Picking apart and reworking my lessons from last year, I noticed that I almost exclusively model worked examples by handwriting “live” rather than pre-typing. I don’t recall making a conscious decision to do this, but following a discussion with my husband this weekend about print vs digital media, I thought I’d think more critically whether either style has any considerable advantage.
There are some interesting studies about reading information via print or digital media; this paper from the Research in Learning Technology journal looks specifically at educational materials and this article from Phys.org summarises some of the preferences people have for different media – generally, we prefer printed media for longer, more involved reads and digital media for shorter reads. I also found this study examining handwriting vs typing inputs for students when solving equations; the authors conclude that the only real difference in this scenario is the speed at which students completed the task, with typesetting algebra taking twice as long as handwriting solutions. However, I’ve not been able to find anything on the modelling of worked examples in Maths, so this blog post is written with the caveat that this is all personal theorising and is not research-based
Advantages of written examples
1) Less time taken to prepare lesson materials
Time-gains while planning are significantly in favour of handwritten examples, particularly as questions become more complex towards the top end of GCSE and into A Level, and when complex algebra is involved. However, as handwriting recognition tools such as Microsoft Office’s Ink Equation Tool are becoming increasingly effective, it might be interesting to see whether this remains true in five or ten years’ time.
2) Teacher’s thought processes are more visible to students
One advantage of working “live” is that the teacher can model thought processes more easily, verbalising each step in the working. This process is also instantly adaptable to the class – the teacher can speed through stages that students are confident with, or elaborate in more detail if a step requires further explanation.
3) Teacher can model layout and presentation of mathematical working
Again of particular importance for top-end GCSE and A Level, as problems and solutions become more complex, students increasingly need guidance with how to present their work in a coherent manner. Watching a teacher model this directly may be easier to translate to their own work than looking at a typed example.
4) Teacher can include student input more easily
When I complete worked examples with a class, I tend to ask for input from students at each step and see what they can work out logically or infer from prior knowledge. The rigidity of a pre-typed example makes it more difficult to include student responses, as many of them may go slightly off-script from the pre-planned work.
Advantages of typed examples
1) More critical selection of illustrative examples
If I’m going to typeset solutions, I make damn certain the example I’m using illustrates a key point or highlights a common misconception. The need to work efficiently means I then select an example which serves many purposes and has decent mathematical weight. It also means that the numbers work properly – while I usually check all the examples I prepare in my lesson materials, occasionally the odd blooper slips in – the star culprit is quadratics that don’t have real roots. While this can form the basis of an interesting discussion, it can also wrong-foot a class and damage confidence in the teacher.
2) Final solution can be presented with greater clarity/use of colour
While there are losses in the step-by-step process, the end product is often more coherent when digitised. There are times when I’ve stepped back from a worked example and actually looked at what’s on the board – and because I’ve documented my thinking process or expanded on certain points, the solution can end up looking mathematically messy or sometimes confusing to students. I also have a tendency to use arrows and random bits of annotation which get very scruffy when handwritten, and my handwriting itself has deteriorated significantly over the last ten years as I get less and less daily practise.
3) Diagrams and graphs can be incorporated more easily
Unless I am specifically modelling graph-plotting with Year 7, I no longer draw graphs by hand. It’s tedious and time-consuming, and incredibly difficult to do when working right up against a whiteboard! Using a tool like Desmos is now pretty commonplace, but if I’m doing this I will often prepare graphs in advance rather than plot “live”, as I can make sure that the graph is scaled appropriately. Diagrams or pictorial representations always look better when done digitally – for example, your standard “counters in a bag” when working with tree diagrams.
This year I’m probably going to carry on handwriting most of my examples – I enjoy the process of working through problems collaboratively with a class, and I think typeset examples lose some of this. However, I am going to reconsider selecting some gold-standard examples for each topic to typeset; at GCSE, our students have small exercise books that are used to take important notes and write down illustrative examples, and these would seem the ideal candidates for a bit of deeper thought and clearer presentation.
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