Risks in Identifying Gifted Students in Mathematics: Case Studies

2.1 A Gifted Student with Dyslexia

Students who display characteristics of the twice-exceptional profile (from the profiles of Betts & Neihart, 1988) are students who have two exceptionalities – giftedness and one or more disabilities (Ronksley-Pavia & Neumann, 2020). Students with this profile can have reduced resilience (impacting on behavioral engagement), limited coping strategies and learning strategies (inhibiting cognitive engagement), and impeded persistence (hindering both behavioral and cognitive engagement) (Ronksley-Pavia & Neumann, 2020).

Twice-exceptional students are divided into three groups: (1) identified gifted students as low-achieving students; (2) gifted students with specific learning disabilities (SLD) and diagnosed with SLD; and (3) students whose SLDs and talents mask each other (Yenioğlu, Melekoğlu, & Yılmaz Yenioğlu, 2022).

Dyslexia is usually specified as difficulties in the area of reading. As stated by Portešová et al. (2014), reading is not the only difficulty for students with dyslexia. Others are, e.g., deficits in the phonological processing of information, a deficit in the ability to quickly recall concepts from long-term memory, working speed, memorization, or insufficient graphomotor speed. In gifted students with dyslexia, various paradoxes arise between cognitive abilities and their difficulties. Therefore, these pupils are often referred to as paradoxical pupils and students (Tannenbaum & Baldwin, 1983). Gifted students with dyslexia tend to have difficulty with timed testing. They can achieve only partial success compared to if they had the opportunity to write the test in nonstressful conditions without a time limit. For that reason, they are very difficult to identify by testing. And, not only with testing that determines IQ or giftedness but also with practically every common test. They may achieve worse grades and may be perceived as weak rather than gifted. One compensatory mechanism that can help students with dyslexia is extended time. The second is to focus on the places where the student has difficulties and teach him to work with these mistakes.

2.2 A Gifted Student with an Extremely High IQ

One of the risk groups of gifted students is students with an extremely high IQ, usually greater than 150 (Hříbková, 2009). These pupils have very different individual profiles of different abilities. They have difficulties with peers and often find themselves socially isolated. Letta Hollingworth, who carried out a case study of 12 pupils with an IQ above 180 (Hollingworth, 1975), saw the reason in the fact that these children had interests, knowledge, and verbal expression corresponding to an adult rather than a child. This was a disadvantage for them because they did not have common interests with their peers. However, they did not feel comfortable even among the teachers, and sometimes they were considered “weirdos” and liars. Hollingworth reports that students with an IQ of 180 were taught by teachers with an IQ of 120, which was reflected in the way they communicated with the students. Hollingworth pointed out that gifted pupils with extremely high IQs waste a lot of time in school, e.g., as a result of not being given sufficiently challenging tasks: “In the ordinary elementary school situation children of 140 IQ waste half of their time. Those above 170 IQ waste practically all of their time” (Hollingworth, 1975, p. 299).

These are some of the reasons why educating gifted students with extremely high IQs is difficult and why these students may see school as a waste of time.

Ronksley-Pavia and Neumann (2020) present ideas on how to support autonomously gifted students in education (the profile according to Betts & Neihart, 1988). This profile perhaps corresponds most to students with an extremely high IQ who, like autonomous students, are able to set their priorities and adapt to the demands of the school, so we will now be inspired by these suggestions. These include, e.g., the following suggestions: giving pupils additional extended help if needed; offering inspirational study resources in their area of interest; strengthening motivation for independent study; and providing feedback in doing so.

The following study focuses on the description of two mathematically gifted students who were not identified as gifted in the school environment. It attempts to capture their academic development in the first years of education, certain moments that can be a sign for teachers that a given student is gifted, and the difficulties that students encounter in school. The parents’ view of their child’s mathematical education is captured. Furthermore, the child’s suggestions for a change in their education, which would enable them to develop better academically at the school level, are presented.

For this purpose, the following research questions were asked:

1. What characteristics in the field of education did the pupil show in preschool and younger school age?

2. What obstacles and difficulties in the area of education of gifted pupils did the student encounter at school?

3. How do their parents assess the student’s school education?

4. How does the student imagine his optimal mathematical education?

In both cases, a 1-year intervention was carried out, which focused on the initial deficiencies in the students’ mathematical procedures. A procedure based on the assignment of nonstandard tasks was proposed, which the students solved independently and then presented their solutions to the researcher. The aim of this intervention was to cultivate their mathematical thinking and expressions.

3.1 Participants

The first participant is a mathematically gifted student with dyslexia. He has been monitored since the fourth grade, and when his parents solved problems with his behavior and performance at school, they became more interested in his dyslexia and the possibilities of mathematical development. Today he is a pupil of the seventh year of elementary school. His mathematical abilities were determined by the test TIM^3–5, which is a standardized test designed to identify the mathematical talent of third and fifth graders (Cígler, Jabůrek, Straka, & Portešová, 2017), according to which the student is at the hundredth percentile. His IQ was determined to be higher than 130.

The second participant is a mathematically gifted student with an extremely high IQ. He has been monitored since the seventh grade when his parents were looking for a tutor who would be able to adequately develop his mathematical abilities, as this did not happen at school. Today, he is a student in the second year of a 4-year high school. His mathematical abilities were determined by standardized tests in a pedagogical–psychological counseling center. His IQ was determined to be higher than 140 (there are no tests in the Czech Republic that would determine an IQ higher than 140).

3.2 Aim of the Study

The aim of the research was to describe the students’ situation, the moments that led their parents to intervene in education outside of school, the characteristics of the gift that the students showed, and the possibilities of cultivating their mathematical thinking. The research has two levels: (1) the level of pupils’ difficulties or adverse circumstances that led to the fact that the pupils’ talents were not identified and the pupils did not develop at the school level; and (2) a long-term view of the pupils, which will make it possible to determine whether and at what point they started to be pupils perceived in school as gifted.

3.3 Data Collection

Semi-structured interviews with parents and participants were used as the data collection technique in this study.

Parents were asked questions such as “How was the development of your son’s math abilities and skills in preschool?” “Was the son independently interested in mathematical concepts in the early years of education? And which ones?” and “Has something hindered the development of your son’s talents at school? And what?”. The participant was then asked about the “ideal teacher” and the “ideal math lesson.”

Interviews were recorded and transcribed.

3.4 Data Analysis

The content analysis technique was used to analyze the data obtained in the study. No software was used for data analysis. Records were searched for predetermined sub-categories. These are presented in tables.

4.1 Context of the Situation

This case concerns a pupil with dyslexia. For the purposes of this case study, we will refer to him by the fictitious name Ron. The fact that Ron has an aptitude for mathematics was already suspected by his parents at preschool age. He was particularly interested in construction kits; he built very complex spatial structures, which required a high degree of spatial imagination. He was interested in numbers and liked solving word problems. However, after he started school, his parents started to deal more with problems with his social and emotional skills, which were greatly reduced. When they used to go to school for the third year to solve various problems with his behavior, the situation seemed hopeless, and they paid for a private psychologist. She reassured them that the son was socially and emotionally at the level of a child three years younger, but that the situation would improve on its own during puberty.

In addition, during the first two years of education, Ron’s dyslexia began to show strongly. The development of mathematical talent could not take place at school because the teachers spent most of the time solving problems with his behavior or problems associated with dyslexia. However, his parents continued to perceive his increased interest in mathematics as well as his high abilities in this subject. He quickly acquired new knowledge and had very flexible thinking with the ability to generalize. Arithmetic was no problem for him; he quickly learned to add, subtract, multiply, and divide. He liked to solve nonstandard problems, and his solutions were often imaginative. In Figure 1, we can see his original solution to the following problem: If you add two numbers, you get 58. One of the numbers is 5 more than the other. What are the numbers? He solved this task in his second year. He used a controlled experiment strategy. His answer is “26 and a half and 31 and a half = 58.”

Figure 1

An original way of solving the problem in the second grade: “If you add two numbers, you get 58. One of the numbers is 5 more than the other. What are the numbers?” The answer is “26 and half and 31 and half = 58.”

Because of dyslexic difficulties, the parents decided to visit the pedagogic-psychological counseling center in the fifth grade. The assessment stated: "Reading at a slow pace with increased error rate (dyslexia), written expression is marked by increased error rate on the basis of weakened phonological awareness; graphomotor clumsiness. Reasoning skills are at a very good level. The ability to perceive spatial relationships, logical reasoning and mathematical judgment is advanced. Verbal skills are average. The work pace is slower. Frequent positive feedback helps him to stay focused.” An individual education plan was drawn up for Ron, with the emphasis placed on strengthening the teaching of the Czech language and also on enriching the mathematics curriculum. This helped him, and he began to do better in school.

4.2 Measures Leading to the Stabilization of Dyslexia Problems and the Development of Giftedness

The diagnosis of dyslexia was crucial for Ron. Many of his problems stemmed from frustration at not being able to complete schoolwork as successfully as his classmates. Once the problem was named, Ron could concentrate on compensating for dyslexic difficulties and developing his mathematical aptitude.

A mathematical intervention was carried out for him in the fifth grade. His initial strategy that he used before starting the intervention was a controlled experiment. Metacognition was manifested in some cases (e.g., evaluating the meaningfulness of the result), but he was not able to manage his strategies. In the course of the one-year intervention, he managed to eliminate these deficiencies, began to consciously use solution procedures, and was aware of individual steps. He expanded the spectrum of solving strategies and began to use arithmetic and sometimes algebraic strategies more.

From the sixth grade, he began to study mathematics outside of school in the mathematics club. He is in seventh grade today. His dream is to become a computer programmer, and he has begun to adapt his actions to this: he thoroughly learns the English language because he knows that he will need it; together with his parents, he decided on a vocational high school focused on programming because he would have problems with academic subjects in high school; strives to have good knowledge and results in mathematics; attends a programming class. Now, in the seventh grade, we can say that his dyslexic problems persist. He switched to freehand writing, which allowed him to write faster and focus more on the rules of spelling. Even so, he often makes mistakes. He is guided by teachers and parents to look for and correct mistakes himself.

In mathematics, he continues to be distinguished by novel and original solution ideas. He understands new knowledge and connections very quickly and is able to explain mathematics to others. But most importantly, he learned to work with his difficulty and found a goal to work toward. The psychologist parents visited in the third grade was right. In adolescence, he socially caught up with his classmates. However, his situation at school and in the classroom improved significantly immediately after the diagnosis of dyslexia and mathematical giftedness.

At school, however, he was never perceived as mathematically gifted, and the teacher described him as an average student.

5.1 Context of the Situation

David started going to kindergarten at the age of 4. The parents were dealing with problems associated with his behavior – e.g., he took apart a model of a human skull, including all the teeth, which he hid in various places around the class and asked the teachers to find the individual parts and assemble the skull. He showed an unusual interest in numbers and had other interests that absorbed him, such as knowledge of dinosaurs.

David started first grade when he was almost 7 years old. He showed difficulties in the social area and a reduced ability to adapt to a new environment more than his mathematical talent. During the first grade, the mother was invited by the class teacher who told her that David is strange, does not fit in with the team, disrupts the lessons, and tries to teach the other children from examples to which they do not know the answers. The teacher was asking the parents to do something about him because if a school inspection comes to her class, it will be hard to explain. The mother perceived a difference in her view of her son, who showed an unusual interest in mathematics as well as unusual knowledge, and the view of the school, where David was seen as more problematic and, therefore, had David tested by Mensa. The test showed highly above-average intellectual abilities. The parents were advised to transfer David to another elementary school, where he can work better with gifted students. The transfer condition was confirmation of mathematical talent by pedagogical–psychological counseling. The investigation showed extraordinary abilities in mathematics, logic, and spatial imagination.

He was average in other subjects and had difficulty in the ability to understand the spoken word and in writing. His weak point was his social skills. During the second grade, he came to the conclusion that “you can also calculate the square of a number by adding the same number to the square of a number once more and then one larger number and you get the square of the number 1 larger” (e.g., 5 2 + 5 + 6 = 36 which is 6 2 ).

There was no acceleration in the gifted school because disciplinary problems were constantly being solved. David was not happy at school, and he was even hospitalized with mental problems. The parents, therefore, decided to change school yet again. Because he was in the fifth grade, they sent him to the entrance exams for a multiyear gymnasium and waited until the end of the school year to change school.

In the sixth grade, he transferred to a multiyear gymnasium. Even here he did not get the desired education he imagined. He was included in the group of average students, even in mathematics, and he did not have any supplementary studies as part of the teaching.

From the seventh grade, he started attending my individual math lessons.

David’s mathematical abilities were demonstrated at a younger age in a pedagogical-psychological counseling center and later, from the seventh grade, were described on the basis of his performance during individual tutoring.

His extraordinary mathematical abilities were obvious – he was able to calculate from memory even complex calculations with fractions or decimal numbers and determined powers and square roots from memory. He solved problems easily and was proficient in generalization and geometry. He had an extraordinary memory, thanks to which he remembered everything the first time.

At the same time, however, the uneven development of mathematical abilities was evident: he could not write down the solution procedure intelligibly, experimented chaotically, and did not use solving strategies or metacognition. In structural geometry, his knowledge and skills were not even at the level of a seventh grader.

There were noticeable gaps in the social area – he usually did not make eye contact, answered sternly, and of all topics, he was only interested in mathematics.

5.2 Actions to Improve David’s Situation at School

David was dissatisfied with the fact that, even at the multiyear gymnasium, teaching mathematics did not enrich him. He was looking for new stimuli, and most of all, he would like to start studying straight university mathematics. But I saw that first, we would have to eliminate the gaps in his ignorance of solving strategies, nonuse of metacognition, and ability to write down the solution procedure.

Initially, David mostly used an undirected experiment to solve problems. This was mainly due to his ability to calculate even difficult examples from memory. I immediately warned him that I would not accept the results unless I understood how he arrived at them. He immediately responded to my request and began to think about how he had actually proceeded. However, this led him to more sophisticated procedures – arithmetic or algebraic. When he had to start thinking about the procedure, he realized that there are more elegant ways of solving things than simply guessing the result.

During the seventh grade, he also learned to write down the solution process clearly. He had his own ways of thinking, and sometimes I had to do a lot of thinking to understand his process. Let us show this phenomenon in the example of David’s solution to one of the tasks: We took some number, multiplied it by its two-thirds, and subtracted a third of that sum from that sum. We got 10. What was the original number? David’s solution can be seen in Figure 2.

Figure 2

Sample of the solution: “We took some number, multiplied it by its two-thirds, and subtracted a third of that sum from that sum. We got 10. What was the original number?” David’s answer: “The original number was 9.”

David recorded some of his essential thoughts, but the notes are atypical, and so one must infer the information that is missing. He proceeded arithmetically with the method from the end. He started by saying that 10 is two-thirds of a number. However, he was unable to write it down (e.g., the algebraic notation 10 = 2 3 x or a verbal description is offered), and he captured his idea at least as the mathematically false “ 10 = 2 3 .” So, he looked for a number that is three halves of 10, converted the three halves to the decimal number 1.5 and calculated 10 ∙ 1.5 = 15 . He then reasoned that increasing a certain number by 2 3 is the same as multiplying it by 5 3 . This is where the last calculation comes from when he multiplied the number 15 by 3 5 or the number 0.6. So, the number we are looking for is 9. At first, David did not do the true-falls tests at all, and gradually I managed to get him to at least perform them by heart.

The fact that David did not perform true-falls tests was related to not using metacognition. This was reflected in cases where he had an error in the result but presented the result as correct. David was not used to checking his procedures and results. If he submitted a wrong result, I had him go back through the solution and find the error, then justify the newly chosen solution and finally perform the true-falls test. In this way, he began to realize that not all of his solutions were correct and that it was necessary to check and also justify the procedures.

During Grade 8, David improved in his use of strategies as well as in his use of metacognition. While in the first months of our interventions, he most often resorted to undirected experimentation, and during the eighth grade, he successfully used arithmetic strategies and gradually began to prefer the algebraic strategy. The tasks that I had to give him in order to fulfill his potential gradually exceeded the level of difficulty of the elementary school curriculum.

During seventh and eighth grade, we covered high school math, but even that was getting too easy for him. In the eighth and ninth grades, we discussed differential and integral calculus. He mastered the subject matter with deep insight and understanding.

Despite his undoubted mathematical ability, he was never considered mathematically gifted throughout elementary school. The teacher let him work together with the other students on the regular curriculum, and he was not interested in preparing stimulating materials for him or discussing them. David was wasting his time in math class. To the teachers, David seemed more like “a weird,” and he often showed it to David and his parents. The change only came when he entered grammar school when the teacher changed. The new teacher immediately recognized his talent and started offering him various activities in the form of assignments, math competitions, correspondence seminars, etc.

As for David’s social situation, he felt isolated in his early years of education. Over time, he was able to find friends with whom he shared hobbies such as ping pong and cycling. However, his interests continued to be fundamentally different from those of ordinary children. For example, he stopped playing computer games because he realized how much time they were taking away from his studies.