# Teaching approaches: Mathematical thinking

Mathematical thinking (and scientific thinking) should encourage pupils to engage mathematical language in reasoning tasks through active learning. The classroom resources associated with this teaching approach are particularly good for encouraging such learning, and the teacher education resources provide some further guidance.

# Relevant resources

 Consecutive Sums Using Prime and Square Numbers - How Old Am I? Last year I was square, but this year I am in my prime. How old am I? This short activity offers opportunity for pupils to engage in mathematical thinking(ta) and higher order(ta) problem solving/reasoning(ta). They should be able to make links between different areas of mathematics and explore their ideas in whole class(ta) discussion(ta) and questioning(ta). ICT Creating Instructional Videos Children create instructional videos to upload to YouTube This activity is a cross-curricular(subject) activity with a literacy focus, involving a collaborative(tool) approach, giving children to opportunity to work together to produce a set of instructional resources. Children were encouraged to engage in group talk(ta) and discussion(ta) in the classroom to reflect on what they should include in their videos. The activity furthers e-skills(topic) through the use of whole class(ta) participation. It develops e-safety(topic) skills through discussion of the issues relating to posting digital content online. Children were allowed to choose their own subject for the video, although this could be set by a teacher with a specific outcome in mind, or could be tailored to cover a particular topic or subject. It could, for instance, be used to explain their mathematical thinking(ta). Investigation Consecutive Sums Can all numbers be made in this way? For example 9=2+3+4, 11=5+6, 12=3+4+5, 20=2+3+4+5+6 By definition, a problem is something that you do not immediately know how to solve, so learning how to solve something unfamiliar is not straightforward. Tackling an extended problem is difficult. This lesson gives pupils an opportunity to engage in mathematical thinking(ta) and develop their higher order(ta) thinking skills on a problem that is accessible but which has interest. For example, the problem is presented in diagrammatic and numerical ways. The plan suggests several visualisation(ta) methods to present the same underlying task. It should be useful for teachers to compare these different presentations and either to select the one that they feel will be most useful for their pupils or explore ways for the pupils to see the links between the different methods. The assessment(ta) ideas, using other pupils' solutions from the NRICH website are widely applicable to other problems too. Patterns Exploring Pattern Exploring patterns in mathematics Each chapter of this tutorial highlights the study skills(topic) required to work through the real world examples and activities given. There are problems to be solved, some of which involve higher order(ta) thinking skills (for example, being asked to correct a set of instructions), and all of which encourage the use of mathematical language(ta) and mathematical thinking(ta). The resource could also be used in class, or as a useful homework(ta) pack. Polygons Exploring properties of rectangles: Perimeter and area. Do two rectangles that have the same area also have the same perimeter? A problem to inspire higher order(ta) questioning(ta) especially in whole class(ta) dialogic teaching(ta) encouraging pupils to engage in mathematical thinking(ta) and language(ta). You could use Geogebra(tool) in this investigation, as an example of same-task group work(ta). Probability Playing with Probability - Efron's Dice I have some dice that are coloured green, yellow, red and purple... Efron's dice provide a discussion(ta) topic for joint reasoning(ta) - whole class(ta) or in group work(ta). Pupils can explore aspects of mathematical thinking(ta) particularly with relation to probability. Simultaneous Equations Love Food, Hate Waste - Simultaneous Equations Using real world data to explore simultaneous equations Using a source that was not intended by its creators as a mathematical resource, pupils are introduced to informal ways of solving simultaneous equations. The lesson starts with an intriguing ‘hook’, pupils are able to use reasoning(ta) skills to find an answer to the problem and can then, later, formalise this in an algebraic context, using their informal work to support the transition to mathematical thinking(ta). whole class(ta) work supports this inquiry(ta) into the data provided. Using a resource not targeted at mathematics specifically encourages pupils to think about maths outside of the classroom. Standard Index Form An Introduction to the Standard Index Form Working out the rules according to which a calculator displays large numbers The Standard Index Form is a key idea for mathematicians and scientists. The notion that we choose to write numbers in this way requires some explanation. So in this activity, pupils take part in an investigation(ta) on how standard index form works. This is a higher order(ta) problem solving context where students are encouraged to engage in mathematical thinking(ta). They may be involved in whole class(ta) or small group work(ta) discussion(ta), so they have a good opportunity to practice using mathematical language(ta) and questioning(ta). This means that students do not need to be able to explain their ideas in full: they can use the calculator's feedback to discover whether their ideas are correct or not. This is also an exciting way for pupils to realise an initial idea that fits the data may need to be extended when new data arises. This resource therefore aims to develop investigative skills, as well as introduce pupils to standard index form in a memorable way. The pupils can later use their knowledge of indices in discussion(ta) and group talk(ta) as they explain what is happening. Statistics Cubic Equations and Their Roots To interactiviley explore and understand complex mathematics with GeoGebra This lesson features a ‘real life’ example for students to explore using visualisation(ta) via GeoGebra. The focus on ‘real life’ increases student motivation. The activity engages pupils in group talk(ta), mathematical thinking(ta) and vocabulary(ta). This open ended(ta) task encourages higher order(ta) thinking, and encourages whole class(ta) discussion(ta)/questioning(ta) and inquiry(ta) projects.