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# Mathematical Modelling

Despite having complete solutions for examples on PixiMaths PowerPoints, I never actually use them. Students often ask me why, and it occurred to me the other day that it is maybe something I should clarify here too.

Mathematical modelling is vital to secure students’ knowledge and understanding of mathematical processes. Emphasis on certain steps and justifications of how or why things work or do not work can be the difference between learning by rote and actually understanding maths. Eggen and Kauchak (2001) defined modelling as “an instructional strategy in which the teacher demonstrates a new concept or approach to learning and students learn by observing”. I take it a step further than this and encourage students to participate rather than just observe, through questioning and use of mini whiteboards. When used effectively, Bandura (1986) commented that “modelling engages students and encourages learning”.

It is important for students to see the thought process we, as teachers, go through in answering a question. I will either use the whiteboard at the side of my SmartBoard, import PowerPoint slides into SmartNote or use OneNote to ensure I can talk through and write down the mathematical processes one step at a time. I’ll often annotate answers as I go with what I’ve done in each step and why, and use different colours to emphasise different stages of working out.

Having worked in a school with no whiteboard, and only a TV screen to project onto previously, I started animating slides carefully as this was the closest I could get to modelling required processes in lessons. I know some teachers find them useful still, especially non-specialists, and when students are using PixiMaths it makes each slide less daunting. But generally, it is far better to model solutions to students rather than just clicking the mouse and asking students to copy.

It’s also important for students to see their teachers make mistakes. For us, it’s innate to check that our answers are reasonable and realistic, and most of us will check as we go. This is something that students often bypass, so it is vital that we model our thought processes in this too. A colleague was explaining recently how he was trying to demonstrate that two linear graphs were parallel, but had plotted the x and y coordinate the wrong way around. He spotted this straight away because he could see that the gradients were not parallel. Instead of being embarrassed or glossing over this mistake, he used this as a great learning opportunity to explain how he knew there was one.

Another reason for not just clicking through the PowerPoints is that I often just use one method in them. I find when teaching, however, that multiple or different methods are often needed. This is another way of adapting teaching to suit the needs of your students – not just using something as it is.

Some things to be careful with when modelling are to take care to not spoon-feed students too much. Include them in your thought processes with rigorous questioning, that is open-ended where possible. Inserting mistakes in your working out will keep students on their toes too, and prevent them from just accepting what you say. They need to keep thinking, and this can be a perfect opportunity to discuss common mistakes and misconceptions.

Several of my revision resources do not have solutions included, the new starter packs in particular. I feel there is no point in using this resource if you do not go though full solutions with students, clearly modelling methods and thought processes as you go.