The metaphor for =

Many physics students struggle with algebra. As physics teachers we are constantly complaining to each other that the maths department hasn’t done their job because our students can not rearrange equations!

I was working with a student recently and he asked for help on rearranging an equation

40 \rm{J} = 0.5 \times 0.025 \rm{kg} \times v^2

I have noticed that many students who have trouble will be aiming to have their unknown variable (v in this case) on the left hand side of their finished rearranged equation.

So I tried to diagnose to this student to see if this was the case.

Teacher: “Would you be ok if when we finished rearranging this equation, the v stays on the right hand side?”

Student: “I dont know what you mean…”

T: “Ok, lets look at a different problem. Are these two equations the same?”



S: “No, they are different”

T: “Really?! Can you explain how you know that?”

S: “If I rearrange the bottom one (6=x) I get x=-6

There are two separate problems here. The first is that the student made a mistake in trying to rearrange the equation in their head, and ended up with a rogue negative.

The second is that this student needed to rearrange the equation in first place to compare the two equations. I look at these two equations and see they are the same. This student didn’t. Why? I think the answer lies in the metaphor they have attached to the “=”.

As human beings, we understand many abstract concepts by attaching meaning through metaphor or analogy. You may take for granted that you understand what “=” means. Its only when you meet someone that has a different understanding than your own that you may reflect on your own understanding.

The metaphor I attach to “=” is a set of balance scales. 


To me “=” means that one side of the equation is balanced with the other side. Both sides may look different, but the “=” sign tells me they are weighted the same.

This student (and many others) don’t attach this metaphor to the “=” sign. The metaphor/meaning my student attaches to “=” is and the answer is. I think this is perpetuated/started by their use of a calculator. There, students type in a series of steps for the calculator to evaluate. Then the student pushes “=”, and the calculator says “and the answer is”.

This is the reason my student couldn’t see the equivalence between 6=x and x=6. They have a fundamentally different understanding of the “=” sign. And this is a very difficult belief to try and dislodge. It isn’t just a “fact” I can easily replace by explaining to them their error. This will require a number of activities to try and start to remedy the situation. To be honest, I am not sure where to start…

Last note: When discussing this with the other physics teacher at our school he mused that this would be the reason kids struggle so much with the proportion sign. Whereas \propto seems a small step from = to us experts, it seems to give students huge trouble. But now understanding that they have a different understanding of = than us, it seems so obvious why they struggle here.




Physics NCEA Review 6: Let’s drop Newton’s Laws.

… from Level 1 Science.

We were recently given an exemplar as part of PD. What is the problem? My colleague and I have a problem with “thrust force”.

cycle forces.png

Let’s start with a quote I’m going to attribute to Colleen Megowan (but she may say it comes from someone else). “All models are wrong, but some models are useful…”

There are four fundamental forces (thrust isn’t one of them…). They are gravitational force, electromagnetic force, strong nuclear force, and the weak nuclear force. Well actually, many theorists believe three of these forces could be unified into a single force…

But this model of forces is not very useful to our kids. Most of these forces are something they can not see, and so their abstract nature makes them hard to comprehend and understand.

And so we create a simpler, more concrete model for our students. Mainly, we split the electromagnetic force into further categories that kids can see: tension force, friction force, normal force (or is it support or reaction?). At a microscopic level, all of these forces just describe electrons repelling electrons, but these names are more concrete and useful for our students.

As I said above, all models are wrong, but some are useful. So I am completely ok with our example of thrust force on the bike being wrong. It’s just a model. But is it useful….?

The issue is that many teachers will tell you that “thrust force” is not the only correct answer. They will also accept engine force, forward force, push force, thrust force, bike force, pedal force and maybe if they are feeling super liberal, a friction force.

And so our students see a variety of names. They see this variety in exams, exemplars, text books, fellow students and often a variety of names in the space of a few sentences from their teachers.


What do you think they make of this? A rational student would think that no one knows that they are talking about. An average student will simply think they have a licence to use their imagination when to it comes to naming forces, just like everyone else.

And so we come to a problem that really tests their understanding.


Newton’s Law’s are notoriously difficult to understand, and are even harder to teach. Our students have a full lifetime of experiences that have led them to developing misconceptions that are incredibly hard to dislodge and replace. (If you disagree with me, I would contend you suffer from the curse of knowledge – have a read of Nobel laureate Carl Wiewan’s brief article on it).

Not only are Newton’s Laws difficult, but by having a plethora of names for our forces, we have handed our students a licence to invent whatever names they want to fit their misconceptions to our current ice skating example.

Our students then predictably draw in their favourite thrust/forward/push/skater force.

And so we arrive back where we started. All models are wrong, but some are useful. Is this method of multiple names for the same force useful? Since it perpetuates difficult-to-remove misconceptions, then it clearly is not useful. Let me reiterate: I contend that our current system, where no one agrees on a small set of force labels, is actually doing more harm than good. 

So, in our curriculum review, we either need to dump forces from Year 11, or come up with a smaller, agreed upon list of types of forces that all teachers use.

edit: I call the force pointing to the right of the bike a friction force. I also make my kids label forces with the on/by notation. Thrust has specific meaning in physics discourse. It describes the equal and opposite reaction to expelling gas backwards, like a rocket. I have a particular bugbear against “thrust”. Students (and teachers) have no definition of thrust, so it becomes the label for “any force I want to add in the forward direction”. It is an enabler. That is, it makes it easier for kids to add in non-existent forward forces, and needlessly perpetuates their misconceptions.

Normal, Support and Reaction Forces

These are three alternative names interchangeably used in New Zealand schools to describe the same force:


normal and reaction1.jpg.png

Marking guide from Level 2 Physics Exam

Why do we have 3 names? To confuse our students of course. Should we cull 2 of them, and just use a single label? You betcha! And let’s do our kids a favour by using this one name across all of our lessons, textbooks and exams.

Normal Force

Advantage: “Normal” means perpendicular, and so this is a great description that is easily applicable to many situations.

Its the accepted term used by many scientists, and wikipedia.


Disadvantage: This is a new definition of “normal” for our students than the one they use in their everyday discourse.

Support Force

Advantage: To the student, this word is the closest metaphor to what they actually see. For anything on a flat surface, the surface does seem to support the object.

Disadvantage: It soon becomes apparent that if you tilt your surface (create a ramp) the support force doesn’t actually support the object anymore…

Reaction Force

Advantage: It seems to be the most commonly used term in NZ (anecdotally).


  • Students (and teachers) confuse this force with a Newton pair (that for every action force, there is an equal and opposite reaction force). In fact, I’ve seen this common misconception being perpetuated by popular textbooks… Calling something a “reaction force” in many other countries implies you are talking about a Newton pair. And the force of gravity on an object, and its normal force, are not a Newton pair!
  • For an object stationary on a slope, the normal force, and the friction force could both be seen as reaction forces! That is, they both occur because they are a reaction of the object to the slope. Therefore, unless you want to start calling the friction force a reaction force, we need to drop this term.
  • Reaction is a verb. It implies an “alive” agent. That is, students may sometimes ask – “how can a table react? It is not alive, so how does it do that?”


I choose “Normal”. You might think differently, that’s fine, but convince me! Either way students will find life much easier if we teachers (and text book writers, and examiners) choose one set of nomenclature, and stick to it.

Beware, the right thing, said once.

This is a blatant rip off, and rewrite of Kelly O’Shea’s blog post. Its more for my benefit, than anyone else’s, as synthesizing and writing down ideas helps me solidify them in my own head. I don’t think there has been an idea I agree with more in teaching than “Beware, the right thing, said once

In summary this phrase operates on two levels:

1. At the classroom level. Often when I am teaching, I end up defaulting back to preaching to the whole class. I ask a question, there is a flurry of hands (well only the same 5 hands), and someone answers my question correctly. Time to move on…

This is obviously an issue because one student does not represent the whole class. But Kelly also introduced me to Clever Hans. If you haven’t read about Clever Hans, I implore you to do so. 

2. At the individual level. As Kelly rightly talks about in her blog, there is another version of this that is often overlooked. The above example shows a teacher who is ready to move on with the entire class based on one student’s answer. However, even if I think about my learners as individuals, instead of the whole class, hearing the correct answer from that one student doesn’t mean that that individual is even ready to move on.

I read an article earlier this year in which researchers followed and interviewed a number of kids learning thermodynamics. The most striking finding I took from this longitudinal study was that kids can hold the correct science concept in their head, while simultaneously holding misconceptions in their head, and thinking nothing of it…

Therefore as the process of mastering a concept is so much more than learning the initial idea, it involves:

  • Learning the ideas
  • Mastering the skills
  • Unlearning misconceptions
  • Making connections between all these things, and all other ideas they have learned up until now.

So hearing the right thing, said once, only gives you a snap shot that a single student may have any one, or more of these dot points. It doesn’t necessarily mean the individual, or your whole class, is ready to move on.

As always, take your time, don’t hurry to the next concept. Your pupils will thank you for it.

Physics NCEA Review Part 5: Lessons learned.

Be careful what you wish for… It looks like there will be substantial review of all the NCEA standards over the next few years. Here I would like to offer my experiences in being part of curriculum change, and what I have learned from this.

I taught for a few years in Melbourne, Australia. Each state develops their own curriculum, although there is a push at the Federal level to develop a nationwide curriculum. The equivalent of NCEA in Victoria (the state containing Melbourne) is the VCE (Victorian Certificate of Education).

In 2013 and 2014 I applied for, and was part of a panel to look at and review the VCE curriculum for Physics, over the year levels of 12 and 13 (Australian Year 11 and 12).

A panel of 10 stakeholders, including teachers, university staff, and VCE staff, met monthly to debate and develop this curriculum.

From our meetings, our two main informal guiding principles were “What content do we think is important in physics?”, and “What are our students interested in?”. In retrospect, this led to us overpacking the curriculum. Each stakeholder at the meeting had their own opinion on pieces of content that just could not be left out! We compromised with each other, so our own pieces of content that we thought were important could be added in. This meant that everyone’s favourite part of physics was added, and we ended with a very busy physics course.

Overall, I think we must have increased the content in the curriculum by maybe 1/4 to 1/3. At no point was there a serious discussion in these meeting about how students learn physics, and how that statement might guide what we include in the curriculum.

So how do these lessons impact my thoughts on the current NCEA redesign? It looks as though there will be a push for less standards (2 internals and 2 externals). I can see the push being to try and stuff our current “default curriculum” (2 internals, 3 externals) into the 4 new standards. That is, same curriculum, with different labels.

Here is a radical idea. What if we kept the current standards, but just dropped an external? What would you do with the extra time?

After all, research shows, that in physics, topics done with more depth, over breadth, lead to better long term outcomes for our students [Link 1], [Link 2].

Without thinking too hard, my first thoughts on a new curriculum would look like:

Level 2:

  • External 1 (6 credits): Mechanics (Current L2, unchanged)
  • External 2 (4 credits): Thermodynamics (You know you want to…)
  • Internal 1 (6 credits): Practical (Current L2, but make it a portfolio spread over the year and interspersed in the thermo and mechanics topics)
  • Internal 2 (4 credits): Modern (Current L2)

Level 3:

  • External 1 (6 credits): Electricity (Current L2 course – yes, its hard enough as it is, they currently dont understand it…)
  • External 2 (4 credits): Waves (Current L3 course)
  • Internal 1 (6 credits): Practical (Again a portfolio, but drop the concept of errors, the cognitive load is too high)
  • Internal 2 (4 credits): Modern/Mechanics/Context/Relativity ????

Radical I know… But chances are, the curriculum will be redeveloped in Auckland, so this is my chance to start sowing some different idea in your heads!

Footnote: I haven’t discussed my thoughts on whether this change will lead to decreased choice (will there only be 20 credits total to choose from?). My issue is, that if they tried to keep choice in the externals (say there are 3 externals, but you can choose 2), I think the inevitable drift will be towards favouring 2 of the externals over the other. Inevitably, one of the externals will be easier (or perceived to be easier), students/teachers will drift towards that, and then universities will require the “harder” two externals to be pre-requisites for engineering etc. Eventually these two externals will become the “default” “proper” physics course. So lets keep that in mind.


m/s or ms-1?

Until recently I wasn’t worried. If anything I tended to side with “ms-1“. Why? It just seems more hi-tech, even sexier. “m/s” just seems old fashioned. “ms-1 is what the scientists use. It’s certainly what the Year 11 General Science exam uses…

So I went with “ms-1“. Of course I prefaced this by showing my kids the mathematical equivalence between the two. But, once that was out of the way, I was determined that “ms-1” had exclusive use in my classroom to prepare my kids for the end of year exam.

However, in the last few years, although I agree both m/s and ms-1 are mathematically equivalent I have come to believe that m/s is a far superior unit for our students. Let me try and convince you.

My main argument here can be summed up by saying both of these units out loud: “metres per second” compared to “m s negative one“.

There is an entire branch of learning science that studies the link between how we use language, and how we think: cognitive linguistics. These cognitive scientists argue that “symbols are, of course, just symbols, not ideas. The intellectual content of mathematics lies in its ideas, not in the symbols themselves.  In short, the intellectual content of mathematics does not lie where mathematical rigor can be most easily seen – namely, in the symbols. Rather it lies in human ideas”. [Lakoff, Nunez. Where mathematics comes from].

Cognitive linguists analyse language, and language is really a window to our brain. Its a way we can see how students are thinking. What are students thinking when they say “metres per second“? What about “m s negative one“?

I think a better question is what idea are students attaching to the symbol/unit when they say “metres per second” or “m s negative one“?

I would argue that more students are attaching the correct concept of speed when they say “metres per second” than “m s negative one“. In fact, I would argue for many students,  “m s negative one” just becomes a slogan, with no meaning, or idea attached.

Here is a video I often show my kids for a laugh. It’s probably staged, but listening to the lady’s use of language (she often says “mph” instead of “miles per hour”), it gives us a window into how she understands (or misunderstands) the concept of speed.

Physics education research has shown that many students, although capable of numerical calculations, have no idea of the concept of speed. How? They often mix up position and speed (that is if two objects have the same position, they must have the same speed).

Therefore we need to use every opportunity we have to help our kids to understand speed. And using “m/s” instead of “ms-1” is a great start.

And there are other educators that go further. They argue that the word “per” has no meaning for students. We should replace “per” with “for every” in our classroom language. Therefore these teachers would argue “metres for every second” > “metres per second” > “m s negative one“. I have tried to encourage my kids to use “for every” much more in classroom discussions.

This year I did a bit of an informal experiment with my year 10 science class (caveat: they are the top class). We did a large physics unit (the whole of term 1) on velocity, acceleration and energy. How did I stretch it that long? We dived deep into pictorial representations of these concepts (graphs, motion maps, energy bar charts), did a number of experiments, and (tried to) facilitate whole-class discussions on these concepts.

Interestingly, I never taught, and these student’s never saw, the standard formula for speed/velocity (speed=distance/time). And yet nearly every single one of these 14 year olds had no trouble answering this question on their end of topic test:

Who runs faster? Elle ran 100m in 12 seconds. Karsen started 20 m in front of her, and ran the remainder of the 100m in 10 seconds.  (Show calculations that lead to your answer).

I like to think this was possible because these students had a solid grasp of what the concept of speed/velocity is, and a large part of that was these student’s continually using the language of “metres per second“.

If NZ teachers ever get a chance to have input into a change of our NCEA standards/exams, then we need to be thinking more than just what content we cover. After all, much of our teaching is guided by the exams. When writing these these exams,  exam writers shouldn’t ignore subject specific pedagogy. That is, how students actually learn science. I think in the case of  “m/s” vs “ms-1“, more thought needs to be given.

Physics NCEA Review 4 (Part 1): What depth of understanding do we want our students to achieve in our courses?

For those of you heading to Christchurch for the NZIP conference, have a great time, and I hope you have plenty of great seminars and conversations about teaching the subject we all love.

Perhaps this topic could be another starter for conversation?

Today I want to talk about the quality of learning that goes on in our NZ physics standards. Because this involves a few ideas, I want to split this into two posts: how we know the quality of learning can be improved, and what we can do about it.

The summary of my argument today is

  • In at least some parts of our physics curriculum, we expect students to gain understanding (mechanics, electricity and waves).
  • I’ll detail an example that shows that maybe the quality of understanding of our student’s isn’t as high as it should be.
  • I currently believe we think we teach too much, and too quickly, for deep learning to occur for the majority of our students.
  • To combat this overloaded curriculum we need to have a lighter curriculum that encourages more depth of learning, AND we should have more overlap between years.


My definition of “understanding” (or deep learning) can be mapped to the NCEA mark schedule as follows:

  • Achieved: Memorized understanding and simple equation hunting.
  • Merit: Simplify familiar situations down and apply physics concepts to them
  • Excellence: Generalised understanding of physics concepts such that students can apply physics understanding to solve a novel problem

What do I mean by a novel problem? In the 2018 L2 electricity there were two novel problems.

smoke question

Figure 1: Q1diii from 2018 Level 2 Paper

ammeter question

Figure 2: Q 3d of the 2018 Level 2 Paper

Let’s concentrate on Q1diii (Fig 1 above). If you haven’t already, try and answer it yourself.

I spoke to someone who marked this paper. And what they have to say about this question I find absolutely fascinating. Although they are reticent to release how they marked the particular question, they are OK with a conversation about the understanding of the physics involved.

  • This person marked around 1000 papers.
  • About 40 kids “could apply their physics knowledge to solve a novel problem”. They answered something like “Turn magnet to create a magnetic field perpendicular to electron path”.
  • A further 20 students showed at least partial understanding “Mag Field lines must be perpendicular to electron path”, but they didn’t know how to achieve this (to turn the magnet).
  • Of the remaining 940-ish students who didn’t understand or answer the problem, most pupils confused the idea of magnetic and electric fields. That is, most pupils answered along the lines of “The electron will be attracted to the north pole of the magnet as opposites attract”, or some tried to use a formula “F=BqL, so increase B will increase the force”.

Just ponder those numbers for a second before you move on.

Do I have anything against the question? Not at all, I think it’s a great question. My issue is that the curriculum is so full that we don’t have the sufficient time to get a reasonable number of our kids to this level of understanding.

And a quick note for the other novel question with the ammeter which was shorted over the bulb (Fig 2 above). I remember a lot of worried teachers on Facebook hoping their kids picked up that it said the ammeter had “negligible resistance”. According to my friend, this wasn’t a problem. The problem was that students didn’t understand the physics and often simplified it down to “No effect as it has negligible resistance”.

I often get told I talk about “increasing student understanding” in physics too much, as though my students have little or no understanding at all. I freely admit, I probably sit at one end of a continuum with how I view physics teaching. However the incredibly high numbers of kids who showed genuine confusion in the question detailed above isn’t helping to convince me that I am wrong…

What should a good physics curriculum do? It should increase student’s knowledge and skills in the area of physics AND it should increase our number of physics graduates (hopefully going off to a STEM related career). I think there are improvements that we could make to increase the number of kids in both these categories.

Part 2 here