# Posts from the ‘schools’ Category

Malcolm Gladwell’s Revisionist History episode Miss Buchanan’s Period Of Adjustment is a fascinating, disheartening tale about the backlash from the USA’s famous Supreme Court ruling – Brown v Board of Education.

Gladwell describes how the end of legally enforced racial segregation in American schools led to the firing of black school teachers. As student bodies were integrated, white school leadership fired black teachers on pretexts ranging from competence to attitude. In many cases, white parents would not accept black teachers for their children. Unlike Brown v Board of Education, their appeals failed.

I had never heard this appalling story. It deserves to be told.

## Task

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.

Diane Ravitch describes the ills of the USA’s education system. There are echoes of the policies of both the last Labour government and the current Conservative government (yes, yes) in what she says. I worry that this is the UK’s future:

More or less, this is what I think, too.

An interesting report on Finland’s education system was on the BBC News website today:

Report on Finland’s Education System

I think there are a few things that they miss out from the report, like a very homogenous population, a smaller wealth gap between rich and poor, and a very low density of population. Still, it does paint the picture of an idyllic scenario of an educational system free from interference from politics. I wish a party in the UK would put that in their manifesto!

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

The Cambridge Primary Review today published their recommendations for how the primary curriculum and classroom environment should be arranged. The briefing is an interesting read, the headlines of which can be found on the BBC News website.

As I read the part on SATs I reflected on the way in which the relationship between politics and education continues to work to this day. The review argues that SATs narrow the curriculum focus and put pressure on children unnecessarily. It argues that the concept of standards that underlies the system of SATs is “restricted, restrictive and misleading”. It further argues that assessment of childrens’ learning should be detached from assessment of schools’ accountability.

It is perhaps inevitable that this is how education and politics interact: governments change the way education works with an agenda justified by their electoral mandate, but often with no educational justification to back it up. It can then take a decade or more for evidence to be gathered, arguments to be made and reports to be compiled before the deficiencies of the system can be established to the satisfaction of politicians, and the scheme can be scrapped. Then, another government can come in with their agenda and try again.

I knew that SATs restricted curriculum, failed to assess students reasonably and were a monumental waste of time and money, years ago. I’ve blogged about it before, years ago. Most of the bright, intrested teachers that I’ve met have known similarly. But education is one of the few things that governments with mandates can interfere with almost at will, and the obvious truths for teachers on the ground are difficult to express to people living in the ivory towers of Westminster. It’s about to happen again. I believe that the best we can hope is that they (whoever they are) make a slightly less-bad set of decisions in this next round of reforms.

A quickie: here’s an interesting game from Canada where users have to find various interesting geometrical properties by eye and are assessed programmatically on their accuracy:

My score as about 3.03, having frustratingly crepty above an accuracy score of 3 with a shocking 9 in my final problem.

The following conversation in Metric Views catches the attention both for the interesting article and the subsequent comments.

Metric Views: Are our schools entrenching the ‘very British mess’?

The gist is that our schools reflect our current social muddle by teaching both imperial and metric measures and their relative magnitudes in school. In the article it is argued that the time and cost wasted on this is horrifying.

I have no love of imperial measures; I find it frustrating to have to remember how many pounds are in a stone, or ounces in a pound, or yards in a mile, and struggle to do so. I also find it difficult to convert between anything other than kilometers and miles. I know my weight in stone, but not in pounds, and certainly not in kilograms. I know my height in both metres and feet-and-inches. I am not sure that I can estimate volume in any unit with any degree of accuracy. It’s a horrible, muddy, confusing mess; that is undeniable.

I think my misgivings about the article are about the underlying idea that we should stop teaching both measures to achieve a feat of social engineering; by removing from the minds of the youth any conception of imperial measures, we would hasten the demise of imperial measures, which would be a Good Thing.

My difficulty is that feet-and-inches is such a good measure of height. I am 1.83m or 183cm, but neither is as satisfying as being 6′ tall, and neither is immediately conjourable in my mind. I don’t like Americans’ removal of ‘stone’ as a measurement either; 13 (and a bit) stone is much easier to remember than… whatever number of kilograms or pounds I am.

Feet, inches, stones and pounds are good measures because they are useful. They give us a scale rooted in humanity and the measurement of humans, and allow us to compare ourselves with others accurately. I am not convinced that the removal of these measures in classrooms will remove their common use.

I should not be confused with someone in defence of a curriculum which monitors and assesses the knowledge of different weights and measures and their conversion. Conversion is a fairly dry arithmetical topic. However, there might be problems that involve imperial or metric measures (or even their conversion) which may contain some good mathematics. I would not want that potential to be excluded from the curriculum any more than I would want their being taught made compulsory.

In an ongoing email conversation within the ranks of the ATM on its purpose and voice within the uk educational establishment, one of our numbers recommended we read Lockhart’s Lament, an article posted on the website of the Mathematical Association of America by Keith Devlin.

Lockhart’s Lament is a a heartfelt plea to the beauty of mathematics, the place of mathematicians as artists, not engineers, and society’s complete miscomprehension of what mathematics *actually is*.

The article opens with a parody: what if society had the attitude towards music that it currently has to mathematics? Lockhart asks us to imagine a world where students learn musical theory without ever grasping what music *is*. In this world, students don’t hear music or feel it, it is a word used to describe a formal system, emotionless and austere. Perhaps a few get to understand, listen to and feel music when they get to university. If they try to describe their joy and amazement, people look at them blankly and conjour up memories of their tests on harmonic scales when they were at school.

For Lockhart, Mathematics is in turns the *art of explanation* and the *music of reason*. However, it is as poorly understood by modern western society as music is in his imaginary music-less world. Lockhart argues that “there is no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum.” Through standardisation and testing which puts the onus on *memorisation* over *understanding and exploration*, the subject is fundamentally undermined.

The breadth of Lockhart’s exasperation is great: from society to schools, to teachers, and universities, but most forcefully to the government and the curriculum. This, written in 2002 is ever more true. It is an unsettling prospect that the USA is further down the road of standardising the maths out of maths than we are in the UK. Perhaps, using them to see into our future we can change it. Reading this Lament strengthens my belief that we must try.

I hope you gain as much enjoyment, and as much fervour from its reading as I did.