TMA: MI5 task

The MI5 task is my favourite of all the Mathematics Apprenticeship tasks. In it, the children have to design two security codes, position laser beams in the Head of Secret Service main office and consider other security features.

I want to focus here on one aspect of the task: writing a 4-digit security code to operate the main door at MI5 HQ. The children are asked to write a formula that each agent can use to work out their personal entry code. The formula must include personal information (so each agent has there own code); it must change with the date; and it must always give a 4-digit product. In the prompt, an example formula is given:

(L x A x 20) – D
L is the number of letters in your first name
A is age
D is day of week (Monday = 1, Sunday = 7)

Early in the task, I show the children the extent to which this formula works. For example, a 25 year-old called James going to work on Tuesday would calculate his code as follows: (5 x 25 x 20) – 2 = 2498. A 62 year-old called Charlotte going to work on Friday would have the following code: (9 x 62 x 20) – 5 = 11,155. However, this is a 5-digit product. Consequently, this formula is not ideal.

The children will then start to create their own formulas. Which variables should they use? How many variables? And how, exactly, should they use them? I tend to scaffold the maths here more than in any other area of TMA, but always giving the children freedom to explore their own ideas.

Here are some typical points to consider. When choosing your variables, think about the range of possible values. For example, ‘day of month’ (1-31) varies more than ‘number of letters in day of week’ (6-9). Also, note that if you multiply you will normally get a wider range of outcomes than if you add or subtract. Some groups may also realise that an easy way to produce a 4-digit code is to use ‘year’ as a variable, as it is a 4-digit number.

Here are some of the examples from my current class:

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A is age
B is 1 less than your age
C is using Greek alphabet code (below)

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We highlighted the strengths of this code: for example the ingenuity of the use of the Greek alphabet and the use of 3 variables. We then reflected on the fact that A and B are essentially the same variable: that B = A – 1. Also, we drew out the maximum and minimum likely products (based on a minimum age of 21, and a maximum age of 65):

Maximum code: 4160
Minimum code: 501

The children were able to figure out that if they added 500 to their code, it would always result in a 4-digit product. This, for me, was a really meaningful learning experience.

My favourite formula was the one below:

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This formula always produces a 4-digit product, and notice that the group had worked out the year in which the formula would give an answer that would be a 5-digit product – 2024! They had chosen their variables carefully, and had toyed with their formula until it was perfected. They were acutely aware of the largest and smallest possible products that their formula would give.

The MI5 task provides a really powerful context for learning. When managed well by the teacher, it results in some amazing learning experiences for the children too.

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TMA: Memories

Here are 3 of my own, in no particular order:
After the gadget task, listening to two children having a hearty (and good-natured) argument well into break time about the relative merits of investing in the glow-in-the-dark marbles as opposed to a remote controlled fridge.

Phoning my own classroom on SKYPE dressed as a secret agent from the MI5, but repeatedly being caught out by confused members of staff who didn’t have a clue what was going on.

Watching a previously timid child convincing the school technician to stop fixing the school’s urgent ICT problems and instead figure out how to download the graffiti font they wanted (needed!) for their task.

And when asked for their memories of The Mathematics Apprenticeship, here are a selection of the children’s responses:

‘I liked the Mathematical Apprenticeship, it’s fun and it gets people working together.’

‘I liked doing the garden project; we had a little puzzle to work out with paving stones but it felt great when we finally worked it out. I thought it was a great experience and we had a few laughs along the way.’

‘Each challenge provided us with the skills and knowledge that we need for the future.’

‘It isn’t like a boring lesson writing answers from a book. It’s more fun, more sophisticated.’

‘I liked the classroom challenge. We designed the classroom layout on a computer, Mr Metcalfe said that it looked very professional. The maths apprenticeship is brilliant – it made me more confident.’

I hope that it gives you and your children great memories too.

TMA: Feedback

There are great opportunities for giving the children meaningful feedback at the end of each Maths Apprenticeship task. I would say that the feedback covers three main areas: the maths content, the way the group communicate their work (in written and spoken forms) and the way in which the group work together.

Maths content
Some of the feedback will be based on the accuracy of the group’s calculations. For example, were they able to work out a 15% discount based on a bulk order? In other cases, the feedback will be more open-ended. For example, in the MI5 task can a group adjust their security code formula to ensure that the product is always a 4-digit number? There is more detail about what to look out for and how to extend each task on the teachers’ USB stick.

Communication of work
However well the mathematics element of the task is carried out, to be successful the children must be able to communicate their ideas effectively. You can look at whether they provided all of the information that the customer required; whether key cost calculations have been included; and how personable they were when meeting the customer (often played by the teacher in role, lots of fun!) to explain their work. Also, the way in which children choose to use technology as a means of communication is key.

Group dynamic
Often, the main determinant in the success of a group is the way in which the children work together. Do the children discuss the strengths and weaknesses of different ideas, with every child involved and nobody dominant? How are the different jobs within a task split up between team members? Are time deadlines met? Which tasks are prioritised? These are all difficult issues to face, and the team manager has to control the overall direction of the task well.

When providing feedback, it is often difficult to separate these three issues. For example, in the sandwich task there may be an error in the food order quantities. However, the underlying issue may be that this part of the task needed prioritising more, and that the person doing the calculations required more support from the team leader.

When the children are given high quality and detailed feedback, the quality of the work that they produce improves enormously from one task to the next. I have so often been blown away by the way in which the children improve from task to task. Their determination to be outstanding apprenticeship candidates has always been clear to see. As a result, the children will develop a great range of skills throughout their mathematics apprenticeship!

TMA: More Than a Number

The skills that you need to be a successful mathematician in school – at least, the skills against which children are assessed in SATS tests – are much narrower than the broad range of skills that you need when using mathematics in the real world. Maths ideas and concepts have evolved to help people find efficient solutions to a range of practical problems; yet mathematics is often seen as being a relatively closed, abstract subject.

The Mathematics Apprenticeship helps to bridge this gap. In it, the children get to ‘do’ mathematics in real contexts, helping them to see the practical application – and importance – of maths as a subject.

Also, like in the majority of real-life contexts, being able to ‘do the maths’ is only one part of what it takes to complete a tasks successfully. The children will have to organise the work between the team members; prioritise tasks so that they meet time deadlines; communicate their ideas in writing; and interact confidently with a range of different customers.

A natural need will arise to write letters, show calculations, use technology and give formal presentations. I remember, for example, one group of children discovering and using Google Sketch-Up so that they could produce a 3D model of the ‘Head of Secret Service office for the MI5 task. Also, it allows children with different skills to make a positive contribution – whether mathematical, organisational or in presenting ideas.

The aims of the project, therefore, are not limited to improving the children’s maths skills, but to giving them the broad range of qualities that they need in order to be successful in the future.

TMA: The Challenge

As a teacher, I am always looking for new ways to challenge and excite the pupils in my class. I want the children to use their imagination, manage extended tasks and engage in activities that are significant to them. What is more, I wanted them to have this experience in maths, a subject that can so often be reduced to abstract, closed-procedure operations. 

Don't misunderstand me - many abstract maths tasks, such as those found at nrich.maths.org, can be excellent for engaging children and developing their understanding of number. The Mathematics Apprenticeship, however, was written to provide a different type of mathematical challenge: one that is open-ended, based in real world contexts and which shows the modern-day relevance of maths.

Among the tasks on TMA, the children will have to write security codes for the MI5, review the TV schedule at the BBC, design an adventure park and set up an ordering system for a market stall. The children will need about 3 hours working on each task until they are ready for the 'hard sell' - presenting their ideas to the customer.

I have just so many memories that I could share from watching the children work on TMA. Like conversations with the school chef about the long-term drawbacks of cost-cutting when making chicken sandwiches; a debate about public liability insurance; how hectares relate to the metric system when drawing diagrams; and which parts of your date of birth you should use when designing a security code for the MI5!

When children are taking part in The Mathematics Apprenticeship, I'm never quite sure what will happen. I do know, though, that they will develop a real emotional attachment to their work, and that they will see how mathematics can be used in the real world, beyond the confines of their school experiences. And what is more, they (and I) will enjoy it!


Potential heavy users of mathematics should experience a rich, rigorous and challenging mathematics education, rather than being accelerated through the school curriculum.' - Advisory Committee on Mathematics Education, 2012