For the first few years of my teaching career, I viewed the maths curriculum as a check list of separate things for students to learn. “This is how you simplify fractions. This is how you calculate angles in a pie chart. This is how you convert currency.” It was probably because I was mostly teaching these things at a surface, procedural level. It was only later on that I started to appreciate quite how interconnected the maths curriculum is. More experienced colleagues pointed this out to me, but it took a couple of times of working through the curriculum myself to see the connections clearly. The more times I re-taught the Y7-Y11 schemes of work, the more it apparent it came that students were doing similar things to work stuff out.
This seems especially so for topics relating to equivalence and proportionality. For example, take this question on finding a missing number in a pair of equivalent fractions:
Two ways to work out the missing number could be:
- to multiply the numbers in the first fraction by 3 to get the numbers in the second fraction;
- to multiply the numerators by 5 to get the denominators.
Now let’s compare this to four other questions:
Proportion and Price:
Speed, Distance, Time:
It almost seems like these last four questions are in fact the first question but dressed in different clothes. They all deal with issues of either equivalence or proportionality and can all be solved by following the same steps. These similarities have been especially emphasised by using the same numbers.
While these five skills tend to be spaced out in time across the curriculum or schemes of work, students may find it useful to see the questions side-by-side at some point too. Then by asking them to discuss what is similar and different between them, it could lead them to decipher some of the bigger, overarching ideas about proportionality and equivalence. Once understood, these ideas might then be applied to other topics:
The lesson that I took from this was that learning mathematics is not about remembering lots and lots of different methods for lots and lots of different topics. Instead, when teachers and students work on understanding bigger principles, and practise identifying situations where apply, then they can learn how to answer many questions using fewer methods. That’s not to suggest that students shouldn’t use other methods or representations for these topics, such as the formula for speed, distance and time. But seeing ideas presented in a similar way can help students make sense of new concepts by relating them to things that are familiar with.
It is also worth bearing in mind that students are unlikely to understand big principles straight from the off; they’ll need to be proficient in enough of the smaller things before they can make any sense of how they’re connected. To build a model, you need to have the pieces ready before you start firing your glue gun!
In my personal experience, students have found it reassuring to know that mathematics isn’t a never-ending list of separate methods to memorise – especially as they approach their exams. But more than anything, I’ve found that looking for opportunities to link different mathematical ideas together makes planning lessons that little bit more interesting!
If you would like to share any of the points or images from this blog with other teachers (e.g. in a staff meeting), feel free to use this Power Point to help you: ‘One Method: Many Topics’
If you found this post interesting, then I would also recommend the following:
- For an illustration of how interconnected mathematics is, go to this post by William Emeny: You’ve never seen the GCSE Maths curriculum like this before…
- For an introduction to SOLO Taxonomy, see this brief summary: About SOLO Taxonomy
- For a far more elegant and sophisticated article about proportional reasoning and using a common method (a different method to the one in this blog), see this article by Andrew Steed: Ratio and Proportion – You may need to be a member of ATM to access this one, sorry.
Previous blog posts:
This series ‘Making Connections’:
‘What I Learned From Shanghai’ Series:
- Not As Complicated As It Looks
- A Shanghai Lesson on Adding Negative Numbers
- Do Sweat The Small Stuff
‘Planning Thought’ Series: