[Maths-Education] Re: Mathematics education and Covid

Sun Aug 23 10:26:57 BST 2020

Psychometrics has been described as "the triumph of statistics over education”.
This is a vivid example, resulting from the priority politicians give to psychometric considerations - and.
And test-retest studies show a result is only good to roughly a grade either way - though the difference can be life-changing.

All the best

Hugh Burkhardt

> On 23 Aug 2020, at 09:38, Ernest, Paul <P.Ernest at exeter.ac.uk> wrote:
> 
> Dear Colleagues
> 
> Ofqual have published their 'formula' for modifying student grades at
> 
> https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/909368/6656-1_Awarding_GCSE__AS__A_level__advanced_extension_awards_and_extended_project_qualifications_in_summer_2020_-_interim_report.pdf
> 
> see also
> 
> https://www.gov.uk/government/publications/awarding-gcse-as-a-levels-in-summer-2020-interim-report
> 
> [https://www.gov.uk/assets/static/opengraph-image-a1f7d89ffd0782738b1aeb0da37842d8bd0addbd724b8e58c3edbc7287cc11de.png]<https://www.gov.uk/government/publications/awarding-gcse-as-a-levels-in-summer-2020-interim-report>
> Awarding GCSE, AS & A levels in summer 2020: interim report - GOV.UK<https://www.gov.uk/government/publications/awarding-gcse-as-a-levels-in-summer-2020-interim-report>
> Report into the grading of GCSE, AS, A level, advanced extension awards and extended project qualifications in summer 2020.
> www.gov.uk
> 
> Here is a selection
> 
> Sorry the formatting collapsed on copying
> 
> Predictive model For simplicity, the procedure outlined above referenced a single relationship between prior-attainment and results that was applied to all students of the simple form: 𝑌𝑖 = 𝛽0 + 𝛽1𝐴𝑖 where 𝑌𝑖 is the mark achieved by student 𝑖 with prior attainment measure 𝐴𝑖 . This predictive model was fitted at Step 1 and then used to generate predictions at Step 3. As discussed in Section 6.1.1, it is recognised that different centres demonstrate different rates of value-added. It would, therefore, be inappropriate to use such a simple single relationship to predict the behaviour of all centres. There is, however, an overall relationship for the cohort between measures of prior-attainment and the final mark that they achieve. In these circumstances where it is necessary to reflect the clustering of data, but also to draw on the statistical power provided by a population-level relationship, multi-level models50,51 can be used. These approaches facilitate the production of individual centre-level estimates reflecting local relationships to inform the student level estimates, but in a more reliable way than would be possible if attempting to analyse each centre in isolation. An example of the sort of relationship that may be established through fitting such a model is shown illustratively in the figure below. In this model, each different coloured line represents the relationship between prior-attainment and the marks achieved for each individual centre. 50 Goldstein, H. (2011). Multilevel statistical models (4th edition). Chichester: John Wiley & Sons. 51 Snijders, T. & Bosker, R. (1999). Multilevel analysis. An introduction to basic and advanced multilevel modelling. London: Sage Publications. Prior attainment Marks Centre specific intercept Centre specific slope Awarding GCSE, AS, A level, advanced extension awards and extended project qualifications in summer 2020: interim report 41 In this instance, the model allows each centre’s relationship to vary in terms of the gradient of the line and where it intercepts the ‘marks’ axis. However, a range of different forms of this model were considered for use. These included random intercept models of the form: 𝑌𝑖𝑗 = 𝛽0 + 𝛽1𝐴𝑖𝑗 + 𝛾𝐶𝑗 + 𝑢𝑗 + 𝑒𝑖𝑗 (F1) Random slope models of the form: 𝑌𝑖𝑗 = 𝛽0 + 𝛽1𝐴𝑖𝑗 + 𝛾𝐶𝑗 + 𝑢0𝑗 + 𝑢1𝑗𝐴𝑖𝑗 + 𝑒𝑖𝑗  (F2) And polynomial forms: 𝑌𝑖𝑗 = 𝛽0 + 𝛽1𝐴𝑖𝑗 + 𝛽2𝐴𝑖𝑗 2 + 𝛽3𝐴𝑖𝑗 3 + 𝛾𝐶𝑗 + 𝑢𝑗 + 𝑒𝑖𝑗  (F3) where: • 𝑌𝑖𝑗 is the uniform mark achieved by student 𝑖 in centre 𝑗 • 𝐴𝑖𝑗 is the student level measure of prior attainment • 𝐶𝑗 is a summarising measure(s) of centre level historical performance (articulated as mean uniform mark, mean grade or cumulative percentage outcomes at key grades) • 𝑢 are the centre level random effects • 𝛽0 , 𝛽1 , 𝛾 and 𝛿 are the fitted regression coefficients • 𝑒𝑖𝑗 is the student level residual During preliminary test testing of these different approaches, it was clear that, applying one of the more complex random slope or polynomial forms of the model was neither productive nor desirable. For the random slope models the computational overhead was significant with the fit failing to converge in a number of cases. The lack of convergence is problematic in the context of needing to apply the standardisation approach across 157 subjects (each requiring their own model) and the desirability of applying the same approach as broadly as possible. Regarding the polynomial models, the increase in variance explained was extremely marginal. For example, when fitting the models for A level biology, the linear variants resulted in 𝑅 2 values in the range 0.54-0.55 and, for the equivalent polynomial forms, the values were in the range 0.55-0.56. This increased complexity was, therefore, not considered to be necessary.
> 
> There are 3 formulas considered. F1 is adopted - a simple linear model. The linear model uses 2 main variables 𝐴𝑖𝑗 Student level measure of prior attainment  and 𝐶𝑗 is a summarising measure(s) of centre level historical performance to construct as an artefact the final grade 𝑌𝑖𝑗  that was to be awarded (i is indiv. student, j is centre)
> 
> In other words, your predicted mark will be scaled down (or up?) to fit the centre's history to give your final mark
> 
> What is most interesting are the grounds for rejecting formulas F2 and F3 (F markers are my insertions)
> 
> F2 is rejected because  the computational overhead was significant  (too much hard work - plus some technical fit problems)
> F3 is rejected because  the increase in variance explained was extremely marginal. (too much hard work - for too little gain)
> 
> Would you want your own children's future (or the students you teach) to have their futures decided on the basis of models chosen in this way? Does It not look like a very disconnected and unethical way to treat our charges?
> 
> There are some justifications, but imposing an ill justified model to achieve norms is undoubtedly going to deprive many students of the grades they would have received (although some may also be promoted). Is balancing the number upgraded with the number downgraded justice?
> 
> Who scrutinised this for its ethical implications? (Nobody)
> 
> Luckily public opinion was so strong that Ofqual decided to withdraw the procedure and persuaded government - who are now busy playing the blame game.
> 
> Just asking
> 
> Keep safe
> 
> Paul
> 
> _________
> Paul Ernest
> Emeritus Professor, Education, Exeter University, Exeter, EX1 2LU, UK
> Homepage <http://www.people.ex.ac.uk/PErnest/> http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/ The Philosophy of Mathematics Education Journal
> 
> ________________________________
> From: maths-education-bounces at lists.nottingham.ac.uk <maths-education-bounces at lists.nottingham.ac.uk> on behalf of Ernest, Paul <P.Ernest at exeter.ac.uk>
> Sent: 16 May 2020 18:00
> To: maths-education at lists.nottingham.ac.uk <maths-education at lists.nottingham.ac.uk>
> Subject: [Maths-Education] Fw: Mathematics education and Covid
> 
> CAUTION: This email originated from outside of the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe.
> 
> 
> Dear Colleagues
> 
> This (below) might seem trivial  --  and you will have spotted it - but I think it is still worth saying and provides us with educational opportunities
> 
> Best wishes to all - and keep safe
> 
> Paul
> 
> _________
> Paul Ernest
> Emeritus Professor, Education, Exeter University, Exeter, EX1 2LU, UK
> Homepage <https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.people.ex.ac.uk%2FPErnest%2F&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702110591&sdata=jK4vLVjwwKCck%2FpzTX%2FiiOXbRGKXEHUzNY7GsR44qks%3D&reserved=0> https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fsocialsciences.exeter.ac.uk%2Feducation%2Fresearch%2Fcentres%2Fstem%2Fpublications%2Fpmej%2F&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702110591&sdata=YqHbtNyvKqguyHapj7AXL2Haf8N5kG50r%2BLUPnzg3Qc%3D&reserved=0<https://eur03.safelinks.protection.outlook.com/?url=http%3A%2F%2Fsocialsciences.exeter.ac.uk%2Feducation%2Fresearch%2Fcentres%2Fstem%2Fpublications%2Fpmej%2F&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702110591&sdata=YqHbtNyvKqguyHapj7AXL2Haf8N5kG50r%2BLUPnzg3Qc%3D&reserved=0> The Philosophy of Mathematics Education Journal
> Memories of a misspent youth  https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fsites.google.com%2Fsite%2Fwitchescauldron60s%2Fhome&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702110591&sdata=Ywew%2FfVTPI8ubAn12naUBn9TASmxZBiB6UdUDEdtPa4%3D&reserved=0<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fsites.google.com%2Fsite%2Fwitchescauldron60s%2Fhome&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702110591&sdata=Ywew%2FfVTPI8ubAn12naUBn9TASmxZBiB6UdUDEdtPa4%3D&reserved=0>
> 
> 
> 
> 
> Why the maths behind covid alert levels makes no sense<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Ftheconversation.com%2Fcoronavirus-why-the-maths-behind-covid-alert-levels-makes-no-sense-138634&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702110591&sdata=leUZyFi4caPdrPAEnrpMAZVnMHvNBOzvDtyCO2nAFTs%3D&reserved=0>
> 
> Will de Freitas, has written a piece explaining why the maths behind covid alert levels makes no sense<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Ftheconversation.com%2Fcoronavirus-why-the-maths-behind-covid-alert-levels-makes-no-sense-138634&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702120586&sdata=2xsWvkRTInIHP3UB4V5g9PTSS5WoTzGeVlSny1cmmbA%3D&reserved=0>. He critiques the Government’s ‘mathematical models' of Corona Virus. He shows that there are problems with the meanings of the 3 slides used to model the dangers of  Covid-19, and our response to it. In the Government’s slides, there is a central equation CAL = R + N.
> 
> As Will de Freitas, correctly points out – the units on the LHS and the RHS do not match up.  CAL is a discrete number in the range 1 – 5, so it is a non parametric statistic. Whatever R + N is, it is not of the same kind, being mostly made up of a (large) number of persons. But there is a deeper problem and misunderstanding in this equation. Rates are fundamentally multiplicative terms, so if the RHS of the equation was RxN it would mean something. If 100,000 people are infected (N = 100,000), and R = 1.5 (or 0.5)  then RxN = 150,000 (or 50,000) meaning that 100,000 people each infect 1.5 (or 0.5. persons, respectively) each leading to 150,000 (or 50,000) new cases.
> 
> But as it stands the value of R + N  has a value (for large N)  about that of N (100,001.5 or 100,000.5  respectively). To misunderstand rates as additive rather than multiplicative is elementary and widely noted among  young children who have not made the step to Logical Reasoning, in Piagetian terms. It is rather alarming coming with the full authority of HM Government and the NHS.
> 
> The misleading use of mathematics to back up the presentation of the Government's Covid Strategy was termed “number theatre” by David Spiegelhalter. My point here is not a political one, just that it is worrying that no one in No. 10 either spotted the errors, or if they did, did not care for their inaccuracy. For it is well within the grasp of a normal school child to identify the errors and absurdities in the presentation. This provides us with an opportunity that can be exploited in education.  The concept of R, the rate of spread of the infection indeed presents a useful concretisation of ratio and proportions which can be exploited in the teaching of these ideas.
> 
> References
> 
> Will de Freitas, why the maths behind covid alert levels makes no sense<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Ftheconversation.com%2Fcoronavirus-why-the-maths-behind-covid-alert-levels-makes-no-sense-138634&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702120586&sdata=2xsWvkRTInIHP3UB4V5g9PTSS5WoTzGeVlSny1cmmbA%3D&reserved=0>, In The Conversation at https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Ftheconversation.com%2Fcoronavirus-why-the-maths-behind-covid-alert-levels-makes-no-sense-138634&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702120586&sdata=2xsWvkRTInIHP3UB4V5g9PTSS5WoTzGeVlSny1cmmbA%3D&reserved=0<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Ftheconversation.com%2Fcoronavirus-why-the-maths-behind-covid-alert-levels-makes-no-sense-138634&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702120586&sdata=2xsWvkRTInIHP3UB4V5g9PTSS5WoTzGeVlSny1cmmbA%3D&reserved=0>, retrieved 16 May 2020.
> 
> Anjana Ahuja, Boris Johnson’s Covid-19 threat alert system is a parody of mathematical precision, New Statesman online, at https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.newstatesman.com%2Fscience-tech%2Fcoronavirus%2F2020%2F05%2Fboris-johnson-s-covid-19-threat-alert-system-parody-mathematical&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702120586&sdata=u43q3aR9QSN2%2BTHjH15Pq6nsjzXRVbQ2V0M1er8dkS0%3D&reserved=0<https://eur03.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.newstatesman.com%2Fscience-tech%2Fcoronavirus%2F2020%2F05%2Fboris-johnson-s-covid-19-threat-alert-system-parody-mathematical&data=02%7C01%7CP.Ernest%40exeter.ac.uk%7Cad22a2ffad1942946fa508d7f9babaa2%7C912a5d77fb984eeeaf321334d8f04a53%7C0%7C1%7C637252452702120586&sdata=u43q3aR9QSN2%2BTHjH15Pq6nsjzXRVbQ2V0M1er8dkS0%3D&reserved=0>, retrieved 16 May 2020.
> 
> [cid:d2d99efa-e54e-42da-b56d-64d37c7124a3] [cid:1364c71c-162e-40a3-b2cb-39b0721c0990]
> 
> ________________________________
> 
> 
> 
> This message and any attachment are intended solely for the addressee
> and may contain confidential information. If you have received this
> message in error, please contact the sender and delete the email and
> attachment.
> 
> Any views or opinions expressed by the author of this email do not
> necessarily reflect the views of the University of Nottingham. Email
> communications with the University of Nottingham may be monitored
> where permitted by law.
> 
> 
> 
> 
> REPLY goes to sender only
> REPLY ALL goes to list members