Make 'Em Think Maths

Welcome to Make 'Em Think Maths. On this website you will find ideas for teaching Mathematics with a focus on conceptual understanding. Time is precious, so I make short informative animations and stills for teachers.

I don't want this website to be just a bank of pdfs and powerpoint resources. I think the best resource we have is our own resourcefulness. I hope this site to give you ideas for you to run with!

Follow on Twitter: @mcguirea499

Subscribe on Youtube: Make'EmThinkMaths


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 4th 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!

Latest teaching ideas:

Fraction Sense - Ratio Table - Fractions of Amounts

Using the ratio table, double number line and bars to develop fraction sense. Fractions of amounts.

Making Links With Trigonometry

Template below

Teaching Trigonometry for Understanding - Part 1

SOH CAH TOA is a trick. A necessary trick for last minute teaching of Trigonometry. What does it look like if we try and teach it for understanding? Trigonometry needs to be visited in year 9. That should give us enough time to teach it properly, surely?

What is the same and what is different?


Making Links With Quadratics NEXT STEPS

Expanding Polynomials with tree diagrams

A tree diagram is a very useful tool. Make 'Em Link!

Slides here to download

A link to an excellent Geogebra file made by @boss_maths

Additive vs multiplicative relationships with exponents

With number first?


Equivalent Fractions Linking to Multiplication

Using the area model to explore equivalent fractions and how the same representation can be used to conceptualise the multiplication of fractions.

Expanding Brackets: exploring the distributive law

I have been planning how I would teach expanding brackets to students who haven't seen it before. Do teachers focus enough on the distributive law? I feel, most students have a sense of this in their mathematical 'tool bag' and from prior learning. Why not tap in to it? Make 'em LINK!

In this clip you will see how I have tried to embrace a connectionist mindset in my planning. Thanks to Chris McGrane's Mathematical Tasks book for making me reflect on this.

Powerpoint here

Thinking about Standard Form

Ideas for learning Standard Form.

I try to borrow knowledge of powers of 10 and also students work on indices - especially reciprocals.

Using the Gattegno chart to construct numbers and then multiplying/dividing by powers of 10 can really help pupils see what is going on.

Powerpoint file here

Substitution Memory Game

Low starting point- high ceiling task. An excellent way to practice algebraic notation whilst learning substitution.

The gives pupils a sense of agency. They have the opportunity to own the Mathematics with tasks like this.

PowerPoint file here

What equations can you make?

Low starting point- high ceiling task. Pupils can make it as complicated as they like. Practice for algebraic notation.#MakeEmthinkMaths

Can you see... y = 1 + 3

x + y = 5z

(5+1)/3 = z

The list is endless.

What equations can you make? Display students work and get others to try and find them.

Deeper understanding task: Build your own picture!


One of my favourite Malcolm Swan tasks. So simple. Pick a value for n and off you go.

What could the value of n be if you arranged in this order?

Put them in an impossible order.