Published September 2010,February 2011.
Often I am asked how we decide what goes on the NRICH website. In the same way that authors find it difficult to answer the question, "Where do you get your ideas", it's not a straightforward question for me to answer. People sometimes send us ideas for problems, and old books are a great source of rich starting points. Many ideas that have been around for a long time are well worth dusting off and revisiting with a fresh twist.
For me, what tends to happen is that I'll play with a piece of mathematics either on my own or with another member of the NRICH team, keeping a note as I go of what questions occur to me and what strikes me as interesting. I work closely with Charlie Gilderdale on the Stage 3 and 4 parts of the site, so once we have an idea that we think can be developed into a problem or game for the site, we think about where it fits into the curriculum and what age of learner it might suit.
Once the problem has been written up with a student audience in mind, we think about how we would use the task in the classroom, and reflect on similar ideas that we have used when teaching that topic in the past. Whenever we have the opportunity, we try out a new idea in a pupil workshop so that we can draw on direct experience of using the task when we write the Teachers' Notes.
This month's site has been developed with the new academic year in mind, so apologies if it's not a new school year where you are, though we hope there will be something here for everyone. We are aware that there are challenges involved in using rich tasks for the first time, so we wanted to publish a collection of engaging resources with no theme other than offering ideas suited to the start of the year. One example is the stage 3 problem Diminishing Returns. The problem begins with an image showing seven nested squares, and we have suggested in the Teachers' Notes that learners draw the image and calculate the proportions of the total coloured in each colour. The task goes on to explore what would happen if the pattern continued into the centre of the page.
I think this makes an excellent task for the start of a new term for several reasons. Firstly, it is an example of a Low Threshold, High Ceiling task - everyone can make a start on it, but there are opportunities for high levels of mathematical thinking. The first part of the problem offers practice on working with fractions. This provides an excellent assessment opportunity for teachers with new classes, to watch how learners tackle the problem and see the what prior knowledge they bring to the task. We suggest in the Notes that the class could discuss different methods they used for working out the fractions - establishing a classroom culture where methods are shared is a great thing to do early on with a new class. Then in discussion on what might happen if the pattern could continue forever into the centre of the page, learners are asked to come up with convincing explanations of their thinking. This is particularly useful in establishing the maths lesson as a place where ideas are explored and proved, and overcoming some learners' preconceived idea of maths as a subject where a method must be learned in order to progress to a single correct answer.
Another problem from our September Collection is the Stage 4 resource Curvy Areas. As with Diminishing Returns, all learners should be able to make a start on this problem. As far as curriculum content is concerned, there are a variety of topics met in the task. The problem begins with constructing a diagram made from arcs, which could be used to give much-needed practice in using a pair of compasses. It goes on to look at calculating areas of the regions of the shape, a possible introduction to finding the area of sectors of a circle, and then leads to some sequences of areas which can be described using algebra. It can seem hard in a busy curriculum to find time for maths enrichment, but a task such as Curvy Areas covers diverse topics from the curriculum in an engaging context, and offers the chance for learners to be surprised by the results they find. Rather than spending separate lessons on construction, area of circles, and sequences, these skills can be taught as learners meet them through the investigation.
Teachers using NRICH for the first time can be daunted by the sheer volume of resources on the site, which is one reason why we have our Curriculum Mapping Documents. Each problem we've identified as having strong curriculum links appears on the document at the point most appropriate to its curriculum content. However, that doesn't mean it's the only place that task can be used - many of our problems could be encountered for the first time in Year 7 and then revisited in Year 10 or 11 to be tackled at a higher mathematical level.
Finally, I'd like to share some of our thinking about the problem Shady Symmetry. This problem originally appeared on the site a few years ago with the title 'Isometrically', and challenged learners to work systematically to find all the examples of a symmetrical pattern on a triangular grid by shading in four triangles. In rewriting the problem, we've chosen to use a smaller grid, and also to offer the choice between triangular and square grids.
Convincing someone else that your solution is complete is an important skill to learn in mathematics. The problem is written with the hope of encouraging discussion of the different ways of working systematically on this problem. Of course another important consideration is that these symmetrical patterns are very pleasing to the eye and can be used to fill empty display boards with students' work at the start of term!
In the Teachers' Notes, we have suggested that learners decide for themselves a line of enquiry to explore. This is a key feature in many recent NRICH tasks, as we want to model the way the maths community works, and to encourage learners in classrooms to consider themselves research mathematicians working to find something out. However, for classes who are working in this way for the first time, it can be very daunting to be asked to come up with conjectures and lines of enquiry, so we have put some suggestions in the problem for teachers to use to prompt their classes.
Our hope is that the problems we chose to publish this month will whet both teachers' and learners' appetites for learning through working on rich mathematical tasks.