The development of mathematics revision resources and economics student demand for these resources

Abstract

1 Introduction

This article describes the development of online mathematics revision (review/refresher) resources, created predominantly for undergraduate students studying economics as a single major or as a joint major at a UK Russell Group university.¹ The article introduces the resources and how they are structured, providing a link to the publicly available version of the resources for interested readers. The article goes on to consider students’ usage of the resources as an indication of their revealed preference for the resources and so also the usefulness of the resources. The resources were initially developed in summer 2021, being made available to students prior to them starting their studies in the 2021–2022 academic year. The resources have since been updated and are still made available each academic year on the university virtual learning environment (VLE). A more basic version of the resources (i.e. without the ability to track students’ performance on the quizzes, guidance on the relevance of specific topics to students on different University of Warwick degrees and without a discussion forum and support) has been made available publicly via The Economics Network website from spring 2022.² This article makes a valuable contribution to the very limited literature on best-practice in the development and use of mathematics revision resources for economics students.

There were several reasons for developing the revision resources originally. We were aware that economics student cohorts come from a diverse range of backgrounds, including many different countries. Approximately 50% of students come from beyond the UK, with the vast majority of international students coming from a broad range of countries beyond Europe.³ As such, students may have different levels of mathematical knowledge and may have studied different topics, even though they are expected to have equivalent mathematics qualifications by the time they complete secondary school. An increasing number of UK students come from a widening participation background.⁴ While these students may face slightly lower contextualized offer entry requirements, they have the same mathematics entry requirements as all other students. This is because statistical analysis within the department of factors contributing to student performance success has indicated that prior mathematics performance is significantly positively correlated with undergraduate economics performance.

Despite the requirement for equivalent mathematics qualifications for these groups of students, studies have shown that they have different levels of mathematical knowledge, see for example Gallimore & Stewart (2014); Shaw & Tranter (2021). The differences in mathematics knowledge were only heightened during the pandemic when many pupils faced disruption to their school/college education. We were particularly concerned that students from widening participation backgrounds might face disruption to their studies, pre-arrival at university. We were also concerned that students might not have practised their mathematics skills for at least 4 months in the summer prior to starting their university studies, and some students may not have studied mathematics for considerably longer. This was the case for joint economics majors for whom the mathematics entry requirements were lower. Single economics majors must have studied mathematics until leaving school/college around the age of 18. Unlike the single economics majors, joint economics majors only need to have studied mathematics until the age of 16.⁵ Two mathematics for economics modules are offered to students in the first year of their undergraduate studies, with the easier module for the joint major students who have more limited mathematics knowledge.

There is an extensive literature that concludes that mathematics knowledge is a key influence of student undergraduate degree performance in economics, as discussed below. Hence, we hoped that students would benefit from being able to access mathematics revision resources prior to starting their degrees, although the revision resources would also remain available throughout their studies. We also hoped that students would benefit from having mathematics concepts explained by economists who would link concepts to their use in economics.

While the focus of this article is a discussion of the development and value of mathematics revision resources, it should be noted that these were developed as part of a broader range of online student academic support resources, including writing resources, resources explaining basic economics concepts to students who had not studied economics previously and resources advising students how to study most effectively in a university environment.

2 Literature review

Many studies have shown that mathematics ability is a predictor of students’ success in economics modules and degrees, for example, see Anderson et al. (1994); Ballard & Johnson (2004); Mallik & Lodewijks (2010); Arnold & Straten (2012); Denny (2014); McAlinden & Noyes (2019). With the exception of the paper by McAlinden & Noyes (2019), these papers all use regression methods to confirm that mathematics ability is associated with success in introductory economics or economics principles modules. Anderson et al. (1994), Ballard & Johnson (2004) are distinct in highlighting the importance of prior calculus knowledge rather than broader mathematics skills. Meanwhile, Denny (2014) extends previous analyses by dividing the student sample into economics majors and non-majors, confirming the link between mathematics ability and performance in an introductory economics module for both groups of students. Very few analyses have challenged the link between mathematics ability and economics performance, Wan & Cheo (2012) being notable in this regard. Again, they use regression methods. The only factor that might explain their different conclusions is the learning experience of the students prior to university as their study focuses on students in Malaysia and Singapore.

Nevertheless, there is also a growing literature indicating increasing concern regarding students’ mathematical skills when they start their undergraduate university studies. Gallimore & Stewart (2014) highlight this in a UK context, while Er (2017) reports a comparable phenomenon in the US. Of particular relevance to the current analysis, Shaw & Tranter (2021) conclude that the attainment gap for mathematics students from a widening participation background skills was increased during the pandemic. These papers indicate a need for mathematics revision resources like the ones described in this paper.

Despite the increasing concern regarding students’ mathematical knowledge, we were surprised to discover how little literature exists on the design of mathematics revision resources, and students’ usage of said resources. Leggett et al. (2022) conclude that repeated mathematics quizzing slightly improves students’ economics performance, but otherwise there appears to be a dearth of literature on the design and benefits of developing mathematics revision resources for economics students. Hence, we hope that this article makes a valuable early contribution to a future literature on the development of mathematics revision resources and their value to students.

3 Resources created

The mathematics revision resources were initially co-created over the spring and summer of 2021 by a team of economics and mathematics academics and economics students at the University of Warwick. The choice and scope of topics was decided based on examination of the overlap of typical GCSE and A-level syllabi, but with the specific aim of supporting students’ understanding of the mathematical content of core year one undergraduate economics modules at the university.⁶ As a result, the scope is not identical to standard GCSE/A-level syllabi. For example, the resources omit some core A-level topics (for example, trigonometry, parametric equations) which are not essential for year one economics modules. Alternatively, the resources cover topics, for example, partial differentiation and implicit differentiation, that are typically studied only in Further Mathematics A-levels but are used in core economics modules early in term one. Table 1 summarizes the topics included.

Each topic, as listed in the second column of Table 1, begins by stating the learning objectives associated with that topic. For example, the learning objectives associated with the rules of differentiation are:

“You should be able to differentiate the following types of function

• power functions like |${x}^4$|

• exponential functions like |${e}^{\left(3x+2\right)}$|

• logarithm functions like |$\mathit{\ln}\left(5x+7\right)$|

You should also be able to apply the following rules of differentiation, possibly using several in combination to differentiate functions made from those above.

• the sum rule (for addition)

• the product rule (for multiplication)

• the quotient rule (for division)

• the chain rule (for functions of functions)”

Each topic is then organized with a set of subsections. Now considering the topic of Integration, the subsections are as follows:

• Introduction: an economist’s perspective

• What is an integral?

• Computing indefinite integrals as antiderivatives

• Definite integrals and the fundamental theorem of calculus

• Economic application: consumer surplus

One important feature of the resources is that they allow students flexibility in tailoring their revision to only the subset of topics which they need to refresh. Instead of being referred directly to a set of explanatory videos, students are expected first to self-assess their understanding of key concepts within a topic by completing a diagnostic quiz. If a student’s performance on this quiz is sufficiently strong, it is suggested that they need no further revision of this topic and should instead move to the next one. Alternatively, they are referred to specific lecture videos and notes that cover concepts whose revision would be beneficial. The diagnostic test feature of the resources is intended to allow for efficient allocation of students’ time, and we see it as particularly relevant in the context of a revision course, where after all, students are expected already to have a good grasp of the key concepts from previous studies. To accommodate this, the development of the main learning resources involved firstly the creation of a set of interactive quizzes for self-assessment. For each topic, these included an initial diagnostic quiz, a quiz on economic examples and a quiz of more advanced questions for interested students, as well as mini quizzes associated with each subsection of each topic. The quizzes were based on the development of a question bank of 327 questions using Numbas.

Numbas is a free and open-source online system for mathematical assessment, developed by mathematicians at the University of Newcastle, UK, who also provide a free and open-source online editor for questions and quizzes (Foster et al., 2012; Perfect, 2015). Some attractive features of Numbas include randomization of parameters (ensuring a virtually inexhaustible set of exercises on a question subject due to parameterization), variety of question types (e.g., the possibility to require the input of an algebraic expression as an answer), use of interactive graphs (through JSXGraph integration), ability to construct multi-part questions (e.g., asking students to fill in a sequence of arguments in a proof), among others. In addition, while Numbas can be integrated with VLEs such as Moodle, each quiz is a stand-alone programme which runs in the student’s browser posing no computational requirements for the VLE server. The latter feature is distinctive for Numbas, relative to other e-assessment systems such as STACK (Sangwin, 2013). Given the expectation that large numbers of students would engage with the revision resources at the same time, one of the key reasons for using Numbas was to make sure that this did not pose a risk of overloading the VLE server.

Second, 84 asynchronous lecture videos were produced (17 videos motivating the significance of the mathematical content of each section by linking it to economics concepts discussed in core year one undergraduate economics modules; 55 tutorial videos introducing core mathematical content; 12 videos discussing economic examples related to the content of specific sections). The videos were recorded by academics from the Department of Economics and the Mathematics Institute who teach key undergraduate economics modules, allowing students to get an initial level of familiarity with some of their future lecturers.

For the development of the bank of core, mini and advanced quiz questions, the academic project team enlisted the support of four undergraduate interns, selected from a pool of 30 applicants studying economics degrees at the university in question. Soon after the end of their summer examinations, the students received training on using the Numbas editor, guidance on producing randomized parametric questions with feedback, and each was allocated a list of sample mathematical problems to program in Numbas, under supervision from academics. The interns were also asked for their feedback on the design and navigability of the resources through their development. The interns worked through the resources to confirm that it was feasible for a student to work through all the resources in the period before starting their formal undergraduate studies.

The initial development of the resources was complete by mid-August 2021, organized in the structure of a course on the university Moodle VLE, and made available to students early in September of that year, approximately 1 month before their studies would formally start. Care was taken to ensure consistency of vocabulary and notation across the resources developed.

Once the resources had been made available, the academic team were able to check that there were no problems with the content nor with accessing it. Student focus groups were organized to obtain initial student feedback on the ease of use of the revision resources and students’ perceived value of the resources. In spring 2022, the basic version of the resources was made publicly available via The Economics Network website.

The revision resources have been updated regularly with an additional statistics section added in advance of the 2023/24 academic year. Note that since the resources were first made available to students in 2021, the VLE has always included a discussion forum for students to ask questions which academics and other students are free to answer. Students can also book additional one-to-one support with a tutor at any time once they start their studies. However, take up of these advice and feedback slots has been relatively low, we expect because first year undergraduate economics students all take a mathematics for economics module in the first term of their studies. Hence, once the first term starts, students are likely to turn to their lecturers and tutors on these modules for additional mathematics support.

4 Research methodology

There are multiple ways of assessing whether the introduction of a new learning resource has been successful. Students (and staff) can be surveyed, and/or focus groups can be utilized. However, with any method that asks students to give feedback on their views re the usefulness and value of a learning resource, there is a risk that students may not accurately report their preferences. Students may tend to respond that any additional resources are valuable, rather than giving more objective feedback. There are also challenges associated with restricting students surveyed to those who have used the resources sufficiently to be able to give an accurate evaluation of the resources’ use.

To avoid these problems, in this article, we use a revealed preference methodology to determine how valuable students find the revision mathematics resources developed. We can view students’ usage of the resources to determine to what extent they find the resources valuable. Given that accessing the resources was recommended but not compulsory at the university at which they were created, and that when provided on The Economics Network website, they were simply added as a resource with little marketing, it is hoped that usage statistics reflect the usefulness of the resources to students. There are precedents in the pedagogy literature for adopting a revealed preference approach, for example see Elliott & Neil (2016).

The article looks primarily at students’ usage of the resources in the first term of the 2023/24 academic year, the choice of year was deliberate as the new students would not have had their last two key years of study at school/college significantly impacted by the pandemic. Hence, it is hoped that these recent data reflect student demand for resources in a ‘regular’ year. In the years immediately prior to the 2023/24, academic year students’ demand for the resources may have been ‘artificially’ increased due to disruptions to learning caused by the pandemic. As such, the 2023/24 usage data should give a more accurate reflection of demand for the revision resources.

The number of joint economics majors on the more basic mathematics for economics module this year was 153: these students were recommended to work through at least all of the Section 1 topics. For these students, it is not compulsory to study mathematics at school/college up to the age of approximately 18. There were 531 students on the more advanced mathematics for economics module, compulsory for single major economics students. It is compulsory for these students to study mathematics until they leave school/college aged around 18. These students were advised to work through all the sections of revision resources.

5 Results

In this section, we discuss students’ engagement with the mathematics revision resources created. Initially, we focus on student usage via the VLE at the university where the resources were created, before going on to comment on the accessing of materials that are publicly available via The Economics Network website. Note that by considering simply the number of views of resources we hope to get an initial indication of students’ revealed preferences for the resources available. However, we accept that this is only a simple measure of student engagement. Nevertheless, we believe that even this preliminary analysis highlights a demand for mathematics revision resources by economics students.

Table 2 first indicates the number of views of the ‘top’ page associated with a sample of the topics covered in the mathematics revision resources.⁷ Topics selected for inclusion in Table 2 are those that are of particular relevance to the first year of undergraduate study. Each topic top page includes the learning outcomes associated with that topic, instructions on the order in which the topic resources should be accessed, a link to the VLE discussion forum where student questions can be answered and crucially the slides, short films on the topic and initial diagnostic test for the topic in question. The number of views associated with each of the five topics in the precalculus section of the revision resources are provided as all students were advised to work through these, regardless of whether the students had A-level mathematics or equivalent, or not. Counts of the views associated with each of the second section topics (univariate calculus) and the first topic of section three (partial differentiation within multivariate calculus) are then provided. Only single major economics students were recommended to work through these later resources, so the views give an indication of demand for these resources by students who had studied A level mathematics or equivalent. Data on the views of two of the initial diagnostic tests are also provided. The test for topic 1.1 (arithmetic) was recommended for all students, and then data associated with the views of the diagnostic test on the rules of differentiation has been selected for inclusion. This test was selected as designed for students who had previously studied A level mathematics, and because these students should have identified this topic as particularly useful and relevant to their future economics studies.

Fig. 1

Arithmetic revision resources accessed over time.

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Note that observations in the dataset have been removed associated with colleagues accessing the resources. Hence, the views are all student views of the VLE pages, and limited to those dates from when the resources were first made available to students starting at the university in autumn 2023.

Table 2 indicates substantial demand for the VLE revision resources. More specifically, over 70% (70.9%) of new students on the single and joint economics undergraduate degrees engaged with the resources.⁸ It is also reassuring to see that students who have studied mathematics until leaving school/college with A level mathematics or equivalent select which topics they need additional information on. Hence, while demand for resources in Section 2, univariate calculus, decreases with successive topics, the number of students accessing the materials on partial derivatives is higher, as we would expect of a topic that not all students will have previously studied. The differences in the numbers in the second and third columns of Table 2 are also encouraging as they indicate that students access materials more than once if they discover that they need further information and practice of particular topics.

Figures 1 and 2 show the number of times samples of the resources were accessed by students. To illustrate the timing of students’ usage of the resources, we have selected a topic from Section 1: pre-calculus and a topic from Section 2: univariate calculus. Recall that joint economics majors are only recommended to review pre-calculus concepts, while single economics majors are recommended to review both sections. The topic from the pre-calculus section that we selected is Topic 1.1: Arithmetic. Figure 1 indicates the number of views of the topic content materials and the assessment for Arithmetic. From the univariate calculus section, we selected Topic 2.3: rules of differentiation. Similar data for that section are displayed in Fig. 2. There are more views of the quizzes than the topic content as students are advised to repeat quizzes in which they initially score poorly.

Fig. 2

Rules of differentiation revision resources accessed over time.

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Even though resources had been made available to students since late August 2023 with a recommendation that students work through the relevant resources prior to starting their university studies, it is notable that views peaked around the start date of teaching, namely 2 October 2023. Demand for the resources rapidly tails off but this can be explained as in the week commencing 2 October 2023 all of the students would have started a mathematics for economics module and so can be expected to start focusing on the VLE learning resources associated with those modules.

We now turn to look at demand for the basic version of the mathematics revision resources developed which are publicly available, see Fig. 3A–C. Although the resources are expected to be accessed largely by academics and students in the UK who are familiar with the work of The Economics Network, the resources are of course available internationally. Hence, it makes sense to look at demand across a 12-month period, rather than restricting attention to the weeks around the start of the UK university year, as above.⁹ Again, data for a sample of topics including the earliest but also some later topics are provided. Although users of the resources may not be located in the UK, it is clear from Fig. 3A that demand for the resources peaks around the start of the academic year in North America, the UK and mainland Europe.

Fig. 3

(A. B) Economics network website views. (C) Website views data.

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The data in Fig. 3A–C indicate a demand for the mathematics revision resources created, beyond the university at which they were developed. Further, we again see that users do not necessarily work through the resources in topic order, maybe with focus waning. Instead, users self-identify which topics are most useful for them. Hence, we see a greater number of unique users accessing the implicit differentiation resources than the algebra resources. The website data offer additional insights as information is provided on the length of time accessing each topic’s resources. Consequently, in Fig. 3C, we can see that users spend significantly longer working through the partial differentiation resources compared to the resources available on other topics.

6 Conclusions

This article outlines a set of mathematics revision resources created to support university economics students, particularly near the start of their university studies. Usage data confirms students’ revealed preferences for the resources created, considering both university VLE usage data and ‘hits’ on the resources provided on the publicly available Economics Network website. Interestingly, demand for the resources includes demand from students who had anyway studied mathematics until leaving school/college. Future research could explore the optimal design of revision mathematics resources, student engagement with revision resources in more detail, as well as estimating statistically any relationship between engagement with the revision resources and student academic performance.

Acknowledgements

We would like to thank the Royal Economic Society for funding provided to support the development of the learning resources discussed in this paper. Thank you to Martin Poulter at The Economics Network for making the resources discussed in this paper available publicly and for providing The Economics Network data discussed below. We would also like to thank participants at the INERME International Network on Mathematics in Economics Education conference, Norway, January 2024 for helpful comments received. Finally, we extend our sincere thanks to the journal editor and two reviewers whose comments were so helpful and insightful.

Footnotes

References

ANDREW BRENDON-PENN Andrew Brendon-Penn is a teaching fellow in the Warwick Mathematics Institute and also teaches mathematical and statistical techniques to students in the Economics department at Warwick. He maintains a keen focus on optimizing teaching practices through technology, supporting neurodiverse students, and facilitating smooth transitions for incoming university students.

CAROLINE ELLIOTT Caroline is a teaching-focused Professor in the Department of Economics at the University of Warwick, and an Alumni Fellow of the Warwick International Higher Education Academy. She is currently Deputy Chair of the Faculty of Social Sciences with responsibility for education. Caroline has a long-standing association with The Economics Network which works to enhance the quality of university economics teaching and learning, and is currently Deputy Director of the network. Caroline is a Principal Fellow of Advance HE. Her research spans pedagogical research with a particular focus on the use of technologies in teaching, empirical education economics and industrial economics research areas.

EMIL KOSTADINOV Emil is a teaching-focused Assistant Professor in the Department of Economics at the University of Warwick, and a Fellow of the Warwick International Higher Education Academy. His research interests are in search theory and its applications to labour economics. He is teaching mathematics for economics and is especially interested in the use of tools for computer-aided assessment and feedback.

JEREMY SMITH Jeremy is a Professor in the Department of Economics at the University of Warwick. His research is primarily in two areas the economics of education and time series econometrics, with a focus on forecast evaluation. Jeremy was Head of Department of Economics between 2016–2022.