## Teaching Magnification Using the Singapore Bar Model

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.

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.

## Lottie and Lorentzian Length Contraction

@_youhadonejob tweeted this textbook picture with the amusing and sardonic comment “Little girl in this textbook is 5 m tall”.

I liked @jim_henderson60’s take on this when he tweeted: “You see. Physics helps us all grow tall.”

But then I started thinking, what if the 5 m measuring stick was in an inertial frame moving past Lottie’s inertial frame at a substantial fraction of light speed? (In my head, I named the girl “Lottie”, although “Alice” would be more in the more usual tradition of SR* pedagogy, I guess.)

The illustration could represent that single instant at which both ends of the 5 m ruler were precisely opposite Lottie’s head and feet as its inertial frame whizzed by hers…

A quick calculation indicated that Lorentz length contraction could indeed account for the relative measurements on the illustration if v = 0.97c

Of course, Lorentz length contraction is a two way street. From the 5 m ruler’s inertial frame, length contraction would make Lottie appear even shorter than her compact 1.2 m. Given that v = 0.97c, I calculate that she would appear only 0.29 m tall.

Correction: not appear. She would genuinely be only 0.29 m tall when viewed from that inertial frame, just as the 5 m rule would genuinely be only 1.2 m long when viewed from Lottie’s inertial frame.

We live in an universe where everything is indeed relative. However, for most of us that takes a fair amount of getting used to…

*SR = special relativity. My brain is currently too small to handle GR (general relativity).