# How To Use Maths Investigations To Develop Mathematical Reasoning At Any Age

When you think of maths investigations, what do you conjure up in your mind? For me it’s about being a great detective, like Sherlock Holmes, and involves lots of consideration, examination, probing, scrutiny, problem solving skills and studying.

The clues are all in the word. Invest + i + gate: I have to go through lots of doors and invest time searching for a solution. For pupils to work on a maths investigation, they need to invest time and energy, alongside plenty of reasoning and collaboration with others.

In this blog, we look at maths investigations across different key stages and how you can nurture a mathematical mindset, regardless of which year group you are teaching!

### What are maths investigations?

Maths investigations are investigations that require pupils to collect and analyse data and use it as evidence to help them solve a problem. The difference between an investigation and a word problem is that finding an answer is secondary to the main purpose of finding a process.

Pupils need to adopt the mindset and persona of a detective (like Sherlock Holmes) when ‘doing’ maths.

When children do this, they take on board some of the essential characteristics needed to understand a ‘case’: they must have a keen eye for details and strong analytical skills. They need to work with the right strategy.

Sherlock was ruthless at collecting, analysing and deducing information and that’s precisely what children need to integrate into their own ways of working.

Finding an answer is one thing but finding a solution is quite another and that means going through a process. An important intellectual skill is the nurturing of cognitive determination and investigations can support this by giving children contexts to express their thinking out loud.

### Maths Investigations: The Detective Mindset

Children need to adopt the mindset and persona of a detective (like Sherlock Holmes) when ‘doing’ Maths.

When children do this, they take on board some of the essential characteristics needed to understand a ‘case’: they must have a keen eye for details and strong analytical skills. They need to work with the right strategy.

Sherlock was ruthless at collecting, analysing and deducing information and that’s precisely what children need to integrate into their own ways of working.

Like him, children need to be ruthless in how they look for evidence and gather what data they can to help them solve a problem. The deductions they make need to be backed up with evidence.

Finding an answer is one thing but finding a solution is quite another and that means going through a process. An important intellectual skill is the nurturing of cognitive determination and investigations can support this by giving children contexts to express their thinking out loud.

### Why are maths investigations important?

Maths investigations provide pupils with clear opportunities to reason mathematically and find evidence to support their ideas and generalisations. On this topic, the national curriculum for mathematics states that one of its aims is to ensure that all pupils:

• Become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately.
• Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
• Can solve problems by applying their mathematics to a variety of routine and nonroutine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

National Curriculum in England (2013)

These are important skills not only for maths, but also lay the foundations for science, English and other humanities subjects through reasoning, problem solving and developing an argument.

### Maths investigation mindset

Mathematicians are nothing special but their styles of thinking can make them appear this way. A key philosophy underpinning ‘maths mastery’ is the growth mindset and this is also a key ingredient of the detective approach to maths investigations.

Maths mindsets have to be ‘open for business’ which means as teachers we need to provide a rich number of opportunities to allow for development. Every child has the potential for developing higher mathematical -logical intelligence.

### How you can nurture a mindset ready to start your maths investigations

In her book, Mathematical Mindsets, Jo Boaler argues that in order to promote a detective mindset for Maths, we need to:

• Support all learners to understand that they can learn anything and that struggle is the best time for brain growth
• Teach multidimensional Maths – ask children to represent ideas visually, numerically, graphically, verbally, algorithmically – colour code and make connections between them
• Put the brakes on speed and celebrate depth and creativity
• Encourage and value multiplicity – all the different ways children think
• Help children to realise that they ‘can do’ Maths and that when they believe in themselves then their brains are more active and grow more.

Her suggestions are a great starting point for many schools, but there are more ways in which you can develop good Maths investigation mindset.

Let's Investigate: KS2 Maths Activities

Over 24 KS2 Maths investigations designed for Years 5 and 6, including open-ended activities similar to NRich, with many possible ways to complete

### Practical ideas to lay the foundations of good maths investigation work in your school

If your primary curriculum already prizes curiosity and an investigative approach to learning then your pupils will already be at an advantage. Paced and supported appropriately you can start introducing maths investigations at Key Stage 1 or even in the early years, for 4 – 5 year olds.

Here are some ways in which you can lay those foundations

1. Inspire awareness of numbers and shapes in the environments and how real-world maths is all around us.
2. Teach and provide a variety of problem-solving strategies
3. Encourage children to work systematically.
4. Provide a range of rich tasks so children can ‘get inside’ the mathematics – go beyond worksheets.
5. Use low-threshold high ceiling tasks
6. Practise ‘guided maths’ akin to ‘guided reading’.
7. Encourage ‘flow-chart’ thinking and have pupils sequence steps when solving a puzzle.
8. Give children opportunities create categories for sorting and classifying.
9. Promote the regular use of Venn diagrams and Carroll diagrams for ‘graphical thinking’.
10. Include lots of strategic maths games.
11. Get children to create and pose their own maths problems (included open-ended problems).
12. Extend knowledge or apply knowledge in new contexts.
13. Make brainteasers a daily feature of maths to train logical thinking, you can easily add these to your usual teaching powerpoints or have handy print outs for students.
14. Let children of all age groups work within intriguing contexts.
15. Provide ideas for personal learning projects where children can explore, predict and experiment for themselves beyond the lesson plan.
16. Devote a part of the classroom to interactive puzzles, problems and posers.
17. Push STEAM (Science, Technology, Engineering, Arts and Maths) all the way so that children see and apply Maths across the curriculum.
18. Provide opportunities for children to see other people being mathematical or see the role of maths within different cultural settings .
19. Promote a conjecturing atmosphere.
20. Set up a range of ‘active assessment’ opportunities for collaborative dialogue rich talking tasks so children can explain, justify, verify, convince and refute.
21. Promote a range of questions and prompts to fuel divergent thinking.
22. Promote analogy, schematising, defining and deduction.
23. Give one or more counter-examples of a given concept.
24. Provide maths challenges and open-ended maths investigations to provoke children to clarify differences between related concepts.
25. Encourage regular reflective ‘journaling’ so children can transcribe and draw their ideas into a learning log.

A cerebral culture that develops and places a high value on investigative intelligence must be the starting point for any classroom hoping to cultivate a detective mindset.

Deductive reasoning is one of the most valuable skills a child can have and so this is an essential component of any maths investigative curriculum.

Mind benders, thoughts and crosses, logic puzzles, riddles  all help to support this critical thinking faculty and helps children see patterns, ask questions, learn and discover.

We’ve got lots more free teaching resources around KS2 maths problem solving investigations and real-life problems based around events like World Book Day available in the blog as well as on our maths hub if you want to explore further.

### What children need to know before they start

The most important thing we can help children to do is learn how to weed out the important information from the superfluous in any maths investigation.

In other words, they need to do what Sherlock would do and learn how to recognise vital facts from incidental facts.

Some investigations contain additional irrelevant information in their narratives that can throw children off the trial and it is this extraneous gunk that children should be trained to spot and bin. The power of observation is everything.

Weaning out the incidental facts from those that are relevant helps children crack the case. Knowing how to spot information and what is needed is the first step to becoming a sleuth Maths detective.

Nothing is useless unless you decide it is but identifying the importance of something is what children need to get good at. They need to consider everything and discard with efficiency.

Knowing how to sieve though the information and create a clear connection between numbers and words is the logical way to find a solution.

Children often mess up a question in their SATs because they aren’t paying attention to the finer details. They are trying to get things done quickly and because of that they either miss important information or focus on the smaller details that are ‘distractors’ or redundant and superfluous information. Cultivating new habits of slowing down and paying attention is vital.

A great habit to get children into is simply asking “Why is that there?” when looking at maths investigations and SATs style questions.

### Practice maths investigations

To build these skills in pupils, try including some of these maths activities and investigative maths questions for KS2 in your next few lessons.

#### Maths investigations Year 5

• There are 29 chapters in a book and 45 pages per chapter. The average number of words per page is 500. How many pages in total?
• A packet of Cheesy Wonder crisps weighs 28g. There are 16 crisps in a multipack costing £4.50. Samraaj buys 8 multipacks for his party. How many grams of crisps has he bought?

#### Maths investigations Year 6

• A pet food lorry makes a 296 km trip 56 times in 2 months. It’s average speed is 55 km/h. How far does the lorry travel in total?
• Miss Dickens, aged 22, needs to save £846 for her holiday to Menorca. She saves £26 a week. How many weeks will it take to save enough money for her holiday?
• A pack of 32 chocolate rolls weighs 1440g and costs £1.49. How much does one roll weigh?

### Strategies for tackling any maths investigation

If pupils struggle to know where to begin with investigative questions, you might want to try these techniques. After all, to successfully solve a problem, children need to know exactly what the problem is!

Children could try to:

• Draw a diagram
• Draw a table
• Act it out or use concrete material
• Guess and check
• Create an organised list
• Look for a pattern
• Create a tree diagram
• Work backwards
• Use simpler numbers
• Use logical reasoning
• Analyse and investigate
• Rephrase the question

Defining a problem and investigating a solution provides children with the chance to ‘do’ mathematics, to ‘make sense’ of their world and to ‘be mathematicians’.

When they discover ideas and invent procedures, children have a stronger conceptual understanding of connections between mathematical ideas.

### Maths investigations are for life, not just for Christmas

When children engage with investigative work they begin to see and appreciate the inner sanctum of ­­ and the true beauty of the subject. They can make inferences, get creative, challenge themselves and work with their peers and upgrade their own knowledge, understanding and skills.

We might assume that investigating, exploring and solving problems are central elements of the maths lessons in every classroom but this sadly isn’t the case everywhere.

I know of some schools that ‘reserve’ the last week of every half-term to a maths investigation but this is frankly not enough.

Investigations aren’t booked in like a table for four in a restaurant as a special treat. Investigations need to be a regular feature of Maths work so that the key Sherlock skills are exercised rather than awakened every now and then as some sort of bolt-on to satisfy an ‘added value’ tick-box.

However, you might want extra ones for Christmas: Christmas maths quiz

There is always time for a maths investigation because they can vary in length from just a few minutes to a whole lesson and beyond.

No time? No excuses – there is always time to build in an investigation and the Maths mind palaces need stocking up as ‘empty shelves’ don’t solve problems.

Investigations come in all shapes and sizes and may be framed variously as: a question to be answered, a problem to be solved; a task to be accomplished, or an issue to be explored.

Who says that an investigation has to be a whole lesson? It doesn’t. It could be something you start in the morning and finish in the afternoon.

Micro investigations can be some open-ended questions that have no one right answer or it could mean finding the same solution using different routes and pathways.

Small-scale investigations might involve similar open-ended questions but as part of a more complicated context. Macro investigations can involve a range of maths skills and concepts and be used as an ongoing teaching tool throughout the term that step by step scaffolds understanding.

Investigations are involving and interesting activities that absorb children. They stimulate holistic thinking, they can be integrated naturally in every part of the curriculum, they promote complex thinking but reinforce basic concepts, and they constitute an essential part of every mathematician’s work.

Maths investigation FAQs

What are maths investigations?

Maths investigations are situations or problems which require sustained mathematical enquiry: collecting, analysing and using information to make generalisations and conjecture relationships. They are different from word problems because rather than the focus being on the answer, the focus is on the process and journey pupils take to find their answers.

Why do we teach math through investigations?

Maths investigations encourage curiosity, debate and communication. They also require pupils to link different subject areas and mathematical topics to approach them.

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