He was particularly indignant against the almost universal use of the word idea in the sense of notion or opinion, when it is clear that idea can only signify something of which an image can be formed in the mind. We may have an idea or image of a mountain, a tree, a building; but we cannot surely have an idea or image of an argument or proposition.
— Boswell’s Life of Johnson
The Singapore Bar Model is a neat bit of maths pedagogy that has great potential in Science education. Ben Rogers wrote an excellent post about it here. Contrary to Samuel Johnson’s view, the Bar Model does attempt to present an argument or proposition as an image; and in my opinion, does so in a way that really advances students’ understanding.
Background
The Bar Model was developed in Singapore in the 1980s and is the middle step in the intensely-focused concrete-pictorial-abstract progression model that many hold instrumental in catapulting Singapore to the top of the TIMSS and PISA mathematical rankings.
Essentially, the Bar Model attempts to use pictorial representations as a stepping-stone between concrete and abstract mathematical reasoning. The aim is that the cognitive processes encouraged by the pictorial Bar Models are congruent with (or at least, have some similarities to) the cognitive processes needed when students move on to abstract mathematical reasoning.
Applying the Bar Model to a GCSE lesson on Magnification
I was using the standard I-AM formula triangle with some GCSE students who were, frankly, struggling.
Image from https://owlcation.com/stem/light-and-electron-microscopy
Although most science teachers use formula triangles, they are increasingly recognised as being problematic. Formula triangles are a cognitive dead end because they are a replacement for algebra, rather than a stepping stone that models more advanced algebraic manipulations.
Having recently read about the Bar Model, I decided to try to present the magnification problem pictorially.
“The actual size is 0.1 mm and the image size is 0.5 mm. What is the magnification?” was shown as:
From this diagram, students were able to state that the magnification was x 5 without using the formula triangle (and without recourse to a calculator!)
The above question was presented as:
Note that the 1:1 correspondence between the number of boxes and the amount of magnification no longer applies. However, students were still able to intuitively grasp that 100/0.008 would give the magnification of x12500 — although they did need a calculator for this one. (Confession: so did I!)
More impressively, questions such as “The actual length of a cell structure is 3 micrometres. The magnification is 1500. Calculate the image size” could be answered correctly when presented in the Bar Model format like this:
Students could correctly calculate the image size as 4500 micrometres without recourse to the dreaded I-AM formula triangle. Sadly however, the conversion of micrometres to millimetres still defeated them.
But this led me to think: could the Bar Model be adapted to aid students in unit conversion? I’m sure it could, but I haven’t thought that one through yet…
However, I hope other teachers apply the Bar Model to magnification problems and let me know if it does help students as much as I think it does.
e=mc2andallthat 
Reblogged this on The Echo Chamber.
Interesting approach! Would you say that the main success of the bar model is helping pupils recognise that the actual size will always be smaller than the image size. When the bar is then taken away, is the embedded idea then: ‘how many times bigger is the image size compared to the actual size?’ Could an actual image substitute the bars in that case? Especially when the 1:1 correspondence no longer applies. If this is the case, then how useful is the bar method? Just some critical questions for a debate! Thanks for sharing your application to the approach.
I think using actual images would fit in the Concrete stage of the Singaporean CPA model. My thought is that the pictorial bar model is a step away from concrete thinking towards
abstract reasoning.
As such, I think using actual magnfied images would be a step backwards towards the concrete. The ideal, of course, is that students eventually dispense with all pictorial aids — but I genuinely think it is a useful halfway house. And what is more, it’s a staging post rather a dead end like formula triangles.
And thank you so much for the comment 🙂
Thanks. I’m using this to introduce the topic next time. I’ve just started using the bar model with my Year 7 physics classes too. I like it.
Great article.
Thank you for the article. Are there any other methods from Singapore maths that you are applying to teaching Physics. I’d appreciate a little more information and if anyone else is adapting anything else from the Singapore scheme or what mileage it has in Physics education.
I am not aware of any specifically Singaporean schemes that have been adapted. However, the Singapore trademark progression of concrete to abstract explanation has been immensely influential to many UK teachers and seems to have become regarded, almost universally, as “best practice” pedagogy.