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 gives you ideas for you to run with and therefore enhance the learning of others.
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Principles and Mantra to teach Mathematics by.
- Concept > Procedure
- Give students agency
- Make the time to remember
- Teach them the next thing
- Make 'em think!
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 the learning of Mathematics are obvious to me. I spend most of my lessons concept building, especially in KS3. Path smoothing rules like -- make a + 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 the 'rule', 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. It is like using deep 6ft concrete foundations when building a house extension rather than short 3ft ones. Strong mathematical foundations are essential (see principle 4). My job is to help build connections in students thinking and to keep testing them 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 in the students minds. Compartmentalising topics does the opposite. For example, think about all the elements of quadratics you teach separately. Expanding brackets, factorising, completing the square, graphing the functions, solving the equation, the quadratic formula…etc all usually taught in discrete learning episodes. If you understand how one relates to the other or how they differ, you can build the concept. see my video and resources on this
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 said ‘methods’ between year groups with different teachers. Isn’t this more confusing?
I think, for most learners, the use of multiple representations 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 learning.
2. Give students agency
Learning mathematics should be an immersive experience. Students should wrestle with, play with and experiment with. I don't sit on either side of the inquiry vs direct instruction debate. When I feel that students just need showing, I model the hell out of it and interrogate the hell out their understanding through whole class interactive discussion and the use of mini white boards. When I want students to 'flex' with what they know, I would go down a more exploratory route, by asking questions that give them more agency. In all, I want students to have ownership of what they are learning. They need to be the ones thinking mathematically.
Mathematics, as mystical as it can be to some students, shouldn’t be kept at arm’s length by only the teacher having possession of it.
Giving the students agency within the context of what they are learning, brings them 'closer' to the Mathematics. Now their schema is more established, 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, 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
Retrieval practice is the act of recalling what you have previously studied. It’s a way of getting information out of your long-term memory. The act of doing this, strengthens your memory of the thing you recalled for future retrieval. 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. Student performance on in a lesson can often masquerade as true learning. As John Mason stated “Teaching takes place in time, but learning takes place over time” (Griffin, 1989). What a waste of time if they’re not going to remember it!
So, we need to ‘surf the forgetting curve’. I get students to retrieve the concepts in a timely manner, with the spacing in between getting further apart. 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. The problem with the 'sweet spot' is that it differs for everyone. I use a sensible estimate and I stick to it.
I am more than willing to dedicate 30 minutes of every lesson to retrieval practice because I want what I have previously taught to be remembered. A four/five question activity is usually my go to. It’s not randomly picked off the internet. It is really important you plan it because not only am I retrieving, I am building schema too. 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.
This idea is to use the retrieval 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 and thoroughly assess the students understanding. Simple, but really helpful strategy to reduce cognitive load.
All in all, I’d always make the time to remember!
4. Teach them the next thing
Find out what they know and teach accordingly. If you find yourself scaffolding so much that the maths is just a set of rules to memorise, it is probably because they are not ready for it. If I tried to learn about astrogeology at my current level of understanding, I would have no chance! I need to learn the next thing. I need to build schema and make connections to the things I already know. As Mark McCourt talks about in his book Teaching For Mastery, we put our children on a conveyor belt based on their age, not their current stage of understanding. Let's take the labels off and find out where they are and teach from there.
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
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?
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.
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.
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.
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.