The Hidden Structure of Mathematics Education

In this post, Rhianna Sookhy and Luke Garner reflect on the hidden structure of Mathematics Education. By analysing prerequisite knowledge structures within a mathematics degree, and exploring the extent to which these structures are made transparent to both students and educators, they identify the importance of acknowledging links between modules’ content and requirements, and outline the need to communicate this clearly to learners and educators.

Most educators recognise that mathematical knowledge is not acquired in isolation, rather it is built up over time, layer by layer, relying on previously acquired concepts and techniques [1, 2]. At university level, this cumulative structure becomes particularly important, as gaps in foundational knowledge can limit students’ ability to engage with advanced material and progress through their degree.

In 2024, the University of Exeter examined a central question in mathematics education: how do the modules within a mathematics degree depend on one another? This led to two broader research questions that underpin this article:

• What are the prerequisite knowledge structures of a mathematics degree?
• How transparent are these structures to students and educators?

Our findings suggest that mathematics degrees are organised as hierarchical systems of interdependent knowledge, rather than as collections of standalone modules. Making these structures explicit has the potential to improve curriculum coherence, support active learning, and ease students’ transition into higher education.

Why Structure Matters

Mathematics education is inherently structured. Progression through the subject depends on students acquiring and consolidating prerequisite knowledge before moving on to more abstract or specialised topics. When this progression is clear and well supported, students are better able to apply mathematical ideas in academic and applied contexts.

Our analysis highlights how core areas such as calculus, linear algebra, and differential equations form the foundation for later study in statistics, analysis, algebra, and mathematical modelling. These relationships are not incidental: advanced topics often assume fluency with earlier material, both conceptually and technically. Understanding these dependencies is therefore essential for both teaching and learning.

One way to represent this structure is through a dependency tree (Figure 1). In this model, broad subject areas are linked to more specific topics, with directed connections indicating prerequisite relationships. For example, vector concepts underpin linear algebra, while differential equations are required for subjects such as fluid dynamics. Visualising these relationships makes the structure of a mathematics degree more explicit and easier to navigate for both students and educators.

Figure 1: Example dependencies tree of topics covered in a mathematics degree.

Prerequisite Structures in Mathematics Degrees

Addressing the first research question, our work shows that mathematics degrees are organised around a relatively stable set of prerequisite structures. While module titles and ordering may vary between institutions, the underlying dependencies between topics are largely consistent. Students are expected to progress from concrete computational skills towards greater abstraction, proof, and application.

However, these prerequisite structures are often implicit rather than explicit. Module descriptions may list assumed knowledge, but they rarely explain how that knowledge will be used or why it is essential. As a result, students may underestimate the importance of earlier material or fail to identify gaps in their understanding until they encounter difficulty later in their degree.

Transparency for Students and Educators

The second research question concerns transparency. Our findings suggest that prerequisite structures are not always visible to students and are not always systematically discussed within departments. This lack of transparency can disadvantage students, particularly those transitioning from school to university or entering with varied educational backgrounds.

Educational research consistently shows that active learning approaches support deeper understanding and longer-term retention [3]. Frameworks such as Bloom’s Taxonomy [4] also emphasise the importance of structured progression, moving from foundational knowledge towards higher-order thinking. When prerequisite structures are made explicit, these pedagogical approaches can be more effectively aligned with curriculum design.

Transparency benefits educators as well. Clear articulation of dependencies supports coordination across modules, reduces unnecessary repetition, and helps ensure that essential skills are taught at the appropriate stage.

Recommendations for Universities

Based on this analysis, several actions could strengthen mathematics education:

• Make prerequisites explicit: Clearly state the knowledge and skills required for each module, including how they will be used.
• Support curriculum coherence: Encourage regular discussion within departments to align expectations and identify key dependency points.
• Use structured active pedagogy: Design teaching activities that reinforce prerequisite knowledge while developing higher-level skills.
• Support transitions into university: Provide targeted resources that help students entering university identify and address gaps in foundational knowledge. This will reduce the “skill gap” that can damage confidence and resilience.

A structured and transparent curriculum supports student confidence, progression, and engagement, while also strengthening the overall coherence of a degree programme. These are small but powerful changes that can make mathematics education more coherent, more supportive, and more effective.

Improving transparency is a practical step towards a more coherent, supportive, and effective mathematics curriculum. To support this, we recommend that educators consider the following two key questions:

For lecturers: What prior knowledge do your modules assume, and how clearly is this communicated?

For departments: How well do your modules align, and how visible are their connections to students?

Future Directions

This work is ongoing. In 2021, Exeter students designed an educational game exploring proof by contradiction, showcasing how innovative teaching methods can make fundamental topics more engaging. Building on this, further strategies could include gamified learning, collaborative projects, or digital visualisations of mathematical dependencies.

There is also scope for wider research:

• How do other Russell Group universities structure their mathematics programmes?
• Can we quantify how prerequisite knowledge influences student performance and retention?

By situating Exeter’s findings within the national (and international) picture, we can better understand how mathematics education could evolve. For instance, imagine a headline figure: “70% of Russell Group universities require knowledge of differential equations before studying fluid dynamics.” Numbers such as these make the hidden structure of mathematics visible and actionable.

This research has shown thus far that mathematics degrees are built on structured systems of prerequisite knowledge. When these structures are clear and transparent, students are better able to engage with their learning and progress confidently, and we need to therefore find ways to clarify and communicate these structures to our learners.

References

1] Ofsted. Research review series: mathematics, 2021. URL https://www.gov.uk/government/publications/research-review-series-mathematics/research-review-series-mathematics. Published 25 May 2021. Applies to England.

[2] Ziying Yang, Xiaobing Yang, Ke Wang, Ying Zhang, Ge Pei, and Bin Xu. The emergence of mathematical under-standing: Connecting to the closest superordinate and convertible concepts. Frontiers in Psychology, 12:525493, nov 2021. doi: 10.3389/fpsyg.2021.525493.

[3] Heiko Dietrich and Tanya Evans. Traditional lectures versus active learning – a false dichotomy? STEM Education, 2:275–292, 11 2022. doi: 10.3934/steme.2022017.

[4] Nancy E Adams. Bloom’s taxonomy of cognitive learning objectives. Journal of the Medical Library Association, 103:152–153, jul 2015. doi: 10.3163/1536-5050.103.3.010.

For more information please contact: