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.


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:

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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!)

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Magnification question.

The above question was presented as:

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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:

Screen Shot 2018-03-18 at 12.22.21

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.





Filed under Education, Science

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).


Filed under physics, Uncategorized

Corinne’s Shibboleth and Embodied Cognition

You can watch a bird fly by and not even hear the stuff gurgling in its stomach. How can you be so dead?

— R. A. Lafferty, Through Other Eyes

In modern usage, a shibboleth is a custom, tradition or speech pattern that can be used to distinguish one group of people from another.

The literal meaning of the original Hebrew word shibbólet is an ear of corn. However, in about 1200 BCE, the word was used by the victorious Gileadites to identify the defeated Ephraimites as they attempted to cross the river Jordan. The Ephraimites could not pronounce the “sh” sound and thus said “sibboleth” instead of “shibboleth”.

As the King James Bible puts it:

And the Gileadites took the passages of Jordan before the Ephraimites: and it was so, that when those Ephraimites which were escaped said, Let me go over; that the men of Gilead said unto him, Art thou an Ephraimite? If he said, Nay; Then said they unto him, Say now Shibboleth: and he said Sibboleth: for he could not frame to pronounce it right.

Judges 12:5-6

The same story is featured in the irresistible (but slightly weird) Brick Testament through the more prosaic medium of Lego:


Sadly, the story did not end well for the Ephraimites:

Then they took him, and slew him at the passages of Jordan: and there fell at that time of the Ephraimites forty and two thousand.

This leads us to Corinne’s Shibboleth: a question which, according to Dray and Manogoue 2002, can help us separate physicists from mathematicians, but with fewer deleterious effects for both parties than the original shibboleth.
Corinne’s Shibboleth

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Mathematicians answer mainly B. Physicists answer mainly A.

This is because (according to Dray and Manogoue) mathematicians “view functions as maps, taking a given input to a prescribed output. The symbols are just placeholders, with no significance.” However, physicists “view functions as physical quantities. T is the temperature here; it’s a function of location, not of any arbitrary labels used to describe the location.”

Redish and Kuo 2015 comment further on this

[P]hysicists tend to answer that T(r,θ)=kr2 because they interpret x2+ y2 physically as the square of the distance from the origin. If r and θ are the polar coordinates corresponding to the rectangular coordinates x and y, the physicists’ answer yields the same value for the temperature at the same physical point in both representations. In other words, physicists assign meaning to the variables x, y, r, and θ — the geometry of the physical situation relating the variables to one another.

Mathematicians, on the other hand, may regard x, y, r, and θ as dummy variables denoting two arbitrary independent variables. The variables (r, θ) or (x, y) do not have any meaning constraining their relationship.

I agree with the argument put forward by Redish and Kuo that the foundation for understanding Physics is embodied cognition; in other words, that meaning is grounded in our physical experience.

Equations are not always enough. To use R. A Lafferty’s picturesque phraseology, ideally physicists should be able to hear “the stuff gurgling” in the stomach of the universe as it flies by….

Dray, T. & Manogoue, C. (2002). Vector calculus bridge project website, http://www.math.oregonstate.edu/bridge/ideas/functions

Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology. Science & Education, 24(5-6), 561-590.

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Filed under Education, Philosophy, Science

Markopalypse Now

AHT VAL: And once you’ve finished marking your students’ books and they have responded IN DETAIL to your DETAILED comments, you must take them in again and mark them a second time using a different coloured pen!

AHT HARVEY: A page that’s marked in only one colour is a useless page!

NQT BENJAMIN: Erm, if you say so. But why?

AHT VAL: It’s basic Ofsted-readiness, Benjamin. Without a clearly colour-coded dialogue between teacher and student, how can we prove that the student has made progress as a result of teacher feedback?

NQT BENJAMIN: But I’ve only got this red biro…


AHT HARVEY: In this school we wage a constant battle against teacher sloth and indifference!

(With apologies to The League Of Gentlemen)

I have been a teacher for more than 26 years and I tell you this: I have never marked as much or as often as I am now. We are in the throes of a Marking Apocalypse — a Markopalypse, if you will.

And why am I doing this? Have I had a Damascene-road conversion to the joy of rigorous triple marking?

No. I do it because I have to. I do it because of my school’s marking policy. More to the point, I do it because my school expends a great deal of time and energy checking that their staff is following the policy. And my school is not unique in this.

Actually, to be fair, I think my current school has the most nearly-sensible policy of the three schools I have worked in most recently, but it is still an onerous burden even for an experienced teacher who can take a number of time-saving short cuts in terms of lesson planning and preparation.

Many schools now include so-called “deep marking” or “triple marking” in their lists of “non-negotiables”, but there are at least two things that I think all teachers should know about these policies.

1. “We have to do deep/triple marking because of Ofsted”

No, actually you don’t. In 2016, Sean Harford (Ofsted National Director, Education) wrote:

[I]nspectors should not report on marking practice, or make judgements on it, other than whether it follows the school’s assessment policy. Inspectors will also not seek to attribute the degree of progress that pupils have made to marking that they might consider to be either effective or ineffective. Finally, inspectors will not make recommendations for improvement that involve marking, other than when the school’s marking/assessment policy is not being followed by a substantial proportion of teachers; this will then be an issue for the leadership and management to resolve.

2. “Students benefit from regular feedback”

Why yes, of course they do. But “feedback” does not necessarily equate to marking.

Hattie and Timperley write:

[F]eedback is conceptualized as information provided by an agent (e.g., teacher, peer, book, parent, self, experience) regarding aspects of one’s performance or understanding. A teacher or parent can provide corrective information, a peer can provide an alternative strategy, a book can provide information to clarify ideas, a parent can provide encouragement, and a learner can look up the answer to evaluate the correctness of a response. Feedback thus is a “consequence” of performance.

So a textbook, mark scheme or model answer can provide feedback. It does not have to be a paragraph written by the teacher and individualised for each student.

Daisy Christodoulo makes what I think is a telling point about the “typical” feedback paragraphs encouraged by many school policies:

[T]eachers end up writing out whole paragraphs at the end of a pupils’ piece of work: ‘Well done: you’ve displayed an emerging knowledge of the past, but in order to improve, you need to develop your knowledge of the past.’ These kind of comments are not very useful as feedback because whilst they may be accurate, they are not helpful. How is a pupil supposed to respond to such feedback? As Dylan Wiliam says, feedback like this is like telling an unsuccessful comedian that they need to be funnier.


Filed under Assessment, Education, Humour, Uncategorized

Whither Edu-blogging?

The task of an author is, either to teach what is not known, or to recommend known truths by his manner of adorning them.

Samuel Johnson, The Rambler, 27 March 1750

I regret to say that, for me at least, blogging has become a habit that has been more honoured in the breach than in the observance. And, judging from a conversation or two on Twitter, I haven’t been alone. A number of edu-bloggers also seem to have hit a dry spell.

Some ask: what’s the point? What have we actually achieved? In a typical school, how many teachers actually read any edu-blogs? Outside the edu-Twitter bubble, has anyone ever changed anybody else’s mind, ever? Humans can generally change their location easily enough, but as Horace observed mordantly many years ago, “Who can change their mind?”

And yet. Reading blogs and engaging in Twitter conversations has changed at least one person’s mind: mine. And one of the most important things it taught me was: I was not alone.

I wasn’t alone in thinking that group work was over-emphasised as a panacea to the point of absurdity. I wasn’t alone in thinking that Learning Styles seemed a bit dodgy. I wasn’t alone in believing that a teacher should, on occasions, be an unapologetic sage-on-the-stage and not a permanently-muted guide-on-the-side.

And, in my opinion, a number of things have indeed changed for the better. Ofsted still has issues but it isn’t the educational Thought Police which brooked no dissent from the One True Path that it was a few years ago. A significant part of the credit for this should go to the edu-blogging pioneers who pointed out that a number of its policies had no clothes, and did this using evidence and reasoned argument rather than merely relying on a set of appeals-to-educational-authority as was the style at the time. I would single out @oldandrewuk, @tombennett71 and @daisychristo as being particularly influential in this regard, but there were many others.

I agree with @larrylemonmaths‘ comment that “When the stonemason hits the rock, the first 99 times, it seems like nothing is happening, then suddenly, on the 100th blow, the rock breaks apart. It’s important to keep blogging and talking and arguing, even if it seems like nothing is happening.”

So if we are to continue blogging, what should we blog about? Whither Edu-blogging? in other words.

If I was to highlight some current issues that I think would benefit from more people blogging about them, they would be:

1. Markopalypse Now: why are most teachers in most schools marking so much? When did insane amounts of over-marking become the new normal? Do people realise that written marking is not the same as feedback and that the majority of marking is being undertaken to comply with school policy and a misguided idea of “what Ofsted wants”.

2. The Bonfire Of The Greyhairs: why are so many experienced teachers leaving the profession? Are some of them being forced out because of budgeting pressures with manufactured “performance issues”? Is there any other profession where the wisdom of long-serving colleagues is not only sidelined as an irrelevance but actively rejected?

3. Accountability Roulette And The Culture Of Fear: research suggests that the “teacher factor” is responsible for between 1 and 14% of educational outcomes. Why, then, are teachers judged as if they are accountable for 100%?

No doubt I will blog on other issues besides the ones above (assuming that I blog at all!), but I will try to contribute to the tap-tap-tap of stonemason’s chisels on the adamantine rock of these problems at least.


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Electrifying Engelmann

It is a long-standing and melancholy truth that, despite the best efforts of many legions of Physics teachers, many students continue to not only dislike electricity, but to hate it with the white-hot intensity of a million suns.

What we have here, I think, is a classic failure to communicate.

A final fact is that samenesses and differences of examples are more obvious when the examples are juxtaposed. This fact implies that the continuous conversion of examples provides the clearest presentation of samenesses and, differences because it creates the changes that occur from one example to the next.

— Siegfried Engelmann and Douglas Carmine, Theory of Instruction (1982) p.46

Looking at my own teaching, I certainly attempt to juxtapose a number of circuits. I really want to highlight the similarities and differences between circuits in order to better develop my students’ understanding. But the problem is that both limited resources and other practical considerations mean that the juxtapositioning cannot happen by continuous conversion, except very rarely.

For example, I would set up (or ask students to set up) a circuit with a single bulb with an ammeter, then I (or we) would disassemble the circuit and rebuild it with the ammeter in a different position, or a second bulb added in series or in parallel . . .

It occurs to me that what we are relying on to thread these juxtapositions together in students’ minds is a sequence of circuit diagrams. I suppose it’s another case of the curse of knowledge writ large: experts and novices think differently.

As a beginning teacher, I remember being genuinely shocked that many students found it easier to interpret a photograph or a 3D drawing rather than the nice, clutter-free, minimalist lines of a circuit diagram.

Without a doubt, many students retain strong visual impressions of many of the circuit diagrams they encounter, but they do not parse and decode the diagrams in the same way as their teachers do.

And that, I think, is the major problem when we are introducing electric circuits.

But what to do?

— R. S. Thomas, The Cure

Can we introduce the important aspects of electrical circuits by continuous conversion of examples?

I think we can. And what is more, I think it will be more effective than the itty-bitty assembly and disassembly of circuits that I have practiced to date.

Conservation of electrical current (and current in parallel circuits) by continuous conversion

Parallel Circuit

This is introduced with a teacher demonstration of the above circuit. Students are invited to note the identical readings on both ammeters and asked to explain why they are identical. They are then asked to predict the effect of adding a second bulb in parallel. The teacher then adds the second bulb by connecting the flying lead. The process is repeated with the third and fourth bulbs, with the teacher testing students’ understanding by asking them to predict the change in current readings as bulbs are added and removed. The teacher also tests students’ understanding of the conservation of current by asking students to predict whether the reading on both ammeters will be the same or different as bulbs are added and removed.

I find it useful to include a bulb that is not identical to the other three. It should be noticeably brighter or dimmer than the other three with the same p.d. so that students do not make the incorrect inference that the current always increases or decreases in equal steps when the circuit is changed.

The teacher could also draw the original circuit on a student whiteboard and ask students to do likewise. The changes that are about to be made could be described and students could be asked could alter the picture/circuit diagram and write their prediction on their whiteboards. They could then compare their version with the teacher’s and their prediction could be quickly tested by making the proposed changes “live” in front of the students.

If resources and time permit, students could then, of course, go on to construct their own parallel circuits as a class practical. However, I think it is important that these vital, foundational ideas are introduced (or re-introduced!) via a teacher demonstration to avoid possible cognitive overload for students.

Series circuits by continuous conversion

Series Circuit

In this demonstration circuit, four of the three bulbs are short-circuited so that they are initially unlit. The teacher asks students to explain only one bulb in the circuit is lit: it is helpful if they have previously encountered parallel circuits and can explain this in terms of electrical current taking the “easier” route (assuming they have not yet encountered the concept of electrical resistance).

Again, the two ammeters allow the teacher to emphasise and test students understanding of the idea that current is conserved.

The teacher then asks students to predict the change in current reading when switch X is opened: will it increase or decrease? Why would it increase or decrease? The process is repeated with switches Y and Z and students’ understanding is tested by asking them to predict the effect on the current reading of opening or closing X, Y or Z.

As before, the teacher would amend her circuit diagram on her student whiteboard and students would do likewise. For example: “I am going to open switch Y. Change the circuit diagram. Show me. What will happen to the reading on the left hand ammeter? What will happen to the reading on the right hand ammeter? Explain why.”

Again, I recommend that at least one out of the four bulbs in not identical to the other three to help prevent students from drawing the incorrect inference that the current will always increase or decrease in identical steps.


Filed under Direct Instruction, Education, Physics, Siegfried Engelmann

When Harold Met William

Legend has it that in 1988, U.S. Presidential candidate Michael Dukakis opened an election rally in front of a huge crowd in a red state with the ringing words: “This joke will appeal to the Latin scholars amongst you…” He went on to lose decisively to George H. W. Bush.

On that note, this joke will appeal to all the Physics teachers (and other aficionados of the dot-and-cross convention).


For the non-physicists amongst you, this is an illustration of the dot-and-cross convention, which allows us to represent 3D objects on a 2D diagram. The dot represents a vector emerging out of the plane of the paper (think of an arrow coming towards you) and the cross represents a vector directed into the plane of the paper (think of an arrow going away from you).


A solenoid (electromagnet) represented using the dot-and-cross convention. From http://www.wikiwand.com/en/Solenoid

I’ll get my coat…

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Filed under Humour, Physics