Building multiple linear regression skills via ‘puzzling’ active learning

1. Introduction

Drawing upon Cuthbertson (2014), Cook and Watson (2024) have promoted the crossnumber puzzle (CNP) as a tool for both developing and challenging empirical skills. This work built upon their earlier research (Cook and Watson, 2023a), which highlighted crosswords as a means to introduce interactive exercises into classroom teaching. Unlike crosswords, which focus on word-based answers, CNPs require precise numerical solutions. In their 2024 analysis, Cook and Watson used CNPs to challenge both operational and interpretative econometric skills through tasks designed with a two-stage structure. The first stage required learners to conduct empirical analysis following specific guidelines, while the second stage tested their understanding by requiring the extraction of particular elements from the wide range of results produced.

Although unit root testing and cointegration analysis were the focus of Cook and Watson (2024), CNPs can be easily adapted to explore other topics. In this paper, we propose that CNPs can enhance the teaching of multiple linear regression, a fundamental component of introductory econometrics. The numerical nature of CNPs provides a solid foundation for a range of tasks, including estimation, identifying specific elements of output, and conducting hypothesis testing.

Before presenting an illustrative CNP to demonstrate its application to multiple linear regression, it is important to reflect on broader pedagogical considerations. As Cook and Watson (2024) point out, the use of CNPs connects to various themes in pedagogical research. Most notably, CNPs introduce active learning into the classroom. Unlike crosswords, which rely on solving ‘clues’, CNPs require ‘tasks’ to be completed to derive numerical solutions, underscoring active engagement. The nature of these tasks can be varied, aligning with discussions on the Expertise Reversal Effect (Kalyuga et al., 2003), which suggests that tasks should be tailored to learners’ prior knowledge and experience, thus promoting different levels of behavioural and cognitive engagement (see Mayer 2004, 2021). Moreover, CNPs offer a way to address the well-documented issue of ‘quantitative anxiety’ (see, inter alia, Dreger and Aiken, 1957; Dowker et al., 2016; Cook and Watson, 2023b). By fostering self-efficacy (Bandura, 1978) through the successful completion of tasks, learners can reflect on their demonstrated skills, thereby mitigating quantitative anxiety and its negative effects, given the established inverse relationship between self-efficacy and anxiety (see Rozgonjuk et al., 2020).

2. Terms and conditions!

The rules for completing this illustrative CNP adhere to those outlined by Cook and Watson (2024). All answers must be numerical and provided to three decimal places. Additionally, decimal points are treated as separate digits, each placed in its own box. For example, if a task yields a result of 9.8765, the required entry in the CNP would be:

To simplify, the CNP below is designed to exclude minus signs, either by selecting specific tasks or by using absolute values. Following Cook and Watson (2024), this CNP also employs ‘blocking’ rather than ‘barring’ to separate answers. Consequently, the CNP includes some empty cells, resembling the blocking format of a traditional crossword (for a detailed discussion on blocking and barring, see Cuthbertson, 2014).

3. A crossnumber puzzle for multiple linear regression

Completion of this CNP requires empirical analysis using the data available in the EViews file MLR.wf1.[1] This file contains artificially generated data for five variables: {Y, X1, X2, X3, X4, X5}. The dataset spans a hypothetical sample period from January 1980 to December 2023, comprising 528 observations.

To complete this CNP, the empirical analysis must be based on the following multiple linear regression model:

The tasks below pertain to this model and require elements derived from its estimation and subsequent hypothesis testing to be accurately placed in the CNP. The required tasks, and the CNP grid, are provided below. Following standard crossword format, the tasks are divided into ‘Across’ and ‘Down’.

Across

1. The answer for this task is the two-sided p-value associated with the t-test of the null
4. The answer for this task is  
6. The answer for this task is the calculated F-statistic for the testing the null
8. The answer for this task is the value of the residual () in March 1980.
9. The answer for this task is the calculated t-statistic for testing the null that

Down

1. The answer for this task is the calculated t-statistic for testing the null that .
2. The answer for this task is the calculated value of the adjusted .
3. The answer for this task is . (Note: an absolute is required).
5. The answer for this task is the calculated F-statistic for the testing the null
7. The answer for this task is the standard error of the regression.

Completion of the CNP should result in the derivation of the answers below.

4. Concluding comments

Building on Cook and Watson’s (2024) introduction of CNPs as a tool for challenging and developing econometric skills, this paper has demonstrated their application in teaching multiple linear regression. The aim has been to show how familiarity with multiple linear regression components and hypothesis testing can be cultivated within the active learning framework provided by CNPs.

CNPs offer significant flexibility. As discussed, the tasks required to complete a CNP can be adjusted to accommodate varying levels of complexity. Additionally, CNPs are versatile in their timing and context of use. They can be incorporated during topic presentations or employed as a recap or revision aid at the end of modules. In terms of context, CNPs can be used in the classroom to foster discussion after task completion. The tasks naturally lend themselves to further exploration of topics such as inference and the relationships between different output elements. For instance, the task in ‘5 Down’ could prompt a deeper discussion on its connection to individual coefficient significance. Alternatively, CNPs can support flipped learning, where students attempt the puzzles outside of class and use classroom time for discussion. This approach aligns with non-didactic flipping, which has been shown to offer advantages over traditional flipping methods (see Cook et al., 2019; Webb et al., 2021).

Notes

[1] The analysis in this paper was conducted using EViews 13.

References

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Cook, S. and Watson, D. 2023. Crosswords and the ‘active learning’ quest. Economics Network Ideas Bank. https://doi.org/10.53593/n3585a

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Dreger R. and Aiken L. 1957. The identification of number anxiety in a college population. Journal of Educational Psychology 48, 344-351. https://doi.org/10.1037/h0045894

Kalyuga, S., Ayres, P., Chandler, P., and Sweller, J. 2003. Expertise reversal effect. Educational Psychologist 38, 23-31. https://doi.org/10.1207/S15326985EP3801_4

Mayer, R. 2004. Should there be a three-strikes rule against pure discovery learning? American Psychologist 59, 14-19. https://doi.org/10.1037/0003-066x.59.1.14

Mayer, R. 2021. Multimedia Learning (3^rd edition). Cambridge: Cambridge University Press.

Rozgonjuk, D., Kraav, T., Mikkor, K., Orav-Puurand, K. and Täht, K. 2020. Mathematics anxiety among STEM and social sciences students: the roles of mathematics self-efficacy, and deep and surface approach to learning. International Journal of STEM Education 7, 46. https://doi.org/10.1186/s40594-020-00246-z

Webb, R., Watson, D., Shepherd, C., & Cook, S. (2021). Flipping the classroom: is it the type of flipping that adds value? Studies in Higher Education 46, 1649-1663. https://doi.org/10.1080/03075079.2019.1698535