Strategies to help all learners think Mathematically

This is a response to #blogsync topic of “A Teaching and Learning strategy intended to elicit the highest levels of student motivation in my subject”. See more at

I’ve always worked in comprehensive schools, mostly 11-18, which means I can move in one hour from supporting a student who has difficult with number bonds up to 10 to helping support a small group of students for Cambridge entrance exams. I love this variety and I love the opportunities it gives me to see links that I might otherwise miss. I think this has helped improve my thinking as a Mathematician and has forced me to find and use strategies to allow all students to develop these skills as well.

I’ve drawn a lot on the work of Malcolm Swan, David Wright, Sally Taverner, Jo Boaler and Andrew Blair.

I’ve split this into two parts – strategies that help thinking and tasks that helps thinking. I’m working on gathering together a more comprehensive list – if you have any obvious ones I’m missing, please let me know!



This is a strategy that is used so much that it almost isn’t worth talking about. Having students think about an answer, share with their partner and then share more broadly helps everyone have a chance to articulate their answers. I find it is a good way of emphasising the use of Mathematical language and the importance of quality explanations.

Self-regulation Cards (inspired by Andrew Blair of

I’ve recently had some of Andrew’s cards made up that students have on their desks – these provide prompts to help them self-direct their learning. They give ideas like ‘Draw a picture’, ‘What easier problem could help?’, ‘What other information might you need?’, ‘Ask a partner’ and ‘Ask the teacher’. Now, when I drop in to chat to a student I’ll ask them which strategy they are using. This has helped them become more independent and self-motivated.

Mini whiteboards

I don’t know what I’d do without these – how else can I quickly ask a question and process 32 responses instantly? I can quickly identify which students need more support and which concepts need more explanation. My only problem is, I keep running out of pens – do others have better organisational strategies for these?


Odd One Out

This is one of the easiest tasks to implement. Students are given a set of 3 numbers, shapes, functions or whatever and explain why each one might be considered the odd one out.

Using two-way tables or Venn Diagrams

For this task, students are given a large set of objects and then have to categorise them. This helps to pull out similarities and differences, developing Mathematical language and explanations. They then can design a two-way table or be given on to complete. I’ve used this for topics as varied as trigonometric graphs and rotational symmetry of 2D objects.

Multiple representations

To try to avoid focusing on technique rather than understanding, interpreting different representations of the same information can help make links that otherwise go unnoticed. This might be matching numbers (fractions, decimals, percentages, number lines), algebraic expressions (words, symbols, area diagrams) or statistical diagrams (frequency table, cumulative frequency curve).

Always, Sometimes, Never

When helping students to evaluate the veracity of their claims, it is helpful to ask is that always true? Sometimes? Providing provocative statements and having students develop their arguments is a really helpful task to identify key errors and misconceptions (eg multiplying always makes a number bigger, continuous graphs are differentiable, if you double the lengths of the sides you double the area).

These are some of my key moves – what are yours?