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.
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.
Read the full story. Coming soon: comments on mastery and NCETM‘s thinking
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…
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.
A paper by Chris Havergal in Times Higher Education. Some quotes:
The first-ever release of Ucas data at institutional level shows that the University of Cambridge admitted only 65 18-year-olds from the UK’s most disadvantaged neighbourhoods in 2015, while it gave places to 1,260 learners from the most advantaged backgrounds. Taken as a proportion of the total size of these groups, this meant that the most privileged students were 16 times more likely to win a place.
The overall ratio for the Russell Group of highly selective institutions was 7.7, but this remained significantly higher than the UK-wide average of 2.45. At providers with the lowest entry standards, the most privileged students were only 12 per cent more likely to get in.
Read the whole article.
Please consider crowdfunding this remarkable project of Natural Math.
I’d like to draw your attention to a new book: ‘Teaching Mathematics for Social Justice: Meaningful Projects for the Secondary Mathematics Classroom’. The aim of the book is to share teaching resources and ideas generated from the TMSJ Research Project (a participatory action research project). The book was published by the Association of Teachers of Mathematics in April 2016.
The book is:
* Aimed at teachers of mathematics who are interested in addressing issues of social justice in their classrooms.
* Based on the premise that conventional approaches to teaching maths do not adequately address the needs of all learners or the needs of society as a whole.
* Suitable for students in Key Stages 3 and 4, those studying the new ‘core mathematics’ curriculum and for those on post-compulsory numeracy courses.
* Written in a style that allows teachers to use the ideas in a flexible, creative and non-prescriptive way.
The book contains:
* Seven projects addressing issues of social justice in the mathematics classroom;
* Twenty task sheets designed to be photocopied for students;
* Teachers’ notes offering ideas for supporting and developing classroom practice;
* Six accessible research articles exploring the theories underlying the teaching ideas.
Further details of the book can be found on:
and on the ATM website:
Dr Peter Gates