Flat teaching design refers to an instructional approach that offers students multiple learning pathways and allows students to progress at their own pace (Hoffmann and Egri-Nagy, 2021). In traditional mathematics class, teaching contents are organized in a linear, progressive and hierarchical structure (Hoffmann and Egri-Nagy, 2021). This structure means that learning progress builds on prior knowledge, linking each new concept to foundational knowledge. However, if students do not adequately comprehend mathematical concepts, they may encounter challenges in learning following courses (Deshler et al., 2001). Therefore, students may experience learning gap anxiety (Piccirilli et al., 2023).

Learning gap anxiety indicates that if students fail to grasp foundational knowledge, then they may struggle to learn complex and abstract mathematical concepts (Furner and Gonzalez-DeHass, 2011). According to flat teaching design, I believe that teachers should incorporate multiple learning pathways into the curriculum instead of focusing on a core concept. In other words, teachers should divide the content into sub-modules, with each sub-module serving as a separate learning unit. These units can help students manage cognitive load and improve learning outcomes (Kellman et al., 2010).

Consequently, two primary educational models have been proposed. These are the multi-entry learning pathway model and dynamic knowledge mapping model (Hoffmann and Egri-Nagy, 2021). The first model provided multiple entry points into the knowledge content, enabling students to select suitable learning progressions (Hoffmann and Egri-Nagy, 2021). By following this model, teachers can divide the mathematics curriculum into distinct learning units (Kellman et al., 2010). As a result of this approach, students could choose different learning sequences according to their own learning interests. Furthermore, teacher should use this model to enhance students’ learning flexibility and stimulating initiative. For example, the function unit can be divided into multiple modules: function images, function properties, practical modelling and interdisciplinary applications (Eisenbart et al., 2017).

Dynamic knowledge mapping model focuses on interconnected and updated knowledge points (Heide and Lis, 2012). By following this model, teachers can adjust the links and order of modules based on student performance. The model structures content into three levels. The first level covers basic concepts, such as function definitions, types, domain, and range (Eisenbart et al., 2017). The second level includes formal operations, such as function transformations, inverse forms, and compositions (Bart, 1971). The third level focuses on the use of functions in problem-solving contexts, such as modelling tasks (Hall, 2014). In this model, students can select the order of modules and change learning paths based on their progress.

The REACT model involves five dimensions: relating, experiencing, applying, cooperating, and transferring (Putra et al., 2023). Each dimension of REACT model will be explained in the following section.

Constructivists believe that learners actively construct knowledge based on their own experiences rather than passively receiving it (Bada and Olusegun, 2015; Loyens and Gijbels, 2008). Consistent with this view, according to the first dimension ‘Relating’, it emphasizes the importance of connecting new concepts to students’ prior knowledge and experiences (Putra et al., 2023). However, research suggests that many students struggle to establish such connections, particularly in mathematics education (Sihite, 2023). This difficulty often stems from the abstract nature of mathematics, which makes it challenging for students to relate theoretical concepts to real-life questions (Sihite, 2023). To address this issue, I believe that teachers should sequence complex knowledge into stage-based learning goals. Meanwhile, instructional materials should be designed to align with students’ prior knowledge.

According to the second dimension ‘Experiencing’, it focuses on students’ hands-on involvement in practical activities to construct knowledge (Da, 2023). This principle aligns with Dewey’s philosophy of “learning by doing” which emphasizes experiential, exploratory, and reflective learning processes (Ord, 2012). In this context, students are not passive recipients of knowledge, but active participants in the construction of understanding (Ord, 2012). Therefore, teachers should foster diverse and interactive teaching models where students could experiment, reflect, and learn through firsthand experiences (Buckley and Sullivan, 2023). In this way, a supportive and low-anxiety learning environment will be fostered to encourage students’ resilience, cultivate a growth mindset, and promote meaningful learning.

In terms of “Applying”, teachers should help students to reinforce their learning by applying acquired abstract concepts to realistic scenario. This aligns with Bloom’s theory of classification of cognitive goals: remembering, understanding, applying, analyzing, evaluating and creating (Conklin, 2005). Specifically, application is learners’ ability to apply knowledge or methods to new situations (Conklin, 2005). Researchers shown that some students develop mathematics anxiety from learning applications and problems irrelevant to real life (Sitorus and Saragih, 2023). I believe that when students apply their mathematical knowledge in concrete situations, they may understand abstract concepts and improve problem-solving skills.

The fourth dimension is ‘Cooperating’. According to Mathias et al. (2024), it includes student collaboration, teacher-student interaction and home-school communication. This aligns with Vygotsky of the zone of nearest development theory. Specifically, “there are two levels of student development: one can independently complete the task; the second is the development of problem solving with the help of others” (Wu, 2025, p. 84). Furthermore, cooperative learning could reduce students’ sense of isolation, ease anxiety, and prevent excessive competition and peer pressure (Ford, 2018). In this process, students could listen to lectures, take notes and participate in discussions. Through group work activities, students should practice teamwork skills, build confidence in expressing ideas. In addition, high parental expectations may put psychological pressure on students, thereby leading to mathematics anxiety (Si et al., 2022). Consequently, I suggest that teachers should maintain positive communication with parents and create supportive learning environment.

The last dimension is “Transfer”. This means that teacher should guide students to apply their existing knowledge to new situations (Deshler et al., 2001). At this stage, students should migrate knowledge into different contexts. In educational theory, migration is categorized as near transfer and far transfer (Perkins and Salomon, 1992). Near transfer refers to migration between similar contexts. On the contrary, far transfer refers to migration between unsimilar contexts (Perkins and Salomon, 1992). Therefore, I believe that teachers should help students start with basic knowledge in similar situations and transition to high level knowledge in complex situations. In this process, students could relate and apply knowledge in realistic situations and internalize their knowledge.

By implementing the REACT model, teachers should focus on students’ psychological state and individual differences during the teaching process, reducing anxiety and improving students’ mathematics performance.

Application-oriented courses usually link abstract mathematical knowledge with college students’ majors and future careers (Oladejo et al., 2023). Research indicates that the training objectives of university education should shift from imparting knowledge toward enabling graduates to possess specific capabilities (Li et al., 2019). These capabilities must meet the demands of both enterprises and future society (Li et al., 2019). Therefore, I believe that teachers should help students develop and apply theoretical knowledge in real life.

In addition, research showed that the disconnect between course content and practice is the cause of negative learning state (Hall, 2014). Traditional mathematics courses focused on derivation and lacked links to professional contexts (Smith and Morgan, 2016). To address this issue, I believe mathematics instruction should incorporate context-specific scenarios drawn from applied fields. For example, teachers may introduce data analysis tasks related to program evaluation or client assessment in classroom activities (Lalayants, 2012). Such integration may help students understand the practical value of knowledge. In the process, students could develop statistical thinking and strengthen learning participation, and reduce anxiety associated with abstract content (Da, 2023).

Meanwhile, teachers should strengthen school-enterprise cooperation such as industrial visits and internship programs (Li et al., 2019). Industrial visits are short term visits to actual workplaces for students to study (Al-Atabi et al., 2013). Internship programs are introductory practicum for students in companies (Al-Atabi et al., 2013). These experiences can help students understand how mathematics is applied in workplace settings. Teachers should also create opportunities for students to transform theoretical knowledge into practical skills. In this way, students can apply what they have learned to real tasks and strengthen their problem-solving abilities (Smith and Morgan, 2016).

In conclusion, existing literature suggests that when teachers use application-oriented courses, they could better meet students’ professional needs and reduce their mathematics anxiety.

The experiential mathematics curriculum is based on the core concept of learning by doing, which emphasizes the active participation of students in concrete situations (Da, 2023). Based on this idea, researchers and educators have proposed experiential learning as a response to persistent challenges in mathematics education, particularly mathematics anxiety (Marshall et al., 2017).

For example, some studies recommend that teachers should use contextualized curriculum design to connect mathematical concepts with practical situations and help students grasp abstract knowledge (Smith and Morgan, 2016). In this way, students could understand and apply abstract mathematical principles from familiar contexts. Secondly, teachers should incorporate hands-on learning and modelling activities (Hall, 2014). These tasks should include immediate feedback, helping students recognize the link between their actions and specific outcomes, which may enhance their sense of progress and reduce anxiety. Furthermore, teachers should promote collaborative learning by designing group-based activities that encourage students to explore mathematical problems together (Ford, 2018).

Additionally, teachers should use experiential courses to develop differentiated teaching practices. For example, teachers can diversify the content and presentation in these courses to respect students’ cognitive differences (Hoffmann and Egri-Nagy, 2021). For example, in geometry and spatial reasoning courses, teachers can introduce tools such as Geometer’s Sketchpad to guide students in constructing shapes (Liu and Kaino, 2007). From my perspective, students constructed knowledge by exploring, verifying, and interpreting mathematical ideas, rather than passively memorizing geometric theorems.

In conclusion, existing literature suggests that when teachers use experiential curriculum, they can promote students’ peer collaboration and facilitate their knowledge construction.