This is the third in a three part series around developing Mathematical Fluency (which has been highlighted as a key aim in the National Curriculum) in teaching and learning. The previous posts have dealt with using technology to enhance pedagogy and allow children to take on the mantle of the expert. The other discussed developing children’s conceptual understanding which would allow children to apply those principles with greater authority.
3. Problem Solving
One of the most potent forms of developing mathematical fluency is giving children the opportunity to apply basic skills in as many problem solving activities as possible. As a profession, I think we have a tendency to go through the basic concept first and then (and only then) introduce the idea of applying these skills into problems that relate. However, recently I planned out a few sessions focus on teaching volume to my Year 6 (a brand new concept for the vast majority of my class). The results have been really pleasing – not because they can just apply a formula I’ve given them, but because they really have grasped the concept.
The Golden Cube
With a variety of Maths resources in front of them, I asked my class to close their eyes and picture the following problem as I verbally told it to them:
The Golden Cube Problem
“You create a larger cube from a number of smaller cubes. You decide to paint the outside of this larger cube all gold, every face. You go to put this golden cube on the windowsill to dry but you trip and drop the cube. When it hits the floor, it breaks into all the original smaller cubes. How many cubes have 3 faces painted gold, how many cubes have 2 faces painted gold, how many cubes have 1 face painted gold and how many cubes have no faces painted gold?”
I gave them no more guidance on how to solve the problem – and of they went. Little did the children know that they were not only developing their understanding of 3D Shape, but they were beginning to become aware of volume. As can be seen below, different strategies were used and we discussed that learning moment too:
So, with the problem solved and some challenges completed along with that, the next session I asked the children how many base 10 cubes (because that was the main resource that was used) made up the cube. 64 cubes. What were the dimensions of the cube? 4x4x4 – what do you notice? The number of cubes used can be calculated by the dimensions of the cube. Visualiser up, we tested that hypothesis with a couple more cuboids. It worked. The children then went into pairs and worked on created cuboids which the partner had to calculate how many cubes were used to create the cuboid by just being given the dimensions by the pair creating the cuboid. I would include photos but they contain children focusing intently on their learning!
Finally, after discovering that we could create different cuboids with the same volume, the children then moved on to calculating the volume (and also a missing dimension when given the volume) on a cuboid given to them not to scale. The children, because of the problem solving activities that had gone before, were so fluent in applying the formula and working with it to suit their needs. These questions also involved converting measurements in the question and subtracting chunks of the cuboids that had been taken out. However, because of their grounding in the concept of volume, their fluency in applying the knowledge to a formula, these did not raise a problem.
Of course, after this three part series, I’m sure others to add ideas to getting children to teach, developing conceptual understanding and using problem solving. Please, if you have done anything in the classroom that has developed mathematical fluency really well, please don’t keep it to yourself, share it!