new zealand math problem solving

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• Manukau Institute of Technology

Mathematics

Problem solving.

• Introduction
• Basic Maths Skills
• Steps for Mathematics Problem Solving One-page guide with steps for solving mathematics problem (downloadable form of this webpage)

Mathematics is full of problems that need to be solved. A problem may be about solving an equation, calculating an amount of something, finding an answer to a word problem, etc. You can use problem-solving strategies to help you work out the correct answer for mathematics problems in your assignments. See the steps below:

Steps for Mathematics Problem Solving

1. Read the question carefully to the end

Example: Find the volume of fluid which can be contained inside a 2.1 m length of cylindrical pipe with diameter 120mm. The walls of the pipe are 2 mm thick.  Write your answer in cubic metres. Round your final answer to 3 decimal places.

2. Identify the question itself (underline or highlight what you are being asked to find)

Find the volume of fluid which can be contained inside a 2.1 m length of cylindrical pipe with diameter 120mm. The walls of the pipe are 2 mm thick. Write your answer in cubic metres . Round your final answer to 3 decimal places .

3. Write down any data you have been given including units

Pipe length = 2.1m

Pipe diameter = 120mm Thickness of pipe = 2mm

4. Identify and write down any formulae or relationships which may apply to the situation

Volume of a cylinder Vol=πr^2L where r = radius, L=length

5. Draw a diagram of the situation or write down the steps required to solve the problem

6. Convert any given data to units required in the final answer

length = 2.1m diameter = 120mm = 120÷1000 = 0.12m thickness = 2mm = 2÷1000 = 0.002m radius = diameter÷2 = 0.12÷2 = 0.06m

7. Check if you have used all the information/data given

length, diameter, thickness

8. Communicate your working clearly (use words to describe your working if necessary)

"How do I use wall of pipe measurement? The question asks for volume of fluid inside the pipe. Therefore the internal pipe radius is needed in the calculation i.e. the pipe radius minus the width of the wall of the pipe." 60mm - 2mm = 58mm = 0.058m (internal radius in m)

Vol = πr^2L = π × 0.058^2 × 2.1 Vol = 0.022m^3 (3 decimal points)

9. Check your final answer

•    Does it make physical or logical sense? •    Did you answer the question asked? •    Did you answer all parts of the question? •    Is your answer in the correct units? •    Is your answer rounded to the required decimal places or significant figures?

• << Previous: Basic Maths Skills
• Last Updated: Apr 20, 2023 4:20 PM
• URL: https://library.manukau.ac.nz/Mathematics

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Create lifelong mathematicians

Take your learners further with the researched-based teaching approach proven to raise attainment and build maths confidence.

A teaching approach you can count on

You strive for excellence in your practice. But when no one can decide what best practice looks like, and half of your students think they ‘ can’t do maths ’ — it’s not easy.

What if you had access to teaching methods proven to develop learners’ problem-solving and critical-thinking skills? Or strategies to enrich advanced learners and support struggling learners? What if you could see your students’ faces light up as they start to believe in their own mathematical capabilities?

Based on methods developed in Singapore and the pedagogy of influential educational theorists, Maths — No Problem! is a teaching framework proven to help students gain a mastery level understanding of maths.

It has already transformed primary maths teaching in schools around the world.

Will your school be next?

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.

Flexible teacher support

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.

How Maths — No Problem! works

Tools to take students further

Close conceptual gaps with evidence-based resources proven to foster a deep understanding of maths in children of all attainment levels.

Strategies for success

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

Support when you need it

Take the uncertainty out of teaching maths for mastery with hands-on PLD courses and workshops, online teacher guides, and online training videos.

[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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Maths — No Problem! resources, step-by-step teaching support, and online PLD videos are a click away. Request a free demo to see how it all works, or email [email protected] for more information.

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Sample Problems

• Algebra - with solutions
• Combinatorics
• Number Theory

Past NZMO Problems

• 2023 NZMO2 with solutions
• 2023 NZMO1 with solutions
• 2022 NZMO2 with solutions
• 2022 NZMO1 with solutions
• 2021 NZMO2 with solutions
• 2021 NZMO1 with solutions
• 2020 NZMO2 with solutions
• 2020 NZMO1 with solutions
• 2019 NZMO2 with solutions
• 2019 NZMO1 with solutions

International Sites and Competitions

• International Maths Olympiad
• International Maths Olympiad Foundation
• Art of Problem Solving
• Australian Maths Trust
• United Kingdom Maths Trust
• Harvard MIT Maths Tournament
• Canadian Mathematical Olympiad
• Asian Pacific Mathematical Olympiad
• Tournament of Towns
• Dutch Mathematical Olympiad
• Singapore Mathematical Society
• Putnam Competition
• International Mathematics Competition
• Simon Marais Competition

New Zealand sites

• Auckland Mathematical Association
• Canterbury Mathematics Association
• Bay of Plenty Mathematical Association
• Northland Mathematics Association
• Otago Mathematics Association
• Waikato Mathematical Association
• Wellington Mathematics Association
• Maths Craft

Problem and Puzzle Pages

• Northwestern University Math Problem Solving Group
• Crux Mathematicorum
• KöMaL Magazine
• The Mathematical Gazette
• Berkeley Math Circle
• The Riddler
• Project Euler
• Purdue Problem of the Week
• Cut the Knot
• Calculus Problem of the Week
• Torsten Sillke Puzzles
• Macalester College Problem of the Week
• Nick's Mathematical Puzzles
• William Wu Riddles
• Math Puzzle
• Mudd Math Fun Facts

Books, Interesting Sites and Other Resources

Problem solving tactics.

Every aspiring olympiad mathematician should own a copy of PST.

Derek Holton's Mathematical Olympiad Series

This book series could be very useful to some of you - I would recommend getting the one at the right level for you rather than all of them. A few libraries in New Zealand hold these books - just search in the national library catalogue .

Evan Chen's Notes

Evan has amassed a nice collection of lecture notes aimed at students training for the IMO.

Advice from Geoff Smith

Geoff is the president of the IMO board, so is intimately familiar with the IMO, and has some excellent advice and resource suggestions.

Bertrand’s Postulate

The fact that, for every positive integer n, there is a prime between n and 2n is known as Bertrand’s postulate. It arises occasionally in Olympiad style problems (usually with the note “You may assume Bertrand’s Postulate …”). Michael Nielsen has a nice post giving an elementary proof at the Polymath wiki.

Number theory

At least a couple of good, comprehensive introductions to elementary number theory are available online. These notes by Jim Hefferon and W. Edwin Clark are nicely written and gently-paced. These ones by Naoki Sato are a bit more Olympiad-focused.

Chrystal’s Algebra

A classic (pre 1900) textbook on algebra by Chrystal has been scanned and made available electronically . Volume 1 is not likely to be of much interest for training purposes but chapters 23 and 24 (on combinatorics), 32 through 34 (continued fractions), and 35 (number theory), of volume 2 are.

Discrete and polyhedral geometry

Problems in discrete geometry (i.e. the border between combinatorics and geometry) have made several appearances on the IMO. A nice book by Igor Pak covers much of the important material in this area — the book is aimed at undergaduate and graduate students in maths, but the first few chapters in particular are suitable for Olympiad level students. As in many cases, the important thing is not the results themselves (though Helly’s theorem is a useful tool in lots of setting) but the “style” of proofs in this area.

The Princeton Companion to Mathematics

The Princeton Companion to Mathematics , an “encyclopedia of modern pure mathematics” is a fantastic book, edited by mathematician Timothy Gowers . It touches on a lot of topics, some of which are quite advanced, however section 1.3 on fundamental definitions and structures is very useful, in particular the parts about limits and continuity, and section 5.13 is a good overview of the Fundamental Theorem of Algebra. You may also find some useful notes on how professional mathematicians go about problem-solving.

Useful list for problem solving

Having the right mental habits is one of the keys to being a successful problem solver. A really good list was posted at a math education blog . The list is supposedly for sixth grade (i.e. year six) students, but is much more universal than that.

Abstract Math

The site Abstract Math by Charles Wells presents some useful insights on writing and doing proofs. The author’s opinions in some areas are arguable (so you should read and think about them, rather than follow the advice verbatum). For instance, I wouldn’t recommend the “Languages of Math” section which is far too detailed at present. However, there are little gems to be found everywhere, and from within that section comes the link to Timothy Gowers article, “The language and grammar of mathematics.” . By the way, the Tricki is alive, although small and incomplete, but it is still a great resource.

How to Write Proofs

Another site that walks through how to write proofs, which could be useful. Again, take things with a pinch of salt.

Recommended Mathematics Literature

This site is now fairly old, but still has some good recommendations for mathematics textbooks, puzzle books and other good reading.

Integer sequences

Neil Sloane, who is the father of the encyclopedia of integer sequences , has produced a paper , which highlights seven of them. There are no problems to solve there (other than some hard open ones!), but lots of interesting stuff. For example, you might like to try and prove the following result before reading it:

Consider the sequence defined as follows a(1) = 1, a(2) = 2, and for n >= 3, a(n) is the smallest positive integer not yet in the sequence such that gcd(a(n), a(n-1)) > 1. Prove that every positive integer eventually appears in the sequence.

The Geometry Junkyard

David Eppstein has put together an interesting collection of random things related to geometry. I guarentee you will find something interesting.

Gusty Games

Our very own Ross Atkins runs monthly programming workshops on this website. There are also some fun and interesting mathematical challenges in the "apps" section, as well as a couple of puzzle apps published to the Android and iOS stores.

Google Shared Drive

Ordering pr1me, case studies.

Principles of Teaching and PR1ME Mathematics

PR1ME is based on a composite of approaches , used by the top-performing nations in mathematics, and based on Singapore Maths used by over 58 nations globally.

If you want to start to understand how the top performing nations in primary maths education get great results, being aware of some of the stated Principles of Teaching used in Mathematics education in Singapore might help.

Principle 1

Teaching is for learning; learning is for understanding; understanding is for reasoning and applying and, ultimately problem solving .

Principle 2

Teaching should build on students’ knowledge ; take cognizance of students’ interests and experiences; and engage them in active and reflective learning .

Primary Mathematics Teaching and Learning Syllabus , Ministry of Education, Singapore, 2012, pp21-22

No doubt, many New Zealand schools and teachers also have these principles central to their maths teaching.

How are these teaching principles delivered so successfully in Singapore?

The Ministry of Education believes that many teachers need support to deliver the curriculum , and acknowledges the role that textbooks play in providing some of the support required. As a result, it assesses all published materials made available to teachers and students in Singapore. It is the way they ensure the integrity of their curriculum is delivered the way they want it to be.

PRIME Mathematics is a collaboration between the Singaporean Ministry of Education and Scholastic . It is designed to give schools outside of Singapore, the benefit of the world-class curriculum and teaching/learning experiences used in Singaporean schools.

Teaching Principle 1: Problem solving

“The learning of mathematics should focus on understanding, not just recall of facts or reproduction of procedures. Understanding is necessary for deep learning and mastery. Only with understanding can students be able to reason mathematically and apply mathematics to solve a range of problems. After all, problem solving is the focus of the mathematics curriculum .”

Primary Mathematics Teaching and Learning Syllabus , Ministry of Education, Singapore, 2012, p21

There is no doubt that for many years, New Zealand teachers have been emphasising the critical importance of problem solving. This is not new. Problem solving is at the heart of the New Zealand Curriculum: Mathematics .

Previous questions

Welcome to the Junior Mathematics Competition!

This year the competition will be held in two parts - the first being a multiple-choice and short answer exam sat entirely online , while the second part will be a paper exam limited to the top 15 percent of participants in the first part.

The first part of the competition will be sat from Monday April 8 to Friday April 12 , while the second part will be held on Wednesday May 15 .

2024 is the 39th year of this highly successful Competition.

This mathematics competition is for students in years 9 to 11 (forms 3 to 5), although younger students may also take part. Every student throughout New Zealand sits the competition on exactly the same days, usually two Wednesdays in April in May (one for each part).

The competition lasts for 50 minutes per part (so one hour, 40 minutes in total). The emphasis is on problem solving, so expect surprises!

Mathematical skills are important, but so is careful reading. Students who read the questions carefully and work out exactly what is being asked generally do better than those who jump in and write down the first thing that springs to mind.

Although many of the same questions are asked for each year level, students only compete against students at the same level as themselves. (However students in years 7 or 8 are regarded as being in year 9 for the purposes of the competition.)

Full details of the competition may be found here , with a note on who can enter here .

Note: This competition should not be confused with Problem Challenge , which is run by a separate unit within the Department of Mathematics and Statistics at the University of Otago. For Problem Challenge enquires please contact the Problem Challenge organisers directly.

© Department of Mathematics & Statistics

Opportunities for Students

On this page is information about competitions and activities for your students

Click the blue headings for more information

Mathex is a challenging and fast-paced mathematics contest where teams of four students compete against each other to answer questions as quickly as possible. AMA run the Mathex competition for years 7 to 10 over two nights in August.

Mathsweek is New Zealand’s premier online maths event attracting an audience of over 250 000 students teachers and parents. Mathsweek takes place in August each year.

This is a team competition for students of year 13 calculus. Calcex is held at venues across New Zealand in the last week of term 3. It is the brainchild of Jamie Sneddon.

Junior Mathematics Competition

This competition is for students in years 9 to 11 and was initiated in 1985 by Professor Derek Holton of the University of Otago Mathematics and Statistics Department. The competition attracts around 10 000 entries from 250 schools each year. It takes place on the same day across New Zealand.

Forging New Opportunities for Problem Solving in Australian Mathematics Classrooms through the First National Mathematics Curriculum

Mathematical knowledge for teaching teachers: knowledge used and developed by mathematics teacher educators in learning to teach via problem solving.

Joanna O. Masingila, Dana Olanoff & Patrick M. Kimani

Part IV: Commentary – Characteristics of Mathematical Challenge in Problem-Based Approach to Teaching Mathematics

Avoid common mistakes on your manuscript.

Introduction

For many years curriculum documents worldwide have positioned mathematics as a problem solving endeavour (e.g., see Australian Curriculum, Assessment and Reporting Authority, 2018 ; Ministry of Education, 2007 ). There is evidence however that even with this prolonged emphasis, problem solving has not become a significant presence in many classrooms (Felmer et al., 2019 ). Research has reported on a multitude of potential barriers, even for experienced teachers (Clarke et al., 2007 ; Holton, 2009 ). At the same time it is widely recognised that beginning teachers encounter many challenges as they start their careers, and that these challenges are particularly compelling when seeking to implement ambitious methods of teaching, such as problem solving (Wood et al., 2012 ).

Problem solving has been central to mathematics knowledge construction from the beginning of human history (Felmer et al., 2019 ). Teaching and learning mathematics through problem solving supports learners’ development of deep and conceptual understandings (Inoue et al., 2019 ), and is regarded as an effective way of catering for diversity (Hunter et al., 2018 ). While the importance and challenge of mathematical problem solving in school classrooms is not questioned, the promotion and enabling of problem solving is a contentious endeavour (English & Gainsburg, 2016 ). One debate centres on whether to teach mathematics through problem solving or to teach problem solving in and of itself. Recent scholarship (and this research) leans towards teaching mathematics through problem solving as a means for students to learn mathematics and come to appreciate what it means to do mathematics (Schoenfeld, 2013 ).

Problem solving has been defined in a multitude of ways over the years. Of central importance to problem solving as it is explored in this research study is Schoenfeld’s ( 1985 ) proposition that, “if one has ready access to a solution schema for a mathematical task, that task is an exercise and not a problem” (p. 74). A more recent definition of what constitutes a mathematical problem from Mamona-Downs and Mamona ( 2013 ) also emphasises the centrality of the learner not knowing how to proceed, highlighting that problems cannot be solved by procedural effort alone. These are important distinctions because traditional school texts and programmes often position problems and problem solving as an ‘add-on’ providing a practice opportunity for a previously taught, specific procedure. Given the range of learners in any education setting an important point to also consider is that what constitutes a problem for some students may not be a problem for others (Schoenfeld, 2013 ).

A research focus exploring what supports beginning teachers’ learning about teaching mathematics through problem solving is particularly relevant at this time given calls for an increased curricular focus on mathematical practices such as problem solving (Grootenboer et al., 2021 ) and recent recommendations from an expert advisory panel on the English-medium Mathematics and Statistics curriculum in Aotearoa (Royal Society Te Apārangi, 2021 ). The ninth recommendation from this report advocates for the provision of sustained professional learning in mathematics and statistics for all teachers of Years 0–8. With regard to beginning primary teachers, the recommendation goes further suggesting that ‘mathematics and statistics professional learning’ (p. 36) be considered as compulsory in the first 2 years of teaching. This research explores what the nature of that professional learning might involve, with a focus on problem solving.

Scoping the Context for Learning and Sustaining Problem Solving

The literature reviewed for this study draws from two key fields: the nature of support and professional development effective for beginning teachers; and specialised supports helping teachers to employ problem solving pedagogies.

Beginning Teachers, Support and Professional Development

A teacher’s early years in the profession are regarded as critical in terms of constructing a professional practice (Feiman-Nemser, 2003 ) and avoiding high attrition (Karlberg & Bezzina, 2020 ). Research has established that beginning teachers need professional development opportunities geared specifically to their needs (Fantilli & McDougall, 2009 ) and their contexts (Gaikhorst, et al., 2017 ). Providing appropriate support is not an uncontentious matter with calls for institutions to come together and collaborate to provide adequate and ongoing support (Karlberg & Bezzina, 2020 ). The proposal is that support is needed from both within and beyond the beginning teacher’s school; and begins with effective pre-service teacher preparation (Keese et al., 2022 ).

Within schools where beginning teachers regard the support they receive positively, collaboration, encouragement and ‘involved colleagues’ are considered as vital; with the guidance of a 'buddy’ identified as some of the most valuable in-school support activities (Gaikhorst et al., 2014 ). Cameron et al.’s ( 2007 ) research in Aotearoa reports beginning teachers joining collaborative work cultures had greater opportunities to talk about teaching with their colleagues, share planning and resources, examine students’ work, and benefit from the collective expertise of team members.

Opportunities to participate in networks beyond the beginning teacher’s school have also been identified as being important for teacher induction (Akiri & Dori, 2021 ; Cameron et al., 2007 ). Fantilli & McDougall ( 2009 ) in their Canadian study found beginning teachers reported a need for many support and professional development opportunities including subject-specific (e.g., mathematics) workshops prior to and throughout the year. Akiri and Dori ( 2021 ) also refer to the need for specialised support from subject-specific mentors. This echoes the findings of Wood et al. ( 2012 ) who advocate that given the complexity of learning to teach mathematics, induction support specific to mathematics, and rich opportunities to learn are not only desirable but essential.

Akiri and Dori ( 2021 ) describe three levels of mentoring support for beginning teachers including individual mentoring, group mentoring and mentoring networks with all three facilitating substantive professional growth. Of relevance to this paper are individual and group mentoring. Individual mentoring involves pairing an experienced teacher with a beginning teacher, so that a beginning teacher’s learning is supported. Group mentoring involves a group of teachers working with one or more mentors, with participants receiving guidance from their mentor(s) (Akiri & Dori, 2021 ). Findings from Akiri and Dori suggest that of the varying forms of mentoring, individual mentoring contributes the most for beginning teachers’ professional learning.

Teachers Learning to Teach Mathematics Through Problem Solving

Learning to teach mathematics through problem solving begins in pre-service teacher education. It has been shown that providing pre-service teachers with opportunities to engage in problem solving as learners can be productive (Bailey, 2015 ). Opportunities to practise content-specific instructional strategies such as problem solving during student teaching has also been positively associated with first-year teachers’ enactment of problem solving (Youngs et al., 2022 ).

The move from pre-service teacher education to the classroom can be fraught for beginning teachers (Feiman-Nemser, 2003 ), and all the more so for beginning teachers attempting to teach mathematics through problem solving (Wood et al., 2012 ). In a recent study (Darragh & Radovic, 2019 ) it has been shown that an individual willingness to change to a problem-based pedagogy may not be enough to sustain a change in practice in the long term, particularly if there is a contradiction with the context and ‘norms’ (e.g., school curriculum) within which a teacher is working. Cady et al. ( 2006 ) explored the beliefs and practices of two teachers from pre-service teacher education through to becoming experienced teachers. One teacher who initially adopted reform practices reverted to traditional beliefs about the learning and teaching of mathematics. In contrast, the other teacher implemented new practices only after understanding these and gaining teaching experience. Participation in mathematically focused professional development and involvement in resource development were thought to favourably influence the second teacher.

Lesson structures have been found to support teachers learning to teach mathematics through problem solving. Sullivan et al. ( 2016 ) explored the use of a structure comprising four phases: launching, exploring, summarising and consolidating. Teachers in Australia and Aotearoa have reported the structure as productive and feasible (Ingram et al., 2019 ; Sullivan et al., 2016 ). Teaching using challenging tasks (such as in problem solving) and a structure have been shown to accommodate student diversity, a pressing concern for many teachers. Student diversity has often been managed by grouping students according to perceived levels of capability (called ability grouping). Research identifies this practice as problematic, excluding and marginalising disadvantaged groups of students (e.g., see Anthony & Hunter, 2017 ). The lesson structure explored by Sullivan et al. ( 2016 ) caters for diversity by deliberately differentiating tasks, providing enabling and extending prompts. Extending prompts are offered to students who finish an original task quickly and ideally elicit abstraction and generalisation. Enabling prompts involve reducing the number of steps, simplifying the numbers, and/or varying forms of representation for students who cannot initially proceed, with the explicit intention that students then return to the original task.

Recognising the established challenges teachers encounter when learning about teaching mathematics through problem solving, and the paucity of recent research focussing on beginning teachers learning about teaching mathematics in this way, this paper draws on data from a 2 year longitudinal study. The study was guided by the research question:

What supports beginning teachers’ implementation of a problem solving pedagogy for the teaching and learning of mathematics?

Research Methodology and Methods

Data were gathered from three beginning primary teachers who had completed a 1 year graduate diploma programme in primary teacher education the previous year. The beginning teachers had undertaken a course in mathematics education (taught by the author for half of the course) as part of the graduate diploma. An invitation to be involved in the research was sent to the graduate diploma cohort at the end of the programme. Three beginning teachers indicated their interest and remained involved for the 2 year research period. The teachers had all secured their first teaching positions, and were teaching at different year levels at three different schools. Julia (pseudonyms have been used for all names) was teaching year 0–2 (5–6 years) at a small rural school; Charlotte, year 5–6 (9–10 years) at a large urban city school; and Reine, year 7–8 (11–12 years), at another small rural school. All three beginning teachers taught at their respective schools, teaching the same year levels in both years of the study. Ethical approval was sought and given by the author’s university ethics committee. Informed consent was gained from the teachers, school principals and involved parents and children.

Participatory action research was selected as the approach in the study because of its emphasis on the participation and collaboration of all those involved (Townsend, 2013 ). Congruent with the principles of action research, activities and procedures were negotiated throughout both years in a responsive and emergent way. The author acted as a co-participant with the teachers, aiming to improve practice, to challenge and reorient thinking, and transform contexts for children’s learning (Locke et al., 2013 ). The author’s role included facilitating the research-based problem solving workshops (see below), contributing her experience as a mathematics educator and researcher. The beginning teachers were involved in making sense of their own practice related to their particular sites and context.

The first step in the research process was a focus group discussion before the beginning teachers commenced their first year of teaching. This discussion included reflecting on their learning about problem solving during the mathematics education course; and envisaging what would be helpful to support implementation. It was agreed that a series of workshops would be useful. Two were subsequently held in the first year of the study, each for three hours, at the end of terms one and two. Four workshops were held during the second year, one during each term. At the end of the first year the author suggested a small number of experienced teachers who teach mathematics through problem solving join the workshops for the second year. The presence of these teachers was envisaged to support the beginning teachers’ learning. The beginning teachers agreed, and an invitation was extended to four teachers from other schools whom the author knew (e.g., through professional subject associations). The focus of the research remained the same, namely exploring what supported beginning teachers to implement a problem solving pedagogy.

Each workshop began with sharing and oral reflections about recent problem solving experiences, including successes and challenges. Key workshop tasks included developing a shared understanding of what constitutes problem solving, participating in solving mathematical problems (modelled using a lesson structure (Sullivan et al., 2016 ), and learning techniques such as asking questions. A time for reflective writing was provided at the end of each workshop to record what had been learned and an opportunity to set goals.

During the first focus group discussion it was also decided the author would visit and observe the beginning teachers teaching a problem solving lesson (or two) in term three or four of each year. A semi-structured interview between the author and each beginning teacher took place following each observed lesson. The beginning teachers also had an opportunity to ask questions as they reflected on the lesson, and feedback was given as requested. A second focus group discussion was held at the end of the first year (an approximate midpoint in the research), and a final focus group discussion was held at the end of the second year.

All focus group discussions, problem solving workshops, observations and interviews were audio-recorded and transcribed. Field notes of workshops (recorded by the author), reflections from the beginning teachers (written at the end of each workshop), and lesson observation notes (recorded by the author) were also gathered. The final data collected included occasional emails between each beginning teacher and the author.

Data Analysis

The analysis reported in this paper drew on all data sets, primarily using inductive thematic analysis (Braun & Clarke, 2006 ). The research question guided the key question for analysis, namely: What supports beginning teachers’ implementation of a problem solving pedagogy for the teaching and learning of mathematics? Alongside this question, consideration was also given to the challenges beginning teachers encountered as they implemented a problem solving pedagogy. Data familiarisation was developed through reading and re-reading the whole body of data. This process informed data analysis and the content for each subsequent workshop and focus group discussions. Colour-coding and naming of themes included comparing and contrasting data from each beginning teacher and throughout the 2-year period. As a theme was constructed (Braun & Clarke, 2006 ) subsequent data was checked to ascertain whether the theme remained valid and/or whether it changed during the 2 years. Three key themes emerged revealing what supported the beginning teachers’ developing problem solving pedagogy, and these constitute the focus for this paper.

Mindful of the time pressures beginning teachers experience in their early years, the author undertook responsibility for data analysis. The author’s understanding of the unfolding ‘story’ of each beginning teacher’s experiences and the emerging themes were shared with the beginning teachers, usually at the beginning of a workshop, focus group discussion or observation. Through this process the author’s understandings were checked and clarified. This iterative process of member checking (Lincoln & Guba, 1985 ) began at a mid-point during the first year, once a significant body of data had been gathered. At a later point in the analysis and writing, the beginning teachers also had an opportunity to read, check and/or amend quotes chosen to exemplify their thinking and experiences.

Findings and Discussion

In this section the three beginning teachers’ experiences at the start of the 2 year research timeframe is briefly described, followed by the first theme centred on the use of a lesson structure including prompts for differentiation. The second and third themes are presented together, starting with a brief outline of each beginning teacher’s ‘story’ providing the context within which the themes emerged. Sharing the ‘story’ of each beginning teacher and including their ‘voice’ through quotes acknowledges them and their experiences as central to this research.

The beginning teachers’ pre-service teacher education set the scene for learning about teaching mathematics through problem solving. A detailed list brainstormed during the first focus group discussion suggested a developing understanding from their shared pre-service mathematics education course. In their first few weeks of teaching, all three beginning teachers implemented a few problems. It transpired however this inclusion of problem solving occurred only while children were being assessed and grouped. Following this, all three followed a traditional format of skill-based (with a focus on number) mathematics, taught using ability groups. The beginning teachers’ trajectories then varied with Julia and Reine both eventually adopting a pedagogy primarily based on problem solving, while Charlotte employed a traditional skill-based mathematics using a combination of whole class and small group teaching.

A Lesson Structure that Caters for Diversity Supports Early Efforts

Data show that developing familiarity with a lesson structure including prompts for differentiation supported the beginning teachers’ early efforts with a problem solving pedagogy. This addressed a key issue that emerged during the first workshop. During the workshop while a ‘list’ of ideas for teaching a problem solving lesson was co-constructed, considerable concern was expressed about catering for a range of learners when introducing and working with a problem. For example, Charlotte queried, “ Well, what happens when you are trying to do something more complicated, and we’re (referring to children) sitting here going, ‘I’ve no idea what you're talking about” ? Reine suggested keeping some children with the teacher, thinking he would say, “ If you’re unsure of any part stay behind” . He was unsure however about how he would then maintain the integrity of the problem.

It was in light of this discussion that a lesson structure with differentiated prompts (Sullivan et al., 2016 ) was introduced, experienced and reflected on during the second workshop. While the co-constructed list developed during the first workshop had included many components of Sullivan’s lesson structure, (e.g., a consideration of ‘extensions’) there had been no mention of ‘enabling prompts’. Now, with the inclusion of both enabling and extending prompts, the beginning teachers’ discussion revealed them starting to more fully envisage the possibilities of using a problem solving approach, and being able to cater for all children. Reine commented that, “… you can give the entire class a problem, you've just got to have a plan, [and] your enabling and extension prompts” . Charlotte was also now considering and valuing the possibility of having a whole class work on the same problem. She said, “I think … it’s important and it’s useful for your whole class to be working on the same thing. And … have enablers and extenders to make sure that everyone feels successful” . Julia also referred to the planning prompts. She thought it would be key to “plan it well so that we’ve got enabling and extending prompts” .

Successful Problem Solving Lessons

Following the second workshop all three beginning teachers were observed teaching a lesson using the structure. These lessons delighted the beginning teachers, with them noting prolonged engagement of children, the children’s learning and being able to cater for all learners. Reine commented on how excited and engaged the children were, saying they were, “ just so enthusiastic about it ”. In Charlotte’s words, “ it really worked ”, and Julia enthusiastically pondered this could be “ the only way you teach maths !”.

During the focus group discussion at the end of the first year, all three reflected on the value of the lesson structure. Reine called it a ‘framework’ commenting,

I like the framework. So from start to finish, how you go through that whole lesson. So how you set it up and then you go through the phases… I like the prompts that we went through…. knowing where you could go, if they’re like, ‘What do I do?’ And then if they get it too easy then ‘Where can you go?’ So you've got all these little avenues.

Charlotte also valued the lesson structure for the breadth of learning that could occur, explaining,

… it really helped, and really worked. So I found that useful for me and my class ‘cause they really understood. And I think also making sure that you know all the ins and outs of a problem. So where could they go? What do you need to know? What do they need to know?

While the beginning teachers’ pre-service teacher education and the subsequent research process, including the use of the lesson structure, supported the beginning teachers’ early efforts teaching mathematics through problem solving, two key factors further enabled two of the beginning teachers (Julia and Reine) to sustain a problem solving pedagogy. These were:

Being involved in complementary mathematics professional development alongside members of their respective school staff (a form of group mentoring); and

Having a colleague in the same school teaching mathematics through problem solving (a form of individual mentoring).

Charlotte did not have these opportunities and she indicated this limited her implementation. Data for these findings for each teacher are presented below.

Complementary Professional Development and Problem Solving Colleague in Same School

Julia began to significantly implement problem solving from the second term in the first year. This coincided with her attending a 2-day workshop (with staff from her school) that focused on the use of problem solving to support children who are not achieving at expected levels (see ALiM: Accelerated Learning in Maths—Ministry of Education, 2022 ). She explained, “ … I did the PD with (colleague’s name), which was really helpful. And we did lots of talking about rich learning tasks and problem solving tasks…. And what it means ”. Following this, Julia reported using rich tasks and problem solving in her mathematics teaching in a regular (at least weekly) and ongoing way.

During the observation in term three of the first year Julia again referred to the impact of having a colleague also teaching mathematics through problem solving. When asked what she believed had supported her to become a teacher who teaches mathematics in this way she firstly identified her involvement in the research project, and then spoke about her colleague. She said, “ I’m really lucky one of our other teachers is doing the ALiM project… So we’re kind of bouncing off each other a little bit with resources and activities, and things like that. So that’s been really good ”.

At the beginning of the second year, Julia reiterated this point again. On this occasion she said having a colleague teaching mathematics through problem solving, “ made a huge difference for me last year ”, explaining the value included having someone to talk with on a daily basis. Mid-way through the second year Julia repeated her opinion about the value of frequent contact with a practising problem solving colleague. Whereas her initial comments spoke of the impact in terms of being “ a little bit ”, later references recount these as ‘ huge ’ and ‘ enabling ’. She described:

a huge effect… it enabled me. Cause I mean these workshops are really helpful. But when it’s only once a term, having [colleague] there just enabled me to kind of bounce ideas off. And if I did a lesson that didn’t work very well, we could talk about why that was, and actually talk about what the learning was instead…. . It was being able to reflect together, but also share ideas. It was amazing.

Julia’s comments raise two points. It is likely that participating in the ALiM professional development (which could be conceived as a form of group mentoring) consolidated the learning she first encountered during pre-service teacher education and later extended through her involvement in the research. Having a colleague (in essence, an individual mentor) within the same school teaching mathematics through problem solving appears to be another factor that supported Julia to implement problem solving in a more sustained way. Julia’s comments allude to a number of reasons for this, including: (i) the more frequent discussion opportunities with a colleague who understands what it means for children to learn mathematics through problem solving; (ii) being able to share and plan suitable activities and resources; and (iii) as a means for reflection, particularly when challenges were encountered.

Reine’s mathematics programme throughout the first year was based on ability groups and could be described as traditional. He occasionally used some mathematical problems as ‘extension activities’ for ‘higher level’ children, or as ‘fillers’. In the second year, Reine moved to working with mixed ability groups (where students work together in small groups with varying levels of perceived capability) and initially implemented problem solving approximately once a fortnight. In thinking back to these lessons he commented, “ We weren’t really unpacking one problem properly, it was just lots of busy stuff ”. A significant shift occurred in Reine’s practice to teaching mathematics primarily by problem solving towards the last half of the second year. He explained, “ I really ramped up towards terms three and four, where it’s more picking one problem across the whole maths class but being really, really conscious of that problem. Low entry, high ceiling, and doing more of it too ”.

Reine attributed this change to a number of factors. In response to a question about what he considered led to the change he explained,

… having this, talking about this stuff, trialling it and then with our PD at school with the research into ability grouping... We’ve got a lot of PD saying why it can be harmful to group on ability, and that’s been that last little kick I needed, I think. And with other teachers trialling this as well. Our senior teacher has flipped her whole maths program and just does problem solving.

Like Julia, Reine firstly referred to his involvement in the research project including having opportunities to try problems in his class and discuss his experiences within the research group. He then told of a colleague teaching at his school leading school-wide professional development focussed on the pitfalls of ability grouping in mathematics (e.g., see Clarke, 2021 ) and instead using problem solving tasks. He also referred to having another teacher also teaching mathematics through problem solving. It is interesting to consider that having positive experiences in pre-service teacher education, the positive and encouraging support of colleagues (Reine’s principal and co-teacher in both years), regular participation in ongoing professional development (the problem solving workshops), and having a highly successful one-off problem solving teaching experience (the first year observation) were not enough for Reine to meaningfully sustain problem solving in his first year of teaching.

As for Julia, pivotal factors leading to a sustaining of problem solving teaching practice in the second year included complementary mathematics professional development (a form of group mentoring) and at least one other teacher (acting as an individual mentor) in the same school teaching mathematics through problem solving. It could be argued that pre-service teacher education and the problem solving workshops ‘paved the way’ for Julia and Reine to make a change. However, for both, the complementary professional development and presence of a colleague also teaching through problem solving were pivotal. It is also interesting to note that three of the four experienced teachers in the larger research group taught at the same level as Reine (see Table 1 below) yet he did not relate this to the significant change in his practice observed towards the end of the second year.

Charlotte’s mathematics programme during the first year was also traditional, teaching skill-based mathematics using ability groups. At the beginning of the second year Charlotte moved to teaching her class as a whole group, using flexible grouping as needed (children are grouped together in response to learning needs with regard to a specific idea at a point in time, rather than perceived notions of ability). She reported that she occasionally taught a lesson using problem solving in the first year, and approximately once or twice a term in the second year. Charlotte did not have opportunities for professional development in mathematics nor did she have a colleague in the same school teaching mathematics through problem solving. Pondering this, Charlotte said,

It would have been helpful if I had someone else in my school doing the same thing. I just thought about when you were saying the other lady was doing it [referring to Julia’s colleague]. You know, someone that you can just kind of back-and-forth like. I find with Science, I usually plan with this other lady, and we share ideas and plan together. We come up with some really cool stuff whereas I don’t really have the same thing for this.

Based on her experiences with teaching science it is clear Charlotte recognised the value of working alongside a colleague. In this, her view aligns with what Julia and Reine experienced.

Table 1 provides a summary of the variables for each beginning teacher, and whether a sustained implementation of teaching mathematics through problem solving occurred.

The table shows two variables common to Julia and Reine, the beginning teachers who began and sustained problem solving. They both participated in complementary professional development with colleagues from their school, and the presence of a colleague, also at their school, teaching mathematics through problem solving. Given that Julia was able to implement problem solving in the absence of a ‘research workshop colleague’ teaching at the same year level, and Reine’s lack of comment about the potential impact of this, suggests that this was not a key factor enabling a sustained implementation of problem solving.

Attributing the changes in Julia and Reine’s teaching practice primarily to their involvement in complementary professional development attended by members of their school staff, and the presence of at least one other teacher teaching mathematics through problem solving in their school, is further supported by a consideration of the timing of the changes. The data shows that while Julia could be considered an ‘early adopter’, Reine changed his practice reasonably late in the 2 year period. Julia’s early adoption of teaching mathematics through problem solving coincided with her involvement, early in the 2 years, in the professional development and opportunity to work alongside a problem solving practising colleague. Reine encountered these similar conditions towards the end of the 2 years and it is notable that this was the point at which he changed his practice. That problem solving did not become embedded or frequent within Charlotte’s mathematics programme tends to support the argument.

Understanding what supports primary teachers to teach mathematics through problem solving at the beginning of their careers is important because all students, including those taught by beginning teachers, need opportunities to develop high-level thinking, reasoning, and problem solving skills. It is also important in light of recent calls for mathematics curricula to include more emphasis on mathematical practices (such as problem solving) (e.g., see Grootenboer et al., 2021 ); and the Royal Society Te Apārangi report ( 2021 ). Findings from this research suggest that learning about problem solving during pre-service teacher education is enough for beginning teachers to trial teaching mathematics in this way. Early efforts were supported by gaining experience with a lesson structure that specifically attends to diversity. The lesson structure prompted the beginning teachers to anticipate different children’s responses, and consider how they would respond to these. An increased confidence and sense of security to trial teaching mathematics through problem solving was enabled, based on their more in-depth preparation. Beginning teachers finding the lesson structure useful extends the findings of Sullivan et al. ( 2016 ) in Australia and Ingram et al. ( 2019 ) in Aotearoa to include less experienced teachers.

In order for teaching mathematics through problem solving to be sustained however, a combination of three factors, subsequent to pre-service teacher education, was needed: (i) active participation in problem solving workshops (in this context provided by the research-based problem solving workshops); (ii) attending complementary professional development initiatives alongside colleagues from their school (a form of group mentoring); and (iii) the presence of an in-school colleague who also teaches mathematics through problem solving (a form of individual mentoring). It seems possible these three factors acted synergistically resulting in Julia and Reine being able to sustain implementation. If only one factor is present, in this study attending the problem solving workshops, and despite a genuine interest in using a problem based pedagogy, the result was limited movement towards this way of teaching.

Akiri and Dori ( 2021 ) have reported that individual mentoring contributes the most to beginning teachers’ professional growth. In a manner consistent with these findings, an in-school colleague (who in essence was acting as an individual mentor) played a critical role in supporting Reine and Julia. However, while Akiri and Dori, amongst others (e.g., Cameron et al., 2007 ; Karlberg & Bezzina, 2020 ), have identified the value of supportive, approachable colleagues, for both Julia and Reine it was important that their colleague was supportive and approachable, and actively engaged in teaching mathematics through problem solving. Having supportive and approachable colleagues, as Reine experienced in his first year, on their own were not enough to support a sustained problem solving pedagogy.

Implications for Productive Professional Learning and Development

This study sought to explore the conditions that supported problem solving for beginning teachers, each in their unique context and from their perspective. The research did not examine how the teaching of mathematics through problem solving affected children’s learning. However, multiple sets of data were collected and analysed over a 2-year period. While it is neither possible nor appropriate to make claims as to generalisability some suggestions for productive beginning teacher professional learning and development are offered.

Given the first years of teaching constitute a particular and critical phase of teacher learning (Karlberg & Bezzina, 2020 ) and the findings from this research, it is imperative that well-funded, subject-focussed support occurs throughout a beginning teacher’s first 2 years of teaching. This is consistent with the ninth recommendation in the Royal Society Te Apārangi report ( 2021 ) suggesting compulsory professional learning during the induction period (2 years in Aotearoa New Zealand). Participation in subject-specific professional development has been recognised to favourably influence new teachers’ efforts to adopt reform practices such as problem solving (Cady et al., 2006 ).

Findings from this study suggest professional development opportunities that complement each other support beginning teacher learning. In the first instance complementarity needs to be with what beginning teachers have learned during their pre-service teacher education. In this study, the research-based problem solving workshops served this role. Complementarity between varying forms of professional development also appears to be important. Furthermore, as indicated by Julia and Reine’s experiences, subsequent professional development need not be on exactly the same topic. Rather, it can be complementary in the sense that there is an underlying congruence in philosophy and/or focus on a particular issue. For example, it emerged in the problem solving workshops, that being able to cater for diversity was a central concern for the beginning teachers. Attending to this issue within the problem solving workshops via the introduction of a lesson structure that enabled differentiation, was congruent with the nature of the professional development in the two schools: ALiM in Julia’s school, and mixed ability grouping and teaching mathematics through problem solving in Reine’s school. All three of these settings were focussed on positively responding to diversity in learning needs.

The presence of a colleague within the same school teaching mathematics through problem solving also appears to be pivotal. This is consistent with Darragh and Radovic ( 2019 ) who have shown the significant impact a teacher’s school context has on their potential to sustain problem based pedagogies in mathematics. Given that problem solving is not prevalent in many primary classrooms, it would seem clear that colleagues who have yet to learn about teaching mathematics through problem solving, particularly those that have a role supporting beginning teachers, will also require access to professional development opportunities. It seems possible that beginning and experienced teachers learning together is a potential pathway forward. Finding such pathways will be critical if mathematical problem solving is to be consistently implemented in primary classrooms.

Finally, these implications together with calls for institutions to collaborate to provide adequate and ongoing support for new teachers (Karlberg & Bezzina, 2020 ) suggest there is a need for pre-service teacher educators, professional development providers and the Teaching Council of Aotearoa New Zealand to work together to support beginning teachers’ starting and sustaining teaching mathematics through problem solving pedagogies.

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Bailey, J. Learning to Teach Mathematics Through Problem Solving. NZ J Educ Stud 57 , 407–423 (2022). https://doi.org/10.1007/s40841-022-00249-0

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Solving the maths problem

Since the 1980s, average maths achievement has trended downward in new zealand - so how do we address this long standing issue.

There is mounting concern over the state of numeracy skills among ākonga in New Zealand – fortunately, there are a myriad of effective math programmes, resources and PLD available to help kaiako.

New Zealand is facing an “ongoing and long-term declining achievement in Mathemathics”. 1

Since the 1980s, despite continued investment in the area, our average maths achievement has continued to trend downward. So, what gives? Why does New Zealand’s maths performance consistently lag other countries in the OECD, and how can we reverse this trend?

Read the Term 2 edition of School News HERE

Research shows that maths specialisation might be required for effective maths teaching. Anecdotally, evidence suggests teachers struggle with teaching maths confidently in New Zealand. The data shows that 79 percent of Year 5 students are taught by non-specialised teachers. Only 59 percent of these had any maths training in the past two years. 2 Reports have recommended professional development programmes for teachers to improve national numeracy standards, by creating more teachers with specialist mathematics knowledge.

In their paper “The big subtract: Can we improve our maths performance?” Wills and Hogan advocate for specialised maths teachers in primary schools. They believe primary school “is a time when some degree of specialisation is likely to be critical to prevent a lack of confidence being passed on from teacher to student to ensure a wider range of skills are developed.”

Though increasing available specialist teachers is a more long-term, structural goal for our education system, there are a number of agencies that provide specialised PLD for teachers and kaiako looking to improve their maths teaching. These programmes are supported by the Ministry of Education, and can help improve outcomes at the primary level.

Apart from PLD, there are a few known strategies that teachers can employ to further learning in any subject, and particularly maths. Effective teachers ensure that students’ current knowledge and interests are at the centre of learning, helping students retain confidence and build resilience in any areas they may struggle with. Student engagement with mathematics principles can also be improved by connecting the skills being learnt to real-world applications.

Other effective strategies for improving maths learning include using and normalising mathematical language; effectively implementing group learning to support individual confidence; effective mathematics tasks (rather than repetitive and menial tasks); supporting connection-making and problem-solving; and using everyday contexts in teaching. These strategies and more have been extensively researched.

The Ministry of Education’s website, Te Kete Ipurangi, NZ Maths also has suggested strategies for improving student engagement with the subject. It notes the first step is to ensure ākonga feel comfortable speaking in class; one suggested method is splitting students into subgroups. Revoicing is recommended, to clarify students’ responses – reiterating what a student has said to bolster confidence and engage the other students. Using partner talk and extending response wait times are also suggested methods.

NZ Maths also has resources for tracking ākonga progress to best plan lessons for individual and group maths learning and skills development.

There are several externally provided maths programmes for students. These providers range from targeted interventions for selected ākonga, to supporting high achievers to reach their full potential. Programmes may also work together with teachers to identify learning needs and develop an effective maths pedagogy within the classroom.

To learn more about how maths teaching could be supported in your school or kura, School News spoke to some providers below, who are on a mission to improve maths learning across the motu.

Janine Trembath, from the IT Education Co NZ said, “Improving mindsets around math is a key starting place for greater engagement in the subject.”

“Students need to be encouraged that they can do math, that it is useful, and to embrace mistakes. Teaching students to embrace mistakes encourages them on a journey of math discovery and enjoyment. Scientific research has shown that synapses in our brains grow when we make mistakes, and that no growth occurs for right answers.”

Mrs Trembath believes that developing a sound understanding of number during primary years would help to improve overall student math skills. “Number is the basis of all mathematics. Students need to be taught and allowed to experience what number relationships are, as well as the language around math.”

Adaptive online tools, Mrs Trembath said, can be useful in assisting teachers to reach the varied needs of their students. Online work should be supported by teacher contact time and wide discussion. “Math programmes should be highly visual, include multiple ways of learning, and aid students to develop a conceptual understanding of math.

“Good teaching programs will provide detailed student progress reports that make it easy for staff to identify the math concepts mastered, and any currently posing a challenge, as well as suggestions staff can use to support students over hurdles as they arise.”

Maryanne Tipler OMNZ and Sue Timperley from CaxEd NZ said that when selecting a maths programme to support students, it must reflect the objectives of the New Zealand curriculum, and meet the needs of the students in the classroom.

“Students will truly understand the concepts being taught by engaging in practical activities and discussing their work and their reasoning. Programmes should be flexible, allowing teachers to adapt activities to meet the needs of their students. Acknowledging the different ways students learn means that a diverse range of tools should be used to cater for different learning styles.

“Student engagement is best achieved when activities and resources being utilised are ones that students and their wider community can relate to. Whanau engagement will enhance student learning and successful outcomes if they can see the relevance of maths to their future lives,” they said.

“There is a wide range of resources available to track student progress, but some are better than others. Ongoing teacher observation and monitoring of students’ work is of more value than a one-off point in time assessment. Resources should allow for teacher-student interaction on a regular basis to achieve a broader view of student progress.

“Ongoing teacher professional development will support teachers in providing a rich learning environment.”

Alex Laurie from Maths No Problem believes one of the difficulties teachers face when teaching maths is the lack of support materials for creating a consistent and high-quality programme for a whole school.

“Often, we expect generalist primary school teachers to be involved in curriculum development, meaning we expect them to create and source their own lessons. We also expect them to decide the correct order to teach concepts in,” Ms Laurie said.

“Autonomy to create a localised maths curriculum that meets the specific needs of learners in different schools is important. In my experience, though, it is easier to change the contexts of expertly designed maths problems that are given in the correct developmental sequence in good support materials.

“Conventionally, children are placed in different groups and given content matched to their perceived ability. From an early age, then, children can be unintentionally labelled as those who can do maths and those who can’t. These labels can become self-fulfilling prophecies and can negatively impact children’s achievement,” Ms Laurie said.

“Whole class teaching offers all pupils access to the full maths curriculum. This inclusive approach and its emphasis on promoting multiple methods of solving a single problem builds self-confidence and resilience in pupils. Whole class teaching builds the expectation that all children will achieve, and be exposed to the learning appropriate for their age.”

1 Morrow, Neil, Elizabeth Rata and Tanya Evans. “A Study of the New Zealand Mathematics Curriculum”. (2021)

2 Wills, Olivia and Sarah Hogan. “The big subtract: Can we improve our maths performance?” (2021).

3 Anthony, Glenda, and Margaret Walshaw. “Characteristics of Effective Teaching of Mathematics: A View from the West”. Journal of Mathematics Education . (2009). 2(2), 147 – 164.

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You are here, problem solving in mathematics.

Not all teachers are comfortable with using the technique of problem solving, which is an attempt to find the answer to a problem when the method of solution is not known. This research shows how problem solving can be used in secondary school mathematics classes.

PROBLEM SOLVING In Mathematics

Derek Holton Department of Mathematics and Statistics, University of Otago

Jim Neyland Mathematics and Science Education Centre, Victoria University of Wellington

Julie Anderson Department of Mathematics and Statistics, University of Otago

WHERE ARE WE?

In 1993, a new national mathematics curriculum which emphasised the importance of problem solving was introduced in New Zealand. Three years later there was evidence that a significant proportion of teachers believe problem solving to be important and many are making some effort to incorporate problem solving into their teaching. However, despite this evident commitment, problem solving does not seem to occupy a regular place in most classrooms.

By problem solving we mean the attempt to find the answer to a problem when the method of solution is not known. In problem-solving situations the solver has to use strategic skills to find appropriate mathematical techniques which will settle the question. Neyland (1995) discusses the similarities and differences between the problem-solving approach and seven other approaches to teaching mathematics.

It is worth noting that what is a problem to one person may not be a problem to another. For a five-year-old determining the number of legs three sheep have will almost certainly be a problem. However, it should not be a problem for most 15-year-olds. The five-year-old may need to draw, use equipment, or employ some other method to solve the problem. The 15-year-old will just say “3 times 4 equals 12”.

Word problems do not necessarily involve problem solving. If a word problem is introduced to practise a technique which has just been acquired, it should be reasonably obvious what technique has to be used. On the other hand, problem-solving situations may be presented as wordy problems, though this is not always the case.

According to the National Council of Teachers of Mathematics:

Problem Solving should be the central focus of the mathematics curriculum. As such it is a primary goal of all mathematics instruction and an integral part of all mathematical activity. Problem Solving is not a distinct topic but a process that should permeate the entire program and provide the context in which concepts and skills can be learned.

We would like to underline the last sentence of the above quote as we find it difficult to differentiate between problem solving on the one hand and mathematics, or its essential nature, on the other.

In Australia, these views are reflected in A National Statement on Mathematics for Australian Schools (1990), where problem solving is one of four sub-headings in the Mathematical Inquiry strand:

The mathematical processes described within this strand cannot be developed in isolation from the work of other strands. They should pervade the whole curriculum.

In New Zealand, the importance of problem solving has been underscored by the Ministry of Education publication Implementing Mathematical Processes in Mathematics in the New Zealand Curriculum (1995):

Problem solving is the first heading in the Mathematical Processes strand; unless the ability to solve problems is developed, there is little point in studying mathematics.

The Mathematical Processes strand in the Mathematics in the New Zealand Curriculum document provides the umbrella for learning in the other five strands of Number, Measurement, Geometry, Algebra, and Statistics. It incorporates three areas: problem solving, developing logic and reasoning, and communicating mathematical ideas. We have decided to focus our attention on problem solving because it can provide the medium for both developing mathematical processes and interlacing the learning of specific knowledge and skills. However, is problem solving valued in our classrooms?

THE STUDIES

The work in this paper is based upon two pieces of research. The first was a questionnaire that was given to 11 secondary school mathematics departments in a provincial city of New Zealand. Some 53 teachers responded to the questionnaire (representing a 100 percent response rate from the schools) and the issues raised were discussed with seven experienced mathematics teachers.

The second study involved observations of problem-solving sessions in five secondary classrooms at the form 3 and 4 level [year 9 and 10 in New Zealand; year 8 and 9 in Australia].

The main questions in the questionnaire were:

How is problem solving being used and how often?

There are a variety of approaches to problem solving in the classroom. The questionnaire avoided questions about specific types of activities (open-ended tasks, investigations, short challenging tasks, projects, or mathematical modelling) and concentrated on broader areas of usage that had been observed in classrooms. The six categories considered were homework tasks, lesson starters not directly related to the content of the lesson, lesson starters related to the content of the lesson, throughout lesson, at the end of a unit of work, and catering for the needs of more able students. See table 1 for results.

Homework tasks

From table 1, we see that problem-solving activities are being used as homework tasks more regularly than any of the other five categories surveyed. However, of those teachers using homework tasks at least weekly, under half (48 percent) used problem solving through all of a lesson at least weekly. This suggests that homework tasks are often independent activities which are not supported by problem solving in the classroom.

Lesson starters

One of the teaching approaches identified by Sigurdson et al. (1994) is the problem-process approach , which involves using simple problems related to the mathematical content of the lesson, solving these problems through student-teacher interaction in a whole class setting and focusing on the processes being used. While using problems as lesson starters seems a good starting-off point for the implementation of a problemsolving approach, over half of the teachers surveyed never used problemsolving starters or used them only sometimes. Less than four percent use lesson starters related to content daily. But, more surprisingly, stand-alone problems are used as lesson starters at least once a week by only 21 percent of respondents. This suggests that teachers should be encouraged to use short challenging tasks to initiate problem solving in their classroom. This might be a useful point to begin teacher professional development in this area.

Throughout lesson

Over one-third of teachers (40 percent) stated that they were using a problem-solving approach throughout all of their teaching of a lesson, at least weekly. This suggests that a regular integrated approach to problem solving is occurring in a number of classrooms. However, evidence from later written responses suggested there may have been an overstatement of the actual situation.

The reason for the relatively small percentage of teachers using a problem-solving approach regularly through all lessons may be found in Burkhardt (1988). He believes that standard teaching methods are largely single track and depend on the method being explained by the teacher. On the other hand, problem solving is multi-track and is led by the methods proposed by individual students. Burkhardt feels that problem solving is difficult for the teacher. He identifies three areas giving rise to this difficulty— mathematical, pedagogical, and personal .

Mathematically the teacher must scan the different approaches that students are using and assess how useful each of these may be. The teacher then has to decide how best to complete the mathematical task from each of these starting points.

Pedagogically the teacher must decide when to help and when not to help, and what support and questioning to provide for each student or each group of students.

Personally there is the problem of teacher confidence. Because teachers may be confronted with approaches that they have not considered before, they need to believe that they can determine, possibly with student help, which approaches lead to solutions and which don’t. This is a potentially non-trivial barrier to tackling problem solving in class.

End of unit

In the past it has been the practice to include word problems at the end of a section in order for students to practise newly learned skills in potentially novel situations. Some 27 percent of the respondents are using problem solving in this way at least monthly. It is worth noting though that all but one of the teachers who used problem solving at the end of a unit of work was also using problem-solving lesson starters or was also using problem solving throughout all of their teaching weekly. Applying skills to problems at the end of a unit of work seems to be only one component of a teacher’s problem-solving repertoire.

Extension students

Many schools encourage their capable students to experience independent problem solving in a variety of ways. These include mathematics or computer clubs, small groups working apart from the normal class, through mathematics competitions, or via individual extension material. In a survey of talented mathematics students, Curran, Holton, Daniel, and Pek (1992) found that such students enjoyed the challenge and the open approach of problemsolving tasks. Burkhardt (1988) highlights a common belief in the need for problem solving for the gifted child but is reserved about how best to balance this need in the gifted students’ mathematical diet.

The survey shows that, at least monthly, individual students are being given specific extension opportunities in the area of problem solving by 43 percent of teachers. Other able students will benefit from being exposed to regular classroom problem-solving activities. However, of the 57 percent of classrooms which are not providing specific problem-solving experiences regularly for their capable students, some 41 percent do not provide opportunities for problem solving in the regular classroom at least monthly. Hence, in over 23 percent of classrooms, bright students are only occasionally being exposed to any form of problem solving.

What strategies are being promoted?

Pólya (1973) stresses the importance of strategies for solving problems. Strategies are just means to discover a solution. They are almost always not the method of solution nor a justification of the answer. They are, however, a mechanism by which the answer may be found and then justified, if necessary, by some other means.

The suggested learning experiences in the problem-solving section of Mathematics in the New Zealand Curriculum include the “devising, using, and modifying of problem solving strategies”. The questionnaire in our study referred to 12 strategies from Mathematics in the New Zealand Curriculum . These were the strategies that we thought would be commonly used by teachers. The results are summarised in table 2.

The predominance of Find a pattern, etc is perhaps to be expected. It is one of the easiest techniques to promote because of the natural occurrence of patterning in students’ environments. In addition, children come to secondary school with a history of patterning experience from primary school for teachers to draw on. Further, the emphasis on patterning is encouraged by Mathematics in the New Zealand Curriculum . Exploring patterns and relationships is an achievement objective for all levels.

We believe that Make a list and Guess and check are also popular because they are easy to teach and can be applied to a range of problems. Equally it may be that these strategies are well suited to the types of problems that are frequently used in the early years of high school.

Surprisingly though, Draw a diagram was only rated as a commonly-used strategy by just over half of the teachers. It is not clear whether this is because teachers feel that this is not “proper” mathematics at the secondary level or whether its value as a tool is not appreciated. In the research project we have seen diagrams being of great value, especially to weaker students. It provides a closer representation of the problem than, for instance, algebra does, and hence may make some problems more accessible to students.

A similar accessibility can be provided through the strategies Make a model or Act out . These heuristics (or the means by which a solution is sought) give more control of the problem to the students and can provide motivation. Lovitt and Clarke (1988) advocate these approaches, referring to them as “kinesthetic”. It is now acknowledged that children learn in different ways and the tactile, physical, pictorial thinkers could well benefit if these strategies and Draw a diagram were used more often. It may be that some teachers avoid these strategies because they view them as being more appropriate for primary schools. They may also find the strategies too difficult to implement because of the resources needed and the time constraints on their lessons. Only a quarter of respondents regularly discuss Thinking creatively . It is difficult for a teacher to model with students that “flash of brilliance” which seems to come from nowhere. However, the fostering of such insights by discussion and brain storming is vital if we are to extend the horizons of students’ thinking and to enable them to strike out in new directions.

Strategies need to be taught

Our research supports the conclusion that strategies need to be taught. They also need to be practised otherwise they become forgotten. This seems to be especially true for young students and students who are weak mathematically.

Generally a number of strategies can be used in one problem. For example:

Greedygrimes Charlie ate a total of 100 jellybeans in 5 days, each day eating 6 more than the previous day. How many jellybeans did he eat on the third day?

This problem was solved by students in form 4 [year 10 in New Zealand, year 9 in Australia] using:

The Australian National Statement on Mathematics says that the methods of good problem solvers are likely to be “idiosyncratic” and that students need to discuss a variety of strategies to increase their “awareness of the range of techniques available”. This is supported by Beagle (1979), Pólya (1973), and Schoenfeld (1992). The first of these authors states:

… problem solving strategies are both problem and student specific often enough to suggest that finding one (or few) strategies which should be taught to all, or most students, is far too simplistic.

It is not clear, though, how heuristics should be taught. Holton (1994) and Neyland (1994) share a concern that teaching should not lead to strategies being employed like another set of algorithms or rules. They also say that students should be encouraged to see how to translate strategies to other problems and situations. Strategy usage and teaching is not straightforward and much more research is required in these areas.

How would you like to use problem solving in the future?

Just under 60 percent of the teachers surveyed want to use problem solving more often, suggesting that a receptive environment for future development does exist. They do identify a number of common difficulties though. Among these are:

Our experience is that problem solving does take more time, especially in the early stages. However, time can be saved later. One of the teachers in the research project commented that using a group investigative approach to construction tasks and locus problems enabled her students to develop a number of concepts in one lesson which usually took her two or three lessons of more structured teaching. This approach also freed her to help individuals who were struggling, resulting in all students achieving more than usual in the time.

There is a real concern for the needs of less-able students. This relates in particular to their reading ability. One of the classes in our research project is a low-ability class. They work effectively at problem solving. This may be because all the students are of comparable ability, because the teacher uses problems that can be completed in ten to fifteen minutes, because the reading level required by the problems is not too high, or because the teacher chooses problems in a single lesson that require the same strategies.

What help do you need with problem solving?

The teachers in the survey overwhelmingly requested pre-prepared resources and problems that would fit in with content strands. Some 74 percent of the respondents asked for resources of this kind. Some typical comments regarding teacher needs were:

As Neyland (1994) and Lovitt (1995) say, problem-solving activities by themselves are not enough to guarantee good problem solving. However, a start in this direction cannot be made without these activities. There is an urgent need to produce them in a form which makes them easily used.

Other comments by teachers included the provision of appropriate physical environments for problem solving. This meant appropriate equipment as well as space. Teachers also felt they needed help and training in scaffolding, heuristics, and metacognition (thinking about thinking—an important aspect of problem solving). Some assistance in extension work for bright students would also be appreciated.

Two valuable suggestions were made which could usefully be explored. These are:

With the change of the political environment, teachers may find the Ministry of Education in New Zealand will no longer provide the continuing services it has done in the past. Resources for teachers are less likely to be seen as a central responsibility and if not contracted out may be left to private enterprise to produce. Until these appear on the scene the next phase of teacher development will be largely left in the hands of the teachers themselves.

A TEACHING PERSPECTIVE

In the short period since the implementation of the new curriculum, it appears that most teachers are making at least some effort to incorporate problem-solving approaches into their mathematics classrooms. At present, problem solving is not a regular feature of most classrooms but it is clear that teachers are prepared to increase their use of problem solving. This increased use can be facilitated by the production of appropriate content-related problems. Indeed the teachers surveyed see this as the first priority to the further implementation of a problem-solving approach to mathematics.

ADVANTAGES OF PROBLEM SOLVING

The advantages proposed for problem solving in school are that it:

Unfortunately simply giving teachers good reasons for teaching problem solving, and giving them good problems to pass on to their students, does not produce good problem solving. As Lovitt (1995) says:

All the early problem-solving efforts were mostly devoted to the creation of suitable problems in the belief that teachers could present these in classrooms and generate effective learning with the same maths they used for expository teaching. It has taken some time to recognise that this is not the case …

Neyland (1994) expresses the same view:

Some well designed [problem-solving activities] do not result in the active learning event intended. There appear to be two main reasons for this. Firstly, the way the activity was originally presented to the class diminished its potential. And secondly, the interactive component of the teaching–learning process was not adequately prepared and unanticipated problems arose. [emphasis in original]

Before starting to teach problem solving, teachers should be clear why, apart from the fact that it is in the curriculum, they intend teaching it. It is important too, to note that the problem-solving approach to teaching mathematics is different from what has become the traditional approach. However, much of the problem-solving approach is not new. Scaffolding, which is not limited to problem solving, has its origins in the Socratic method. Heuristics too have a long pedigree. So how is problem solving different from what has become the standard “chalk and talk” method for teaching mathematics?

The main difference is one of attitude or philosophy on the part of the teacher. The shift is from the so-called “sage on the stage” to the “guide by the side”. Philosophically, the teacher needs to change from a giving role to an encouraging role, from “here is how to solve a linear equation” to “how might we solve this linear equation? ”

Naturally the students will not be able to invent all the mathematics they need for themselves. There will still be things that they need to be told. They will still need to practise both skills and problemsolving processes. However, more time needs to be spent by the students exploring mathematics with the teacher as a guide. During this exploration, seeds will be planted and students will develop connections between various parts of their learning which will increase learning, understanding and retention later ( see Hiebert and Carpenter, 1992).

So, instead of teachers taking students down well-trodden paths in a sequential unfolding of mathematical structures, there should be more emphasis on the students themselves structuring mathematical knowledge for specific problem contexts. Certainly the structures that are formed this way have to be justified but not necessarily in the one way shown in the textbook. Where possible, students should be given the opportunity to provide their own justifications. Many of these will be correct and may well be different from the text-book proof.

This is not to say that the development of mathematics during the year is anarchic. Teachers should know from the start what material is to be covered. However, there should be some flexibility in the manner and order in which the content arises. Of course, in highstakes years, when there are external examinations, the course may have to be slightly more structured. Nevertheless, it is important even on these occasions for teachers’ questioning to be open rather than closed in order to stimulate students’ thinking.

In the problem-solving approach then, there is less emphasis on students applying rules to problems which have been carefully chosen to fit those rules. There is greater emphasis on the creative construction of mathematical structures and solutions in non-routine situations and over a range of contexts.

Incorporating process and model

In order to give the following five-part overview of the problem-solving process we combine the model of problem solving given in Holton and Neyland (1996) with the heuristics and metacognition discussed in that paper ( see figure 1).

We recall that heuristics are means by which a solution is sought. Most of the heuristics of Begg (1994) are covered in this overview of the problem-solving process. We also show that certain heuristics tend naturally to appear at certain stages. In what follows, heuristics are emphasised in italics.

I Getting started

This is the first part of problem solving and includes “problem” and “experiment” from figure 1. It also covers the first two phases of Póya’s four-phase model.

1. Understand the problem

The problem will either be posed by the students themselves, presented in writing, or given verbally. If the problem is not one posed by the students, a first and crucial step is for them to make the problem their own; to become familiar with its conditions, characteristics, and variables. If it is in writing the solver has to read the problem and read it carefully. Many a solver has begun by solving a problem they thought was there but was actually not.

As the solvers read the problem they should be asking the various questions posed by Pólya. They should also be looking for key words . These are words which, if changed, will lead to a quite different problem. They are words or phrases which are essential in the solution of the problem. For novice problem solvers, underlining key words is a useful strategy. It is also a good idea for the problem solver to restate the problem for themselves . This helps to emphasise the key aspects of the problem. Other strategies such as experimenting with special cases also help the student to understand the problem.

Consider the problem below.

Peter is 18 years older than his daughter Sue who is 7 years old. How old will Peter be when he is twice as old as Sue?

In this problem, “18”, “7”, and “twice as old” all appear to be key words or phrases. The question, and more importantly the answer, is altered if key words are changed. (In actual fact, “7” is an extraneous piece of information here.) On the other hand, “Peter” could be “Peta” and “Sue”, “Sam” without changing the answer or the method of solution.

In this particular problem it is obvious that both Peter and Sue age at the same rate, one year at a time. This is nowhere explicitly stated but it is an hidden assumption . Many problems have implicit conditions that turn out to be important in finding a solution. So it is vital to make sure that hidden assumptions have been noted.

One of the metacognitive aspects of problem solving is to know when to come back to the original problem and read it again. This may be necessary to see if you are on the right track of a solution or if you have inadvertently started to solve another problem. It may also help you to try another approach if you have made no progress with a particular method of attack or change the point of view . Once an answer has been obtained it is important to reread the question to ensure that the solution you have obtained does indeed solve the original problem.

Apart from the metacognition (or “thinking about thinking”) at the start of a problem—which largely relates to monitoring and controlling the problem-solving processes—it is necessary for solvers to go through a mental list of heuristics in an effort to find a few which will get them started.

It is worth asking the following Pólya questions:

In addition it is worth considering:

3. Experiment

This is often an early stage in the problem-solving process. It has two main functions. The first is to get a feel for the problem. The second is to start to produce some evidence for a conjecture. The experiments may be calculations or measurements or a variety of things depending on the problem in hand.

The experiments should be systematic . The aim is to get central, useful information that can be collated in some way, rather than an unconnected jumble of, say, numbers. And the results of experiments should be recorded in some logical fashion— in a list, table , or diagram . When appropriate, consider the case n = 1, then n = 2 and so on. It is worth keeping these experiments until the problem has been completely solved. They may well be useful to provide or inspire a counterexample later.

Another problem that is useful to illustrate some ideas here is the frog problem.

Four spotted frogs (S) and four green frogs (G) are sitting on lilypads as in figure 2. Frogs can move to the next lilypad if it is free or they can jump over another frog if there is an empty lilypad on the other side. What is the smallest number of moves which can interchange the green and the spotted frogs?

There is a certain amount of symmetry in this problem. Whatever happens next, certainly it doesn’t matter whether a spotted or a green frog starts first.

This is a good problem too, to illustrate solving a simpler problem first . It’s easier to do the problem above once you’ve tried moving one frog of each type, two frogs of each type, and so on.

This is a common feeling at the start of a new problem. Success and experience will give the solver confidence to continue. However, even expert problem solvers face panic on occasions. Be prepared to think “there is no way I will ever do this problem” and then get on and realise you can.

II Conjecturing

The next process in problem solving involves conjecture.

This can be the most enjoyable and interesting part of the problemsolving process. In more difficult problems it is necessary to find a pattern, rule , or relation in order to produce a conjecture.

Some problems give up the correct conjecture immediately, other problems take more time. In harder problems it may be necessary to discard a number of conjectures before obtaining the right one. Some people find some problems easier to make conjectures about than others. And conjectures require practice. So it is necessary to start with simple problems and work up.

Does the conjecture make sense? Is it consistent with previous knowledge and the data that has been assembled in the experimental phase? Does the conjecture imply anything that feels wrong? Intuition and common sense can and should be used here. A conjecture which implies that the height of a mountain is five centimetres must surely be wrong. Any conjecture should be justifiable . Even though it cannot be proved at this stage, there should be good reasons for choosing one conjecture over another.

III Proof/Counter example

The more difficult problems will require the solver to trade off conjecture against proofs against counterexamples. The process is a dynamic one which is only completed when the final step in the solution is written down. Frequently, trying to prove a conjecture gives an idea for a counterexample and looking for a counterexample can give an idea for a proof. Whether starting down the proof or counterexample trail the solver may see that the conjecture needs adjusting and the process starts again.

It is difficult to see whether to first try to justify a conjecture or show it is wrong. The decision as to which to try first depends on the solver and the problem. If testing a few cases will cover all possible counterexamples, then the solver should go that route. This will either produce a counterexample or strongly confirm the conjecture. In the latter case, of course, a proof will still be required.

Some problems look like others and this may suggest a way to proceed to a proof. A slight change of a known proof might work. Or the problem may be one which suggests a well-known proof technique such as proof by contradiction or mathematical induction .

1. Extreme cases

In trying to find counterexamples it is often valuable to try extreme examples or extreme cases . These examples are somehow at the edge of the spectrum of values that are being used. For instance, what happens if a number is very small or what happens as “n” approaches infinity? What happens if the triangle is isosceles? Extreme cases are usually easier to handle than more general situations so it is worth testing a few before trying a proof. Even if they do not provide a counterexample they may well give the solver some useful information which can be used later. Here is a problem which can be quickly solved by looking at an extreme case: Do all one-litre milk containers have the same surface area? The answer, no, can be easily obtained by imagining a one-litre box-shaped milk container with a base the size of a tennis court and a height a fraction of a millimetre. Clearly this one-litre container has a surface area much larger than the one in the fridge.

2. Special cases

A conjecture can also be tested against special cases rather than extreme cases. Special cases are typical cases such as the ones during the experimental phase. Where possible, cases would be tested because they are straight forward or easy to test. But sometimes more complicated cases are forced on the solver.

3. Simpler problems

Sometimes problems are far too difficult to solve the first time. They may involve far too many variables to be able to understand easily. In that case reduce the problem to a simpler problem in some way. For instance, in the frog problem using fewer frogs is a good way to start. Similarly a gambling problem involving five dice might be reduced at first to one involving only two. Insight gained from the two dice case may well lead to the solution of the original problem. Here the aim is to keep as many of the essentials of the problem as possible while reducing the problem to a manageable size. Using simpler problems is also something that is useful at the experimental stage, as we saw earlier. Sometimes problems can be simplified in more than one way and students should learn to choose the best from the range of available simplified situations.

4. Exhaust all possibilities

One of the very simple methods of proof is to exhaust all possibilities . If there are sufficiently few cases to handle (under 50, say), then by looking at each one in turn, the solver can learn about the entire problem. This way any conjecture about the situation is immediately verified or disproved. Simple combinatorial problems such as the behaviour of two dice are often open to this approach. For instance, how many different ways are there for two dice to give a sum of four?

When a solution has been obtained by older children by the exhaustive approach, it is worth asking the Pólya phase four question “Can you derive the result differently?” They should be looking for more sophisticated approaches. There are times, however, when no other method is available. Certainly this will be the case with younger children. And finding a justification, any justification, is better than no justification at all.

5. Guess and check

One of the simplest ways of tackling a problem is guess and check . This is sometimes also called trial and error . In the problem of Peter and Sue, an answer can be found by guessing and checking. It is therefore a good strategy for simple problems. There are drawbacks, however. First, if the problem involved has a large number of cases, for instance the five dice gambling game we alluded to earlier, then it may not be easy to guess the correct answer in a reasonable time. Second, guessing and checking will give an answer. However, the method cannot tell you whether the answer is unique. It may well be that the problem has a number of answers. Only an exhaustive search will be able to determine all answers in such a problem. Finally, the guess and check method in complicated situations may only reveal the conjecture required; it may not justify that conjecture.

The efficiency of guess and check can be increased with metacognition by using guess and improve . Here subsequent guesses are improved using the data of past guesses. Again in simple problems where a unique answer is almost certain, this is a useful strategy. In Peter and Sue’s problem, we could first guess that Peter was 30, in which case Sue would be 12. The check 2 × 12≤ 30, shows that we have guessed incorrectly. We could then guess, say 40. Now 2 × 22 = 44 which is more than 30. The next guess should then be directed between 30 and 40. Guessing and thinking allows us to hone in on the correct answer.

To make sure that we don’t keep making the same guesses, the results could be recorded in a table such as that in figure 3. Another advantage of the table is that it enables us to move our guesses in the right direction; it enables us to see how best to use guess and improve.

We actually came across a nice solution of this problem where the student had made a model . On the edges of two pieces of paper he wrote the numbers 1 to 30. He then put the edges of the paper side by side and moved one of them until corresponding numbers were 18 higher on one edge ( see figure 3). He then looked along the edges to try to find where one number was twice the other. (Actually he didn’t put down enough numbers, but what he had was enough to enable him to see the right answer.)

6. Work backwards

Work backwards is often useful in problems which involve a sequence of moves, events, or operations leading to a known end point. By reversing the process the starting configuration can be obtained; this is a bit like running a video in reverse. It is useful for investigating two-person games. Start from the last move and work back to the starting position. Work backwards is the basis of many commonly used methods for solving equations. This strategy is also useful when dealing with proofs of trigonometric identities.

6. All information

If the solver is unable to make any headway it is often worth re-reading the question in an attempt to ensure that all of the information given in the problem has been used. Sometimes there is an implicit detail that is not mentioned specifically but which is vital to the solution. Knowing general facts such as the sum of the dots on opposite sides of a dice is seven may well be required to solve a problem. (See hidden information in the experimental stage.)

Having obtained a complete solution the solver should go through the justification and check every step. Even if the proof passes the check test, sometimes the solver still has a nagging doubt. In all cases it is a good idea for the solver to get someone else to check their solutions. At this stage too, sometimes a solver may see a quicker, more efficient, way to solve the problem. The new method needs to be written down and checked too.

IV GENERALISATION/EXTENSION

1. generalisation.

A generalisation is a problem which contains the original problem as a subset or special case. Generalisations can often be found by increasing the number of digits (in some number problems), stepping up the dimension (in a geometry problem), or by increasing the number of variables in some way. For instance, you may want to do the frog problem with an arbitrary number of frogs (n, say) on either side. This is a generalisation of the four frog case.

The reason for considering generalisations is that they give a result which is true for a much wider class of objects. One difficulty with generalisations is that they may require a quite different justification from the justification used in the original problem.

2. Extension

An extension of a problem is one which is related to the original problem but not by way of a generalisation. These can be found by changing one of the conditions of the original problem in some way. It may be that a problem can be extended by changing addition to multiplication. As with generalisations, the proof of an extension may not be linked in any way to the proof of the original problem.

The final part of the problem-solving process may in fact be to give it up!

Some problems are too difficult to be solved right now. So, at times, students will have to abandon some problems. If they are forced to abandon all problems they tackle then the problems are too difficult.

This will depend on the teacher and the student. It becomes a matter of priorities. However, any reasonable problem will take more than 10 minutes and less than two hours (unless it is an extended investigation). In general, we suspect that students can go further than we usually expect. This is especially true of the better students. And it should be remembered that expert problem solvers like to sleep on a problem if no solution is at first forthcoming.

In cases where the problem is too hard for both the teacher and the student, outside help should be summoned. This may be from another teacher, a book, or a local university.

Implementing problem solving

If you are a teacher who has been convinced philosophically of the importance of problem solving in mathematics, or if you see it in the curriculum and feel a responsibility to teach problem solving, the next question to be resolved is how ? The first thing to point out is that there is no unique way to teach problem solving. Teachers and schools will need to develop their own styles. Each teacher will need to develop an individual style which will no doubt develop and be a function of the school, the students, and many other things. An easy way to start is by using a series of one-off problems that may not necessarily be related to the content of the remainder of the lesson. This allows the teacher to gain confidence and gather together a string of useful problems. However, the danger is that students will not realise the relevance and importance of problem solving that stands alone, not integrated with the rest of the curriculum.

Our research suggests that problem-solving lessons work well around a three-stage format. The first stage is a whole-class format, where the problem is discussed by teachers and students, and students may suggest possible heuristics.

In the second stage, students work in small groups. The aim of this stage is to enable students to become involved with the problem and attempt to solve it for themselves. During this period, the teacher is able to go from group to group to provide scaffolding. However, our research suggests that most scaffolding during group work is undertaken by the students themselves as they give peer tuition to their group members.

The third stage is a reporting back stage which gives the students an opportunity to say what they have done and how they did it. Here the spotlight is again on the students who are reporting on their work. Different methods of solution are to be encouraged during this stage. It is important that student thinking and learning takes place in this final stage. One reason for the reporting is to improve students’ ability to communicate but even more important is the chance for students to see alternative approaches to a problem. These alternative approaches should be useful on future occasions. In this stage too, there is an opportunity for students to see links between different parts of mathematics which is again another aid towards understanding.

We have observed these three stages in a variety of problem-solving classes. They seem to work equally well with weak and strong students and with young and old students. A general rule of thumb is that the three-stage cycle should be repeated with young students and weak students. For these children the three stages should take no more than 15 minutes. Very good students may only go around the cycle once in a one-hour lesson.

Problem solving is not easy for many teachers as it involves a rethinking of their whole approach to teaching mathematics. The difficulties arise for many reasons. Some relate to potential loss of control in going from a closed to an open environment where it is never clear in which direction the students will head next. Some relate to the wider range of teaching skills required to handle the situation. Other difficulties are due to social phenomena, including student expectation. But it is our experience that most of these difficulties are surmountable by many teachers, even though at times they may be close to giving up. In our current research, one teacher was in despair after what she thought was a bad lesson. The next problem-solving lesson she taught was extremely successful, with the students seeing links that she herself had not previously seen. From that moment she has been sold on problem solving.

Another teacher had worked solidly doing a competent job for over two terms with a weak-streamed class. In a regular third-term exam (not based on problem solving) her students were 10 percent above similar students in another class. Her students had gained considerably on word problems. The practice they had had in problem-solving situations had given them confidence to successfully tackle the word problems in the exam.

DEREK HOLTON is Professor in the Department of Mathematics and Statistics, University of Otago. E-mail: [email protected]

JIM NEYLAND is Lecturer of Mathematics Education in the Mathematics and Science Education Centre, Victoria University of Wellington. E-mail: [email protected]

JULIE ANDERSON is in the Department of Mathematics and Statistics, University of Otago, and St Hilda’s College, Dunedin.

We would like to thank the Ministry of Education, Research Section, for the financial support which allowed us to engage in the research on which this paper is based. Our thanks also go to the many teachers and students who allowed us into their classrooms.

For more details of this research, see:

Holton D., Spicer, T., & Thomas, G. (1995). Is problem solving too hard? Proceedings of Mathematics Education Research Group of Australasia , 345–351.

Holton D., Spicer T., Thomas G., & Young, S. (1996). The benefits of problem solving in the learning of mathematics (Report to the Ministry of Education). Dunedin: Otago University, Department of Mathematics and Statistics.

The New Zealand curriculum documents are:

Ministry of Education. (1992). Mathematics in the New Zealand curriculum . Wellington: Learning Media Ltd.

Ministry of Education. (1995). Implementing mathematical processes in the New Zealand curriculum . Wellington: Learning Media Ltd.

The Australian curriculum documents are:

Australian Education Council. (1990). A national statement on mathematics for Australian schools . Carlton, Vic: Curriculum Corporation.

That problem solving should be the central focus of the mathematics curriculum is from page 23 of:

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematic . Reston, VA: Author.

The problem-process approach is described by:

Sigurdson, S. E., Olson, A. T., & Mason, R. (1994). Problem solving and mathematics learning. Journal of Mathematical Behaviour 13, 361–388.

That talented mathematics students enjoyed the challenge and the open approach of problem-solving tasks is noted by:

Curran, J., Holton, D., Daniel, C., & Pek, W. H. (1992). A survey of talented secondary mathematics students . Dunedin: Department of Mathematics and Statistics, University of Otago.

A common belief in the need for problem solving for the gifted child is highlighted in:

Burkhardt, H. (1988). Teaching problem solving. In H. Burkhardt, S. Groves, A. Schoenfeld & K. Stacey (Eds.), Problem solving: A world view (pp. 17–42). Nottingham, UK: Shell Centre.

The importance of strategies for solving problems is noted in:

Pólya, G. (1973). How to solve it (3rd ed.). Princeton, NJ: Princeton University Press.

The kinesthetic approaches are advocated by:

Lovitt, C., & Clarke, D. J. (1988). Mathematics curriculum and teaching program, activity bank (Vols. 1 and 2). Canberra, ACT: Curriculum Development Centre.

That the methods of good problem solvers are likely to be “idiosyncratic” is supported by:

Beagle (1979) pp. 145–146, see above.

Pólya (1973), see above.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, meta-cognition and sense making in mathematics. In Douglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.

That students should be encouraged to see how to translate strategies to other problems and situations is stated by:

Holton, D. (1994). Problem solving. In J. Neyland (Ed.), Mathematics education: A handbook for teachers (Vol. 1, pp. 18–30). Wellington: Wellington College of Education.

Neyland, J. (1994). Designing rich mathematical activities. In J. Neyland (Ed.), Mathematics education: A handbook for teachers (Vol. 1, pp.106–122). Wellington: Wellington College of Education.

That problem-solving activities by themselves are not enough to guarantee good problem solving is stated by:

Lovitt, C. (1995). Personal communication.

That time needs to be spent by the students exploring mathematics with the teacher as a guide is noted by:

Hiebert, J. & Carpenter, J. P. (1996). The benefits of problem solving to the learning of mathematics (Report to the Ministry of Education, Wellington). Dunedin: University of Otago, Department of Mathematics and Statistics.

For more details of the heuristics used in the problemsolving process see:

Begg, A. (1994). Mathematics: Content and process. In J. Neyland (Ed.), Mathematics education: A handbook for teacher (Vol. 1, pp. 183–192). Wellington: Wellington College of Education.

Ministry of Education. (1995). Implementing mathematical process in mathematics in the New Zealand curriculum . Wellington: Learning Media, Ltd.

National Council of Teachers of Mathematics. (1980). An agenda for action . Reston, VA: Author.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics . Reston, VA: Author.

Neyland, J. (1995). Eight approaches to teaching mathematics. In J. Neyland (Ed.), Mathematical education: A handbook for teachers (Vol. 2, pp. 34–48). Wellington: Wellington College of Education.

For a good example of problem-solving teachers include:

Stacey, K. & Groves, S. (1985). Strategies for problem solving . Camberwell, Vic: Lattude Publications.

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What maths teaching could and should learn from cognitive science

10 April 2024

Education and society , Faculty of Science

Cognitive science has shown us that information stored in our long-term memory is instrumental in facilitating thinking and learning, and we should apply that knowledge in the way we teach maths, says Tanya Evans

In the arena of educational policy, where decisions shape the future of our students, the plight of mathematics education in New Zealand has become a drawn-out match of political football.

As head of the mathematics education unit in the mathematics department at the University of Auckland, I’ve spent the last three years dissecting the factors contributing to the steady decline in mathematical proficiency across our nation’s schools. What I’ve observed is a troubling dissonance between the prevailing educational theories that have shaped policies for decades and the empirically grounded insights offered by cognitive science. The widening disconnect at the academic level has led to what I perceive as an undeclared paradigm war.

In New Zealand, the educational landscape has a multitude of approaches, each rooted in various educational philosophies and theories – project-based learning, inquiry-based learning, and traditional teacher-led instruction to name a few.

As a mother of three children navigating the New Zealand school system, I’ve witnessed the diverse methods used in their classrooms. My children have shown a distinct lack of enthusiasm for the project-based learning environment embraced by their intermediate school. Having spent two years immersed in this approach, they felt they haven’t progressed beyond the primary school curriculum. The teacher’s role seems confined to that of a facilitator – a telltale sign of adherence to a particular educational philosophy. My son recounted his frustrations: “Our teacher doesn’t offer much explanation; she simply assigns problems for us to solve in groups, and every third lesson, she checks in to see if we have any questions.”

While these approaches reflect differing perspectives on education, some of them neglect to fully integrate the findings of cognitive science, which offer invaluable insights into how we learn and retain information.

Well-organised information that can be easily accessed (without thinking twice) from long-term memory means the working memory operates without apparent constraints. The more we remember, the easier it is for us to think and learn.

One damaging consequence of this paradigm war is the false dichotomy between ‘rote learning’ and ‘learning with understanding’. Mathematics education in New Zealand, as well as in other English-speaking nations, has fallen prey to a misguided notion: that prioritising the memorisation of key mathematical facts, such as multiplication tables, is detrimental to students’ comprehension and enthusiasm for the subject. This belief rests on the misguided notion that long-term memory plays a minimal role in fostering understanding. In reality, the information stored in our long-term memory is instrumental in facilitating effective thinking and learning.

When encountering and processing new information, the thinking process occurs primarily in the working memory. However, the capacity of the working memory is limited in terms of the number of elements it can handle simultaneously and the duration of this processing. We can only process a finite amount of novel information at any given time within our working memory. Exceeding this limit leads to cognitive overload, hindering the ability to absorb and comprehend new material.

These limitations essentially vanish when the working memory processes information that is easily retrieved from long-term memory. Well-organised information that can be easily accessed (without thinking twice) from long-term memory means the working memory operates without apparent constraints. The more we remember, the easier it is for us to think and learn. Neither Google nor ChatGPT are substitutes for long-term memory; the novel information found on Google is subject to the limitations of our working memory – it adds to the count of the elements to be processed. Thus, the metaphor of an external memory stick expanding memory reserves fails to accurately depict the intricacies of human cognitive architecture.

Another detrimental fallout of the paradigm wars is the vilification of the teacher-led classroom model. Teacher-led instruction is often said to stifle students’ innate curiosity, extinguish their interest, and obstruct conceptual comprehension. This conflicts with the science. A well-crafted explanation of a mathematical concept or procedure by a proficient educator serves to channel learners’ attention towards incoming information, fostering active cognitive engagement crucial for selecting pertinent information for further processing in the working memory. This, in turn, enhances our capacity to comprehend new mathematical concepts.

Scientific research has clearly demonstrated the efficacy of explicit instruction when students are introduced to novel concepts and procedures. Teacher-led instruction involves the provision of clear explanations, demonstration of examples, and immediate corrective feedback during initial practice attempts.

Subsequently, as proficiency increases to a satisfactory level of accuracy, students should transition to a phase characterised by extensive practice – a phase marked by repetition and reinforcement. It is through copious and meticulous practice that students achieve the requisite speed of execution, paving the way for the development of automaticity – the ability to do things without thinking too hard, such as riding a bike or driving a car.

The speed is not merely ‘nice to have’, it is a crucial catalyst for mathematical generalisation. Without the attainment of automaticity (mathematical fluency), the ability to generalise becomes arduous. After all, the capacity to generalise is intricately linked with what is commonly perceived as conceptual understanding – the oft-cited pinnacle of learning objectives.

There is much more that we have learned about mathematical cognition (for further reading I’d recommend An Introduction to Mathematical Cognition) and the most effective ways to develop mathematical thinking based on decades of research. I hope policymakers will take a break from the ongoing political football match, which seems to be progressing without on-field referees, who are left yelling from the sidelines, often completely ignored.

I urge them to instead prioritise the science of learning over political expediency. It is imperative that we heed the insights provided by research to inform policy and shape recommendations for a revitalised mathematics curriculum and a common practice model for mathematics teaching and learning.

Dr Tanya Evans is a senior lecturer in the Department of Mathematics, Faculty of Science

This article reflects the opinion of the author and not necessarily the views of Waipapa Taumata Rau University of Auckland.

This article was first published on Newsroom,  Maths teaching fails pupils by ignoring cognitive science findings , 10 April, 2024

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Margo White I Research communications editor Mob 021 926 408 Email margo.white@auckland.ac.nz