Written by Chris McGrane Tuesday, 02 April 2019
Today we launch a new feature to be added to Complete Mathematics: our Dynamic Worksheet Generation tool. This initial launch is focused on the creation of worksheets for the factorisation of quadratics. We have created a tool which captures some of the ‘teacher logic’ inherent in worksheet design and have incorporated intelligent variation and sequencing in the questions generated.
The worksheet generator can produce worksheets with questions which come in a sensible teaching order for this topic. Intelligent variation sequences, are periodically included at appropriate places in the worksheets. These sequences of questions help pupils to attend to the key variants and invariants. One such example is shown:
x2 + 14x + 24 = (x + 12)(x + 2)
x2 + 11x + 24 = (x + 8)(x + 3)
x2 – 11x + 24 = (x – 8)(x – 3)
x2 – 14x + 24 = (x – 2)(x – 12)
x2 + 10x – 24 = (x – 2)(x + 12)
x2 – 10x – 24 = (x + 2)(x – 12)
x2 + 5x – 24 = (x – 3)(x + 8)
x2 – 5x – 24 = (x + 3)(x – 8)
Teachers have the freedom to toggle on and off which variations of quadratics to include on the worksheet. This is particularly useful for the implementation of the mastery cycle. Based upon formative assessment a teacher can identify where individual pupils need to focus their attention. The teacher can then create a worksheet specifically for this pupil in moments for them to remediate and practise on these sub-skills.
At La Salle we talk about a learning episode as opposed to lessons. Learning and teaching can’t be neatly contained in sixty-minute slots, instead a learning episode spans a series of lessons and may include a number of related learning objectives. We consider a learning episode in the context of the following phases:
During the Teach phase, the idea is entirely novel to pupils. It is just beyond their current knowledge and understanding. The teacher will instruct the pupils, tell them key facts, pass on knowledge, show and describe, use metaphor and model, all in order to bring about connections in the pupil’s current schema so that they can ‘meaning make’. This phase is often described as explicit teaching. It is a crucial phase – after all, the teacher knows things and the pupil does not; so tell them!
The end of the 'Teach' phase does not result in learning. It is merely the first step. At this stage the new knowledge is ‘inflexible’, and it is our job as teachers to bring meaning and understanding to the knowledge so that it becomes ‘flexible’ (more on inflexible and flexible knowledge later).
We now ask pupils to Do. At this stage, they do not yet know or understand the new idea, they are replicating what the teacher has told or shown them. The 'Do' phase has two important purposes. Firstly, the teacher is able to observe whether or not the pupils have made meaning of the model, example, metaphor or information they have been given or shown. The teacher can see and act; are the pupils able to replicate what I have demonstrated? If not, the teacher can change their model, example or explanation, perhaps making stronger and more explicit connections to previous knowledge and understanding. The second reason for the 'Do' phase is to give pupils a sense that the idea or task is surmountable – that they, quite literally, can do what they are being asked. Well structured 'Teach' and 'Do' builds pupils’ confidence and shows them there is nothing to be afraid of, the new idea is within their reach.
The Complete Mathematics platform supports teachers in delivering the 'Teach' and 'Do' phases by including pedagogical notes, typical misconceptions, key vocab, and example questions. Additionally, on our CPD days we explore evidence-based techniques to enhance the effectiveness of these phases.
Once both teacher and pupil are clear that the pupil is able to ‘Do’ – that is to say, they can perform – the teacher now segues the pupil to the Practise phase
During 'Practise', we wish to move beyond simply performing. We want the pupil to gain a confidence in working with the new idea, to see its underlying relationships and to assimilate the new idea into their schema of knowledge. In order to achieve these more meaningful goals, the pupil needs to be able to attend to a higher level. In other words, as described earlier, the pupil needs to have achieved fluency at the performing level first, so that they may attend to connections, relationships and a deeper conceptual appreciation.
So, we shall define the point at which the pupil moves from 'Do' to 'Practise' as the point at which they achieve fluency. This is where the dynamically generated worsksheets can be particularly useful. They give pupils ample opportunity to build procedural fluency but also develop conceptual understanding through the built-in intelligent variation as described above.
The final phase, Behave, is the most important phase. This is the phase that brings about understanding. At this stage, teachers create opportunities for pupils to behave mathematically. Complete Mathematics has tasks which support this phase, however, the pedagogical actions are perhaps the most important factor in this phase. You can learn more about this on our CPD days: Deeper Understanding in Mathematics, Embedding Problem Solving in the Mathematics Classroom, Designing Mathematical Tasks and Curriculum Design.
This worksheet generator is our first experiment in this area and we would love to hear any feedback you have either by commenting below or talking to us on twitter. There are a multitude of other ideas we want to explore. For instance, allowing teachers to implement their own variation logic in these worksheets and allowing pupils to answer the questions online. Also, in addition to these further experiments on dynamic worksheets, we are working on dynamic versions of the example questions found on every objective in the Complete Mathematics platform.
Complete Mathematics members will be able to see all these developments and more appearing on the platform in the coming weeks and months. In fact members can use this worksheet generator right now in the 'Factorising Quadratic Expressions' objective in unit 10.5. If you don't want to miss out on these developments, or you want to take advantage of free attendence to the CPD courses mentioned above, you can find out more about becoming a Complete Mathematics member here.