Misha Gavrilovich’s paper Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups, available from
- http://arxiv.org/abs/1608.05927, or
is a follow-up to his paper in The De Morgan Gazette,
M. Gavrilovich, Point-set topology as diagram chasing computations, The De Morgan Gazette, 5 no. 4 (2014), 23-32
The paper raises important questions about optimal approaches to exposition of elementary group theory: quite a number of group-theoretic concepts (for example, solvable, nilpotent group, p-group and prime-to-p group, abelian, perfect, subnormal subgroup, injective and surjective homomorphism) can be expressed as diagram chasing in the category theoretic language.
P. Ransom, Some recollections of early experiences with mathematics, The De Morgan Gazette, 8 no. 3 (2016) 19-26.ISSN 2053-1451. bit.ly/2bM0RyS
As this table from Education Data Lab shows, show, retakes are of low value.
Read the whole article: Repeat After ‘E’: the treadmill of post-16 GCSE maths and English retakes, by Rebecca Allen
New mathematics artwork from Hamid Naderi Yeganeh.
This fractal is constructed by an Africa-like octagon and its lateral inversion.:
The height of the biggest octagon is φ times longer than the height of second octagon; where φ is the golden ratio. (see the following image) It is a necessary and sufficient condition to see the white triangles in this fractal.
- A new vacancy for a candidate with a PhD in mathematics and I am sending this email in the hope that you might have recommendations for potential applicants to this job. The job title is “Senior Mathematics Curriculum Designer” and the salary bracket is £36,000–48,000. Full details about the job and a link for interested applicants to apply is available athttp://www.chestnut.com/en/careers/952105480268/.
- A two year postdoc in Mathematics Education, with a focus on problem
solving, reasoning and educational design, is announced in Umeå, Sweden. The full announcement:https://umu.mynetworkglobal.com/en/what:job/jobID:111103 Information about Umeå Mathematics Education Research Centre: www.umerc.umu.se Information about Umeå University: www.umu.se/english
- A PhD position on embodied, extended and embedded cognition from a philosophical or mathematics educational perspective, http://www.academictransfer.com/35679http://www.academictransfer.com/35679 or 3 positions as PhD students (1,0 FTE) <http://www.peoplexs.com/Peoplexs22/CandidatesPortalNoLogin/Vacancy.cfm?PortalID=4063&VacatureID=849205>
The Digital Turn in Epistemology (DigTEp)
DigTEp is a collaboration between the faculties of Science and of Humanities of Utrecht University (UU), and the faculty of Philosophy of Erasmus University Rotterdam (EUR). The multidisciplinary project DigTEp lies on the cross roads of:
- philosophy of mathematics (PhD 1)
- logic (PhD 2)
- ICT development and mathematics education (PhD 3)
All three PhDs will share an office at the Freudenthal Institute<http://www.uu.nl/en/research/freudenthal-institute> in the faculty of Science of Utrecht University.
Embodied, Embedded and Extended Cognition (E3C) marks the recent Turn in Epistemology, the philosophy of knowledge, as well as in Cognitive Science. The three Es indicate that our knowledge and capacities are not located in our skulls, but extend to, and are distributed over:
- our bodies, which are always embedded in their environment they interact with;
- the artifacts we use, varying from paper notebooks to ICT.
The mediation of knowledge acquisition and application by ICT is becoming so dominant and ubiquitous (smartphone, tablet, laptop, computer, world-wide web) that Epistemology must take a further, Digital Turn. DigTEp concentrates on the following epistemological questions: How to make sense of mathematical knowledge after the Turn in epistemology to E3C, given its abstract character? Does the Digital Turn affect the genesis and the essence of mathematical knowledge? When practical knowledge is primary and propositional knowledge is derivative (E3C), how does this ‘derivation’ work in a digital environment? To answer these pressing philosophical questions, an empirical case study in epistemology will be performed of the acquisition of mathematical knowledge and skills in a controlled ICT-embedded environment, by secondary-school pupils. For a more detailed description of the contents of DigTEp, and of the individual PhD projects, please send an e-mail to the supervisor of the PhD project you wish to apply for.
You may import a “style of teaching”, but cannot import the social environment of teaching — this is a key, and, perhaps, impassible obstacle to development of a coherent mathematics education policy in England. But attempts continue regardless: DfE has no other options.
Thousands of primary schools in England are to be offered the chance to follow an Asian style of teaching maths.
More from BBC:
The government is providing £41m of funding to help interested schools to adopt this method, which is used in high performing places like Shanghai, Singapore and Hong Kong.
The money will be available to more than 8,000 primary schools in England.
This approach to maths is already used in some schools, but the cash means it can be taken up more widely.
The Department for Education says the mastery approach to maths teaching, as it is known, involves children being taught as a whole class and is supported by the use of high-quality textbooks.
More detailed explanations of the NCETM’s thinking in this developing area can be found in several posts on the blog page of our Director, Charlie Stripp, in a document entitled The Essence of Maths Teaching for Mastery, published in June 2016, and in an earlier NCETM paper from autumn 2014.
A recent paper: Anders Eklunda, Thomas E. Nicholsd, and Hans Knutssona, Cluster failure: Why fMRI inferences for spatial extent have inflated false-positive rates.
doi: 10.1073/pnas.1602413113, bit.ly/29j7dKf
sends a pretty grim message:
The most widely used task functional magnetic resonance imaging (fMRI) analyses use parametric statistical methods that depend on a variety of assumptions. In this work, we use real resting-state data and a total of 3 million random task group analyses to compute empirical familywise error rates for the fMRI software packages SPM, FSL, and AFNI, as well as a nonparametric permutation method. For a nominal familywise error rate of 5%, the parametric statistical methods are shown to be conservative for voxelwise inference and invalid for clusterwise inference. Our results suggest that the principal cause of the invalid cluster inferences is spatial autocorrelation functions that do not follow the assumed Gaussian shape. By comparison, the nonparametric permutation test is found to produce nominal results for voxelwise as well as clusterwise inference. These findings speak to the need of validating the statistical methods being used in the field of neuroimaging.
Implications are very serious:
Functional MRI (fMRI) is 25 years old, yet surprisingly its most common statistical methods have not been validated using real data. Here, we used resting-state fMRI data from 499 healthy controls to conduct 3 million task group analyses. Using this null data with different experimental designs, we estimate the incidence of significant results. In theory, we should find 5% false positives (for a significance threshold of 5%), but instead we found that the most common software packages for fMRI analysis (SPM, FSL, AFNI) can result in false-positive rates of up to 70%. These results question the validity of some 40,000 fMRI studies and may have a large impact on the interpretation of neuroimaging results.
Alas, too many people think that everything is normal…
From the BBC:
Official data shows just over half (53%) of 11-year-olds made the grade in reading, writing and mathematics. […]
Department for Education statistics show:
- 66% of pupils met the standard in reading
- 70% in maths
- 72% in grammar, punctuation and spelling
- 74% in the teacher-assessed writing
The overall figure of 53% relates to the number of pupils who reached the expected standard in all three subjects.
Read the full story.
A post on Farnam Street. A quote:
Feynman knew the difference between knowing the name of something and knowing something. And was often prone to telling the emperor they had no clothes as this illuminating example from James Gleick’s book Genius: The Life and Science of Richard Feynman shows.
Educating his children gave him pause as to how the elements of teaching should be employed. By the time his son Carl was four, Feynman was “actively lobbying against a first-grade science book proposed for California schools.”
It began with pictures of a mechanical wind-up dog, a real dog, and a motorcycle, and for each the same question: “What makes it move?” The proposed answer—“ Energy makes it move”— enraged him.
That was tautology, he argued—empty definition. Feynman, having made a career of understanding the deep abstractions of energy, said it would be better to begin a science course by taking apart a toy dog, revealing the cleverness of the gears and ratchets. To tell a first-grader that “energy makes it move” would be no more helpful, he said, than saying “God makes it move” or “moveability makes it move.”
Read the full story.