__first__exploring a concept through inquiry, then being introduced to the theory, rather than the other way around. The study advocates "flipping flipped learning", so that instead of showing students a video or having them read text prior to a lesson, and then exploring that concept, you have them explore first. In this way the students are able to make their own connections and this is more likely to lead to actual understanding, rather than just memorising theories and processes taught.

This is what I started to do when putting together our current Year 8 unit on Probability. Instead of starting the lesson by introducing theory and going straight into how to calculate probabilities (with thousands of practice questions centred around coins, dice and cards) we played a game called Three Dice (ok, admittedly using dice), an activity from the Nuffield Foundation. Each student had to fill a 3 by 3 card with 9 different numbers of their choosing. In pairs, they then rolled 3 dice. If the sum of the three dice was on their card, they could cross it off. The winner was the first person to cross all 9 numbers of their card first.

Students had to write a reason explaining why they chose their 9 numbers before I gave them the dice to play. I went around checking their explanations, not giving any feedback at this point, as I just wanted to get a sense of their initial thinking. Some students said they chose numbers like 10,11,12,13 as they were in the middle of the possible sums (from 3 to 18 - although I did not specifically tell them they could only choose from 3 to 18). Others just said they chose randomly, which I accepted as an explanation, as these students obviously did not have an idea initially that some numbers were more likely than others.

Once they played the game with a partner twice, they could then improve their board and change their numbers, but again had to give an explanation as to why they made the changes they did.

Without any direct instruction some concepts of probability began to emerge, and I could question and prompt individual groups depending on how far they were getting. I began to see some terminology of probability being used, such as likely and impossible, which was great as I hadn't introduced any of that myself. Some students started to keep a tally of how many times each number came up to see which numbers were most likely. They then started to explore different combinations for each sum, and wondering if 1+1+2 was the same as 1+2+1?

Those that got furthest started to see patterns with the combinations. For example, the number 4 can only be made with a combination of 1, 1 and 2. There are 3 ways of arranging these two numbers:

1+1+2

1+2+1

2+1+1

However, a number like 6 can be made with 1, 2 and 3. And because there are three different numbers, there are 6 ways of arranging them:

1+2+3

1+3+2

2+1+3

2+3+1

3+2+1

3+1+2

Once they had discovered this, they could work out the total amount of combinations for each sum, without listing them all. We could then go about calculating the probability of getting each sum, although they needed to have worked out the total number of combinations in order for this to happen.

Now, not all students got to this point, and not all got to this point on their own. However, this was a mixed ability class, and we covered far more topics within probability that I would have normally if I was just introducing the theory and getting them to practice. They had to think about combinations and look for patterns, and also work systematically to ensure that they hadn't missed any combinations.

Since starting to write this post, I have been reading about Ramsey Musallam's model 'Explore-Flip-Apply'. In this model students are given the chance to explore concepts and make their own connections before watching video tutorials that covers content. The students then apply their learning following the video instruction in lessons. The most important thing about the 'Flipped' phase is that they only cover content that is low-level on Bloom's Taxonomy: recalling facts, definitions, straight-forward examples, etc. The applying, evaluating, analysing and creating are done in class, during the 'Explore' or 'Apply' phases of flipped learning.

Last week I met up with the head of maths at our school's other campus, who has implemented a similar model (without the flip) in Year 7 this year. At the start of each unit, the students spend a week 'Exploring' before they are taught any formal mathematics, and they are finding it hugely successful. In a few weeks I will visit their campus to observe what they are doing and talk to some of the teachers who have been involved in the planning of this. We are planning to work more closely together between the two campus, trying to combine their model of exploration before theory with what we have been doing with Flipped Learning.

For the last four weeks of the school year the students move up to the next year level, in what is called our Early Commencement Programme, or ECP. My plan is to have some time for the maths department to collaborate and plan units for the Year 8s and 9s using the Explore-Flip-Apply model. We then have a few days before the start of summer break after the students have finished where we can review what we have done and refine the model for the start of the 2014 academic year.