http://vimeo.com/1908224?hd=1

As the comments underneath the video allude to, there is some utility here to attempt to consider the size of magnification that occurs during a portion of the video, and how to express magnifications of such a magnitude.

Cell Size And Scale

From a scientific perspective it is interesting to reflect upon the relative sizes of elements, from a Coffee Bean to a Carbon Atom (through various things such as a human Ovum, Sperm, various viruses, compounds and so on).

From a mathematical perspective it is also interesting for the way in which the relative scales are measured in the top-left. Exploring the different notations for small sizes would be a useful exercise in place-value for all levels of Key Stage 3 and 4.

This article is a little heavy on the politics for an average maths classroom, but is perhaps useful for A-level students, and is definitely useful for any teachers teaching the IB, as it has excellent cross-over with theory of knowledge. Worth a look

Cosmic Quest

Cosmic Quest This fabulous narrative history of human understanding of the Cosmos tells one of the greatest stories in the history of ideas. It is pleasingly compact, and easy to listen to. All the episodes are available to listen to from the BBC website.

In Our Time – Probability

Melvyn Bragg’s excellent In Our Time broadcast and podcast on probability last week was an excellent discussion of the history of probability with, among others, Prof. Marcus du Sautoy, who is always worth listening to! The podcast can be found here.

In this month’s Mathematics Teacher magazine is an article about prime factorisation by me. It discusses an idea for teaching and learning about prime factorisation that minimises ‘telling’ and maximises students mathematical exploration.

Dave Hewitt, a lecturer in Mathematics Education from the University of Birmingham (and my PGCE mentor a few years ago), has written a series of articles about separating the arbitrary (or contingent) and the necessary and mathematics, and teaching by ‘telling’ only those things which are arbitrary. The idea is that students need to engage and discover for themselves the necessary connections and patterns in mathematics, but the arbitrary are not discoverable in the same way, and so need to be told to people.

This position has influenced my thinking about mathematics education, particularly with respect to algorithms. My conjecture is that using an algorithm involves no mathematical thought; at best, it is an exercise in arithmetic. However, where an algorithm exists there is likely to be a kernel of really interesting mathematics, and creating an algorithm to perform a particular function involves a great deal of mathematical thought. The goal of the investigation was to capture the interesting maths in an interesting way, that the students can engage with, and which they can learn from.

The accompanying software can be found in the primitives section of this website.

For maths teachers, KS3 and KS4 students will enjoy and learn from the discussion between Melvyn Bragg, Professor Marcus du Sautoy, and others. It is well worth a listen.

Daniel Tammet has an extraordinary relationship with numbers. Born with Asperger’s Syndrome, Daniel has an extremely kinaesthetic relationship with number, and has a host of other impressive mental abilities, such as his extreme aptitude for learning languages rapidly. His book Born on a Blue Day is a wonderful memoir of his life to this point, and well worth a look both because it is a great read, and because it offers an interesting insight into Asperger’s Syndrome.

Mathematically however, it is even more interesting. Daniel suffered at school because his mind was creating associations between numbers and other concepts, and between the numbers themselves in creative and unexpected ways. He and his parents had the courage – or perhaps just the necessity – to persist in working in and with those associations. They have served him well; his numerical mental dexterity is far beyond what almost anyone else could muster.

Schools are inevitably ‘one size fits all’ institutions to some extent. Schools are mass-education institutions; it is not possible to ‘personalise learning’ precisely for individual students. That is not to say that the Personalised Learning initiatives from the government are bad things, far from it; merely that because often a teacher is dealing with about 30 students, it is not possible to tailor learning exactly to the needs of every student.

In mathematics classrooms, the tendency to make uniform what is not becomes more apparent. In maths teaching there is a tendency towards drill and rote. It can be seen clearly in textbooks, and it is confirmed by asking any reasonable cross-section of society to recount their experiences of maths at school.

It would be absurd to argue that allowing students more freedom to explore the associations between numbers and to create their own understandings of them will most students become as adept at maths as Daniel. Nevertheless the root of his understanding of number comes across as creatively based rather than based in algorithms and routines. To me, Daniel’s story is compelling evidence in support of the theory that creativity and imagination are central to the development of young mathematicians.

As a foot-note, after reading Daniel’s book I began to reflect on the ways in which I perceive numbers. This directly led me to expand my own concept of the structure of numbers by creating the Primitives concept.

Displayed on screen this image could be used to motivate an activity in which students practice finding angles by creating their own plat diviseurs. It can be quite a fun activity, especially if you buy paper plates for them to use (though I recommend that they practice on paper first!).

They could also be asked which numbers are absent; they hopefully notice that 2,4 and 8 are missing. Discussing why can get into thinking about halving and fractions.

The reciprocity of 3 and 1/3, 5 and 1/5 can also be brought up, by discussing how often the image of a divided circle is often produced as a visualisation of proper fractions.

Since 5 and 7 are not factors of 360 this is also an interesting discussion point. While discussing factors, they should note that points for 3,6 and 9 are coincident at two points, and that if 2,4 and 8 are added what other numbers would have coincident points. I remember asking whether 10 and 12 would have coincident points if we had them as divisions too, which caused some thought.

Note that there are very few good images of plat diviseurs on the internet. I originally got the idea from Problem Pictures.

It’s a lovely problem that offers the opportunity to discuss methods of calculation, and methods for avoiding mistakes. It links to combinations and permutations, as well as triangle numbers, and is a really simple question to pose. An intermediate question is to ask how many dominoes there are in a set, and establish that solution before continuing to the original question.

What a domino is

Dominoes are tiles divided into two. On either side of the tile there is a number of spots from zero to six. You could write a domino as 1|6, which is a domino with one spot on one side, six dots on the other side. Every possible combination of domino is in a set including ‘doubles’ such as 0|0 and 6|6. Note that 1|6 and 6|1 are considered as the same; since tiles can be turned around freely, order is not important.

Extension

What if the maximum number of spots in a domino set changed? How does the number of dominoes change? How does the total number of spots change?

Enter Flickr, a great online photo sharing tool. Though you will not get questions related to each photo, the database is huge. Typing into the Flickr search engine ‘geometry’ turns up 10,810 photos at the time of writing. Most of them are really relevant; some of them are utterly spectacular. Here is a random sample:

I have found that putting an image full screen on an interactive whiteboard when students come into the room is a great way of capturing the students’ attention straight away.