You have just taught your class a new concept. You have thought hard about when the best time to introduce this new concept would be in your sequence of lessons. You have checked for prior knowledge, and have crafted an elegant and powerful explanation of this new concept.
You then show them an example of this new concept in action, and then things start to fall apart.
How could it have gone so wrong after showing them an example? Surely the example should have been the culmination of all of that hard work you put in to get them to this stage.
But you didn’t think the example through carefully enough.
That first example you show a class can either help to solidify the students’ understanding of that new concept, or it can help to introduce a misconception in their minds. Depending on the concept, that first example can sometimes need just as much thought as everything else in your lesson and overall sequence planning.
I will discuss four cases from teaching Physics where I have changed my first examples over the years.
Relative atomic mass & atomic number
You have explained that the ‘smaller number’ on the periodic table is the atomic number and that it tells you the number of protons in an atom of that element. You have explained that the relative atomic mass is the ‘bigger number’, and that it represents the sum of the protons and neutrons.
From this, you have explained that to find the number of neutrons therefore, you must find the difference between the relative atomic mass and the mass number. But have you thought carefully about the first example that will allow them to see this concept in action?
It would seem a good idea to start with Helium or Carbon. They are familiar elements and the numbers look easy to work with. The cognitive load on the students is low and they just need to think about applying this new concept they have learnt.
However, the problem with these examples is that the answer of the number of neutrons is in the question itself.
There are 2 neutrons in an atom of Helium, and the atomic number is also 2.
There are 6 neutrons in an atom of Carbon, and the atomic number is also 6.
The students might start to now think that the atomic number, or the ‘small number’ is equal to the number of neutrons. The examples of Helium and Carbon certainly suggest that. All of that hard work you put into your explanation is now in jeopardy because of the choice of first example.
Instead I recommend using Lithium or Sodium.
Lithium: 7 – 3 = 4 neutrons. There is no 4 in the question.
Sodium: 23 – 11 = 12 neutrons. There is no 12 in the question.
By using these examples instead, the students are immediately shown that they have to do some calculations to find the number of neutrons. You are reducing (hopefully eliminating) the possibility that they might think the number of neutrons is directly given on the periodic table.
Helium and Carbon are of course important examples, but these are ‘special cases’ which I would save for a little bit later on in the lesson.
You have explained to your class that the resultant force is the overall net force acting on an object, and you go to show them an example of what that means in context.
This seems like a reasonable example, the numbers are nice round numbers and they will have no problem finding the difference between 20 N and 10 N. You and the students perform the calculation and successfully find that the resultant force must be 10 N to the right.
The students might now be thinking “oh there is a 10 there on the left, my answer must therefore be one of the numbers on the diagram”. The answer to this particular example is in the question itself. The resultant force is the magnitude of one of the forces given in the question.
It is easy for teachers to fall into the trap of always using nice numbers like multiples of ten. We think that we don’t want to make the first example too difficult, so we pick numbers that we perceive to be easy to work with.
However this line of thought can introduce misconceptions in your students.
I propose here a simple tweak, which is to just subtract 1 from a number you are using. A small change like this will reduce the chances of your students learning incorrect ideas.
In this example, the resultant force of 11 N is not a number in the question, and so again the students are shown that the resultant force is something that they must calculate.
Principle of moments
You have taught moments and the students have had lots of practice in a previous lesson. Now you move on to the principle of moments and the idea that they need to take clockwise and anticlockwise moments into account.
I have always gone straight to the see-saw as a classic example of two forces causing two moments in different directions, but I have found that some students struggle with the idea that the pivot is in the middle, and that on first glance it looks that the two forces are causing a moment to act in the same direction. In this example, while the moments are in different directions, the forces are both going downwards. While I still go over this example, I now use a different one as a first example.
I now use the example of having the point about which to take moments to the far left. The two forces are now causing moments in different directions, and they are also ‘pointing’ in different directions as well. To me this makes the idea clearer.
For contextualisation, this is a good example to go through with the class after showing moments being applied to a door, with the hinge being the point about which to take moments. This example shows the idea that a weaker force further from the hinge can cause a greater moment than a larger force closer to the hinge.
Conservation of momentum
In my experience of the specifications, this is generally reserved for the triple award students, or indeed for years 16-18. While conservation of momentum comes later on in the mechanics teaching sequence, it nevertheless requires very careful consideration.
After having taught momentum, next up is the conservation of momentum, with collisions and explosions being two specific cases. When people think of objects colliding though, many would say the objects are approaching from different directions and would therefore show this as a first example:
However the problem with this as a first example, is that they have just learnt about the conservation of momentum and already they are having to deal with vectors. The velocities are changing both magnitude and direction. Similar to the moments example earlier, while this is an example that they definitely need to know how to deal with, as a first example I believe it is a little too much.
I would instead go with this example:
The masses are the same, and the objects are both going in the same direction, collide, and then continue in the same direction with velocities where only the magnitude has changed. Do not jump straight in to positive and negative velocities, instead build up in difficulty to then include those situations.
Some general tips
Some general tips that can be applied to any first example that you work through:
- When introducing a new formula, make sure your first example uses that formula exactly as you have presented it, with no rearrangement required.
- Use simple values to start. This means no standard form or prefixes to begin with as this increases the cognitive load on the students. Just use straightforward numbers (decimals are okay).
- Avoid multiples, for example 10 N and 20 N on a resultant force question, or 50 m and 100 m on a moments question. Just a little tweak to your numbers will make a big difference
- Try to make sure the answer to your question is not in the question itself, you want your numbers to all be different to each other.
- Avoid the number 2 as much as you can (3 is better!). The reason is that you can do many things to the number 2 where the answer is the same:
- 2 + 2 = 4
- 2 x 2 = 4
- 22 = 4
The big thing to always consider is this: does this example clearly show what I want to show? Next time you are considering your first example, think it through and put yourself into the mind of the student.
We want our students to feel success at every opportunity, and the increase in demand on them should be considered very carefully. Don’t throw too much at them. Remember the specific thing you are teaching them, and only focus your example on that.
Photo by Marcel Eberle on Unsplash
One thought on “Rethinking First Examples”
Nice post Fabio. I never use N14 or Ca40 for the same reason. A chemist in my dept uses PEN (proton, elecron, neutron) and bottom, bottom, top-bottom. I remind them what they used in Chemistry but stay away from electrons.