# Posts from the ‘number’ Category

99% invisible is a wonderful podcast about design, made in Oakland California. I don’t have much to do with the world of mathematics education anymore (at the moment?) but as I listened to this, I heard echoes of students’ questions about the relevance of learning unit conversion in the computer age. Well, NASA can give you \$100millions of reasons! It’s really worth a listen.

99% Invisible, Episode 280: Half Measures

Choose any 4 digit number except for 1111, 2222, 3333, 4444. I’ll choose 1502.

Rearrange the digits to give you the biggest and the smallest numbers you can. 5210 and 0125.

Find the difference of these two numbers. 5210 – 0125 = 4995.

Repeat for the new number you get (using zeros to supplement any missing digits if necessary; you must always have 4). 9954 – 4599 = 5335.

Keep repeating until you have a good reason to stop.

## Example

5210 – 0125 = 4995
5533 – 3355 = 2178
8712 – 1278 = 7434
7443-3447 = 3996
9963 – 3699 = 6264
6642 – 2466 = 4176
7641 – 1467 = 6174
.

I’m not going to repeat any more because 6174 => 7641 – 1467 which is the calculation I just did!

## Discussion

• So every number we’ve tested came to 6174. Do you think all numbers will come to 6174? Why? In maths we can’t say we’re sure about something unless we have a good reason, so unless you want to go through every number and check it, we can’t say that we know that every number sequence we choose will go to 6174! (This is how mathematics differs from science, we don’t just make hypotheses and wait until the next time they break, we find ways to be certain!)
• Would we have to check every different number to be certain or are there shortcuts we can take? (should we check both 1234 and 1243 separately?)
• My rules stipulated that you couldn’t choose all the same number: 1111 or 2222. What would happen if you did use those numbers?
• What about two digit numbers or three digit numbers?
• Could we frame the problem as (1000a + 100b + 10c + d) – (1000d + 100c + 10b + a)?
• What about five or six digit numbers (or more…)? You may need a computer to help!

## Thoughts

I just came across this interesting bit of mathematics via Twitter: http://plus.maths.org/content/os/issue38/features/nishiyama/index. It offers the opportunity for a beautiful open-ended task for secondary maths classes of all abilities. The initial task is easy and produces a startling result that feels like a trick; that hook can lead to discussion and further work about probability, permutations, algebra and programming.

An incredible rendering of the MandelBrot Set, which drives home the notion of infinite complexity!

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.

I stumbled across this lovely page from the University of Utah today, which is extremely simple, but nevertheless well done:

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.

In the BBC Website Magazine today is an article about proportions and magnitudes. It made me reflect that we often spend time teaching students how to express numbers in different forms, but rarely attempt to give students an understanding of how the numerical forms differ, and what they represent.

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

Below I review two media resources that are well worth a listen, for teachers, interested adults, and perhaps older students. These are not resources in themselves, but I am sure that educators will find stories and examples in these programmes that can have direct application in the classroom.

## 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.

In Our Time is a BBC Radio 4 presented by Melvyn Bragg. It is an intellectual talking-heads discussion programme about philosophy, science, mathematics and so on. This week the discussion was aobut the Fibonacci Sequence. The podcast can be downloaded here.

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.

A plat diviseur is a plate designed to make portioning cakes easy. Click here to see one. It works by putting dots at appropriate angles around the rim of the plate, so that when someone wants to cut, for example, five portions they merely cut from the center of the plate to wherever there is a five. For neatness one cut is always common, and tends to have just the number 0!

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.