Building on my recent post about a taxonomy for deep learning in maths, I have been trying to think a bit deeper myself about what each type of ‘deep learning link’ might look like. In particular, I have been researching and putting a lot of thought into what effective ‘visual models’ look like for the ‘key nodes’ I have previously identified as the most important foundation maths knowledge for students to master before starting their GCSE maths course. These are principally number topics.
Last year I became aware of the Singapore Maths Bar Modelling approached have recently found the time to research it further. I bought some Singapore textbooks and read about the work of Dr Yeap Ban Har. This video, featuring Dr Ban Har shows an exemplification of the approach for a typical functional maths problem:
In short, I really like the approach and am convinced it could enhance my own practice significantly by giving students powerful, but simple visual models they can draw upon and use to solve problems. I have been experimenting with some of the models in my lessons this year and have seen the positive effect they have had on student understanding of topics. What these visual models give you is an entry point when teaching a topic that all students seem able to grasp. It presents the concept in its rawest, simplest form without the distraction of lots of words or mathematical notation. The diagrams don’t replace the eventual algorithmic methods, but they provide an entry point where students seem to understand what it is they are trying to solve; something that often gets clouded when algorithms are presented to early on.
In primary education in Singapore, maths teachers follow a Concrete-Pictorial-Abstract (CPA) sequence when teaching maths topics. They start with real world, tangible representations, move onto showing the problem using a pictorial diagram before then introducing the abstract algorithms and notation.
The particular power of the bar modelling pictorial approach is that it is applicable across a large number of topics. Once students have the basics of the approach secured, they can easily extend it across many topics.
I have spent some time putting together some pictures showing how the approach can be used for different topics. They are not teaching slides, but rather ‘notes for teachers’ to demonstrate how this single model can be adapted to be the diagrammatic entry point for many topics.
To start with students are given blank (bar) rectangles (on plain paper) and then get used to dividing the bars into halves, thirds, quarters etc:
They can then calculate a fraction of a quantity by first drawing the fraction in the bar, showing the length of the bar to be the quantity and then calculating the length of the shaded part:
Again, you’ll end up at ‘divide by denominator, multiply by numerator’ eventually, but this does show the concept of what’s going on very nicely and is a good route into showing where the algorithm comes from.
Next up, equivalent fractions:
Then simplifying fractions:
A ‘fraction wall’ (as many teachers use traditionally in England) can be used for ordering fractions:
Adding fractions with the same denominators:
Adding fractions with different denominators:
Dividing by fractions (works ok so long as you have integer answers). That’s enough to get across what is going on… Then you can lead into the method…
Converting mixed numbers and improper fractions:
Next up, understanding place value in decimal numbers. This approach lets you deal with lots of misconceptions like 0.62 not being larger than 0.7 etc. Importantly, this is now taking the bar model and putting a decimal number line onto it. This forms the basis for many topic models that follow:
Now they have an understanding of how the decimal number line works, and they can draw bar models for fractions, they can combine the two on one diagram to convert between fractions and decimals:
Next they can learn that percentages are hundredths, and in doing so can put a percentage number line under the bar model:
They can now combine the fractional bar model with both the decimal and percentage number lines directly underneath it to convert between fractions, decimals and percentages. They draw the fraction bar first, then put on the decimal increments (by dividing 1 by the denominator) and finally put on the percentages (by dividing 100 by the denominator):
Similarly to fraction of an amount earlier, you can use this approach to introduce percentage of an amount. Starting with showing how to find 10% of a number (and thus why you divide by 10), it serves as a nice way into multiplicative reasoning approaches to ‘build off the 10%’ to find other percentages:
Other percentage topics then follow, such as percentage increase and decrease:
By putting both the decimal and percentage number lines on the bar model for this, you can clearly show where percentage multipliers come from, including the ones less than unity for percentage decreases:
You can introduce calculating a percentage change using the bar model approach:
It’s a particularly nice diagrammatic way in to teaching reverse percentages:
Once the above techniques have been mastered, it can be used for showing how compound interest works as follows. Particularly nice is if you turn the bars so they are vertical, it shows the exponentially increasing relationship:
It works for ratio too:
I’ve used fraction walls for years, but it is the inclusion of decimal and percentage number lines built onto the fraction bars that is new for me. It opens up diagrammatic routes into so many topics and in such a coherent, simplistic way. The universality of the approach is what particularly impresses me; from humble beginnings of shading rectangles, the same model leads all the way up to reverse percentages and compound interest. If done in the correct order, there is a beautiful journey of progression all using one simple model. Each topic builds off the last logically as the model is manipulated in different ways.
The visual models won’t ever replace the algorithmic approaches, but instead will I hope provide my students with a better understanding of ‘what is going on’ when we are at the early stages of learning a topic. I hope their conceptual understanding is improved and this in turn enhances their procedural understanding through it giving it a purpose and something visual to hook onto. If they can ‘see the bar picture’ for problems with simple numbers it is my hope that the algorithmic approaches that follow that will enable them to solve problems with more challenging numbers with stick better. If they can represent a problem by drawing one of these models, they may have a better understanding of what the problem wants them to do.
I plan to develop resources to support teacher explanations and student activities in these topics in the coming months with the support of keen beans in my department and will share them with you when they have been tried and tested. There are no silver bullets in education, but this does represent a decent step forward for my teaching. Much to learn still!