From I.Biza at uea.ac.uk Thu Aug 6 07:19:37 2020 From: I.Biza at uea.ac.uk (Irene Biza (EDU - Staff)) Date: Thu, 6 Aug 2020 06:19:37 +0000 Subject: [Maths-Education] IJRUME - Calculus SI - Call for Abstracts In-Reply-To: <4FE80958-CCD3-42DE-B5CB-CDE639BB530A@uea.ac.uk> References: <4FE80958-CCD3-42DE-B5CB-CDE639BB530A@uea.ac.uk> Message-ID: <934A9EA7-984A-4DCA-B66B-DAD8F9739381@uea.ac.uk> Dear Colleagues, I hope you and those close to you are well. This is a reminder of the coming deadline (31th August 2020) for the abstracts for the IJRUME special issue on Calculus. Please, do not hesitate to disseminate this information through your networks. Best wishes and stay safe! Irene, Alejandro and Alon [cid:image001.jpg at 01D66BC1.F038C200] Dear colleagues, We are delighted to announce a Special Issue on Calculus at the intersection of institutions, disciplines and communities at the International Journal for Research in Undergraduate Mathematics Education, guest-edited by Irene Biza (University of East Anglia, United Kingdom), Alejandro S. Gonz?lez-Mart?n (Universit? de Montr?al, Canada) and Alon Pinto (Weizmann Institute of Science, Israel). The Call for Abstracts is attached to this message. We look forward to receiving you abstracts and please do not hesitate to contact us with queries. With our warmest wishes for your health, strength and creativity in these unprecedented times, Irene, Alejandro and Alon ---------------------------------------------------------- Dr Irene Biza Associate Professor in Mathematics Education - Director of the Doctor of Education (EdD) programme School of Education and Lifelong Learning, Room LSB-1.30, University of East Anglia, NR4 7TJ, Norwich, UK, Tel: +44 (0)1603 591741 url: http://www.uea.ac.uk/education/people/profile/i-biza MathTASK: @mathtask, https://www.uea.ac.uk/education/mathtask, https://youtu.be/gt0HZBfBBGI NEW: Biza, I., Hewitt, D., Watson, A., & Mason, J. (2020). Generalization Strategies in Finding the nth Term Rule for Simple Quadratic Sequences. International Journal of Science and Mathematics Education, 18(6), 1105-1126. Open Access. Available at https://rdcu.be/bO14K Biza, I. & Nardi, E. (2019). Scripting the experience of mathematics teaching: The value of student teacher participation in identifying and reflecting on critical classroom incidents International Journal for Lesson and Learning Studies, 9(1), 43-56. Available at publisher page and UEA repository. ------- Any personal data exchanged as part of this email conversation will be processed by the University in accordance with current UK data protection law and in line with the relevant UEA Privacy Notice. This email is confidential and may be privileged. If you are not the intended recipient please accept my apologies; please do not disclose, copy or distribute information in this email or take any action in reliance on its contents: to do so is strictly prohibited and may be unlawful. Please inform me that this message has gone astray before deleting it. Thank you for your co-operation. 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Name: IJRUME Special Issue Call for Abstracts (Calculus at the intersection of institutions, disciplines and communities) 21Apr2020[2][2].pdf Type: application/pdf Size: 726404 bytes Desc: IJRUME Special Issue Call for Abstracts (Calculus at the intersection of institutions, disciplines and communities) 21Apr2020[2][2].pdf URL: From P.Ernest at exeter.ac.uk Sun Aug 23 09:38:11 2020 From: P.Ernest at exeter.ac.uk (Ernest, Paul) Date: Sun, 23 Aug 2020 08:38:11 +0000 Subject: [Maths-Education] Re: Mathematics education and Covid In-Reply-To: References: <88C0A882-EAEF-444E-93D4-683C414F3C82@unb.ca> <6E4384C7-4025-4854-8769-4F05D810B630@unb.ca>, , , Message-ID: 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] Awarding GCSE, AS & A levels in summer 2020: interim report - GOV.UK 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: ?? = ?0 + ?1?? where ?? is the mark achieved by student ? with prior attainment measure ?? . 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: ??? = ?0 + ?1??? + ??? + ?? + ??? (F1) Random slope models of the form: ??? = ?0 + ?1??? + ??? + ?0? + ?1???? + ??? (F2) And polynomial forms: ??? = ?0 + ?1??? + ?2??? 2 + ?3??? 3 + ??? + ?? + ??? (F3) where: ? ??? is the uniform mark achieved by student ? in centre ? ? ??? is the student level measure of prior attainment ? ?? is a summarising measure(s) of centre level historical performance (articulated as mean uniform mark, mean grade or cumulative percentage outcomes at key grades) ? ? are the centre level random effects ? ?0 , ?1 , ? and ? are the fitted regression coefficients ? ??? 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 ? 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 ??? Student level measure of prior attainment and ?? is a summarising measure(s) of centre level historical performance to construct as an artefact the final grade ??? 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://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 on behalf of Ernest, Paul Sent: 16 May 2020 18:00 To: 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%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 Why the maths behind covid alert levels makes no sense Will de Freitas, has written a piece explaining why the maths behind covid alert levels makes no sense. 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, 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, 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, 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. From Hugh.Burkhardt at nottingham.ac.uk Sun Aug 23 10:26:57 2020 From: Hugh.Burkhardt at nottingham.ac.uk (Hugh Burkhardt) Date: Sun, 23 Aug 2020 09:26:57 +0000 Subject: [Maths-Education] Re: Mathematics education and Covid In-Reply-To: References: <88C0A882-EAEF-444E-93D4-683C414F3C82@unb.ca> <6E4384C7-4025-4854-8769-4F05D810B630@unb.ca> Message-ID: <1D297BDD-1B46-4423-BFE0-53525964472A@exmail.nottingham.ac.uk> 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 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] > Awarding GCSE, AS & A levels in summer 2020: interim report - GOV.UK > 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: ?? = ?0 + ?1?? where ?? is the mark achieved by student ? with prior attainment measure ?? . 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: ??? = ?0 + ?1??? + ??? + ?? + ??? (F1) Random slope models of the form: ??? = ?0 + ?1??? + ??? + ?0? + ?1???? + ??? (F2) And polynomial forms: ??? = ?0 + ?1??? + ?2??? 2 + ?3??? 3 + ??? + ?? + ??? (F3) where: ? ??? is the uniform mark achieved by student ? in centre ? ? ??? is the student level measure of prior attainment ? ?? is a summarising measure(s) of centre level historical performance (articulated as mean uniform mark, mean grade or cumulative percentage outcomes at key grades) ? ? are the centre level random effects ? ?0 , ?1 , ? and ? are the fitted regression coefficients ? ??? 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 ? 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 ??? Student level measure of prior attainment and ?? is a summarising measure(s) of centre level historical performance to construct as an artefact the final grade ??? 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://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 on behalf of Ernest, Paul > Sent: 16 May 2020 18:00 > To: 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%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 > > > > > Why the maths behind covid alert levels makes no sense > > Will de Freitas, has written a piece explaining why the maths behind covid alert levels makes no sense. 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, 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, 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, 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 From johnbibbyjohnbibby at gmail.com Sun Aug 23 13:25:59 2020 From: johnbibbyjohnbibby at gmail.com (John Bibby) Date: Sun, 23 Aug 2020 13:25:59 +0100 Subject: [Maths-Education] Re: Mathematics education and Covid In-Reply-To: References: <88C0A882-EAEF-444E-93D4-683C414F3C82@unb.ca> <6E4384C7-4025-4854-8769-4F05D810B630@unb.ca> Message-ID: Thanks Paul. I think this is the same as the one that came out 10 days ago Paul. It has been discussed on the Radstats Jiscmail list. The RSS offered to advise, but were turned down as they would not agree to a gagging clause. 319 pages of slog! John BIBBY On Sun, 23 Aug 2020, 09:38 Ernest, Paul, 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: ?? = ?0 + ?1?? where ?? is the > mark achieved by student ? with prior attainment measure ?? . 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: ??? = ?0 + ?1??? + ??? + ?? + ??? (F1) Random > slope models of the form: ??? = ?0 + ?1??? + ??? + ?0? + > ?1???? + ??? (F2) And polynomial forms: ??? = ?0 + ?1??? + > ?2??? 2 + ?3??? 3 + ??? + ?? + ??? (F3) where: ? ??? is > the uniform mark achieved by student ? in centre ? ? ??? is the > student level measure of prior attainment ? ?? is a summarising > measure(s) of centre level historical performance (articulated as mean > uniform mark, mean grade or cumulative percentage outcomes at key grades) ? > ? are the centre level random effects ? ?0 , ?1 , ? and ? are the > fitted regression coefficients ? ??? 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 ? > 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 ??? Student level measure of > prior attainment and ?? is a summarising measure(s) of centre level > historical performance to construct as an artefact the final grade ??? > 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://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 > > 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 > > > 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. > > > > >