Providing young people with mathematical knowledge, not just procedures
23 May 2018
Author: Vicci Williamson, Director of Research & Development at Hungerhill School
Regardless of which key stage you teach, you will have or will be experiencing significant changes in formal assessments. As a secondary teacher, gone are the days of the first question on the foundation paper requiring the students to write 386 in words. Rarely will students simply be told to multiply two numbers. As a result we need to ensure that we have a class of thinkers and knowers, rather than simply doers and followers. This links closely with the EEFs second recommendation based on manipulatives and representations, and the related section in my blog which looked at the importance of the concrete, pictorial and abstract. It is so important to fully understand the mathematics behind a topic in order to apply it to more challenging questions and it also helps the students to plug the gaps if they have ‘slept since then’.
It is nice to have a report where the different aspects complement each other. This second recommendation is also linked to the third one in the report, involving problem solving strategies. Both recommendations suggest encouraging students to compare and use different strategies and this was reflected in my last blog where I mentioned encouraging the class to find different methods for the same question. One way I implement this is to have different coloured board pens at the ready. When a student has completed a specific question, they can write their working out on the board in one colour. Then, if somebody else used a different method, they write in on the board in a different colour. If targeted at the right type of question, it can produce some wonderful variation in approaches. Students can reflect on the most efficient approach, have valuable discussions, which stretch their thinking, determine when certain methods wouldn’t be appropriate or even spot patterns within the working out.
A few years ago I watched a Shanghai teacher deliver a lesson to a year 7 class on division. The biggest surprise was the limited amount of questions they completed. However, as my understanding of pedagogy has adapted, this is now the norm. Instead of the teacher rushing through several examples using the same method, I believe that it is often better practice to explore one example in various ways. One question which the Shangai teacher looked at was along the lines of 408 ÷ 12. Whilst there was use of short and long division in the lesson, the students were developing their understanding of division through another method. They were encouraged to think about these questions: ‘If I know that 10 x 12 = 120, then 30 x 12 = 360. I am missing 48. 48 is 4 x 12. Therefore 30 x 12 + 4 x 12 = 34 x 12 = 408, and so 408 ÷ 12 = 34’. There also needs to be an emphasis on the language we use. Whilst writing 30 x 12, we can say ’30 lots of 12’, which clarifies the process to the students. The teacher then went on to take suggestions of other ways of reaching 408 using this strategy.
In fairness, it’s no surprise that both of these key points of the fourth recommendation link so closely to other points from the guidance report. After all, the report is about improving mathematics. Clearly one of the strongest ways to do this is to enable students to develop a rich network of knowledge. My last point isn’t something I had explicitly linked to developing students’ knowledge until recently – the use of calculators. I’ve come to realise that I put too much emphasis on mental and written methods for questions, and when I produce questions, I use numbers which means I can encourage them not to use a calculator. Although this works wonders for their non-calculator skills, I’m also putting the students at a disadvantage. Not only am I taking away a safety blanket for those who struggle in confidence in the subject, I’m also taking away a tool which can help them to make connections themselves. I remember (just!!) being at school and using my calculator, and realising some of the patterns which were taking place in my calculation and discovering key concepts for myself.
Therefore, whilst we develop the students’ mathematical knowledge through a plethora of different strategies which delve in depth into questions, let’s not forget that there is also an advantage to using a calculator to support their learning.
If you have anything you would like to share on this, or any other aspect of the report, then please do get in touch. My next blog, looking at independence and motivation, will be out soon.Posted on 23 May 2018
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