Pick it up and move it around!

This is a brief reflection on my experiences this week in thinking about the merits of physical Vs virtual manipulatives. Working in a school with a one to one laptop programme always invites You to think about what a computer can add to an experience. The number of virtual manipulatives available is staggering and some of them have really helped the evolution of teaching methods in mathematics. Having that programme really allows us to take advantage of them. With that in mind, please don't consider this blog post an 'anti technology' entry.

Task design happens in a number of different ways. It is probably fair to say that most of us start by thinking about what it is we want to teach. At that point we either go looking for existing resources or start thinking of new ways to do it. Being that it is the start of a new term, I am prone to the latter given the energy I have after a summer break. I would like to think that I am disciplined enough not to overlook some fabulous existing ones either. The trouble with coming up with new ideas is that they usually involve the creation of new materials! As an optimist, I will always go and look on the internet to see if what I am looking for is already out there, but this is really the wrong way round. The internet is like a garage sale - go in search of something in particular and you are likely to be disappointed, go with some money in your pocket and you will probably find something useful!

Having done some work on introducing the concept of tree diagrams, I decided that what I wanted was some online, interactive tree diagrams where the probabilities were listed but not put in the right place so that the task was to move them into the right places! This would just remove one area for possible error and in the early stages of an idea, I find it a very useful checking mechanism to know that if you have one left over that doesn't make sense then you may well have made a mistake. Anyway, I looked and I looked and I couldn't find it anywhere - I wondered about how long it would take me to program something like this using flash, but resolved that this was not the best use of my time at this time of year. I then decided that I really believed that in this case the 'physical manipulative' would be better. (I am not sure it saved me any time). A consequence of having technology at our disposal is that the benefits of the physical manipulative can be overlooked.

The result was the creation of this resource  Probability trees in which students work in groups with cut out bits of paper to solve problems where they have the answers and just need to put them in the right order. The physical manipulative really helped the group work aspect because more than one person can be involved in the arranging, and the absence of the computer screen allowed both more space and encouraged conversation and reasoning between the group members. In my search I had hoped that I would find something that was 'self-checking' so that students would get instant feedback on their efforts. This is a principle that can be very helpful, but is not without its faults. When no answer is instantly available, students need to reason with each other and reach some kind of consensus before settling on an answer. None of this is to say that the physical manipulative was 'better' in this context, but rather to say that there are lots of benefits to experiences based around physical manipulatives. Below are some pictures of the bits of paper!

On this note, the following are just a few examples of similar activities where physical manipulatives are used.

 Meeting Functions

A classification exercise with different functions, domains and ranges.

 OXO

Using Multilink cubes to relate algebraic sequences to physical situations.

 Quadratic Links

Linking the different representations of a quadratic function together.

Using multilink cubes to look at linear sequences!