Although there is a consensus on the domain-specificity of mathematical giftedness, research literature still lacks a common definition of mathematical giftedness (Singer et al., 2016). Consequently, it is not surprising that programs for mathematically gifted students rely on a variety of criteria in order to select participants. However, an analysis of different definitions reveals at least some common ground. Most theories conceptualize mathematical giftedness as potential for mathematical performance (e.g., Leikin, 2010; Singer et al., 2016; Zehnder, 2022). This article adopts the definition proposed by Ulm and Zehnder (2020). It characterizes mathematical giftedness as the individual potential to develop mathematical abilities, which themselves form the basis of mathematical performance.

Several models describe the process of the development of abilities and/or performance based on giftedness either in general or specifically within the domain of mathematics (e.g., Beumann et al., 2025; Gagné, 2015; Heller, 2013). Most models agree on the relevance of personality traits or states and of environmental factors—such as interest, motivation, school, or peers—for the realization of individual potential. According to Ulm and Zehnder (2020), the main process underlying the development of mathematical abilities based on mathematical giftedness is learning. For this reason, it is necessary to have a closer look at learning and its design, particularly in institutional settings.

Taking a moderately constructive view of learning (Dubs, 1995), we understand learning in institutional settings as an interplay between instruction and construction (Reinmann and Mandl, 2006). Ulm and Zehnder (2020) propose a model of learning environments (see Figure 1) that describes the processes of teaching and learning occurring in such settings. It extends other concepts of learning environments, such as those by Wittmann (1995, 2001) or Dubs (1995).

In the model, the learning environment forms a link between teacher and student. It is designed by the teacher and offered to the learner as an educational opportunity. If s/he decides to work within the learning environment, the teacher gets a feedback on both the learning environment and the learner. Ulm and Zehnder (2020) list six components to further describe the learning environment. We will use them later to structure our findings on the design of educational programs.

When analyzing a learning environment, each of its components may be more or less explicit. For example, in a learning environment based on the task “Discuss different ways of solving a quadratic equation with your partner”, both the content and partners are specified more clearly than in the task “Plan a math project on sustainability”.

Teachers design learning environments as educational opportunities for students. In doing so, they may follow certain principles to ensure high quality. Prediger et al. (2022) conducted a comprehensive review and, combining different perspectives, identified five subject-specific core principles for high-quality mathematics teaching (see also Holzäpfel et al., 2024).

These five principles are by no means distinct, but are interwoven. In other models of teaching quality they are even combined into broader factors (e.g., “cognitive activation” in Praetorius et al., 2018).

We grouped the publication dates of the studies by decade (see Figure 3A). Between 1971 and 1980, there was only one publication (2%). Between 1981 and 1990, there were six publications (12%). Between 1991 and 2000, there were 10 publications (20%). Between 2001 and 2010, there were seven publications (14%). Between 2011 and 2020, there were 15 publications (31%). The last decade is not over, yet there have already been 10 publications (20%) between 2021 and 2024. Therefore, we see an increasing number of studies that meet our criteria in more recent times with about half of them being published in the last 15 years.

We also identified the countries that the people studied come from (see Table 3 for a detailed overview). Since educational systems can differ substantially in some cases, this information is helpful when interpreting the results. The majority of the publications, 21 (43%), studied persons in the USA. Three (6%) studies each collected data on people from Turkey, Norway, and South Korea. Participants in two (4%) publications each originate from Australia, Germany, Canada, and Poland. Finally, the dataset contained one (2%) publication each analyzing people from the following countries: the Czech Republic, Finland, India, Israel, Japan, Malaysia, the Netherlands, Sweden, the United Kingdom, and the United Arab Emirates and one with participants from multiple countries.

More insight into the studies is provided by the percentage of studies that consider students at different grade levels, see Figure 3B for more information. If the study only contained information about participants' ages but not their grade levels, we used this information to determine their grade level in terms of the United States' school system. The age groups are distributed relatively evenly, so there is no indication that any age group is being examined more closely than another.

Another piece of quantitative data is the sample size of the studies. These numbers are difficult to compare because the methods used in the studies vary widely. We can distinguish between different types of trial participants, including gifted students, teachers (e.g., Freedberg et al., 2019), social workers (e.g., Haataja et al., 2020), principals and other administrative personnel (e.g., Campbell, 1988), and parents (e.g., Burns et al., 2017). The sample size varies from two to 236,938 participants. Clearly, quantitative approaches require a larger number of participants, while qualitative studies can be conducted with smaller numbers. On average, the studies had 5,026 participants, regardless of their role. The median number of participants is 42, so the high average is due to some studies having an unusually large number of participants (e.g., Vargas-Montoya et al., 2024; Vock et al., 2022). If we consider only the studies that included students in their sample, we find that there were an average of 5,448 students participating and a median of 42. If we consider only the teachers, the average number of participants is 26 and the median is 13. For other school employees, the average is nine and the median is ten, and for parents, the average is five and the median is three. Of the 49 studies, 45 included students, seven included teachers, three included other school personnel, and three included parents.

We also coded the definition of mathematical giftedness used in each study. Based on Zehnder (2022), we distinguished six categories: one based on potential; one based on ability; one based on intelligence; one based on performance; one category for other definitions; and one for studies where the term is not defined. Most of the studies, 27 (55%), do not provide a definition of mathematical giftedness. Of the studies providing a definition, 12 (24%) use an ability-based definition, followed by four (8%) using a potential-based definition. These two ways of defining mathematical giftedness both conceptualize mathematical giftedness as a basis for performance. Two (4%), Miller et al. (1995) and Zedan and Bitar (2017), use a definition based on intelligence and only one (2%) (Vargas-Montoya et al., 2024) uses a definition based on performance. There are also five (10%) studies which use a definition that does not fit into any of the other categories. Two studies use multiple definitions: Kamarulzaman et al. (2022) use a potential and an ability definition and Vargas-Montoya et al. (2024) use a performance based definition and a characterization using personality traits that does not match any of the categories.

We also documented the methods used to select students for the respective programs and, therefore, identify them as mathematically gifted. As most of the studies do not make explicit the programs' definition of mathematical giftedness, the identification process allows conclusions about the underlying definition. We found four distinct selection methods which partly correspond to the categories of definitions of giftedness. Identification was based on tests of intelligence, tests of abilities, performance, and nomination. As for the definitions, we also introduced a category for other types of selection. A total of 23 studies (47%) used ability tests, 21 (43%) used performance tests, 14 (19%) used nominations, nine (18%) used intelligence tests, and 26 (53%) used other identification methods. A total of 30 studies (61%) reported using multiple methods to identify mathematically gifted students.

Since a main task of schools is education, we considered whether programs took place in or outside of school. About half of the studies, 25, considered programs that took place in school, 19 considered programs that took place outside of school, and three considered a mixed version. Two studies did not provide this information.

Different institutions were responsible for the programs' realization. Most of the programs, 29 (59%), were organized by schools, 18 (37%) were organized by universities, and one (2%) was organized by a mixed entity. One study did not provide this information.

Most of the studies (78%) mention collaborative working as a design feature. Some of them just list it as one feature of the respective program (e.g., Campbell, 1988; Rabijewska and Trad, 1985), others describe positive effects of student collaboration. Collaboration enables students to discuss with each other and gain new insights on problems (e.g., Choi, 2013; Kamarulzaman et al., 2022; Tretter, 2003), refine arguments (van Schalkwijk et al., 2000), create more creative products (Kim et al., 2016), or advance on modeling activities (Kim and Kim, 2010). It also keeps students motivated (Haataja et al., 2020) and persistent (Reid and Roberts, 2006). However, programs only seem to benefit from collaborative work if certain conditions are met. Studies mention challenging tasks (e.g., Simensen and Olsen, 2024; Diezmann and Watters, 2001) and students with comparable abilities (Barron, 2000; Mingus and Grassl, 1999; Sharma, 2010; Zedan and Bitar, 2017) as relevant context factors.

The second most frequent found design feature (47%) is the composition of the learning group. Studies describe homogeneous ability-grouping as positive (e.g., Basister and Kawai, 2018; George, 1976), having a positive influence on the quality of learning opportunities (Simensen and Olsen, 2024), the groups' social support (Burns et al., 2017), or the motivation and perceived challenge (Mingus and Grassl, 1999). Heterogeneous ability-grouping, in contrast, is largely seen as negatively influencing the success of programs. Various reasons are mentioned for the negative influence of a heterogeneous grouping: teachers tend not to focus on gifted students in class (Freedberg et al., 2019) or while planning lessons (Sharma, 2010), less able students slow down the gifted (Zedan and Bitar, 2017), and sometimes gifted students even have to act as assistant teachers, not being able to focus on their own development (Gadanidis et al., 2011). Other grouping factors that seem to have a positive influence are gender (Reid and Roberts, 2006) and multi-grade groups (Campbell, 1988).

Teachers' expertise is another factor influencing the success of programs. The most often cited factor (41%) to this end is their pedagogical and didactic qualification. Teachers should know methods to teach mathematically gifted students (e.g., Basister and Kawai, 2018; Miller et al., 1995; Smedsrud et al., 2022), in particular, they should be able to deal with diversity (Gallagher et al., 1997; Haataja et al., 2020; Smedsrud et al., 2022). They should also have a positive attitude toward mistakes (Haataja et al., 2020; Mun and Hertzog, 2018) and care for the needs of mathematically gifted students (e.g., Choi, 2013; Smedsrud, 2018). Finally, teachers of gifted students seem to be better at diagnosing and giving feedback than those in regular classes (Vock et al., 2022). The second most cited factor (29%) is teachers' subject-related qualification. It is necessary so that teachers can challenge gifted students (e.g., George, 1976; Smedsrud et al., 2022), understand their solutions and thought processes (e.g., Budínová, 2024; Özdemir and Işiksal Bostan, 2021), and provide help (e.g., Basister and Kawai, 2018; Sharma, 2010). Consequently, it makes sense to match the program's topics with the expertise of teachers (Matsko and Thomas, 2014). Few studies (16%) also describe other qualifications for teachers of mathematically gifted students, most importantly the ability to encourage them (Szabo, 2017b). Other studies just describe teachers in a general way as being “superior” (Tyler-Wood et al., 2000, p. 269) or “competent” (Basister and Kawai, 2018, p. 1231).

In some cases (14%), studies reported the inclusion of subject matter experts in programs. Most of the time, these experts were mathematicians (e.g., Mun and Hertzog, 2018; Rabijewska and Trad, 1985) or college students (Burns et al., 2017; Campbell, 1988). Including experts to work with gifted students made students feel proud (Mingus and Grassl, 1999).

The same number of studies (14%) cited mentors as a beneficial influence on the gifted. These mentors can be older students in or alumni of the program (e.g., Campbell, 1988; Choi, 2013), sometimes they are people who share characteristics with their mentees, such as gender (Reid and Roberts, 2006).

The least reported feature (8%) is working in teacher tandems. Team-teaching is used for different purposes like making use of complementary subject-specific expertise when preparing lessons (Tyler-Wood et al., 2000) or developing teachers' expertise through peer feedback (Basister and Kawai, 2018).

The subcategory found most often (59%) is a curriculum modification. When analyzing the respective segments, there are two main themes emerging. First, studies mention acceleration, i.e., going through the content at a faster pace (e.g., George, 1976; Ravaglia et al., 1995; Ysseldyke et al., 2004). While some describe it as a good way of fostering the gifted (e.g., Smedsrud, 2018), others argue that it is not challenging enough (e.g., Smedsrud et al., 2022; Tyler-Wood et al., 2000). Second, studies mention enrichment, i.e., discussing additional content outside the regular curriculum (e.g., Basister and Kawai, 2018; Choi, 2013; Kulm, 1984). When this design feature is taken into account, limitations imposed by the regular (school) curriculum are avoided and gifted students are more likely to be challenged (e.g., Mingus and Grassl, 1999; Zedan and Bitar, 2017; Özdemir and Işiksal Bostan, 2021; Özçakir et al., 2020). However, vertical enrichment appears to be more suitable for this purpose than horizontal enrichment (Freedberg et al., 2019; Tretter, 2003). Some programs, though, avoid overlapping with the school curriculum whenever possible (Wagner and Zimmermann, 1986).

About a third (35%) of the studies report a focus on applications of mathematics (e.g., Freedberg et al., 2019; Mingus and Grassl, 1999) or its relevance in everyday life (e.g., Matsko and Thomas, 2014; Tretter, 2003). However, only some mathematically gifted students need applications or real-world relevance (Sharma, 2010). One way of implementing this design feature is to use modeling tasks (e.g., Kim and Kim, 2010; Ozdemir and Isiksal Bostan, 2021). Another way of pointing out applications of mathematics is through interdisciplinary design. A total of 18% of the studies report features in this subcategory (e.g., Tyler-Wood et al., 2000; Ozdemir and Isiksal Bostan, 2021). A variety of disciplines is combined with mathematics, such as other STEM subjects (e.g., Hersberger and Wheatley, 1989; Kim et al., 2016), social sciences (Reid and Roberts, 2006), or foreign languages (Rabijewska and Trad, 1985).

An interest-based selection of content is one of the more frequently (24%) mentioned features that should be included in a program's design (e.g., Mun and Hertzog, 2018; Wagner and Zimmermann, 1986; Özdemir and Işiksal Bostan, 2021). Gifted students appreciate having a choice and being able to delve deeper into interesting topics (e.g., Tretter, 2003; Zedan and Bitar, 2017).

Formal-abstract work is a central characteristic of mathematics. Some of the studies (20%) report on the integration of related content or activities (e.g., Burns et al., 2017; Sharma, 2010). Two of these activities are generalizations (e.g., Duda, 2011) and proofs (e.g., van Schalkwijk et al., 2000).

Studies addressing a conceptual focus as a design feature (18%) mostly agree that it is more important for gifted students to see why things happen or work than just learn about how they work (e.g., Almarashdi et al., 2023; Robinson and Stanley, 1989). They also report that students enjoy this kind of focus (e.g., Gadanidis et al., 2011; Smedsrud, 2018; Tretter, 2003).

Few studies (6%) have examined the subcategory of higher-order thinking, so the results are limited. However, they list an implementation of higher-order thinking as one of reported programs' design features (Miller et al., 1995; Tyler-Wood et al., 2000; Ozdemir and Isiksal Bostan, 2021).

The majority of studies (57%) mention problem solving or posing as characteristics of the respective programs (e.g., Duda, 2011; Hersberger and Wheatley, 1989; Mun and Hertzog, 2018). In some cases, a goal is to stimulate creative mathematical processes (e.g., Kamarulzaman et al., 2022; Rabijewska and Trad, 1985). Few studies report a relationship between collaborative work and problem solving. Some just try to foster collaboration using problem solving (Tretter, 2003), others document a positive effect of collaboration (Barron, 2000). However, gifted students also report that they prefer to work on problems on their own (Zedan and Bitar, 2017).

About one third of the studies (33%) report the use of open tasks (see Yeo, 2017, for a framework to characterize the openness of a mathematical task). Most of these tasks are open-ended problems (e.g., Basister and Kawai, 2018; Campbell, 1988; Kulm, 1984), others allowed for students to work on them in their own way (Simensen and Olsen, 2024; Ozdemir and Isiksal Bostan, 2021).

Few studies (8%) describe math puzzles or riddles as design features of the respective programs (e.g., Freedberg et al., 2019; Mun and Hertzog, 2018). As with problem solving, math puzzles are intended to foster creative thinking and teamwork (Tretter, 2003).

One reason for using these kinds of tasks may be to generate challenging tasks for the programs. About half of the studies (55%) list challenging tasks as a design feature (e.g., Duda, 2011; Gallagher et al., 1997; van Schalkwijk et al., 2000). Many of them describe a connection between the challenge a task presents and the motivation of gifted students (e.g., Matsko and Thomas, 2014; Smedsrud, 2018; Thompson, 2023). In regular school lessons, challenging gifted students rarely seems successful, which is also due to external conditions (Mingus and Grassl, 1999). Rather than being challenged, gifted students have to repeat easy tasks (Özdemir and Işiksal Bostan, 2021) or help other students (Smedsrud et al., 2022), neither of which they necessarily enjoy.

Another way to improve motivation is to incorporate tasks for entertainment into programs (e.g., George, 1976; Özdemir and Işiksal Bostan, 2021). However, only few studies (8%) report this design feature.

The most common reported method (37%) in gifted education is independent learning. This design feature is implemented in programs in at least two different ways. Either gifted students are independent of the teacher (e.g., Furney et al., 2014; Hersberger and Wheatley, 1989) or they are independent of other learners (e.g., Choi, 2013; Mun and Hertzog, 2018). Regardless of the type of independence granted to gifted students, effects of independent learning are inconclusive. While some studies note that many gifted students prefer a curriculum that allows for independent learning (Sharma, 2010) and teachers consider this method essential for gifted education (Freedberg et al., 2019), other findings suggest that independent learning only benefits some gifted students (e.g., O'Shea et al., 2010; Ravaglia et al., 1995). When it comes to problem solving, working independently of other gifted students can even have negative effects on success (Barron, 2000).

Differentiation is a general term for methods used to account for the diversity of learners (Ulm and Zehnder, 2020). In our review, we found studies describing differentiation in learning pace (33%). They agree that a learning pace adapted to individual (gifted) learners has a positive effect on their development (e.g., Burns et al., 2017; Miller et al., 1995; Ysseldyke et al., 2004). Differentiation in learning pace is not only beneficial in heterogeneous groups (Zedan and Bitar, 2017), but also appears to be necessary in seemingly homogeneous groups (Vock et al., 2022).

We also found studies describing differentiation of the level of difficulty as a design feature of educational programs for gifted students (20%). For this aspect of differentiation, too, the studies agree that it has a positive effect on the learning of gifted students (e.g., Freedberg et al., 2019; Kalchman and Case, 1999). While it is especially relevant in heterogeneous groups to guarantee an adequate challenge (e.g., Sharma, 2010), it also seems relevant for homogeneous groups (Vock et al., 2022).

Some studies (18%) consider presentations as part of learning environments. There are different reasons why it is beneficial to allow students to present their solutions, ideas, or projects. While preparing a presentation, students have to reflect on their own work (e.g., Basister and Kawai, 2018) and they are given the opportunity to benefit from the work and expertise of others (e.g., Kalchman and Case, 1999).

Few studies (14%) mention projects as a design feature of gifted education. They can help students engage in research in a field of interest (e.g., Reid and Roberts, 2006) or combine several disciplines (e.g., Freedberg et al., 2019).

A small number of the reviewed studies (10%) list homework as an essential part of the respective programs. It allows students to delve deeper into mathematical content (Wagner and Zimmermann, 1986). However, in order to have positive effect, homework has to be adapted to gifted students. Instead of simply giving them more tasks, the focus should be on the quality of the tasks (Özdemir and Işiksal Bostan, 2021).

Only one study (2%) reports on the method of experimenting. The term experimenting is not typically associated with mathematics teaching. However, one student used it to describe program activities in the context of geometry (Gallagher et al., 1997). As it plays a more important role in other STEM fields, it is more likely to appear in general STEM programs or interdisciplinary math programs.

A frequently found design feature (45%) in the category of classroom structure is the role of the teacher in the learning process. Although most studies argue for a student-centered learning process for gifted students, they list a variety of different roles the teacher can take on. Some studies suggest that teachers should simply appear as facilitators, guides, supervisors or mentors (e.g., Campbell, 1988; Choi, 2013; Hersberger and Wheatley, 1989; Özçakir et al., 2020), while others describe a more active involvement in the learning process (e.g., Mingus and Grassl, 1999; van Schalkwijk et al., 2000). Teachers might even take on the role of a research partner, working with students on a shared problem (e.g., Thompson, 2023). Overall, it seems important to gifted students that they are neither held back (e.g., Smedsrud, 2018) nor given too much support by their teachers (e.g., Gadanidis et al., 2011).

Some studies (16%) discuss the assessment of student performance, arriving at two different conclusions. Some programs refrain from the use of a grading system, for multiple reasons, for example, to focus on the content or the task itself (e.g., Hersberger and Wheatley, 1989) or to avoid fostering rivalry (e.g., Duda, 2011). Other studies argue that grading is necessary to keep students disciplined (e.g., George, 1976) or to give the students an individual scale to work on their improvement (e.g., Furney et al., 2014).

Studies that mention the length of work phases (12%) agree that the typical durations are not suitable for the gifted. On the one hand, they argue that the time needed for explanation could be shorter (Zedan and Bitar, 2017). On the other hand, they suggest extending the time allotted for working on a task or problem, or making it unlimited (e.g., Budínová, 2024; Duda, 2011).

We found two subcategories regarding media. Thirteen studies (27%) considered digital-supported teaching and nine (18%) considered technical tools used by gifted students. These two subcategories therefore distinguish between the general use of digital tools and the use of specific media by the students.

In general, digital-supported teaching seems to be beneficial for gifted students (e.g., Vargas-Montoya et al., 2024). They use digital media, for example, to pace the learning progress according to their needs (e.g., Ysseldyke et al., 2004; Ozdemir and Isiksal Bostan, 2021). The students also enjoy working with computers (e.g., Robinson and Stanley, 1989).

Studies report on specific technical tools that have been implemented in educational programs. The use of dynamic geometry software helped deepen the understanding of gifted students (e.g., Ozdemir and Isiksal Bostan, 2021; Özçakir et al., 2020), programming on a computer had a positive effect on problem solving abilities (Hersberger and Wheatley, 1989), and a graphic calculator enabled students to reach their zone of proximal development (Duda, 2011).