This post carries on from my previous one where I gave some examples of how I would introduce some topics in comparison to a session by Kris Boulton during #mathsconf23.
The introduction of a topic is a complex beast that is difficult to make meaningful. An argument that Kris used in his session was that pupils aren’t in a good position to understand why in the beginning of a lesson. If pupils just switch off and only start paying attention when you tell them how to answer the questions then we might as well just have a cold open with direct instruction for a task.
There are some topics where this is true but I think a vast majority can be introduced in a way that promotes understanding. It is also my experience that a lot of teachers don’t like to just have a cold open. They don’t like the idea of just having pupils trust them in the beginning that a process works. Kris’ session has convinced me that sometimes the explanation doesn’t need to start the lesson so we need to be able to justify when it is and isn’t best to start with why.
Components that can make up a good explanation are:
- Start with knowledge they already have
- Involve pupils doing something, if the pupils are passive they aren’t concentrating hard enough
- Be more likely for pupils to remember than the final algorithmic process so they can switch back to it if they are unsure
- Be able to be referenced when supporting pupils in a lesson
The last part is based on an issue I have with what Kris said, that he never references the “why” again. I think you should always make reference to the “why”. Especially when pupils make mistakes. How else are they supposed to know why it is wrong?
Let’s say at a later date after the topic a pupil writes that
You could say to the pupil, “no, that is incorrect. Do you remember when we did this before and you had to add the indices” and the pupil will nod and get a similar question correct. Because of course they will. You can then admonish yourself for not correctly applying Ebbinghaus’ forgetting curve and make sure that you do it again soon in another starter/drill/do now.
You haven’t corrected the child because you haven’t corrected their thinking.
Make them expand out y3 and y2.
Make them explain to you why it isn’t y6.
If pupils have forgotten then you need to take it back to the underlying maths because you need to model for them how to approach/read the question. If you just tell them the rule at this point then you have explicitly just now turned it into an isolated piece of knowledge.
Different styles of introducing topics: Concretes
I haven’t got enough experience yet of using concretes. There definitely needs to be more use of these during Secondary school and now that I am going to be a Head of Department I am definitely going to invest in two sided counters for negatives and algebra tiles. From what I’ve read these seem like game changers. They seem to be a really great way for pupils to get a feel (pun intended) for the underlying maths before reverting to written methods and a great way to support pupils if they make mistakes. I highly recommend Visible Maths by Peter Mattock and Teaching for Mastery by Mark McCourt for practical examples of how these can be used in the classroom.
Making it visual
I fucking love bar models. My birthday card from the department this year contained no less than 4 different bar model based jokes. Last year I changed our SoW so that the beginning of Year 7 was mostly just using them in a variety of different topics: proportional reasoning, fractions, percentages, ratios and equations. They demonstrate the underlying maths in an easy to understand way and are great at helping you communicate to the pupils what they should be picturing in their minds. The other useful thing about bar models is that pupils have a high success rate using them; so when they switch to written methods and get stuck or make a mistake it’s easy to support them by just asking them to draw the bar model. Once you start using them you start noticing many situations where they become a useful way of communicating the process to pupils.
I’m also a big fan of the grid method of expanding brackets/factorising (sometimes called the area model). As well as having very clear links to multiplying numbers it’s a much more flexible process for expanding and demonstrates the underlying maths in a very clear way. It also has a seamless transition from algebra tiles to the grid method.
Linking it to prior knowledge
This is what I demonstrated in my previous post. When planning for the start of a topic you should be mindful that pupils aren’t coming to us with an empty slate. For a vast amount of Secondary maths, the pupils will need to have prior knowledge to successfully understand the topic that you are teaching them, not just have the prerequisite skills to be able to access the work. If the maths can’t be done with concretes or visuals then I aim to start off with a question that they are confident in answering and build connections from that into the current topic. This is quick whiteboard work with no pupils opting out. You could argue that in these situations that starting with “how” first could be quicker as it relies less on the prerequisite knowledge and skills but if there are gaps I would rather know that so it can be addressed as soon as possible.
This is not discovery learning; I would say that it is more deducing (I prefer to call it “building on top of your current schema” as it sounds fancy). I am not hoping that pupils will spot a pattern and come up with the process themselves but to draw attention to the underlying maths. We build this up, guided step by guided step, and I will clarify their thinking afterwards using more traditional direct instruction. This is also not starting with a proof of why something works as it is more about having pupils doing maths than watching it.
I try to be concise and not use many words during the guidance, more hinting than being explicit. I want pupils to get a feel for what they are doing before I help to tighten it up and add clarity with the correct mathematical terminology through the direct instruction.
This can also be tricky to pull off as it depends on you knowing how pupils were taught previously and there can be issues if previous teachers haven’t been able to nail the “why” part of teaching (wherever in the learning that they put it). For example, when dividing terms with indices my approach would be based around pupils having been taught how to “cancel out” fractions with prime factor decomposition. If they haven’t been taught that it might not be worth starting with the why.
Teaching it “the long way” before the quicker process
I personally love doing this as pupils tend to find it easier to remember, it allows more understanding than jumping straight to the quicker process and pupils are motivated themselves to try to use their understanding to find a quicker way.
For example when teaching pupils to find averages from frequency tables I start with having them write out the full list and then finding the averages.
This ensures that they can read the tables properly and reduces the amount of misconceptions. They often quickly see how to do it without me being explicit (phrasing the explanation to not create misconceptions is very hard so it’s much easier if they already have some idea of how to do it) and if they forget they can always go back to this method. This year I did have a Year 11 pupil find the median during a mock by writing out the list which I found very satisfying.
Another example would be teaching how to expand brackets with repeated addition. It feels like a roundabout way to get to the process but usually works really well to ensure that they read the maths properly.
In my next post I will be discussing ways to introduce topics that are less helpful at developing understanding