numeracy problem solving task

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• Authentic tasks
• F - 10 Resources

Authentic tasks are designed to help students see mathematics as worthwhile and important. When students understand the purpose of a given problem in mathematics, they are more likely to persist when challenged. Authentic tasks generally have an ‘open middle’ which means that students can use different representations and solutions to communicate their knowledge and reasoning.

These curated links provide MAV members with access to nine authentic tasks from some of our primary consultants’ favourite resources. The 11 criteria provide MAV members with a research-informed context to consider each task’s potential impact on student thinking, ways of working, attitudes towards mathematics, their knowledge and understanding.

The following criteria was used to select the tasks based on their potential:

Used with permission © Martin Holt Educational Consultant 2017

If you would like to learn more about this approach to assessing or using tasks contact [email protected]

Statistics and probability

Measurement and geometry, number and algebra.

These support pages were produced using Strategic Partnership Program funding from the Department of Education and Training.

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These incredibly powerful, flexible activities can be used with a variety of content and contexts.

These games are chosen for their simplicity and depth.

Problems that resist easy solutions while encouraging perseverance and deeper understanding.

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Selecting this shows all the free lessons in our library.

Every student deserves to have the opportunity to problem-solve and engage in genuine mathematical thinking. Rich tasks are designed to make these rich learning experiences possible. We’ve written these tasks to launch quickly, engage students, and promote the habits of mind mathematicians need: perseverance & pattern-seeking, courage & curiosity, organization & communication.

This free video PD series will help you get the most out of the tasks below.

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Problem Solving Activities: 7 Strategies

• Critical Thinking

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

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

I was so excited!

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

It was a proud moment for me!

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

Genius right? 

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

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

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

When I Finally Saw the Light

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

Problem Solving Activities

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

Seasonal Problem Solving

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

Cooperative Problem Solving Tasks

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

Notice and Wonder

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

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

Math Starters

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

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

Calculators

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

Three-Act Math Tasks

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

Getting the Most from Each of the Problem Solving Activities

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

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

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

Shametria Routt Banks

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2 Responses

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

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

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Khan Academy Blog

Free Math Worksheets — Over 100k free practice problems on Khan Academy

Looking for free math worksheets.

You’ve found something even better!

That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

Just choose your grade level or topic to get access to 100% free practice questions:

Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

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Statistics and probability

High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

• Addition and subtraction
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Frequently Asked Questions about Khan Academy and Math Worksheets

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Solving Linear Equations: Rich Tasks

You cannot beat a good rich task! For me, a rich task is one that both stimulates and challenges students of all ages and abilities. Here is a selection of some of my favourites.

Median Rich Tasks and Purposeful Practise keyboard_arrow_up Back to Top

There are no words to express how much I love Don Steward and his amazing resources on his Median Blog . They allow for purposeful practise, to enable students to gain fluency in key skills. But they also have more than a sprinkling of richness, that leads students merrily along the path towards hypotheses and generalisations. Amazing.

Forming and Solving

Solving practice, nrich keyboard_arrow_up back to top.

NRICH is simply one of the best websites for rich maths problems in the whole wide world.

Other Rich Tasks keyboard_arrow_up Back to Top

A selection of some of my favourite rich tasks, including – I must confess – some of my own. For my complete collection, including my thoughts on what makes a good rich task, please visit the Rich Tasks page .

Peter Liljedahl

Professor, faculty of education, sfu, numeracy tasks.

… efforts to intensify attention to the traditional mathematics curriculum do not necessarily lead to increased competency with quantitative data and numbers. While perhaps surprising to many in the public, this conclusion follows from a simple recognition that is, unlike mathematics, numeracy does not so much lead upwards in an ascending pursuit of abstraction as it moves outward toward an ever richer engagement with life’s diverse contexts and situations. Robert Orrill

What follows are collections of numeracy tasks organized according to grade bands – b ut these grade bands are only meant to be guideline. My experience is that these tasks tend to be upwardly applicable. That is, the tasks work well with students older than the band the task was designed for.

Many of these tasks were co-constructed with, and piloted by, teachers from Coquitlam (sd43), Prince George (sd57), Kelowna (sd23), and Mission (sd75). Watch for NEW tasks all the time. Enjoy.

Primary (k-3)

• Sharing Cookies  (there is a nice book to accompany this)
• Sharing Cupcakes
• Going Bowling
• Going Canoeing
• Going Swimming
• Kindergarten Snack Sharing

Intermediate (4-6)

• Building a Garden
• Planning a Tea Party
• The Class Pet
• Celebrity Travel Planning
• The Paper Route
• Designing a Planner Cover
• Activity Day
• Trip to the Waterslides
• Planning a Class Party
• Water Conservation
• Selling Cookies
• Earning Screen Time
• Gagner le screen time
• Unhappy Meals
• Building a Fence

Junior High School (7-8)

• Gwen Stefani Itinerary
• Concert in Toronto
• Design a New School
• 2004 Summer Olympic Results
• 2006 Winter Olympic Results
• New School Schedule I
• New School Schedule II
• Playland Adventure
• Nine Hole Golf Course
• Terry Fox Fundraiser
• Basketball Tournament
• Trouble at the Tournament
• Shopping Spree
• The Two Bike Race
• The New Publishing Room
• Cell Phone Plans
• Giving Out Bonuses
• Ski Trip Fundraiser
• Race Around the World

Senior High School (10-12)

• The Lost Crab Trap
• The Shoe Sale
• The Unresolved Wager
• Joint Investment
• Buying a House

Quote of Day

• Wisdom has two parts: 1) Having a lot to say, and 2) not saying it. (Unknown)
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Contact Information

Peter Liljedahl, PhD Professor Faculty of Education Simon Fraser University 8888 University Drive Burnaby, BC, V5A 1S6, Canada Voice: +778-782-5643 Email: [email protected]

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Problem-Solving, Open-Ended Tasks and Parallel Tasks

Problem solving.

“The world is changing. We no longer need students to just be fast calculators… in fact, technology now does that for us. We need students who can think and develop mathematical models and reason and problem solve…” (Boaler, 2014)

Types of Problem Solving

Worded mathematical problems (sometimes referred to in primary school as ‘story problems’);

Open-ended maths problems;

Rich and contextual tasks; and

Parallel tasks.

Problem Solving Strategies

Problem-solving involves more than being able to use the strategy that has been taught that week. It involves choice.

The ability to choose the most effective and efficient strategy to solve a task or an investigation. To interpret accurately what the task is asking, by modelling and applying mathematical skills in unfamiliar and/ or real world situations. It is this skill of problem-solving that prepares our students to integrate mathematical learning by recognising its value in everyday life.

Open-Ended Tasks

Closed or open tasks?

All mathematical problem types have their place in the classroom. Teachers should make instructional decisions about the type of task they use based on what they are trying to achieve. Closed tasks allow students to demonstrate a specific strategy, piece of knowledge or procedure. Conversely, open tasks allow students to demonstrate understanding of a range of strategies, problem-solving capabilities, mathematical reasoning and application of mathematics learning in a variety of contexts

Consider the information that the teacher is trying to elicit from students' using the following tasks:

Closed task :

There are 10 chickens, some are brown and some are yellow. How many are yellow and how many are brown? (Give counters to assist)

Closed tasks typically:

have one correct answer

can be completed quickly

assess one specific piece of knowledge, or a specific skill or procedure

provide limited information about student thinking

provide limited opportunity for students to demonstrate higher levels of understanding.

Open-ended tasks typically:

have a range of appropriate responses

take longer to complete

assess a range of knowledge and skills

provide information about problem-solving strategies and thinking

provide opportunity for students to demonstrate higher levels of understanding. AAMT, 2022 https://topdrawer.aamt.edu.au/Fractions/Assessment/Designing-assessment-tasks/Closed-or-open-tasks

Open-ended Tasks Readings and Learning

Making Tasks Open-Ended

Open-ended Maths Activities

Using Photos in the Mathematics Classroom

Robert Kaplinsky Problem Solving Tips

Open Ended and Problem Solving Tasks to Inform Teaching and Learning Sequences

Primary Maths Puzzles and Investigations

YouCubed - Tasks

Open-Ended Tasks Used for Assessment

Robert Kaplinsky 70 Real World Problems

Problems with Patterns and Numbers

The Mural Task

Problem Solving Classroom

Open-ended Tasks

10 Lessons for Algorithmic Thinking

Rich Learning Activities

ReSolve Open-Ended & Problem Solving Tasks

Open Ended and Problem Solving Maths Activities - Teacher Resources

It provides practical advice on how teachers can create their own open-ended and problem-solving questions, and use them effectively in the classroom.

This Oxford Open-Ended Maths Activities Teachers Reference explores the idea of open ended questions in a learning environment.

Engaging Maths: Everyday Investigations for Grades 3 to 6 is a combination of a book, a set of photo cards and a disc with these photos. It is a collection of rich, open-ended tasks.

This full-colour volume aims to enrich the mathematical experiences of primary school students (and their teachers) through enjoyable, challenging and active lessons.

Parallel Tasks

These are two or more tasks that focus on the same big idea at different developmental levels but which are quite similar.

They are designed to suit the needs of different students, but so that the whole range of students can participate in a discussion about them.

Newman's Error Analysis Prompts

Australian educator Anne Newman (1977) found that when a person is faced with a written Mathematical problem they are faced with a number of hurdles.

She developed the Newman's Errr Analyse Prompts which assists teachers to...

Identify where errors occur in a students attempt to solve written problems.

There are 5 steps to problem solving successfully.

Comprehension

Transformation

Process Skills

Newman's Error Analysis

References:

The home of mathematics education in New Zealand.

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Problem Solving

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

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

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

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

• Level 1 Problems
• Level 2 Problems
• Level 3 Problems
• Level 4 Problems
• Level 5 Problems
• Level 6 Problems

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

Teaching methods: Engaging students with problem solving tasks in maths

Thanks for downloading this podcast from Teacher magazine. I’m Dominique Russell.

Research shows that challenging problem solving tasks have a positive impact on student learning, but there is little evidence on student attitudes towards problem solving when it comes to doing this in the maths classroom.

My guest today is education consultant at Love Maths , Michael Minas. He works in primary schools to help improve learning in mathematics through professional development, classroom modelling and work with parents.

His areas of interest include problem solving and student engagement, and during his time working in primary school maths classrooms, he’s noticed anecdotally that students respond really positively to being presented with challenging problem solving tasks.

So, to formally investigate this, he conducted a study to assess student attitudes towards problem solving in maths alongside Dr James Russo from Monash University. This study focused on 52 students in two classrooms – a Year 3 and 4 class and a Year 5 and 6 class – in a primary school in Melbourne. Michael led a number of lessons in each classroom which presented challenging problem solving tasks to students. The classroom teachers observed these lessons, and then led these same tasks with the students. The lesson structure used was launch-explore-discuss/summarise, which Michael will go over in more detail in the episode.

After these lessons, the students completed a questionnaire to assess their opinions on the task. The results found that three-quarters of students reported unambiguously positive attitudes towards problem solving, the others were ambivalent, and no student expressed a negative attitude.

So, if you’re interested in implementing challenging problem solving tasks in your classroom, keep listening to hear Michael explain in detail the structure of these tasks, and what elements students enjoyed most. Let’s jump in.

Dominique Russell: Thanks for joining me Michael. I just thought it would be good to get a bit of background on the work you’re doing at the moment to start things off and why this research was important for you to conduct?

Michael Minas: Yeah I guess a lot of my work at the moment is in classrooms and one of the things that a lot of the schools are interested in is trying to get more problem solving happening in their mathematics classrooms. So a lot of the work that I do is in classrooms modelling problem solving lessons, working with teachers to sort of develop their sort of approach, their level of comfort with that style of teaching.

And so for me this was really interesting because, you know, I know anecdotally through my sort of experience with working with hundreds and hundreds of students, that I can see the positive responses. But, you know, it’s obviously an area that hasn’t had a lot of research done into it, so it’s good to be able to have, you know, the start of looking at it in a more formal way, of how do students actually feel in these types of lessons? What’s the experience like for them?

DR: And so the research obviously looks at two classrooms in particular in a primary school in Melbourne, looking at those middle years in primary school, which like you say, hasn’t really been looked at in much detail in the literature. So can you describe for me a bit about the school context of this particular school that you were doing the research in?

MM: Yeah, so I mean it’s a typical sort of primary school. It wasn’t, it didn’t have sort of anything outstanding in terms of the cohort of students, the size of the school – you know, 300 kids – it was a pretty sort of, demographically, a regular mix of students.

In terms of mathematics, it was philosophically quite a traditional environment for students to work in with a lot of sort of teacher directed work – you know, ‘I’ll show you how to do it, and then you go back to your tables and you reproduce what I’ve put on the board and maybe answer a series of questions using the approach that I’ve shown you’. So this style of lesson and learning was quite different for both the staff and the students at the school.

DR: And so obviously this style of learning that you exposed them to was received very positively from the students involved which we’ll talk about in a bit more detail soon, but I’m interested then in what the students opinions were of maths before this problem solving task was introduced to them. Do you have any concept of how they viewed maths in general? Did they enjoy it or were they enthusiastic about it?

MM: Yeah, so we had a couple things. So, you know, fairly informal, but when I arrived there one of the first things I did was I did some surveys with all of the students from Year 3-6. And the surveys were around their attitude to maths and also their self-perception (so, how they saw themselves as maths learners) and there were some really clear negative trends there.

So that was a starting point, you know, working with the leadership in the school to say, there are some issues here and you know, you can clearly sort of see that there are some issues here from the survey data.

But beyond that, I mean, anecdotally, my very first day at the school, I distinctly remember this. I was walking into one of the rooms at the school and a little Grade 3 girl said to me ‘oh, who are you?’ and I said, you know, ‘I’m Michael, I’m going to be here, I’m going to be working with you guys on maths’ and whatever. And her friend that was sitting with her, who wasn’t part of the conversation, inserted herself into the conversation to say, ‘oh, we hate maths. Both of us hate maths.’ Like really wanted to make a point of letting me know that she hated maths.

So that type of interaction was probably the most memorable, but I had lots of those types of interactions where people said ‘oh, you’re working with maths? Yeah I don’t like learning maths. I’m not interested in maths. I hate maths. Maths is my least favourite subject.’

So I had lots of those interactions with the kids and, you know, with that girl on the very first day and I said to her ‘well, you know, hopefully if we have this same conversation in November, that you will have shifted the way that you see maths. But that’s my job to do that, that’s not your job.’

DR: And so can you talk me through really the structure of these problem solving tasks that you led in the classrooms? Because I know you were leading them for a little while and the classroom teacher was observing the lessons that you were conducting. So what’s the structure of these tasks?

MM: Yeah, so one of the things is that the structure is a very, sort of central feature of this approach. And the idea is that, that structure is meant to be very predictable for both the staff and for the students.

You know, so we’d start with a warm up activity and the central idea of that is that you want that to be an engaging warm up to have the kids starting the lesson with, you know, a lot of energy and enthusiasm. And that would be followed by the launch of the problem. And, ideally, most problems, we want them to be launched with some of narrative link, some sort of connection to the real world. And we want that to be done in a concise way.

So the idea there is it’s not like a mini lesson of ‘let me get up the front and tell you everything I know about division for the next 15 minutes’. The idea is that we’re giving them a task that – maybe the task lends itself to multiplicative thinking and division, but we’re leaving space for the students to approach the task from their own perspective. So that, I will say to teachers ideally I want that launch to be sort of somewhere around the five minute mark. And for a lot of classroom teachers that’s a challenge, that sort of directly conflicts with the way they’re currently taking their maths lessons.

And then by extension, the shorter that launch time is, the more time the students have to be exploring, engaged with the task. So in order for, you know, students to stay working on a task for 35, 40 minutes, the task needs to be challenging, it needs to be cognitively engaging for them.

And so that explore time starts with five minutes of silent, independent work. And it’s really important that it is silent and is independent. And then from there, I’m a big advocate for actively encouraging collaboration in the classroom. So, not just saying ‘if you want to work with someone, you can’, but actively encouraging the kids; say: ‘hey, why don’t you go over and talk to Megan and see what she’s doing because she’s got some similar thoughts to you, but she’s approaching it a bit of a different way’.

And then the lessons will always finish with some sort of summary of what we’ve done and that again is student-centred. So the idea is that we’re (myself or whoever the teacher is) is looking for student examples to sort of showcase at the end of the lesson to say: ‘hey, you know, talk to me Dominique about what you’ve done’, and getting you to explain your thinking, but being really strategic about it who you select. So it’s not like ‘everyone come to the floor, okay, who’d like to share their work?’ and the same three kids put their hands up every day. It’s you as a teacher being really strategic about who you select and why.

DR: And so something that I found really interesting in the report, just as a bit of an example of how this structure plays out, the example of the chessboard tournament problem. So, the problem was launched with a short story about a family holiday and there was a big chessboard where they were staying. So you displayed a photo of one of the children playing on the chess board at the front of the classroom, and then you gave the task, which was if six children wanted to have a round robin tournament, how many games would need to be played? And then you had a prompt, which asked students to draw a diagram to show how they’d work this out, then you also provided some extending prompts. So can you give a quick run through of how that played out in the classroom? And did the students respond well to the extension prompts?

MM: Just to give you a bit of an idea about the narrative side of things – that task was based around a photo that I shared with the kids of my own family when we were on holidays playing chess on a chessboard. And I’ve had some really positive interactions with kids around that, where some kids will come and say to you ‘oh I play chess lots,’ and ‘I’m a big fan of chess’ so it’s building relationships there where they can say, you might share a common interest.

I mean I’ve had the other experience where I’ve been at a school and I’ve told the kids: ‘this giant chess board was at this particular holiday place in Queensland, and then I’ve had a student come back to me like two months later over the summer holidays and saying: ‘guess where I went over summer? We went and stayed at Paradise Resort and we played chess on that chessboard’ and the kid being really excited to share that with you.

So the narrative, that was the true part of the narrative. I mean the made-up part was – it talks about us having a round robin chess tournament. Now, we didn’t have a round robin chess tournament, we were actually there trying to enjoy, we didn’t spend all day at a giant chessboard playing chess.

So I think it’s teachers being able to feel comfortable taking parts of their life – you know, some real-world application – but also feeling free to be able to sort of elaborate, add to it, and make it work for the maths.

The task – yeah, it’s a really fantastic task because it’s got quite a low entry point, in that you could work on that task just sort of saying – you know it’s like the old problem where you say ‘there are eight people in a room and they all shake hands with each other. How many handshakes would there be?’ But it’s much more – I mean, the idea of playing chess against each other, students can visualise that a lot better and can sort of conceptualise it to say: ‘well if Nash plays against Isaiah, and then next Nash would play against Genevieve…’ so they can sort of work through all the combinations of who Nash would need to play.

Nearly every student you give that task to can enter the task and can have some level of success. But at the higher level it’s a very cognitively engaging task. I mean, the extension task is asking for them to basically find a formula of how to work out any triangular number. And so I used that task with Year 3/4 students and I’ve had students I’ve worked with in the Year 3/4 cohort who are able to sort of show you, ‘I can work out any triangular number and this is how you do it’. And they can show you visually how the formula works.

So I think that’s the beauty to this approach to teaching in that you’re really allowing for true differentiation. You’re presenting a task and there’s scope there for students to work there at a number of different levels.

DR: And as we mentioned very briefly before the classroom teachers were observing you first conducting these lessons before conducting the lessons themselves. Why was that important to do than just instructing teachers on how to run this and getting them to launch straight into it. Why was the observation element quite critical?

MM: I’m a big believer of if you want to get change happening within an organisation, it’s important to have buy-in from people. It’s important for people to actually believe that what you’re doing is going to be doing is beneficial. And for teachers, the vast majority of teachers, when they see that something is effective with their own students, you’ve won them over. So if they can see their own students being challenged in a way they previously haven’t been challenged.

I mean I had this experience yesterday when I was at a school in the western suburbs of Melbourne and I was working in a prep classroom and there was a prep student who, you know, traditionally didn’t really have a lot of success in the maths class and then this student produced some work and this classroom teacher was literally speechless. He was just blown away, he was like, ‘I cannot believe that he’s just done that, I’ve never seen him do that before’.

Now when I go back to work with that teacher in a fortnight’s time or whenever I’m back out there to work with them, they’re going to be much more receptive to this approach because they can see that it works.

And I think you’re also setting teachers up for success then. Because if they’ve seen that lesson structure a few times, the idea that it is very repetitive as a structure, it gives them something that they can sort of say, ‘right, now if I’m going to have a go at taking a lesson using this approach, these are the things that I want to do’. And it’s very easy to reproduce because they’ve seen it done a number of times.

So it’s both about supporting the teachers so they can have success, but also about generating that buy-in and I think that that comes – it’s one thing to deliver PD and to say ‘this is great’. It’s another thing for teachers to see it working with their own students.

DR: And so another big part of the study was how you actually measured the attitudes that students held towards these problem solving tasks and they were overwhelmingly positive. You’d mentioned before that this was kind of what you were expecting to happen because anecdotally you knew that students responded really well to these kinds of tasks. But something that I found really interesting was that they really enjoyed the challenging aspect of these problems and also the collaborative nature. So can you talk me through what the students said and wrote in their questionnaires about those two particular aspects?

MM: Yeah. I guess, I mean one thing that did surprise me was – I expected the results to be positive because that’s what I kind of see when I work not just at this school, but at lots of schools – I was surprised in the fact that of all the students that were involved in the study, that there was no one that expressed, like, negative attitude. Which, you know, was quite sort of gobsmacking for me.

But in terms of what they identified that made it enjoyable, engaging for them. Like you said, there was a couple things they touched on. So one was the idea of challenge. And I think this is something that sometimes teachers struggle with, this idea that: ‘if I make the work more challenging, the kids will disengage. They won’t persist, they won’t enjoy tackling the task’.

And I actually think that’s counter to everything we know about humans. If we think about ourselves as adults, if we’re given some sort of routine, mundane task to perform over and over again it’s every chance that we might do it if we have to do it, but we’re not going to enjoy it. But people love a challenge, people love being pushed cognitively and trying to see if they can be the first one to figure things out. I think humans love a challenge and if I enjoy a challenge as a 43-year-old, there’s no reason to think that like a six-year-old or a 12-year-old wouldn’t enjoy a challenge. So that’s come through to me anecdotally, you know, time and time again over the years, so it was good to see that come in through formally in the study that we did.

The other really big – and again, in some cases this really contrasts with the regular classroom practice – this idea of allowing the students to collaborate. And like I said before, not just allowing, but actively encouraging it. I think a lot of classroom teachers are concerned that if they let the kids move around the room and talk to each other, they’re going to lose control and it’s going to descend into chaos. But I think the two ideas that you’ve just asked about are connected. Like if they’re working on something that they think is worthwhile and challenging, they’re much more likely to stay on task.

And again, humans enjoy collaborating. Humans enjoy socialising, talking, sharing ideas. So if that’s the way, if I was to present PD [professional development] at a school and I was to do five hours of me talking and there’d be no opportunity for staff to actively engage and collaborate with each other, I mean, I would never be invited back to the school.

So then the question would be, well why do we get our students to do this? Why is a maths lesson me talking for 20 minutes telling you everything I know about place value, and then you working on a worksheet by yourself for half an hour and not being allowed to talk?

That’s not going to be enjoyable for us as adults. Why would it be enjoyable for an eight-year-old in a Year 2 class? So I think that in some cases the success that we have when we go and work in schools is partly because it’s such a sharp contrast to the regular practice in the school about the way maths is learned. And that if we can make mathematics more social, then we have much more chance of having students being engaged and wanting to learn.

DR: And is part of that as well – like you mentioned before – the fact that they have that five minutes at the beginning to concentrate on the problem as an individual and silently, but then they open up to the collaboration. Is that balance quite good and quite important?

MM: Yeah. It’s really crucial. And I always tell teachers that I’m working with that one’s not more important than the other, that they’re equally important. But if you let kids collaborate straight away then what you might find is that kids will just straight away – say you and I are working together, and you’re a stronger student in terms of your current performance in maths, well I might just be led by you, and you’ll just be telling me, ‘do this, do that’.

Much more likely if I’ve had some time to think and ponder on the task, that A, when I come to you, maybe I’ll have some questions about what I’m doing and you can guide me and direct me, rather than telling me what to do. But, B, there might be the chance that I may choose not to work with you, even thought we may be best friends, because I may see that someone else is approaching the problem with a similar mindset, a similar approach to me. Or I may choose to say in this instance: ‘I’m going to keep doing this by myself because I feel like I’m getting some momentum here. I can see that I’m making some progress’.

So I think that five minutes silent time is really crucial, and then it becomes really crucial (this becomes a classroom management thing) as a teacher, you have to be able to make sure that it is truly five minutes and it is truly silent and it is truly independent and also truly productive. Because it’s no good them just sitting silently looking at the clock, you know, looking at the stop watch counting down before they bang go into talking to each other.

So the way you know that’s productive is when you see the kids are on task. When you see the tops of their heads looking down at their page, and they’re thinking, and they’re gathering materials. And you can tell really clearly as a teacher when that’s not happening.

DR: And so just finally then, I’m thinking now for teachers who are listening to this episode who are thinking they want to implement a similar approach in their maths classroom for students of a similar age, is there anything that we haven’t covered already that would be good to keep in mind? Or perhaps some good first steps to take?

MM: Yeah, look I think that the model that I see that works really well is I think what we spoke about before. Is that you have to have people that are able to model what it should look like to be able to win teachers over, for them to be able to say ‘I can see the benefit of this, I can see how this works’. So whether that be – I mean, I’m definitely not trying to spruik for work – but whether that be internally – you know, like a lot of schools have really great classroom teachers. Some of those people have moved into learning specialist roles.

But whether that be internally with those people, like give them the time to go into other people’s classrooms and to be able to model this type of approach and to show the classroom teachers how it works and to be able to answer those questions. Or whether it be externally, by bringing in consultants who have the skill and expertise to do it, I think that’s really important. I think it’s important that people see it in practice first before they try to do it.

And it’s also really important, as well as seeing lessons, that people have time to then unpack the lesson and talk about it together. So if you’ve got a learning specialist at your school that’s modelling this type of lesson for, say a graduate teacher, there needs to be some time allocated for them to sit. Because the graduate teacher may walk away saying, ‘that was a great lesson’. But the next step is them being able to identify why was it a great lesson? What worked? And what can they do to plan a similar great lesson the following week?

Because, you know, if you just say ‘well, that was a great lesson, but I can’t do that lesson again because my kids have already done it, so where do I go with it?’ Whereas if you can identify and say: ‘oh I see what worked well. The thing that worked well is they were engaged with the problem.’ Why were they engaged in the problem? ‘It had a real world link’. Why was your questioning effective during the lesson? ‘Well, it was because you knew, you had a clear focus of what the content was’. What are we focusing here? What’s the mathematical concepts we’re focusing on?

So, as a classroom teacher you know the right question to ask and the right student at the right time and there’s a lot of work that goes into that, but like I said, it’s definitely something that’s attainable for all classroom teachers with the right support.

That’s all for this episode. Thanks for listening. Be sure to subscribe to our podcast channel on Spotify , Apple podcasts or SoundCloud , so you can be notified of any new episodes. While you're there, we'd love for you to rate and review the podcast in your podcast app.

Russo, J., & Minas, M. (2020). Student Attitudes Towards Learning Mathematics Through Challenging, Problem Solving Tasks: “It’s so Hard– in a Good Way”. International Electronic Journal of Elementary Education, 13 (2), 215-225. https://doi.org/10.26822/iejee.2021.185 .

Michael Minas says he believes sometimes teachers struggle with the idea that: ‘if I make the work more challenging, the kids will disengage. They won’t persist, they won’t enjoy tackling the task’.

Reflect on a recent lesson you taught. How challenged would you say students were? How do you know the level of challenge was appropriate? Do you think you could have challenged students further? Were there opportunities for students to participate in extension tasks?

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Maths Problem Solving Booklets

Subject: Mathematics

Age range: 11-14

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23 August 2022

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Maths problem solving booklets covering a wide range of mathematical problems designed to improve problem solving strategies as well as numeracy and mathematical ability.

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Students’ numeracy skills in solving numeracy tasks: Analysis of students of junior high schools

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Cholis Sa’dijah , Heri Purnomo , Abdul Halim Abdullah , Hendro Permadi , Lathiful Anwar , Ety Tejo Dwi Cahyowati , Mukhtamilatus Sa’diyah; Students’ numeracy skills in solving numeracy tasks: Analysis of students of junior high schools. AIP Conf. Proc. 10 January 2023; 2569 (1): 040011. https://doi.org/10.1063/5.0113664

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The Ministry of Education and Culture of Indonesia has set the Minimum Competency Assessment (MCA) to focus on, among others, numeracy skills, namely the ability to use mathematics to solve everyday problems in various contexts. This study aimed to determine students’ numeracy skills in completing numeracy tasks using a quantitative approach with a descriptive method. The researchers involved 150 students of junior high schools in East Java by asking them to solve some numeracy problems online via the Google Form (for those from regular schools) and offline through a paper-based test (PBT) (for those from boarding schools). The results showed that of all the students, 62%, 30%, and 8% were at the surface numeracy skills (SNS), the intermediate numeracy skills (INS), and the deeper numeracy skills (DNS) levels, respectively. These figures found showed the poor abilities of junior high school students to complete numeracy problems. Teachers, therefore, should apply numeracy task-oriented learning experiences, such as constructivist teaching methods, to improve students’ numeracy skills.

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Google Circle to Search can now solve maths problems for students: Here's how the feature works

Google's circle to search focuses on educational support by aiding students in understanding problem-solving techniques, with plans to expand to more complex equations using ai model learnlm..

Tech giant Google has unveiled a significant update to its Circle to Search feature during the annual developer conference, Google I/O. This enhancement aims to empower students by providing assistance with math and physics problems directly from their Android smartphones or tablets.

The revamped Circle to Search now offers a specialized tool tailored to guide students through solving homework problems in mathematics and physics. Unlike merely providing answers, Google emphasizes that the feature is designed to foster a deeper understanding of problem-solving techniques, aiding students in learning how to tackle challenges independently.

Through Circle to Search, students can activate AI-driven assistance for mathematical word problems, which systematically breaks down the problem and provides step-by-step instructions for finding the correct solution. Google underscores that the feature is intended as a supportive educational tool, rather than a shortcut for completing assignments.

This announcement comes amidst ongoing discussions surrounding the integration of AI tools in education, with concerns raised about potential misuse by students seeking to expedite their homework. Google addresses these concerns by positioning Circle to Search as a resource focused on facilitating learning and skill development.

Looking ahead, Google revealed plans to further expand the capabilities of Circle to Search. Soon, the functionality will be upgraded to tackle more intricate mathematical equations, encompassing symbolic formulas, diagrams, and graphs, suggests the American company.

Google will utilize LearnLM , its most recent AI model optimized explicitly for educational tasks, to drive these advanced capabilities.

In a blog post announcing the update, Google stated, "Starting today, Circle to Search can now help students with homework, giving them a deeper understanding, not just an answer — directly from their phones and tablets. When students circle a prompt they’re stuck on, they’ll get step-by-step instructions to solve a range of physics and math word problems without leaving their digital info sheet or syllabus. Later this year, Circle to Search will be able to help solve even more complex problems involving symbolic formulas, diagrams, graphs, and more. This is all possible due to our LearnLM effort to enhance our models and products for learning."

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Group-working Skills

In her article Group-worthy Tasks and Their Potential to Support Children to Develop Independent Problem-solving Skills , Jennie Pennant outlines the benefits of working on mathematics as a group, arguing that group work also supports children's individual problem-solving abilities.  Almost all NRICH tasks could be tackled by a group of learners, as opposed to an individual or pair, but the Developing Group-working Skills feature is a collection of activities which have been designed especially to develop children's group-working skills .  How can we help children get better at group work? We cannot expect learners to be able to work well in groups overnight.  We must help them develop the skills that are needed for successful group work and give them many opportunities to put these skills into practice.  In her article Developing Good Team-working Skills , Jenny Piggott offers the following list of skills related to working collaboratively, based on those found in Elizabeth Cohen's book 'Designing Groupwork':

• Asking questions - making sense of your own understanding
• Explaining by telling how and why
• Helping others - by responding to their needs
• Helping others - to do things for themselves
• Sharing knowledge and reasoning
• Finding out what others think - asking for, listening to and making sense of their ideas
• Reflecting on and making use of what has been said
• Being concise - communicating thinking
• Giving reasons for ideas - communicating reasoning
• Allowing everyone to contribute
• Pulling ideas together - sharing, listening, valuing all contributions
• Finding out if the group is ready to make a decision - consensus making.

Jenny suggests that these collaborative working skills can be developed through particular group activities and she offers six categories of team-building activities that can be used to focus on a range of the different skills.  Jenny's article links to several classroom activities within each category and we have chosen a subset of these in our Group Work feature which exemplifies the full range of skills. The featured activities For more details about the kinds of task and further example activities, please see Jenny's article . Number Match , a Stage 1 task, is only complete as an activity when every member of the group has completed their own part.  The task is undertaken in silence which helps group members respond to the needs of others.  Fraction Match , aimed at Stage 2 and above, is done in exactly the same way. In Counters in the Middle , a 'designer' makes an arrangement of counters without the team seeing it. The team has to agree on the final pattern by asking the minimum number of questions, which requires them to listen to each other, give reasons for their opinions and pull ideas together. In En-counters , each learner completes a picture themselves, based on the designer's instructions, but with support and advice from other members of the team.  This therefore encourages children to respond to the needs of others, help others do things for themselves and explain by telling how. Guess the Houses , a Stage 1 activity, depends on learners sharing reasoning, listening to opinions, reflecting and pulling ideas together. The team is required to guess the rule in the minimum number of questions. In What Shape? , one member of the group is trying to find out what is on their chosen card (the unknown) by asking as few questions as possible.  This task therefore depends on group members being concise, asking questions, listening and reflecting on what has been said. Arranging Cubes requires the group to recreate a 2D arrangement of cubes which matches all the information on their cards without showing each team member's information to anyone else.  Among other skills, learners must allow everyone to contribute, share knowledge and reasoning, reflect on (and make use of) what has been said and come to a consensus. Submitting solutions Due to the nature of these tasks, it will be a little difficult for children to submit solutions in the usual sense to the live problems in this feature.  However, we would love to hear about how the activity helped them work better as a group.  It could be that you as the teacher summarise your observations or it might be that the learners themselves can articulate their thoughts.  We would be delighted to hear from you.  You may find it helpful to use the list of skills above as an assessment 'checklist' (see Skills.doc or Skills.pdf ).  What next? Having tried these activities which aim to build learners' group-working skills, why not have a go at other NRICH activities using a group-work approach?  In May 2010, we created several group-worthy tasks, based on Jo Boaler's research on Complex Instruction, and in February 2010, all our problems were designed with collaborative mathematics in mind. Of course, as we mentioned at the start of this article, almost any NRICH activity could be worked on by a group, so you could encourage children to tackle any of our tasks using their group-working skills.

Taking the tasks home

We have rewritten some of these tasks so that they are suitable for just one child to do together with an adult. We've collected these here . References Cohen, E. G. (1994) Designing Groupwork - Strategies for the Heterogeneous Classroom. Second Edition, Teachers College Press. Here is a PDF version of this article.

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Welcome to the daily solving of our PROBLEM OF THE DAY with Yash Dwivedi . We will discuss the entire problem step-by-step and work towards developing an optimized solution. This will not only help you brush up on your concepts of Number Theory but also build up problem-solving skills. In this problem, we are given an infinite number line. You start at 0 and can go either to the left or to the right. The condition is that in the ith move, you must take i steps. Given a destination d, find the minimum number of steps required to reach that destination.

Input: d = 2 Output: 3 Explanation: The steps taken are +1, -2 and +3

Give the problem a try before going through the video. All the best!!! Problem Link:  https://www.geeksforgeeks.org/problems/minimum-number-of-steps-to-reach-a-given-number5234/1