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Addressing the five ‘big questions’ in problem-solving with NRICH

The importance of ensuring learners acquire the problem-solving skills which will enable them to thrive both socially and economically in their increasingly automated world is widely recognised (Luckin et al., 2017). Nevertheless, government inspectors have reported serious concerns about the quality and quantity of problem-solving in our schools (Ofsted, 2015). This summer schools were challenged to reflect on ‘Five big questions for problem-solving’ (EEF, 2021). In this blog, we will consider each of those five questions and explore the ways that the NRICH team is supporting schools to address them.

Question one: Do teachers in your school select genuine problem-solving tasks for which pupils do not already have a ready-made method available?

Too often, learners are presented with routine word problems which merely require the application of a known algorithm. ‘Genuine’ problems enable them to make their own problem-solving decisions by choosing their own strategies and enabling them to compare their approach with those of other learners, thus developing their problem-solving efficiency and flexibility. At NRICH , our award-winning activities allow learners to develop these key skills alongside the confidence to tackle genuine problems. Moreover, our ‘ low threshold, high ceiling ‘ approach enables everyone to get started on the problem while ensuring a suitable level of challenge too, making them ideal for whole-class teaching.

Question two: Are pupils given the opportunity to see – through multiple worked examples – to use, and to compare different approaches to solving a problem?

Many problems can be explored in more than one way. Working flexibly, making connections between different areas of the curriculum and reflecting on various problem-solving approaches are key steps towards becoming a more fluent mathematician. NRICH encourages learners to develop these skills in these two ways:

Our primary , secondary and post-16 Live Problems invite learners to explore and submit their ideas to the team. We review each submission that we receive and publish a selection on our website showcasing different approaches and the reasoning behind them.

Our NRIC H online activities sometimes feature ‘hide and reveal’ buttons showcasing different starting points towards a solution for learners to explore further for themselves. This approach enables learners to widen their range of strategies for solving unfamiliar problems and develop alternative approaches to explore when they get stuck using their first-choice strategy.

Question three: Are pupils encouraged to use visual representations to support them to solve a problem?

One of the most important approaches towards solving an unfamiliar problem is drawing a good diagram. Learning to draw diagrams is a skill which we encourage learners of all ages to develop alongside their other mathematical skills and knowledge. From sketching graphs to drawing a bar model, good diagrams can help learners clarify their understanding and identify possible ways forward.

Our four steps towards problem-solving feature highlights the importance of drawing a diagram to enable young learners to get started on a problem. We often highlight a useful diagram, table or sketch graph in the solutions chosen for publication. As learners progress through their learning, the team model more specific drawing skills, such as sketching a graph to help solve a STEP problem.

Question four: Are pupils supported to monitor, reflect on, and communicate their reasoning and choice of strategies, possibly through the use of prompt questions?

NRICH  encourages learners to reflect on their learning using this approach inspired by the Strands of Mathematical Proficiency model introduced by Kilpatrick et al. (2001).

Our approach uses child-friendly language that teachers and parents can share with students five key ingredients that characterise successful mathematicians. At NRICH , we believe that learning mathematics is about much more than just learning topics and routines. Successful mathematicians understand the curriculum content and are fluent in mathematical skills and procedures, but they can also solve problems, explain their thinking and have a positive attitude about themselves as learners of mathematics.

With this in mind, we have created  this self assessment tool  to help learners recognise where their mathematical strengths and weaknesses lie. We hope learners will explore NRICH activities and then take time to reflect on their own mathematical capabilities using our model.

Question 5: Is professional development time allocated to develop teachers’ pedagogical understanding of problem-solving, with particular support for early career teachers?

NRICH supports teachers to maximise the potential of our activities by offering free, regular professional development for teachers .  Each session is delivered online, enabling teachers to access the support wherever they are based, reducing teacher travel and cover costs for schools. We also record the sessions and upload them to our website so that schools can access them for future professional development days or staff/department meetings in their settings.

The live sessions are led by NRICH team members and they link directly to our latest primary , secondary and post-16 Live Problems. This approach enables teachers to consider the possibilities of the activities with the NRICH team before exploring them the next day with their own classes. Later, they are invited to share their classwork with our team for possible publication on the NRICH website.

The five ‘big questions’ provide excellent starting points for evaluating the teaching and learning of problem-solving in different settings. I hope that this blog shares an insight into the different ways that NRICH can support schools to address the five questions for themselves by engaging with our activities, Live Problems and teacher webinars.

Dr Ems Lord FCCT

Director of NRICH

Centre for Mathematical Sciences

University of Cambridge

Selected references

EEF. (2021). EEF Blog: Integrating evidence into maths teaching – guiding problem-solving. Accessed from https://educationendowmentfoundation.org.uk/news/eef-blog-integrating-evidence-into-mathematics-guiding-problem-solving /

Kilpatrick, J. Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics (Vol. 2101). J. Kilpatrick, & National research council (Eds.). Washington, DC: National Academy Press.

Luckin, R., Baines, E., Cukurova, M., Holmes, W., & Mann, M. (2017). Solved! Making the case for collaborative problem-solving. Accessed from http://oro.open.ac.uk/50105/1/solved-making-case-collaborative-problem-solving.pdf

Ofsted. (2015). Better Maths Conference Spring Keynote 2015. Accessed here https://www.slideshare.net/Ofstednews/better-mathematics-keynote-spring-2015

NRICH launches new Problem-Solving Schools initiative

Our NRICH programme has launched a new initiative to help schools prioritise problem-solving in maths. The NRICH Problem-Solving Schools programme will offer free resources, advice and teacher professional development training. 

Problem-solving is a critical skill when it comes to empowering students for the future. It opens up a world of possibilities as students have to apply their knowledge, think laterally and use their creativity to solve challenges. Problem-solving is also a crucial part of helping students to discover the joy of mathematics. "Imagine only engaging with football through practice drills and never seeing Messi score, or learning music through scales and never hearing an orchestra play," say the NRICH team.

For over 25 years NRICH has been at the forefront of developing mathematical problem-solving resources, reaching millions of users each year through the free NRICH website. The Problem-Solving Schools programme builds on this expertise and experience in supporting schools, teachers and students to help schools embed problem-solving in the curriculum.

Along with fluency and reasoning, problem-solving is central to the National Curriculum for maths, and the launch of NRICH's Problem-Solving Schools programme offers timely additional support for schools. A new report published by Ofsted in July 2023 encourages schools to focus more consistently on teaching problem-solving, and ensure that all students have opportunities to practise applying their knowledge in a range of problem-solving contexts.

To find out more about the programme, and how to register your school, see the NRICH Problem-Solving Schools information.

You can read more about the Problem-Solving Schools initiative in this University of Cambridge news story .

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Or search by topic

Number and algebra

• The Number System and Place Value
• Calculations and Numerical Methods
• Fractions, Decimals, Percentages, Ratio and Proportion
• Properties of Numbers
• Patterns, Sequences and Structure
• Algebraic expressions, equations and formulae
• Coordinates, Functions and Graphs

Geometry and measure

• Angles, Polygons, and Geometrical Proof
• 3D Geometry, Shape and Space
• Measuring and calculating with units
• Transformations and constructions
• Pythagoras and Trigonometry
• Vectors and Matrices

Probability and statistics

• Handling, Processing and Representing Data
• Probability

Working mathematically

• Thinking mathematically
• Mathematical mindsets
• Cross-curricular contexts
• Physical and digital manipulatives

For younger learners

• Early Years Foundation Stage

Advanced mathematics

• Decision Mathematics and Combinatorics
• Advanced Probability and Statistics

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InfoQ Homepage News How to Scale Agile Software Development with Technology and Lean

How to Scale Agile Software Development with Technology and Lean

May 30, 2024 3 min read

Ben Linders

Agile software development can be done at scale with the use of technology like self-service APIs, infrastructure provisioning, real-time collaboration software, and distributed versioning systems. Lean can complement and scale an agile culture with techniques like obeyas, systematic problem-solving, one-piece-flow and takt time, and kaizen. Fabrice Bernhard spoke about how their company uses technology with lean thinking for doing agile software development at scale at FlowCon France .

The agile manifesto doesn’t apply to large organizations, Bernhard stated. Leaders looking for principles to keep their culture agile while scaling their software organization will need to look elsewhere. And unfortunately that "elsewhere" is now crowded with options called "agile at scale", many of which are very bureaucratic and therefore not in the spirit of the agile manifesto, he mentioned.

Agile can scale, Bernhard said; there are many examples of organizations that scaled while maintaining an agile culture. In the body of knowledge of lean thinking, they found the principles they were looking for to scale their organization while staying true to the agile manifesto.

In the book The Lean Tech Manifesto that Bernhard wrote with Benoît Charles-Lavauzelle, he explores principles, systems, and tools that lean thinking provides to extend the principles of the agile manifesto. He mentioned some examples:

Value models, obeyas, and value streams, to scale "Customer Collaboration" by ensuring "Value for the Customer"’ becomes the North Star of the whole organization Systematic problem-solving with PDCA and 5S, supported by team leaders and enabled in our digital world by collaboration technology, to scale "individuals and interaction" and transform the organization into a "tech-enabled network of teams" Jidoka, dantotsu, poka-yoke, pull, one-piece-flow and takt time, to implement "right-first-time and just-in-time" and scale "working software" Standards, kaizen, skills matrix and communities of practice, to scale "responding to change" with "building a learning organization"

Bernhard mentioned that they felt that lean thinking didn’t fully explain how some large agile organizations were succeeding. He decided to explore how the Linux open-source project and its community scaled from 1 to 55,000 contributors, where they used technology to address the scaling issues that they faced along the way:

The first scaling crisis happened in 1996, when Linus wrote that he was "buried alive in emails". It was addressed by adopting a more modular architecture, with the introduction of loadable kernel modules, and the creation of the maintainers role, who support the contributors in ensuring that they implement the high standards of quality needed to merge their contributions.

The second scaling crisis lasted from 1998 to 2002, and was finally addressed by the adoption of BitKeeper, later replaced by Git. This distributed the job of merging contributions across the network of maintainers and contributors.

In both cases, technology was used to reduce the amount of dependencies between teams, help contributors keep a high level of autonomy, and make it easy to merge all those contributions back into the main repository, Bernhard said.

Technology can help reduce the need to communicate between teams whenever they have a dependency on another team to get their work done. Typical organizational dependencies, such as when a team relies on another team’s data, can be replaced by self-service APIs using the right technologies and architecture, Bernhard mentioned. This can be extended to more complicated dependencies, such as infrastructure provisioning, as AWS pioneered when they invented EC2, offering self-service APIs to spin up virtual servers, he added.

Another type of dependency is dealing with the challenge of merging contributions made to a similar document, whether it’s an illustration, a text, or source code, Bernhard mentioned. This has been transformed in the last 15 years by real-time collaboration software such as Google Docs and distributed versioning systems such as Git, he said.

Bernhard mentioned that he learned a lot from how the Linux community addressed its scaling issues. And where the first agile methodologies, such as Scrum or XP, focus on a single team of software engineers, lean thinking has been battle-tested at scale for decades in very large organizations, Bernhard said. Anyone trying to scale an agile organization should study lean thinking to benefit from decades of experience on how to lead large organizations while staying true to the spirit of the agile manifesto, he concluded.

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Rate this article, this content is in the culture & methods topic, related topics:.

• Culture & Methods
• Agile at Scale
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