Let’s say there are only 2 parts to a plane, the head and the body. All the planes that come back have bullet holes in the body. None of them have bullet holes in the head.

Does that mean you need to put more armour on the bodies? That was the army’s conclusion. The statistician realised that it actually means that when a plane is shot in the head, it just doesn’t come back. So it’s the head that needs more armour.

If the students don’t believe the teacher, he could illustrate it on a made up situation with fatality probabilities of certain hits and made up data from the returned planes that may seem to suggest something different than what they actually do.

In the second world war, the British Airforce had heavy airplane losses. They needed a way to better protect the fighter airplanes (by armour plating). They could not cover the wole airplane in armour (as it would be too heavy to maneuver to be used as a fighter plane), but instead the aim was to cover the most critical parts. They collected data from damaged planes that returned and it turned out most hits were on the wings of the airplanes.

Does that suggest the best way to protect the planes is to put armour plating on the wings?

It doesn’t, of course, the important part of the data is that which is missing – the planes that did not make it back. And those were the planes hit in the fuel canisters at the middle part of the plane. The army actually went to a statistician and expected him to say the wings need more plating. He gave them the right answer though and it helped considerably.

Also perhaps a good asumption is that the enemy fighters cannot aim directly on specific parts of the plane (quite realistic in WW2 conditions I think), so they cannot adapt to the fact, that some parts now have more plating.

I posted a question on MSE that is sleeping for a while on predicting batsman’s runs. Also another problem that I wrestled with is if the Fall of Wickets of all matches approach any graph.

But I also think that it is not realistic to expect that any question is going to interest everybody, and maybe some people will dislike the entire concept. However, the best can be the enemy of the good here. I think the right criterion is whether a course like this would be better for some people than an attempt to cram into their heads some more conventional mathematics. If, for example, 40% of pupils disliked this course intensely and 70% dislike more conventional mathematics intensely, then this course would be doing well.

Is this similar to your philosophy Tim?

Bin packing or putting objects of differing sizes into boxes wasting as little space as possible is one of these. Also network problems such as the Konigsberg bridges and can you draw a house shape without taking your pen off the paper? Route problems such as a postman having to go down each street at least once and wanting the shortest total distance or even how does a satnav calculate the shortest/best route from A to B?

However, I would make two further points – 1) Real world questions need to be an integral part of mathematics teaching through all ages and 2) The syllabus can still come first, with the questions defining the learning objectives. An open ended question and personal study, much like the IB syllabus is also a good thing and as the student poses the question herself, the question will be relevant to her.

The truth is that maths is taught from the Platonic philosophy where mathematical objects “exist” independent of the real world. I think a discussion of that philosophy is outside the scope of this post, but I do believe that it is the fundamental reason many students are alienated by maths. In many ways, by simply adopting a historical or humanistic philosophy in the way we teach maths, we can communicate the abstract ideas more easily.

I think it is great that Dan Meyer has been mentioned already, but his Ted speech and blog are worth looking at.

http://www.ted.com/talk/dan_meyer_math_curriculum_makeover.html

Take this discussion on Online maths: http://blog.mrmeyer.com/?p=15398. Getting student involved in the question should be key to every lesson that is taught. The difficulty is that there is not enough investment going into the design of teaching tools. With the right teaching, you can deliver any syllabus and we should use technology to better standardise and monitor our teaching of maths.

The other travesty, that makes much of the post-16 education discussion problematic, is the huge gap in achievement in the 11-16 age range, discussed at length in the Vorderman report. Without basic numeracy, achieving rigourous learning outcomes from the types of questions you discuss is near impossible.

The other thing mentioned in this discussion is the Computer Based Maths movement. I agree with the objectives of making maths less about computing and more about doing. However, I again think this hinges on the learning up to the age of say 13-14. By this stage, a student 1) needs to understand how computation works, and 2) needs to understand how to algebraically express basic ideas. At this point more computer tools could be introduced alongside algebra to solve real-world problems or answer real world questions.