George Boole is credited with many things, including
Founding invariant theory.
Invention of boolean algebra.
Intorduction of mathematical probability.
Boole’s first mathematical interest was is the realm of mathematical analysis and differential equations. He published two textbooks that also included soem original research contributions concerning thr algebraic approach to differential equations. Broadly speaking we denote the derivative as the application of an operator \(D\). So that if \(D(f) = u \) then \(f = D^{-1}(u) = \int u \). For example we could have the following Differential equation (DE) \( \frac{\mathrm{d}^2f}{\mathrm{d}x^2} + 3\frac{\mathrm{d}f}{\mathrm{d}x} + 2f = \sin (x) \) can be written in operator notation as \( D^2(f) + 3D(f)+2f = \sin (x) \) or \( (D^2+3D+2)(f) = \sin (x) \). For his contributions in this field he was awarded The Royal Medal of the Royal Society.
In 1841 Boole published one of the very first papers on invariant theory, this paper is credited by Arthur Cayley in his paper of 1845 which is often credited as being the start of invariant theory.
Of course Boole is most famous for his contributions in the world of logic. He first pulished a book on logic in 1847 whilst he was still in Lincoln, before publishing again once he had moved to Cork.
I was interested to learn that Boole only allowed the operation of Union when the sets were disjoint, Evegeny then presented a nice table comparing modern notation with that of Boole.
Boole is also famous for his contributions to mathematical probability, most notably his name lives on in Boole’s inequality.
In conclusion “Boole made a giant step towards mathematics as a truly abstract discipline, causing a paradigm shift, giving mathematics enormous scope and potency” (Khukhro, 2015). Evegeny has made his PowerPoint slides available here.
The poll is now closed and the winner is…… “Contrasts in Mathematical Challenges in A-Level Mathematics and Further Mathematics, and Undergraduate Examinations” by Ellie Darlington, and published in the IMA Teaching Mathematics and Its Applications journal. This article achieved almost 50% of the votes, and I am really looking forward to discussing it.
We will be discussing this at 8pm on Monday the 7th of December. I know it will be the run up to Christmas, but I hope lots of you will still be able to make the discussion.
As usual, about a week before I will post some possible topics of conversation or things to think about. I hope you enjoy the article.
Yesterday evening I went to the excellent Festival of The Spoken Nerd in Derby, but before thsat I had the pleasure of seeing Alex Bellos speak at a school in Leicestershire. This post is just going to mention a few interesting things from both of these events.
Alex Bellos was a very entertaining speaker to listen to as he is evidently so passionate about people understanding the beauty of mathematics.
Alex opened his talk with a logic puzzle that he had featured in one of his early Guardian Puzzle blog posts: Find the odd one out in the symbols below
This puzzle was orignally due to Tanya Khovanova who had written about it here. It is intended as a piece of fun to emphasise the intrinsic issue of odd-one-out puzzles. Namely, that they tend to be focussed around a particular way of thinking (Alex provides a nice example in his post) when in fact many could be seen as the odd one out for different reasons. In the puzzle above, the odd one out is actually the one on the left hand side by virtua of not being able to be called the odd one out – an interesting philosophical dilemma there!
One thing I found particularly interesting in Alex’s talk was his discussion of the Sieve of Erastothenes. If I do this in school I tend to have a 10 by 10 grid of the numbers 1 to a 100. I had never thought of arranging them instead in 6 rows, as shown below. This leads to some very interesting patterns when you cross out numbers.
He then described the Ulam Spiral, devised by Stanislaw Ulam where numbers are arranged in a spiral and primes highlighted generating a pattern where primes lie on diagonal lines. I’m going to write some MATLAB code to generate these I think and write a bit more about them in the future.
Alex also talked about the results of his internet survey to find the world’s favourite number; which turns out to be 7. He showed a nice annimation about this which is available on Youtube
Following Alex’s session I had a fairly mad dash to Derby in horrendous weather to go and see Festival of The Spoken Nerd – if you haven’t seen them live before I can’t recommend it enough. It was a great evening. I really liked the visual demonstration of the modes of vibration of a metal plate. These are known as Chladni figures after the scientist Ernst Chladni who first published his work in 1787.
Festival of The Spoken Nerd have plenty more dates in their UK tour, I urge you to see if there is one local to you on their webpage and if there is go. You won’t be disappointed, I’ve seen them a few times now and it is always a very entertaining evening – writing about the show can’t do it justice.
Following on from our second successful discussion last week it is time to vote on the article for the next discussion. The next discussion will take place at 8pm on Monday the 7th of December. I know everyone gets busy in the run up to Christmas but I hope that you can all still take part.
Three article from the last poll have been rolled over to this one as two of them were tied in the number of votes. As usual, the titles and abstracts are below and the poll is available here
Contrasts in Mathematical Challenges in A-Level Mathematics and Further Mathematics, and Undergraduate Examinations; Ellie Darlington (Teaching Mathematics and its Applications) – This article describes part of a study which investigated the role of questions in students’ approaches to learning mathematics at the secondary/tertiary interface, focussing on the enculturation of students at the University of Oxford. Use of the Mathematical Assessment Task Hierarchy taxonomy revealed A-level Mathematics and Further Mathematics questions in England and Wales to focus on requiring students to demon- strate a routine use of procedures, whereas those in first-year undergraduate mathematics primarily required students to be able to draw implications, conclusions and to justify their answers and make conjectures.While these findings confirm the need for reforms of examinations at this level, questions must also be raised over the nature of undergraduate mathematics assessment, since it is sometimes possible for students to be awarded a first- class examination mark solely through stating known facts or reproducing something verbatim from lecture notes.
“‘Ability’ ideology and its consequential practices in primary mathematics” by Rachel Marks (Proceedings of the BSRLM 31 (2)) – ‘Ability’ is a powerful ideology in UK education, underscoring common practices such as setting. These have well documented impacts on pupils’ attainment and attitude in mathematics, particularly at the secondary school level. Less well understood are the impacts in primary mathematics. Further, there are a number of consequential practices of an ability ideology which may inhibit pupils’ learning. This paper uses data from one UK primary school drawn from my wider doctoral study to elucidate three such consequential practices. It examines why these issues arise and the impacts on pupils. The paper suggests that external pressures may bring practices previously seen in secondary mathematics into primary schools, where the environment intensifies the impacts on pupils.
“Train Spotters Paradise” by Dave Hewitt (Mathematics Teaching 140) – Mathematical exploration often focuses on looking at numerical results, finding patterns and generalising. Dave Hewitt suggests that there might be more to mathematics than this.
“Relational Understanding and Instrumental Understanding” by Richard Skemp (Mathematics Teaching 77)
“Knowing and not knowing how a task for use in a mathematics classroom might develop” by Colin Foster, Mike Owlerton and Anne Watson (Mathematics Teaching 247) – Participants at the July 2014 Institute of Mathematics Pedagogy (IMP14) engaged in a wide range of mathematical tasks and a great deal of pedagogical discussion during their four days last summer. Towards the end of IMP14 a conversation began regarding how much knowledge about a task a teacher needs to have before feeling comfortable taking it into the classroom.
I’m looking forward to seeing which article is selected as I haven’t read all of these yet!
I’m quite sad that I missed this on pi day this year but I thought I should share it now..
For this years (special) pi day Wolfram Research have produced a web page mypiday.com that enables you to find the location of your birthday in the digits of pi – it is well known that any date will appear within the digits. Here is the location of my birthday:
Stephen Wolfram also wrote a fairly interesting blog post about the creation of this site using the capabilities of the new Wolfram Language.
I’ve been doing preparation for the Oxford MAT exam with a couple of my Year 13s and in the 2008 paper I came across this very nice question about the Trapezium Rule.
I really like this question as to answer it you need to have more understanding about how the trapezium rule actually works than standard A-Level questions on this subject. Invariably they are just pure “plug some numbers in and crunch them” questions which lead to the wide perception that numerical methods are boring. Of course being a numerical analyst I really don’t agree with this perception, but agree that their presentation in the current A-Level course doesn’t help with this.
The above question is nice in that it combines knowledge of the performance of the trapezium rule with graph transformations.
I will leave you to work out the answer, the small Geogebra file I have written may help you visualise what is happening…
Last night we discussed “Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts” (available online here).
Like last time it was a very fast paced, interesting discussion. I really enjoyed the discussion and some great points were made by lots of people. Have a look through the storify below – I have hopefully put in all the points discussed.
Stephen Cavadino (@srcav) wrote a blog post after the discussion trying to categorise études which I recommend reading. James Pearce (@MathsPadJames) also wrote a post before hand (which unfortunately I didn’t see until about half way through the chat) looking at commonalities between different types of mathematical études – give that I read too!! I would have promoted it more during the discussion if I had seen it earlier.
Thank you to everyone who took part, and I hope you can make the next one which will be on Monday the 7th of December. The poll for voting will open for a week next Monday. If you would like to submit an article please tweet the details to me by this Friday 23rd October.
Following on from last months excellent collection of posts in the Carnival of Mathematics 126 from Stephen at cavmaths it is my turn to host this blog carnival.
The Carnival of Mathematics is a monthly blogging round up that is organised by The Aperiodical (check it out if for some reason you haven’t before) and has a different host each month.
It is tradition to start the post with some interesting facts about the number 127 so here goes….
The most interesting fact about 127, I think, is that it is a Mersenne Prime. This means that 127 is a prime number of the form \(2^n-1\) with \(n\) here being 7. Since 7 is also a Mersenne prime 127 is known as a double Mersenne prime, that is, it is of the form \(2^{2^n-1} -1 \).
127 is a number that appears in the list of centred hexagonal numbers. Coincidentally I have written a post concerning these numbers when I discussed nRich’s Steel Cables problem here.
Since 127 is a centred hexagonal number and prime it is Cuban prime.Namely it is a solution of the equation \( \frac{x^3-y^3}{x-y} \) where \(x=y+1\) and \(y>0\). This can be simplified so that Cuban primes are prime numbers of the form \(3y^2+3y+1\).
127 is a cyclic number. This means that the Euler totient of 127 and 127 are co-prime. (Clearly all primes are cyclic.)
127 is the 8th Motzkin number. For a particular value of \(n\) the \(n-\)th Motzkin number is the number of different ways of drawing non-intersecting chords between \(n\) points on a circle.
127 is a polite number as it can be written as \(127 = 63 + 64 \).
To start on the posts that make up this carnival I thought I would start with the following comic produced by Manan of MathMisery. This made me laugh a lot!
His blog is one that I routinely keep an eye on and I also enjoyed his recent post entitled Executive Education
Ganit Charcha submitted this really interesting article on Montgomery Modular Multiplication. I hadn’t heard of this algorithm before and I need to get round to coding it up soon – this article is very clear and written in a style ideal for being able to code it up.
Shecky R shared a short post looking at a curious finding (made without modern calculating machines) of Pierre de Fermat in his post Get A Life!
I enjoyed reading Tracy Herft’s post Unique Lesson… Polar Clocks. In it she describes a lesson she has developed for Year 7 students which she uses to link the topic of angles with fractions.
The mathematician Nira Chamberlain has shared his poster about the black heroes of mathematics. He has created this for Black History Month (which happens to be October).
I’m ashamed to say that I hadn’t heard about any of these before!
This post by Evelyn Lamb (@evelynjlamb) concerning cutting letters of the alphabet featured a video by Katie Steckles of The Aperiodical. I love how Evelyn describes the “peculiar laziness” of mathematicians – it is certainly true!!
R.J. Lipton has this great accessible post “Frogs and Lily Pads and Discrepancy” discussing Terence Tao’s recently announced proof of Paul Erdös’ Discrepancy conjecture. If you fancy some more of the detailed maths take a read of Terence’s post announcing his papers (and the other related posts). Tao is pretty rare amongst professional mathematicians in that as well as publishing in academic journals he also discusses his research in an open blog.
Herminio L.A. submitted an interesting discussion of a calculation paradox in “Sabotage in the Stores”.
I had recently seen this video on Youtube and it was nice to learn that someone had submitted it for inclusion in this carnival (I would have myself if no one else had). The animations that go with some of the (many) patterns in Pascal’s triangle are fantastic.
Edmund Harriss has this post of beautiful animations of eigencurves of various matrices that are dependent on parameters in one entry. Edmund has also produced a colouring book with Alex Bellos titled “Snowflake, Seashell, Star” that taps in to the colouring craze at the moment. I can’t colour at all, but I am tempted to get this book, if only for the mathematical notes. He discussed the book here.
Christian Perfect (@christianp) shared this article by Lior Pachter. This article concerns something that I am very passionate about – the discussion of unsolved mathematical problems at school. Lior has taken each year in the American Common Core and selected an unsolved mathematical problem whose description is accessible to students of that age. I particularly like how each unsolved problem is accompanied by a starter problem that is accessible by the students. I think discussion of unsolved problems in school is very important as far too often students think that everything in mathematics is known. It’s sad that students s of school age don’t typically get exposed to current research or unsolved problems.
Diane has shared this problem from part of The Center of Maths’ Advanced Problem of the Week series. It’s quite a nice problem involving partial differentiation. This series provides a nice selection of problems to try, however some do require more area specific knowledge than others.
Peter Rowlett (@peterrowlett) recently wrote this fascinating post about Mathematical Myths. E.T. Bell did much to create a series of myths about mathematicians and some of these feature in this post. There are extensive links in this post so it may take a while to read.
Personally my most involved blog post in the last month was the write up of my NQT advice workshop at #msthsconf5.
This brings to an end this issue of the Carnival of Mathematics. The next edition is being hosted by Mike at Walking Randomly. If you haven’t before check out Mike’s blog, he regularly posts good articles about mathematical software.
I think it is great that Ada Lovelaceis beginning to get the recognition she deserves. I can remember reading Ada’s notes on Babbage’s difference engine when I was in sixth form and being completely fascinated by them.
The University of Nottingham have produced this great video about her:
And there is a good article on the Guardian about why Ada Lovelace is important for women in STEM subjects.
Hannah Fry (@FryRsquared) has also written a nice article that goes alongside her programme about Ada Lovelace. If you haven’t seen that yet it is definitely worth a watch.
Next Monday (19th October) at 8pm we have the second #mathsjournalclub discussion. This time (in case you don’t already know) we are looking at Colin Foster’s paper on “Mathematical études: embedding opportunities for developing procedural fluency within rich mathematical contexts”.
I read the article through again on the train on Saturday night and enjoyed the read.
From reading it here are a few things to think about in advance of next Monday.
Would you be comfortable with removing all “drill” practice and relying on richer tasks to develop procedural fluency?
In what other settings could you use a task similar to this one?
I feel that Colin makes an interesting point regarding students defaulting to a favoured approach when tackling “problem solving” type questions. Do you agree?
There is still time to give it a read and take part of you hadn’t considered it before – you can get the article online here.
I hope you can get involved in the discussion on the 19th.
Boole is also famous for his contributions to mathematical probability, most notably his name lives on in Boole’s inequality.
