# Mathematics Education Is Not an Enigma - Part 2

By Williams, Doug | Mathematics Teaching, November 2012 | Go to article overview

# Mathematics Education Is Not an Enigma - Part 2

Williams, Doug, Mathematics Teaching

Doug Williams uses practical experience working with teachers and learners to prove his point Continued from MT230

In Ian Pegram's class, who gives a 'tinkers' about how many people you need in a room to have a good chance of having two with the same birthday? I suggest pretty much nobody. You will never need to know that fact. But, on the other hand, who gives a 'tinkers' about the Birth Month Paradox investigation? Well, for a start, and apparently unexpectedly, all of lan's Year 10 and 11 classes!

First give me an interesting problem also works in primary school. I will let Jacki Healey, Margate Primary School, tell this story of Poly Plug Values as it unfolded in her class. It is recorded in the Calculating Changes section of Mathematics Centre.

I was working with Place Value with my Grade 2 class and I wanted to do something to extend their understanding. I decided to use Poly Plugs. The first time I introduced the Poly Plugs I told them it would be interesting to use them to represent numbers. How could they do this?

Initially I used low numbers, eg: 14, 17, etc. I discovered that given low numbers, most children would assign each Poly Plug a value of 1, ie: 1 to 1 correspondence.

The challenge lay in giving them larger numbers to represent, eg: 96, 43, 85, 74. Large numbers and odd numbers were harder, and required more ingenuity to represent. At the end of the first one hour session, some children had considered giving, for example, a red plug the value of 1, a yellow 2, a blue 5 or 10.

Over the next 3 weeks I repeated the exercise, giving each group the opportunity to swap partners and share ideas. It was tremendously exciting to watch as the children's confidence and ingenuity increased each week.

They were also asked to record their ideas, so that they could share with the whole group. By the end of the month the children's ability to 'skip count' in 2s, 3s, 4s, 5s, 6s and 10s was amazing. Some children even assigned value to the empty places in the plug board. They also thoroughly enjoyed the whole process.

Jacki is using a teaching method here that has been devised by teachers in the Calculating Changes network (HREF4). It is called Threading. The same rich task with a familiar structure is used for small amounts of time, often, over several weeks. The intellectual energy of the learner is invested in the mathematics of the activity, rather than the structure of the activity, and developing learning can be easily tracked. Jacki's story continues:

One day I suggested that they had worked so hard they deserved a break. They could play on the brand new playground equipment, then come back to finish their Poly Plug Values. They unanimously refused and chose to complete the task first!

What is of primary importance here? The mathematics content, or the classroom environment Jacki created, and the way the children felt about themselves as learners?

All it takes to develop a classroom with a Working Mathematically core curriculum is a teacher, you, who decides to adopt a change of attitude; and that adoption is possible, independent of any externally impressed curriculum force.

Sphinx

This task is the most extensively documented of any (HREF5, 6, 7). However, some may not be familiar with it, so a quick background. In the original form, students were given four wooden pieces, shaped as shown, and challenged to arrange them into a larger 'Sphinx' shape.

As it stands the task is a spatial challenge of moderate difficulty. At a teaching day in 1996, which initiated the project to trial the use of tasks in an urban Indigenous situation, two Year 5 Aboriginal and Torres Strait Islander students, Tyler and Michael from Norn's Road School, accepted the challenge. They solved it quickly, but then were challenged to investigate further into the iceberg.

Their solution showed that the base, height and side lengths of the new Sphinx were twice as long as the corresponding lengths in a single Sphinx shape. …

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