students. It may be valuable to offer 3-week intensive courses somewhat similar to the one offered by this author—it should be possible to carve this time out of traditional advanced mathematics courses and still have gifted students do well on standardized tests. If this practice were to become a norm, gifted students may discover the challenge and beauty of mathematics that many indicated was lacking through their regular high school courses.

To have an even greater impact than a separate short course, incorporating the pedagogical characteristics appreciated and highlighted by these gifted students into regular coursework is not only possible, but also recommended by national mathematics standards. These students indicated that a similar experience would be appropriate and helpful for all kinds of students, not just the most academically capable. This philosophy of providing enriching mathematics for all students, termed the equity principle, is a centerpiece of the national mathematics principles and standards (NCTM, 2000, pp. 12–14).

The comments from the students can be synthesized into three broad recommendations for modifications and changes in emphasis to existing traditional mathematics courses:

1. more emphasis on why, instead of how, something is done;
2. more depth, less breadth of coverage; and
3. more challenge.

One aspect of this experience that was greatly appreciated and recommended by these students was the focus on having them understand why certain mathematical approaches or algorithms are used as opposed to merely how. This recommendation is directly aligned with the learning principle espoused by the national mathematics standards (NCTM, 2000, pp. 20–21) that states that students must learn mathematics with understanding. This emphasis should be incorporated into any mathematics course. Although it undoubtedly takes more time to delve into the why, the payoff comes in student interest and understanding of the underlying mathematics. Even if such an approach couldn’t be done for every topic in a course’s curriculum due to time constraints, perhaps it could be done occasionally. The impression given by these students is that this is a rare to nonexistent approach in their mathematics experiences.

A second student theme that deserves more attention in many mathematics courses is to focus more on depth and less on breadth of coverage. This point is also made in several well-respected documents, including national science standards that point out that “the present curricula in science and mathematics are overstuffed and undernourished” (AAAS, 1989, p. 14); a Third International Mathematics and Science Study (TIMSS) report from a comparison of mathematics and science programs worldwide that states, “[The United States’] curricula, textbooks, and teaching all are ‘a mile wide and an inch deep’” (McNeely, 1997, p. 161); and national mathematics standards that advocate for a coherent, focused, and well-articulated curriculum (NCTM, 2000, pp. 14–16.) Many students seemed to appreciate the opportunity to investigate a few mathematical ideas in depth, in the process uncovering connections between seemingly disparate mathematical ideas. Again, this modification requires time, a precious commodity in a classroom, but one that should be spent in the most productive ways. Such a change in focus away from breadth toward depth seems likely to be a productive use of this finite resource. There is much to recommend in sophistication of content over broad coverage of it.

Thirdly, these students relished the challenge presented. Mathematical ideas that were presented in ways that were nontrivial both engaged and frustrated the students—a situation ripe for learning. Part of the national mathematics standards’ teaching principle highlights the need to challenge students (NCTM, 2000, pp. 16–19.) The collaborative nature of classroom discussions of mathematical problems and presentations of students’ disparate ideas on how to attack these various problems seemed to both alleviate the frustration and increase student engagement. In the process of wrestling with various challenging problems, students came to appreciate the rich tapestry of mathematical ideas. They appreciated the value of building new concepts on existing ones and looking for connections between problems. A pedagogical approach highlighting connections and encouraging wrestling with challenging problems over an extended period of time should be incorporated at the appropriate level into any mathematics class. Once students realize that they will not be penalized for not immediately knowing how to solve a given problem, they may enjoy the prospect of tackling challenging problems in collaboration with their peers.

These three recommendations from the students, which are also emphasized by current reform documents, are actually mutually supporting. If a curriculum were to focus more on depth and less on breadth of coverage, this would provide time to emphasize the why of something more than merely how. This emphasis, in turn, would lead to more challenging content for the students. These recommendations are therefore not unattainable; they do, however, require some serious restructuring of the mathematics curriculum.

Some research indicates that having participated in a gifted program in high school does not result in increased university achievement. Grayson (2001) studied the achievement of students over the course of 4 years at a university in Ontario and compared achievement of those students who graduated from high school having participated in gifted programs with those who had not. He found no differences in university achievement between the two groups and attributed this lack of difference to the possibility that the selection process for identifying gifted students may be flawed. An alternative explanation not discussed by Grayson is that perhaps the differentiated instruction received by students identified as gifted was ineffective. Student insights and comments reported in this article suggest that mathematics instruction they received in their high school courses may not be helping them reach their high academic potential. Perhaps educators need to rethink how and why we provide instruction for our gifted high school students in order to have the most powerful impact on their learning.

Educators of gifted students need to be strong advocates for practices that best serve these academically capable students. The best guidance on what is important often comes from the students themselves. The congruence between recommendations of the national mathematics standards and recommendations of these students for effective mathematics curricula serves to emphasize the need to enact these reforms. Students are telling us what they want and need in order to be challenged and stimulated; it is up to us to listen to them and act. M. L.’s closing remark in the introduction to her mathematics journal clearly states this desire: “I have always wanted to have someone instruct me on the ways of the mind.”

Can we fulfill such a request?

American Association for the Advancement of Science (AAAS). (1989). Science for all Americans. Washington, DC: Author.

Coleman, M. R. (2001). Curriculum differentiation: Sophistication. Gifted Child Today, 24(2), 24–25.

Diezmann, C., & Watters, J. (2001). The collaboration of mathematically gifted students on challenging tasks. Journal for the Education of the Gifted, 25, 7–31.

Erickson, F., & Schultz, J. (1992). Students’ experience of the curriculum. In P. W. Jackson (Ed.), Handbook of research on curriculum (pp. 465–485). New York: Macmillan.

Gallagher, J., & Gallagher, S. (1994). Teaching the gifted child (4th ed.). Boston: Allyn & Bacon.

Garduno, E. (2001). The influence of cooperative problem solving on gender differences in achievement, self-efficacy, and attitudes toward mathematics in gifted students. Gifted Child Quarterly, 45, 268–282.

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