problem solving nz math

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Problem Solving

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
• Level 3 Problems
• 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.

What is Problem Challenge?

Most schools with year 7 and 8 children, and some with year 6 children, are on our mailing list and will automatically receive an invitation, in mid-February, to take part in the competition.

The competition has been organised by John Curran and John Shanks, retired members of the Department of Mathematics and Statistics at the University of Otago, with huge administrative help from Leanne Kirk. However John Curran and Leanne retired from the competition at the end of 2023; John Shanks will attempt to run it from 2024, with help from Sarah Stewart handling the book orders.

The value of such problem solving competitions is well recognized overseas. For example, similar schemes are run in Australia, Britain and the United States. Here the New Zealand Curriculum, Mathematics Standards (years 1-8), considers various ways in which effective mathematics teaching can provide quality programmes. Amongst other components, problem-centred activities are highlighted. The document states Cross-national comparisons show that students in high-performing countries spend a large proportion of their class time solving problems. The students do so individually as well as co-operatively. The problems we pose allow children to practise and learn such simple strategies as guessing and checking, drawing a diagram, making lists, looking for patterns, classifying, etc. Although children answer the questions individually on our sets there is ample opportunity for co-operative practise using our resources.

How does it work?

Children participating in the competition attempt to answer five questions in 30 minutes on each of five problem sheets, which are done about a month apart. They do the problems individually but they can share their answers and strategies in small groups afterwards.

Note that all three levels (years 6, 7 and 8) attempt the same problem set although there are separate awards for each of those levels.

The problems are generally aimed at more able children. However, we hope to keep the first question or two reasonably straightforward, so that all children entered can have some success. Many schools that have taken part before will have a good idea of the standard involved. Here are two recent example sets as a guide.

As a general rule, teachers may wish to enter children for whom they feel a score of say 3 out of 5 is an attainable goal. We felt the problems set last year were about the right level of difficulty, so we will be aiming for much the same standard this year. For schools that want more information, there are five books available that give questions and solutions from the first 24 years of the competition. These books can be obtained by completing the order form .

What must the teacher do?

For each of the 5 problem sets that you receive, you will have to photocopy (or otherwise make available) sufficient copies of the problem sheet for the participants from your school, and administer the challenge on the day specified (or as near as possible).

You must mark the pupil responses (using the solutions provided) and return collated results to us, as well as keeping a record of your results (using your own spreadsheet or on a form provided).

• Results are returned to us on-line . Further details of this together with a log-in code for your school will be supplied with the first set of problems.
• All competition material, including sets, solutions and letters, will be emailed to schools or made available on the website .
• Certificates will be provided in electronic form for schools to print.

How does your school benefit?

The problem sets may be used later as a resource for other children in any way the teacher wishes. For example, small groups could solve the problems co-operatively together, talking through the various strategies that could be applied to each question.

For each set you will receive a summary of the overall results, so that you can evaluate your pupils’ progress. In the past we have received very favourable feedback on the benefit of this. (Individual school results will not be collated or publicised so will remain strictly confidential to you.) Overall results from previous years can be seen here .

All children taking part will receive a certificate of participation. Those in about the top 10% in each year will receive certificates of excellence and those in the next 25% or so will receive certificates of merit. Where schools have provided on-line results, the childrens’ names will already be on the certificates.

Each year $25 book tokens are awarded to children in the top 1% or so of the competition. Note that book tokens are normally given to a maximum of 20% of the entries from each school.

When is Problem Challenge held?

As in previous years there will be a Problem Challenge each month from April to August, spaced at about five week intervals. This year’s administration days can be found here. However, as in the past, there is some flexibility in these dates and no school is precluded from entry on account of the timing. This is explained more fully if you enter.

How much does it cost and how do you enter?

The entry fee consists of $20 per school plus $0.40 per child entered (including GST). We will be mailing all Intermediate schools each February asking for entries: at that stage, if you wish to take part, you will need to register on-line and arrange to pay the registration fee (by credit card or University invoice).

Online registration is available between 19 February and 11 April.

The final challenge.

Children who do particularly well in Problem Challenge during the year are invited to enter a final multi-choice competition in late October. Note that, because of limited resources and in fairness to all, we regret that only those who reach a specified total number of problems correct (regardless of absence, sickness, etc) will be eligible to enter.

Final Challenge provides a great challenge for the very able, and there are more substantial prizes for the best performers at both Years. The competition consists of 10 multiple-choice questions, with five options per question, together with 10 questions that require explicit answers. The problems are similar in style to the usual Problem Challenge questions but generally of a standard comparable to question 5 on the Problem Sets or harder.

• Children have one hour in which to attempt the questions.
• The use of calculators is not permitted.

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Resources designed for the NZ learner

Keep your students on the right track with resources designed for the New Zealand Curriculum and maths problems based on real-world situations. Lessons are directly linked to NZ National Curriculum objectives, acknowledge Aotearoa NZ’s bicultural foundations, and cover all number and non-number topics.

Consistency from start to finish

Give your learners the right learning experience at the right time. Using a tried-and-tested spiral methodology, topics build on one another to help learners develop mathematical fluency. Content is covered in an age-appropriate order and revisited to close conceptual gaps and enrich every learner’s experience.

Assessment in every lesson

Get a clear view of progress. During lessons, non-negotiable learning objectives give you the information you need to support struggling learners and stretch advanced ones. Varied activities encourage students to challenge their thinking through collaborative discussion while independent exercises help learners build on their existing knowledge.

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Teach maths for mastery with confidence. From daily lesson structure to extension activities and common misconceptions, online Teacher Guides support you in every lesson. Not sure when to introduce concrete manipulatives or where to differentiate content? The answers you need are all in one place.

Meet the characters

Illustrated characters act as learning companions, guiding students through content and providing familiarity as problems become more complex. Characters from New Zealand, as well as local flora and fauna help learners understand maths’ place in the world around them.

What to expect when adopting a mastery approach

Here’s an inside look at some of the ways mastery has shifted attitudes towards primary maths in this teacher’s classroom.

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Develop your learners’ metacognition with strategies like the CPA approach, bar modelling, number bonds, and effective questioning.

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[Students] are having fun with maths. They can talk about what they’re doing, [show] deeper understanding… and clearly explain their thinking… This is working for all kids. – Marie Nelson, Maths Lead, St Paul's Primary School Auckland, New Zealand

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Problem Solving

Problem Solving and the New Curriculum   Age 5 to 11

Developing a Classroom Culture That Supports a Problem-solving Approach to Mathematics   Age 5 to 11

Developing Excellence in Problem Solving with Young Learners   Age 5 to 11

Using NRICH Tasks to Develop Key Problem-solving Skills   Age 5 to 11

Trial and Improvement at KS1   Age 5 to 7

Trial and Improvement at KS2   Age 7 to 11

Working Systematically - Primary Teachers   Age 5 to 11

Number Patterns   Age 5 to 11

Working Backwards at KS1   Age 5 to 7

Working Backwards at KS2   Age 7 to 11

Reasoning   Age 5 to 11

Visualising at KS1 - Primary Teachers   Age 5 to 7

Visualising at KS2 - Primary Teachers   Age 7 to 11

Conjecturing and Generalising at KS1 - Primary Teachers   Age 5 to 7

Conjecturing and Generalising at KS2 - Primary Teachers   Age 7 to 11

• Mathematical Problem Solving in the Early Years
• Low Threshold High Ceiling - an Introduction
• What's All the Talking About?
• Group-worthy Tasks and Their Potential to Support Children to Develop Independent Problem-solving Skills
• Developing the Classroom Culture: Using the Dotty Six Activity as a Springboard for Investigation

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Google DeepMind’s new AI systems can now solve complex math problems

AlphaProof and AlphaGeometry 2 are steps toward building systems that can reason, which could unlock exciting new capabilities.

• Rhiannon Williams archive page

AI models can easily generate essays and other types of text. However, they’re nowhere near as good at solving math problems, which tend to involve logical reasoning—something that’s beyond the capabilities of most current AI systems.

But that may finally be changing. Google DeepMind says it has trained two specialized AI systems to solve complex math problems involving advanced reasoning. The systems—called AlphaProof and AlphaGeometry 2—worked together to successfully solve four out of six problems from this year’s International Mathematical Olympiad (IMO), a prestigious competition for high school students. They won the equivalent of a silver medal.

It’s the first time any AI system has ever achieved such a high success rate on these kinds of problems. “This is great progress in the field of machine learning and AI,” says Pushmeet Kohli, vice president of research at Google DeepMind, who worked on the project. “No such system has been developed until now which could solve problems at this success rate with this level of generality.” 

There are a few reasons math problems that involve advanced reasoning are difficult for AI systems to solve. These types of problems often require forming and drawing on abstractions. They also involve complex hierarchical planning, as well as setting subgoals, backtracking, and trying new paths. All these are challenging for AI. 

“It is often easier to train a model for mathematics if you have a way to check its answers (e.g., in a formal language), but there is comparatively less formal mathematics data online compared to free-form natural language (informal language),” says Katie Collins, an researcher at the University of Cambridge who specializes in math and AI but was not involved in the project. 

Bridging this gap was Google DeepMind’s goal in creating AlphaProof, a reinforcement-learning-based system that trains itself to prove mathematical statements in the formal programming language Lean. The key is a version of DeepMind’s Gemini AI that’s fine-tuned to automatically translate math problems phrased in natural, informal language into formal statements, which are easier for the AI to process. This created a large library of formal math problems with varying degrees of difficulty.

Automating the process of translating data into formal language is a big step forward for the math community, says Wenda Li, a lecturer in hybrid AI at the University of Edinburgh, who peer-reviewed the research but was not involved in the project. 

“We can have much greater confidence in the correctness of published results if they are able to formulate this proving system, and it can also become more collaborative,” he adds.

The Gemini model works alongside AlphaZero —the reinforcement-learning model that Google DeepMind trained to master games such as Go and chess—to prove or disprove millions of mathematical problems. The more problems it has successfully solved, the better AlphaProof has become at tackling problems of increasing complexity.

Although AlphaProof was trained to tackle problems across a wide range of mathematical topics, AlphaGeometry 2—an improved version of a system that Google DeepMind announced in January—was optimized to tackle problems relating to movements of objects and equations involving angles, ratios, and distances. Because it was trained on significantly more synthetic data than its predecessor, it was able to take on much more challenging geometry questions.

To test the systems’ capabilities, Google DeepMind researchers tasked them with solving the six problems given to humans competing in this year’s IMO and proving that the answers were correct. AlphaProof solved two algebra problems and one number theory problem, one of which was the competition’s hardest. AlphaGeometry 2 successfully solved a geometry question, but two questions on combinatorics (an area of math focused on counting and arranging objects) were left unsolved.   

“Generally, AlphaProof performs much better on algebra and number theory than combinatorics,” says Alex Davies, a research engineer on the AlphaProof team. “We are still working to understand why this is, which will hopefully lead us to improve the system.”

Two renowned mathematicians, Tim Gowers and Joseph Myers, checked the systems’ submissions. They awarded each of their four correct answers full marks (seven out of seven), giving the systems a total of 28 points out of a maximum of 42. A human participant earning this score would be awarded a silver medal and just miss out on gold, the threshold for which starts at 29 points. 

This is the first time any AI system has been able to achieve a medal-level performance on IMO questions. “As a mathematician, I find it very impressive, and a significant jump from what was previously possible,” Gowers said during a press conference. 

Myers agreed that the systems’ math answers represent a substantial advance over what AI could previously achieve. “It will be interesting to see how things scale and whether they can be made faster, and whether it can extend to other sorts of mathematics,” he said.

Creating AI systems that can solve more challenging mathematics problems could pave the way for exciting human-AI collaborations, helping mathematicians to both solve and invent new kinds of problems, says Collins. This in turn could help us learn more about how we humans tackle math.

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A.I. Can Write Poetry, but It Struggles With Math

A.I.’s math problem reflects how much the new technology is a break with computing’s past.

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By Steve Lohr

In the school year that ended recently, one class of learners stood out as a seeming puzzle. They are hard-working, improving and remarkably articulate. But curiously, these learners — artificially intelligent chatbots — often struggle with math.

Chatbots like Open AI’s ChatGPT can write poetry, summarize books and answer questions, often with human-level fluency. These systems can do math, based on what they have learned, but the results can vary and be wrong. They are fine-tuned for determining probabilities, not doing rules-based calculations. Likelihood is not accuracy , and language is more flexible, and forgiving, than math.

“The A.I. chatbots have difficulty with math because they were never designed to do it,” said Kristian Hammond, a computer science professor and artificial intelligence researcher at Northwestern University.

The world’s smartest computer scientists, it seems, have created artificial intelligence that is more liberal arts major than numbers whiz.

That, on the face of it, is a sharp break with computing’s past. Since the early computers appeared in the 1940s, a good summary definition of computing has been “math on steroids.” Computers have been tireless, fast, accurate calculating machines. Crunching numbers has long been what computers are really good at, far exceeding human performance.

Traditionally, computers have been programmed to follow step-by-step rules and retrieve information in structured databases. They were powerful but brittle. So past efforts at A.I. hit a wall.

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Coding camp helps middle schoolers in Montgomery Co. recover from pandemic setbacks

by Kellye Lynn

MONTGOMERY COUNTY, Md. (7News) — Navika Duvey is a rising seventh grader who is expanding her understanding of coding at summer camp.

"Problem-solving, those skills you learn in math is also in coding," she told 7News.

It's an important connection as Montgomery County Public Schools students and those around the country make slow progress in recovering from pandemic-related academic losses.

READ| Delta struggles with baggage, delays after global IT outage leaves travelers stranded

The national decline in math scores has created what education experts call a data literacy crisis.

"A lot of our students are coming in at different levels, and basically the levels that they’re coming in at they might not have the skills to show their level of performance," explained educator and camp on-site coordinator Yvette Edwards.

For three weeks this summer, more than 800 Montgomery County Public Schools students are strengthening essential critical thinking and problem-solving skills through the Montgomery Can Code camp.

"To have an introduction to Apple Swift coding and learning app design," said Kimberly Bloch-Rincan, Director of ignITe Hub which runs Montgomery Can Code.

The free camp offered at three Montgomery College campuses is also inspiring these middle school students to pursue tech careers.

READ| Target stops taking checks; business expert thinks fighting fraud is the reason

11-year-old Veer Amin hopes to someday work as a robotics engineer.

"I love coding and I think coding’s really important to what you can do ahead in life," he said.

"A lot of computer and AI things are going to be very important in the future so I just want to get ahead of the game," student Wilson Dilone told 7News.

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