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Principles and Mantra to teach Mathematics by.

1. Concept > Procedure

2. Give students agency

3. Make the time to remember

4. Make 'em think!

1. Concept > Procedure

Fake it until you make it. I can bake a cake by following instructions, no problem. However, diverging from a recipe can lead to disaster. When I don't understand the concept around proving dough or tempering chocolate, I can't adapt. I can't correct and my confidence is lost. The parallels in Mathematics learning are obvious to me. I spend most of my lessons concept building, especially in KS3. Path smoothing rules like -- make + and formula triangles seem to be a convenient vehicle for the teacher to get from one end of the lesson to other without any stress. In this instance, I feel the students have been handed the 'quick setup' guide rather than the full manual. Of course, I'd by wrong to say I haven't done this. I'm only human, but time spent watching students 'learn', then forget, on repeat, has changed my outlook.

I want to build up a concept enough so that when challenged by an unexpected issue or a cognitive conflict, students can navigate around it and not crumble because their “method” doesn’t work. Like using 6ft concrete foundations when building a house extension, not 3ft. Strong mathematical foundations are essential. That’s the difference between being mathematical and doing maths, isn’t it? My job is to help build connections in students thinking and to keep testing to see if they topple. Testing through task or questioning.

Focus on making connections rather than isolating facts - A lovely mantra, that I overheard, that I have adopted in my pedagogical thinking. The more connections we make explicit in our teaching the closer together the mathematics becomes to the students. I feel separating and compartmentalising does the opposite. For example, think about all the elements you focus on that are about quadratics. Expanding brackets, factorising, completing the square, graphing the functions, solving the equation, the quadratic formula…etc all taught separately. If you understand how one relates to the other, see my video and resources on this, you can build the concept.

Representations play a big part in building concepts. The ‘formal’ representation is not the only way to see the mathematics and if we dive straight-in exclusively with notation, we can lose a lot of students. Using multiple representations is not a thing to confuse matters, as some might say, but to give various perspectives of the same concept. They paint a clearer picture of a concept.

They’re not ‘methods’ either. “I don’t teach that method” is often a phrase heard in maths departments. A phrase that divides opinion and eventually leads to the reteaching these ‘methods’ between year groups with different teachers. Isn’t this more confusing?

For most learners, multiple representation bridges the gap. Concrete to pictorial to abstract (CPA). Abstract is the end goal and there is a journey to get there, it’s called building a concept.

2. Give students agency

Learning mathematics should be an immersive experience. Something that the students get hold of, play and experiment with. Lecture-style presentation of maths has its place, but I feel that students should have ownership of what they are learning. Mathematics, as mystical as it can be to students, shouldn’t be kept at arm’s length by only the teacher having possession of it.

Giving the students agency with what they are learning, brings them closer to the maths. Now the concept is stronger, let them explore. Widen the path. I am not saying let them have free reign, but think about how you can loosen up the constraints to allow exploration. Some would say this a great form of differentiation.

It could be in a lesson about substitution. Instead of ‘if a = 2, b = -4, c = 10 and d = 0.5, evaluate the expression 2a – 2b’ could be ‘write down expressions that evaluates to 12’. Constraints are needed here because some students will try and do some ridiculously complex answer. So maybe say ‘you can only use two letters or two operations or you must include addition.’ See my ‘memory game’ video for this.

If this is… then this is… because… A very simple prompt I use to encourage students to use what they know to find something else. For example, when using a ratio table, students are able to ‘tell me something else…’. ‘If 100% is £340, what else can you tell me?’ I let them make the decision of what comes next themselves, rather than having them panic about finding 31.4% of the number. They're in control, they have ownership and they're the ones being mathematical. See my videos on the ratio table for this.

Give an example of… is my favourite. The great work of John Mason and Anne Watson with regards to questioning prompts really rings true with me. Give me an example of 2 numbers that sum to 3…. (Students have the agency to use what they know – decimals? directed number? Fractions?) Give me an example of a prism with a volume of 120cm³… Give me an example of a nth term of sequence that would have 8 as its 4^th term… The possibilities are endless with this.

3. Make the time to remember

The word retrieval seems to be very fashionable in modern teaching. Retrieval practice is the act of trying to remember what you have previously studied. It’s a way of getting information out of your long-term memory. Sounds like revision? It’s not, it’s learning strategy. A very important learning strategy.

Learning vs Performance

I have had lessons where the students are absolutely smashing a task and are really understanding the concept being studied. Then three weeks later some, if not most, are hopeless at it. Did they learn it or was it performance? I think it was the latter. ‘Pupil performance’ can often masquerade as true learning. As John Mason stated “Teaching takes place in time, but learning takes place over time” (Griffin, 1989). I agree with this. What a waste of time if they’re not going to remember it!

So, we need to ‘surf the forgetting curve’. Revisit the concepts in a timely manner. Forgetting is good too, believe or not, as long as you don’t leave it too long. On the edge of forgetting is the sweet spot to induce the maximum potential of this strategy.

I am more than willing to dedicate 20 minutes of every lesson to retrieval practice because I want what I have previously to be remembered. A four/five question starter is usually my go to. It’s not randomly picked off the internet. It is really important you plan it. Here are two starters I used in one week.

Following a last lesson, last week, last term and last year plan works well for me. Don’t shy away from high order thinking questions too. It’s not to pass the time, it’s to secure actual learning.

Other reasons why it works:

• Formative Assessment – You know what needs to be taught again.

• Making connections to the lesson content and previously studied material

• Confidence – connections build confidence. I know this, so therefore…

• Students worry less because they know you will keep coming back to it. They have more chances to get it right.

• Students are retrieving previously ‘performed’ learning and turning it into actual learning over time.

• You are modelling a revision technique that students can use independently.

Trickle-in teaching

This idea is to use the starter platform to ‘plant the seed’. To 'trickle-in' the teaching. If you are going to teach something that involves a lot of fraction work, write a series of questions with a gradient of difficultly that will help with fractions and display them the week before. Simple, but really helpful strategy to reduce cognitive load.

All in all, I’d always make the time to remember!

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