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Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

Shametria Routt Banks

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2 Responses

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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The Problem-solving Classroom

• Visualising
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• Conjecturing
• Working systematically
• Looking for patterns
• Trial and improvement.

• stage of the lesson 
• level of thinking
• mathematical skill.
• The length of student response increases (300-700%)
• More responses are supported by logical argument.
• An increased number of speculative responses.
• The number of questions asked by students increases.
• Student - student exchanges increase (volleyball).
• Failures to respond decrease.
• 'Disciplinary moves' decrease.
• The variety of students participating increases.  As does the number of unsolicited, but appropriate contributions.
• Student confidence increases.
• conceptual understanding
• procedural fluency
• strategic competence
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Problem solving

Part of Mathematics and Numeracy Processes in mathematics

The ability to problem solve and make decisions for ourselves is a key thinking skill that is hugely important throughout life.

The greater your skill in this area, the better you are at searching for meaning; making predictions; generating possible solutions; justifying and understanding how you solved something; coping with challenges and making connections to things you have learnt in the past.

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Luckily, numeracy provides a wonderful opportunity to constantly sharpen these skills and to put them into practice. When faced with a problem in maths, there are four key steps to think about:

• What I know (Think) : read the problem and think about what you are being asked.
• What I need to know (Identify) : decide what maths strategies you will need to approach the problem. Do you have all the information? What steps are needed?
• What I need to do (Employ) : use your maths strategies to solve the problem.
• What I did (Review) : Were you successful in solving the problem? Did you have any difficulties?

How to use column subtraction

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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

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• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

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The resources on this page will hopefully help you teach AO2 and AO3 of the new GCSE specification - problem solving and reasoning.

This brief lesson is designed to lead students into thinking about how to solve mathematical problems. It features ideas of strategies to use, clear steps to follow and plenty of opportunities for discussion.

The PixiMaths problem solving booklets are aimed at "crossover" marks (questions that will be on both higher and foundation) so will be accessed by most students. The booklets are collated Edexcel exam questions; you may well recognise them from elsewhere. Each booklet has 70 marks worth of questions and will probably last two lessons, including time to go through answers with your students. There is one for each area of the new GCSE specification and they are designed to complement the PixiMaths year 11 SOL.

These problem solving starter packs are great to support students with problem solving skills. I've used them this year for two out of four lessons each week, then used Numeracy Ninjas as starters for the other two lessons.  When I first introduced the booklets, I encouraged my students to use scaffolds like those mentioned here , then gradually weaned them off the scaffolds. I give students some time to work independently, then time to discuss with their peers, then we go through it as a class. The levels correspond very roughly to the new GCSE grades.

Some of my favourite websites have plenty of other excellent resources to support you and your students in these assessment objectives.

@TessMaths has written some great stuff for BBC Bitesize.

There are some intersting though-provoking problems at Open Middle.

I'm sure you've seen it before, but if not, check it out now! Nrich is where it's at if your want to provide enrichment and problem solving in your lessons.

MathsBot  by @StudyMaths has everything, and if you scroll to the bottom of the homepage you'll find puzzles and problem solving too.

I may be a little biased because I love Edexcel, but these question packs are really useful.

The UKMT has a mentoring scheme that provides fantastic problem solving resources , all complete with answers.

I have only recently been shown Maths Problem Solving and it is awesome - there are links to problem solving resources for all areas of maths, as well as plenty of general problem solving too. Definitely worth exploring!

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Engaging Maths

Dr catherine attard, promoting creative and critical thinking in mathematics and numeracy.

• by cattard2017
• Posted on June 25, 2017

What is critical and creative thinking, and why is it so important in mathematics and numeracy education?

Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.

The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies : Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities , one of which is Critical and Creative Thinking – there’s no excuse!

Critical and creative thinking need to be embedded in every mathematics lesson . Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.

How and when do we embed critical and creative thinking?

There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys  are also very worthwhile tasks. For good mathematical problems go to the nrich website . For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300   (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.

Will critical and creative thinking take time away from teaching basic concepts?

No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:

• higher-level thinking within authentic and meaningful contexts;
• complex problem solving;
• open-ended responses; and
• substantive dialogue and interaction.

Who should be engaging in critical and creative thinking?

Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.

Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.

Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.

How will you and your students engage in critical and creative thinking?

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Numeracy for all learners

Numeracy is the knowledge, skills, behaviours and dispositions that students need in order to use mathematics in a wide range of situations. It involves recognising and understanding the role of mathematics in the world and having the dispositions and capacities to use mathematical knowledge and skills purposefully.  (Literacy and numeracy strategy version 2).

Number, measurement and geometry, statistics and probability are common aspects of most people’s mathematical experience in everyday personal, study and work situations. Equally important are the essential roles that algebra, functions and relations, logic, mathematical structure and working mathematically play in people’s understanding of the natural and human worlds, and the interaction between them. 

Why numeracy is important

A child's first years are a time of rapid learning and development. Babies and toddlers can recognise number, patterns, and shapes. They use maths concepts to make sense of their world and connect these concepts with their environment and everyday activities. For example, when playing, children may sort or choose toys according to size, shape, weight or colour.

While much of the teaching of concepts and skills to support numeracy happens in the mathematics learning area, it is strengthened as students take part in activities that connect their learning in the mathematics classroom within the context of other curriculum areas.

As they move through their years of schooling, students are exposed to mathematical:

• understanding
• problem solving

These capabilities allow students to respond to familiar and unfamiliar situations by employing mathematics to make informed decisions and solve problems efficiently (VCAA, 2017).

There is also evidence that other areas of development, such as resilience and perseverance, support achievement in numeracy.

Mathematics gives students access to important mathematical ideas, knowledge and skills. Numeracy connects this learning with their personal and work lives.

Numeracy has an increasingly important role in enabling and sustaining cultural, social, economic and technological advances.

Numeracy development

For an overview of numeracy development see  mapping the numeracy focus areas . Resources in the guide are organised by levels:

• Birth to Level 2
• Levels 3 to 8
• Levels 9 to 10

Numeracy across the curriculum

Being numerate involves more than mastering basic mathematics. Numeracy involves connecting the mathematics that students learn at school with the out-of-school situations that require the skills of problem solving, critical judgement, and sense-making related to applied contexts.

Conceptual framework

Learning activities presented draw upon the conceptual framework of Goos, Geiger, and Dole (2014; also discussed in Goos, Geiger, Dole, Forgasz, and Bennison, 2019). In this framework, numeracy is conceptualised as comprising four elements and an orientation: 

Element 1:  Attention to real-life contexts (citizenship, work, and personal and social life)

Element 2:  Application of mathematical knowledge (problem solving, estimation, concepts, and skills)

Element 3:  Use of tools (representational, physical, and digital)

Element 4:  The promotion of positive dispositions towards the use of mathematics to solve problems encountered in day-to-day life (confidence, flexibility, initiative, and risk)

Orientation:  A critical orientation to interpreting mathematical results and making evidence-based judgements

The resources highlight what numeracy is with respect to each learning area, and outline why it is important to develop students' numeracy capabilities within the learning area. Guidance is provided for teachers on the following:

• how to embed numeracy in their learning area
• how to assess numeracy learning
• how to deal with challenges and dilemmas using strategies recommended by experts.

The activities are described in terms of subject-specific learning intentions and content descriptors. The numeracy content and skills are highlighted and explained, with particular focus on how the numeracy links enhance the learning area's specific concepts. Direct links to the Victorian Curriculum: Mathematics highlight the connections between the activity and the students' previously developed mathematical skills and understandings. The VCAA have detailed information regarding the numeracy demands of the Victorian Curriculum on the Numeracy page of the website.

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Early childhood numeracy and mathematics resource

Mathematics is everywhere.

We all use mathematics to navigate our everyday decisions successfully. Children begin to experience and explore mathematical concepts from birth. With support, they participate in mathematical thinking and use mathematical concepts to organise, record and communicate ideas about the world around them.

Understanding and using mathematical concepts, and being numerate, helps children know and describe the world around them and make meaning of these encounters. It is, therefore, an essential skill for successful daily life. Research and practice evidence suggest that mathematics and numeracy skills will support children to be confident and capable learners as they navigate the increasingly complex global community of the 21st century.

Children who are confident and involved learners have positive dispositions toward learning, experience challenge and success in their learning and are able to contribute positively and effectively to others children’s learning. . . .They develop and use their imagination and curiosity as they build a ‘toolkit’ of skills and processes to support problem solving, hypothesising, experimenting researching and investigating (VEYLDF, 2016)

Families and educators play a critical role in introducing children to mathematics and encouraging them to be curious and enthusiastic about mathematics. From a very young age, adults invite children to use mathematics to understand and participate in their world.

Would you like another piece of toast? We need to find the other shoe – we need one for each foot! How old are you today – three – happy birthday! How many plates do we need? We live at number 36.

Building children’s confidence in understanding and using mathematics to explore and know the world will benefit everyone. Children benefit from many opportunities to generate and discuss ideas, make plans, exercise skills, engage in sustained shared thinking, generate solutions to problems, reflect and give reasons for their choices. Children who are confident and involved learners have positive dispositions toward learning, and experience challenge and success in their learning.

Numeracy in early childhood

Numeracy is the capacity, confidence and disposition to use mathematics in daily life. Children bring new mathematical understandings through engaging in problem-solving. The mathematical ideas with which young children interact must be relevant and meaningful in the context of their current lives. Spatial sense, structure and pattern, number, measurement, data argumentation, connections and exploring the world mathematically are the powerful mathematical ideas children need to become numerate (EYLF p. 38).

When educators consider including mathematics and numeracy in early childhood programs, there is often confusion about the relevance of concepts such as algebra or statistics. Children are active learners, exploring the world and beginning to develop explanations for observed phenomena from a young age. With encouragement, guidance, experience and learning, children further develop their capacity to reflect on their own thinking processes, approaches to learning and using mathematics in their everyday engagement with their world..This resource illustrates the variety of ways that educators, working with children from birth to age five, can support numeracy learning and development.  Presented across three key mathematical concepts; Number and Algebra; Measurement and Geometry; Statistics and Probability (reflective of the Victorian Early Years Learning and Development Framework and the Victorian Curriculum) and organised to consider children's learning from birth to age five; early childhood educators are offered ideas for learning experiences, ways to engage families and opportunities for intentional teaching.

The suggestions included in this resource represent only some recommendations to help educators strengthen and enhance numeracy learning in programs for young children. Educators will have their own ideas that will complement this collection and are encouraged to work with their colleagues, as well as children and families, to expand their ideas and resources. Links to a range of resources are included that offer additional materials for further consideration.

Number and Algebra

Number and Algebra for young children involves exploring mathematical concepts such as patterns, symbols, and relationships. A large part of learning in this area involves using numbers in everyday contexts, counting objects and understanding how the numbers combine and connect to describe the world and help us to make meaning.

Children are engaging with number and algebra when they: 

• use mathematical words to describe the world. E.g. ‘lots of’, ‘more than'
• use numbers to count and refer to objects and people in their lives. E.g. 'I'm three years old, 'I have two trucks at home'
• use numbers to solve problems. E.g. ‘I need another glass for the table’
• begin to count objects in a sequence and recognise the way numbers work. 

Measurement and Geometry

Measurement and Geometry for young children involves exploring mathematical concepts such as the size, shape, position and dimensions of objects. A large part of learning in this area involves becoming familiar with and using numbers and words to describe objects and know the difference between objects.

Children are engaging with measurement and geometry when they:

• feel different shaped items
• sort objects according to their shape
• draw shapes in their art
• describe the world around them using concepts such as ‘I like the circle one’ or ‘I put my hat in the big basket’ or ’the snake was really long.’

Statistics and Probability

Statistics and Probability for young children involves sorting, understanding and presenting information from groups of objects in order to understand what is happening.   

Probability is about understanding the chance of something occurring and making decisions based on that thinking.

Children are engaging with statistics and probability when they:

•  collect and sort ideas or groups of objects into categories
• talk about whether they need to take a coat with them when they go on a walk. E.g. ‘Is it going to rain?’

Early childhood educators' beliefs on mathematical learning

Educators’ own beliefs and attitudes towards mathematics and numeracy have a significant impact on the way these ideas are incorporated into programs for children. Increasing numbers of studies (Anders & Robbach, 2015) (Australian Mathematical Sciences Institute, 2018) have identified that many early childhood educators have had negative mathematics experiences in their schooling and therefore believe they will not be able to support children in this area adequately. It is important for adults to reflect on their anxiety in relation to mathematics and shift their perception towards the potential that mathematics provides to make their lives more meaningful. Many early childhood educators are competent users of mathematical concepts, and their numeracy skills are excellent however, these are not always recognised as a positive and necessary part of their daily lives.

Families play a crucial part in the development of children's mathematics and numeracy learning. As is the case for educators, family members’ own beliefs and attitudes towards mathematics and numeracy influence the way that children feel about engaging with and developing their mathematics and numeracy skills. Since numeracy in the early years is so highly connected to daily life and the way we make meaning of the world, families can provide opportunities to explore mathematics and support children to become confident about their mathematics and numeracy learning.

Educators can encourage families to recognise their role in supporting children’s mathematics and numeracy learning in many ways; from formal communication with families (in a family handbook for example or newsletters) about  how they can support children at home to informal conversations that promote positive attitudes and reinforce responses to children that help build their confidence. When educators maintain a commitment to sharing ideas with families about children’s mathematics and numeracy, learning outcomes are more likely to progress.  

Throughout this resource, learning experiences have been identified that are specifically designed for families to try at home. Educators are encouraged to share these ideas with families in their regular communications.  

• The Victorian Numeracy Learning Progressions - helps schools and teachers, in all learning areas, to support their students to engage with the numeracy demands of the Victorian Curriculum F –10.
• F-10 Victorian Curriculum Mathematics glossary - definitions and examples of mathematical vocabulary.

Human Capital Working Group, Council of Australian Government. (2018). National Numeracy Review Report. Canberra: Commonwealth of Australia.         

Jonas, N. (2018). Numeracy practices and numeracy skills among adults. Paris: Organisation for Economic Co-operation and Development.         

Shomos, A., & Forbes, M. (2014). Literacy and Numeracy Skills and Labour Market Outcomes in Australia. Canberra: Productivity Commision Staff working paper.

Attard, C. (2020, Jan 21). Mathematics education in Australia: New decade, new opportunities? Retrieved from Engaging Maths: https://engagingmaths.com/2020/01/21/mathematics-education-in-australia-new-decade-new-opportunities/

Buckley, S. (2011). Deconstructing maths anxiety: Helping students to develop a positive attitude towards learning maths. Retrieved from ACER: https://www.acer.org/au/occasional-essays/deconstructing-maths-anxiety-helping-students-to-develop-a-positive-attitud

Church, A., Cohrssen, C., Ishimine, K., & Tayler, C. (2013). Playing with maths: Facilitating the learning in play-based learning. Australasian Journal of Early Childhood Volume 38 Number 1 March 2013, 95-99.

Cohrssen, C. (2018, June 6). Assessing children’s understanding during play-based maths activities. Canberra, ACT, Australia. Retrieved from http://thespoke.earlychildhoodaustralia.org.au/assessing-childrens-understanding-during-play-based-maths-activities/

DEEWR. (2009). Belonging, Being and Becoming: The early years learning framework for Australia. Canberra: Commonwealth of Australia.

Department of Education and Training . (2012). Integrated Teaching and Learning Approaches Practice Principle Guide 6 . Melbourne : Department of Education and Early Childhood Development).

Department of Education and Training. (2016). Victorian Early Years Learning and Development Framework . Melbourne: Department of Education and Training.         

Knaus, M. (2016). Maths is All Around You: Developing Mathematical Concepts in Early Years . Blairgowrie: Teaching Solutions .        

NAEYC. (2020). Math Talk with Infants and Toddlers. Washington, USA. Retrieved from https://www.naeyc.org/our-work/families/math-talk-infants-and-toddlers

Vogt, F., Hauser, B., Stebler, R., Rechsteiner, K., & Urech, C. (2018). Learning through play – pedagogy and learning outcomes in. EUROPEAN EARLY CHILDHOOD EDUCATION RESEARCH JOURNAL, 589-603.

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Problem Solving

The Ministry is migrating nzmaths content to Tāhurangi.             Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz).  When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024.  e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available. 

For more information visit https://tahurangi.education.govt.nz/updates-to-nzmaths

This section of the nzmaths website has problem-solving lessons that you can use in your maths programme. The lessons provide coverage of Levels 1 to 6 of The New Zealand Curriculum. The lessons are organised by level and curriculum strand.  Accompanying each lesson is a copymaster of the problem in English and in Māori. 

Choose a problem that involves your students in applying current learning. Remember that the context of most problems can be adapted to suit your students and your current class inquiry. Customise the problems for your class.

• Level 1 Problems
• Level 2 Problems
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• Level 4 Problems
• Level 5 Problems
• Level 6 Problems

The site also includes Problem Solving Information . This provides you with practical information about how to implement problem solving in your maths programme as well as some of the philosophical ideas behind problem solving. We also have a collection of problems and solutions for students to use independently.

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Mathematical Challenges for Able Pupils in Key Stages One and Two

This book of challenges for more able students from the National Numeracy Strategy contains puzzles and problems. These are accessible to a wide range of students. There are four separate files covering Years One and Two , Years Three and Four , and Years Five and Six  and the solutions.

The problems are intended to challenge students and extend their thinking. While some of them may be solved fairly quickly, others will need perseverance and may extend beyond a single lesson. Students may need to draw on a range of skills to solve the problems. These include: working systematically, sorting and classifying information, reasoning, predicting and testing hypotheses, and evaluating the solutions.

Many of the problems can be extended by asking questions such as: ‘What if .....?’. Problems can also be extended by asking students to design similar problems of their own to give to their friends or families.

Learning objectives appropriate to each problem are indicated so relevant problems can be integrated into the main teaching programme.

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Please note you do not have access to teaching notes, developing numeracy and problem-solving skills by overcoming learning bottlenecks.

Journal of Applied Research in Higher Education

ISSN : 2050-7003

Article publication date: 21 January 2019

Issue publication date: 18 June 2019

The purpose of this paper is to present an educational approach to elevating problem-solving and numeracy competencies of business undergraduates to meet workplace demand. The approach is grounded in the theory of constraints following the Decoding the Discipline model. The authors investigated a cognitive bottleneck involving problem modeling and an affective bottleneck concerning low self-efficacy of numeracy and designed specific interventions to address both bottlenecks simultaneously. The authors implemented the proposed approach in an introductory level analytics course in business operations.

Design/methodology/approach

The authors use an empirical study to evaluate the effectiveness of the proposed approach in addressing deficiency in numeracy and problem-solving skills. Cognitive and affective learning interventions were introduced in an undergraduate core course in analytics. The perceived effectiveness of the interventions was evaluated with the use of a survey at the end of the course. To further investigate the effectiveness of the proposed interventions beyond self-reporting, the impact of the interventions on actual learning was evaluated by comparing the exam scores between classes with and without the interventions.

Students who underwent the interventions successfully overcame both learning bottlenecks and indicated a positive change in attitude toward the analytics discipline as well as achieved higher exam scores in the analytics course.

Research limitations/implications

This study succeeds in strengthening the body of research in teaching and learning. The authors also offer a holistic treatment of cognitive and affective learning bottlenecks, and provide empirical evidence to support the effectiveness of the proposed approach in elevating numeracy and problem-solving competencies of business undergraduates.

Practical implications

The proposed approach is useful for business educators to improve business students’ quantitative modeling skill and attitude. Researchers can also extend the approach to other courses and settings to build up the body of research in learning and skill development. Educational policy makers may consider promoting promising approaches to improve students’ quantitative skill development. They can also set a high standard for higher education institutions to assess students’ numeracy and problem-solving competencies. Employers will find college graduates bring to their initial positions the high levels of numeracy and problem-solving skills demanded for knowledge work to sustain business growth and innovation.

Social implications

As students’ numeracy and problem-solving skills are raised, they will develop an aptitude for quantitative-oriented coursework that equips them with the set of quantitative information-processing skills needed to succeed in the twenty-first century society and global economy.

Originality/value

The proposed approach provides a goal-oriented three-step process to improve learning by overcoming learning bottlenecks as constraints of a learning process. The integral focus on identifying learning bottlenecks, creating learning interventions and assessing learning outcomes in the proposed approach is instrumental in introducing manageable interventions to address challenges in student learning thereby elevating students’ numeracy and problem-solving competencies.

• Problem solving
• Theory of constraints
• Decoding the Discipline
• Learning bottleneck
• Workplace skills
• Analytics education

Lee-Post, A. (2019), "Developing numeracy and problem-solving skills by overcoming learning bottlenecks", Journal of Applied Research in Higher Education , Vol. 11 No. 3, pp. 398-414. https://doi.org/10.1108/JARHE-03-2018-0049

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Teens come up with answer to problem that stumped math world for centuries

A s the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson: So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.