{"id":640,"date":"2025-01-15T17:59:36","date_gmt":"2025-01-15T17:59:36","guid":{"rendered":"https:\/\/blog.lboro.ac.uk\/cmc\/?p=640"},"modified":"2025-01-23T08:24:35","modified_gmt":"2025-01-23T08:24:35","slug":"believability-and-mathematical-reasoning","status":"publish","type":"post","link":"https:\/\/blog.lboro.ac.uk\/cmc\/2025\/01\/15\/believability-and-mathematical-reasoning\/","title":{"rendered":"Believability and Mathematical Reasoning"},"content":{"rendered":"\n

This blogpost was written by Lara Alcock, a professor in the Department of Mathematics Education at Loughborough University. Click on the link at the bottom of this blogpost to read more about Lara\u2019s work, and the work of her collaborators, Dr Ben Davies and Prof Matthew Inglis. Edited by Dr Bethany Woollacott.<\/em><\/p>\n\n\n\n

In this blogpost, Lara links logical reasoning in mathematics to psychological research on reasoning about everyday concepts.<\/p>\n\n\n\n

Introduction<\/h2>\n\n\n\n

How believable would you say this is?<\/p>\n\n\n\n

                  If \ud835\udc65 is less than 2, then \ud835\udc65 is less than 5.<\/p>\n\n\n\n

How about this?<\/p>\n\n\n\n

                  If \ud835\udc60\ud835\udc56\ud835\udc5b\ud835\udc65 is greater than 0, then \ud835\udc50\ud835\udc5c\ud835\udc60 \ud835\udc65 is less than 1.<\/p>\n\n\n\n

Both are true but, even if you know all the relevant mathematics, you almost certainly found the second less believable.  How might that affect your reasoning? <\/p>\n\n\n\n

Conditional Inference<\/h2>\n\n\n\n

I came to this question because I am interested in conditional inference<\/em>, that is, in inferences from statements of the form ‘if \ud835\udc34 then \ud835\udc35’.  Conditional inferences take four standard forms, as shown below.  For which ones would you say that the conclusion follows in a logically valid way from the premises? <\/p>\n\n\n\n

\n
If \ud835\udc65 is less than 2, then \ud835\udc65 is less than 5.
\ud835\udc65 is less than 2.<\/td><\/tr><\/tbody><\/table>
Conclusion:<\/em> \ud835\udc65 is less than 5.<\/figcaption><\/figure>\n\n\n\n
If \ud835\udc65 is less than 2, then \ud835\udc65 is less than 5.
\ud835\udc65 is less than 5.<\/td><\/tr><\/tbody><\/table>
Conclusion:<\/em> \ud835\udc65 is less than 2.<\/figcaption><\/figure>\n<\/div>\n\n\n\n
\n
If \ud835\udc65 is less than 2, then \ud835\udc65 is less than 5.
\ud835\udc65\u00a0is not less than 2.<\/td><\/tr><\/tbody><\/table>
Conclusion:<\/em> \ud835\udc65 is not less than 5.<\/figcaption><\/figure>\n\n\n\n
If \ud835\udc65 is less than 2, then \ud835\udc65 is less than 5.
\ud835\udc65\u00a0is not less than 5.<\/td><\/tr><\/tbody><\/table>
Conclusion:<\/em> \ud835\udc65 is not less than 2.<\/figcaption><\/figure>\n<\/div>\n\n\n\n

In mathematical logic, modus ponens<\/em> (MP, top left) and modus tollens<\/em> (MT, bottom right) inferences are valid, i.e., mathematically correct.  Denial of the antecedent <\/em>(DA, bottom left) and affirmation of the consequent<\/em> (AC, top right) inferences are invalid, i.e., mathematically incorrect1<\/sup>.  Evaluating validity might seem fairly simple with this elementary mathematical content; but students are immersed in everyday reasoning, where things are not so clear-cut.  Compare the inferences above with those below, which have the same forms.<\/p>\n\n\n\n

\n
If you mow the lawn, then I will give you \u00a35.
You mow the lawn.<\/td><\/tr><\/tbody><\/table>
Conclusion<\/em>: I give you \u00a35.<\/figcaption><\/figure>\n\n\n\n
If you mow the lawn, then I will give you \u00a35.
I give you \u00a35.<\/td><\/tr><\/tbody><\/table>
Conclusion<\/em>: You mowed the lawn.<\/figcaption><\/figure>\n\n\n\n
If you mow the lawn, then I will give you \u00a35.
You do not mow the lawn.<\/td><\/tr><\/tbody><\/table>
Conclusion<\/em>: I do not give you \u00a35.<\/figcaption><\/figure>\n\n\n\n
If you mow the lawn, then I will give you \u00a35.
I do not give you \u00a35.<\/td><\/tr><\/tbody><\/table>
Conclusion<\/em>: You did not mow the lawn.<\/figcaption><\/figure>\n<\/div>\n\n\n\n

For this everyday content, you might feel inclined to accept the invalid inferences (top right, bottom left)2<\/sup>, and you would be reasonable in doing so.  The conditional ‘If you mow the lawn, then I will give you \u00a35’ would usually be interpreted as the biconditional<\/em> ‘I will give you \u00a35 if and only if<\/em> you mow the lawn’.  This reflects flexibility in everyday uses of \u2018if\u2019: when interpreting everyday \u2018if-then\u2019 sentences, people take account of context and conceptual content. Mathematics, in contrast, demands a single consistent interpretation3<\/sup>.  This means that mathematical logic must be learned, and students\u2019 reasoning will likely be influenced by experience with everyday language4<\/sup>. <\/p>\n\n\n\n

Fortunately, everyday interpretations are well studied: cognitive psychologists have conducted extensive research on conditional inference 1,5<\/sup>.  One thing that caught my attention is that people accept more inferences from conditionals that are more believable<\/em>6<\/sup>.  I wondered whether believability would also affect conditional inference in mathematics. <\/p>\n\n\n\n

Could believability affect conditional inference in mathematics?<\/h2>\n\n\n\n

To find out, I needed mathematical conditionals that varied in believability, so my colleague Ben Davies and I set up a comparative judgement<\/em> study7<\/sup>.  This involved asking mathematics education researchers to generate mathematical conditionals that they thought would vary in believability8<\/sup>.  We then asked these researchers, along with some mathematics undergraduates, to take part in an online study in which they were shown pairs of mathematical conditionals and asked which of each pair they thought more believable9<\/sup>.  From their collective judgements, we used standard comparative judgement analyses to generate a believability score for each conditional. <\/p>\n\n\n\n

We found that the researchers and undergraduates broadly agreed about believability, and that \u2013 as would be expected \u2013 true conditionals were on average judged more believable.  However, truth and believability did not perfectly align.  For instance, both researchers and undergraduates judged these true conditionals highly believable:<\/p>\n\n\n\n

\n
\n

If<\/em> \ud835\udc65 is less than 2, then <\/em>\ud835\udc65 is less than 5.<\/em><\/p>\n<\/blockquote>\n<\/blockquote>\n\n\n\n

\n

If<\/em> \ud835\udc5b is a multiple of 6, then <\/em>\ud835\udc5b is a multiple of 3.<\/em><\/p>\n<\/blockquote>\n\n\n\n

But some false conditionals were also judged relatively believable:<\/p>\n\n\n\n

\n

If<\/em> \ud835\udc65 is an integer, then<\/em> \ud835\udc652<\/sup> > \ud835\udc65.<\/p>\n<\/blockquote>\n\n\n\n

\n

If line <\/em>\ud835\udc3f is tangent to curve<\/em> \ud835\udc36, then<\/em> \ud835\udc3f intersects <\/em>\ud835\udc36 at only one point<\/em>.<\/p>\n<\/blockquote>\n\n\n\n

Indeed, these two false conditionals were judged more believable than some true conditionals:<\/p>\n\n\n\n

\n

If quadrilateral<\/em> \ud835\udc44 has a reflex angle, then it will tesselate.<\/em><\/p>\n<\/blockquote>\n\n\n\n

\n

If <\/em>\ud835\udc65 is greater than 0, then<\/em> \ud835\udc50\ud835\udc5c\ud835\udc60\ud835\udc65 is less than 1.<\/em><\/p>\n<\/blockquote>\n\n\n\n

This was helpful for our research plans, because it meant that we could design a conditional inference task using true conditionals with substantially varied believability.  With our colleague Matthew Inglis, we are now studying which inferences mathematics undergraduates accept as valid from relatively believable and relatively unbelievable conditionals.  Early results indicate that there is a mathematical believability effect, and that there are interesting and educationally relevant individual differences in students\u2019 responses.  These will be the subject of a future blog.<\/p>\n\n\n\n

Acknowledgements<\/h2>\n\n\n\n

This work was supported by Leverhulme Trust Research Fellowship RF-2022-155 entitled \u2018Does Mathematics Develop Logical Reasoning?\u2019.<\/p>\n\n\n\n

\n
Link to learn more about Lara Alcock’s work<\/a><\/div>\n<\/div>\n\n\n\n
\n
Link to learn more about Ben Davies’ work<\/a><\/div>\n<\/div>\n\n\n\n
\n
Link to learn more about Matthew Inglis’ work<\/a><\/div>\n<\/div>\n\n\n\n

References<\/h2>\n\n\n\n

[1] Evans, J.St.B.T. & Over, D.E. (2004). If<\/em>. Oxford University Press.<\/p>\n\n\n\n

[2] Cummins, D.D., Lubart, T., Alksnis, O., & Rist, R. (1991). Conditional reasoning and causation. Memory & Cognition<\/em>, 19<\/em>, 274\u2013282.<\/p>\n\n\n\n

[3] Alcock, L. (2013). How to study for a mathematics degree<\/em>. Oxford University Press.<\/p>\n\n\n\n

[4] Epp, S. (2003). The role of logic in teaching proof. American Mathematical Monthly<\/em>, 110<\/em>, 886\u2013 899.<\/p>\n\n\n\n

[5] Oaksford, M. & Chater, N. (2020). New paradigms in the psychology of reasoning. Annual Review of Psychology<\/em>, 71<\/em>, 305\u2013330.<\/p>\n\n\n\n

[6] Evans, J.St.B.T., Handley, S.J., Neilens, H., & Over, D. (2010). The influence of cognitive ability and instructional set on causal conditional inference. Quarterly Journal of Experimental Psychology<\/em>, 63<\/em>, 892\u2013909.<\/p>\n\n\n\n

[7] Jones, I. & Davies, B. (2023). Comparative judgement in education research. International Journal of Research & Method in Education<\/em>, https:\/\/doi.org\/10.1080\/1743727X.2023.2242273.<\/p>\n\n\n\n

[8] Alcock, L. & Davies, B. (2024).  Believability in mathematical conditionals: Generating items for a conditional inference task.  In S. Cook, B. Katz & D Moore-Russo (Eds.), Proceedings of the 25th<\/sup> Annual Conference on Research in Undergraduate Mathematics Education<\/em>, pp.360-368. SIGMAA on RUME.  http:\/\/sigmaa.maa.org\/rume\/RUME26_Proceedings2024-letter.pdf<\/a><\/p>\n\n\n\n

[9] Alcock, L. & Davies, B. (2024).  Believability in mathematical conditionals: A comparative judgement study.  In Proceedings Fifth Conference of the International Network for Didactic Research in University Mathematics<\/em>, pp.721-730.  INDRUM.  https:\/\/indrum2024.sciencesconf.org\/data\/pages\/Proceedings_INDRUM2024.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

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