# Teaching approaches: Mathematical thinking

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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 Monsters using Scratch Children using a computer programming language to create moving monsters This activity developed the specific e-skills(topic) of programming and digital animation. It could be considered the first step towards enabling children to design and create their own games(tool) using sprites and user-input controls. Computer programming helps to develop investigation(ta) skills as it requires the use of a previously unknown language(ta) to execute commands, which also develops the skills of mathematical thinking(ta). Computer programming also involves the use of modelling(ta) and planning(ta) techniques. Because Scratch is an open source programming language, this also creates opportunities for homework(ta), as the children are able to download the software for themselves at home. 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. Language Exploring shape and its mathematical language through sorting activities Using mathematical language to discuss shapes of objects either printed or hidden in 'feely bags'. Can you feel the forks? The Investigation(ta) of shapes and geometry can be very rewarding. A practical approach using objects from the pupils’ environment can increase their motivation and interest. In this unit, you will be using everyday objects to help pupils develop geometrical skills, such as recognising, visualisation(ta), describing, sorting, naming, classifying and comparing. Through games(tool) on the properties of shapes, 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 could form the basis of whole class(ta) discussion(ta)/questioning(ta) and inquiry(ta) projects. It can be used as a lesson extension, or as a preliminary task. 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 Analysing the performance of Olympic runner, Usain Bolt Exploring real world statistics using the GeoGebra program Using real world data to engage pupils in mathematical thinking(ta), language(ta) and reasoning(ta). This resource provides opportunities for group work or whole class exercises, with engaging questioning(ta). 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. Visualisation GeoGebra STEM Exploration Develop 'real world' GeoGebra mathematical modelling applications which reach out to a wide range of users both students and teachers The half-term activity consists of 3 half-day workshops interspersed with home-working and on-line collaboration. Each workshop is part tutorial and help in GeoGebra, part development, presentation and feedback on their emerging work. The three half-day sessions become gradually less structured as students become more confident taking the initiative in developing their own work: An initial GeoGebra tutorial session features ‘real life’ examples such as mathematical modelling(ta) and visualisation(ta) from photographs of patterns and structure in flowers and architecture; exercises such as “math aerobics” where students model algebraic functions kinaesthetically; and data analysis and exploration such as from astronomy (Kepler's 3rd law) and athletic performance (Usain Bolt’s 100m sprints). Realistic examples such as these, or from students’ previous work, are essential to get the ball rolling. Following this, the onus is very much on the student’s own initiative. The focus on ‘real life’ and student ownership of ideas and project development increases student motivation. The activity engages pupils in group talk(ta), mathematical thinking(ta) and vocabulary(ta). This open ended(ta) task develops higher order(ta) reasoning(ta), and encourages whole class(ta) discussion(ta)/questioning(ta) and inquiry(ta) projects. Visualisation Flying paper planes Very visual and interactive and simple to understand 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. Visualisation Solar and Lunar Eclipse To show and explain how a Solar and Lunar eclipse occurs 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. Visualisation Radioactive Decay and Carbon Dating Using 'real life' data to explore exponential graphs 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.