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Rich Mathematical Tasks

What is a rich mathematical task?

Why would I want to use rich tasks in my maths lessons?

Where can I find rich mathematical tasks for primary children?

I wonder whether you have ever asked yourself any of the above questions.  I am hearing from more and more primary teachers who would like to inject something 'extra' into their maths lessons.  They each have an underlying reason or reasons to get in touch with NRICH:

  • Some feel that they need a change of approach to reinvigorate their mathematics teaching generally;
  • Some report that the children in their school do well, but find it difficult to apply their mathematical knowledge and skills to new situations;
  • Some would like their pupils to enjoy mathematics more;
  • Some are worried that they are not stretching the higher-attaining children;
  • Some are concerned that the lower-attaining children are 'turned-off' maths, lack confidence and have almost given up. 

Of course this is not an exhaustive list.  What you might find surprising is that the professional development we offer at NRICH for all of the above scenarios has a common focus: rich mathematical tasks. 

In this article, I will describe the start of a project, which began in the spring term of 2010.  Pete Hall, the NCETM East of England Regional Coordinator, contacted me to tell me about a number of £1000 grants on offer to schools who wanted to develop their understanding and use of rich mathematical tasks. The application form was relatively straightforward to complete, requiring some detail about the theme (what it was and why it had been chosen); who would be involved and a commitment to contribute to the NCETM rich tasks community.  Schools were required to give a breakdown of how the money would be spent and they promised to submit a short written report to NCETM on completion of the project.

Four schools in the east of England were successfully awarded a grant:  Clover Hill Infants' School in Norwich, Harrold Lower School near Bedford, Lakenham Primary School also in Norwich and St Philip's Primary School in Cambridge.  Each school decided to spend at least some of their money on professional development run by NRICH and I hope by describing what we have achieved so far, you may feel able to lead one or more staff meetings in your own school without necessarily paying for NRICH support!

At all four schools, I have led an initial workshop, varying in length from a staff meeting to a half day.  In all cases, we have begun with having a go at an activity altogether.  I feel it is important for everyone to engage in some mathematics - it reminds us what it is like to be a learner and it gives us a common experience (to some extent), which aids subsequent discussion.  The problem that I have used in all four schools is Magic Vs .  (Do have a go at it if you do not know it.  The approach I took with the teachers is exactly the same as that suggested in the teachers' notes on the website.)  We spent anything from about thirty to forty-five minutes actually working on the problem itself, with me taking the role of 'teacher', just as I would if I was with a group of children.

Having reached a suitable pausing point, we reflected on what we had done.  What mathematical 'content' knowledge did we use as we tackled this problem? By this I mean the aspects of number, calculation, shape and space, data handling and/or measures I needed to know, or I came to know.  In terms of the Magic Vs problem, the following list reflects the range of suggestions:

  • Odd/even numbers
  • Addition/subtraction
  • Number bonds
  • Consecutive numbers
  • Multiplication/division (perhaps)
  • Factors/multiples (perhaps)

Next, we asked ourselves what problem-solving strategies we found useful.  Here are those that came up frequently:

  • Using trial and improvement
  • Noticing and explaining patterns
  • Working systematically
  • Making conjectures
  • Tweaking/altering/varying
  • Testing ideas
  • Generalising
  • Talking to each other

I often find it helpful to reflect on mathematical activities in this way, that is considering the 'content' and processes separately.  In terms of the Magic Vs problem, it is interesting to note that the 'content' we used was fairly basic, possibly not going beyond that usually met in Key Stage 1.  However, we used a vast range of strategies to solve the problem, some of which are rather sophisticated. 

So, this led on to further discussion:  what makes this Magic Vs problem so 'rich'?  Suggestions included:

  • It is easy to get started but also has the potential to be taken to high levels of mathematics (what NRICH terms ' low threshold high ceiling ')
  • It has more than one answer
  • It is 'open-ended', in the sense that although there are some answers, you can go on asking, and pursuing, your own questions
  • The way to go about solving the problem is not immediately obvious
  • It can be approached in many different ways
  • It requires you to use a range of knowledge and skills
  • It leads to generalisations
  • It might deepen our understanding of odd/even numbers
  • It is non-threatening (perhaps linked to the fact everyone can begin to have a go)

By specifically talking about these characteristics, the idea is not to suggest that every problem we use in the classroom should tick all these boxes.  Instead, by raising awareness of a set of characteristics, we can understand how resources we already use might be tweaked to make them 'richer'. 

This in turn leads to another important point.  Although in each session, the participating teachers came up with reasons for Magic Vs being a rich mathematical task, these are not necessarily inherent in the problem itself.  Would the teachers have thought it was rich if I had simply handed each one of them a piece of paper with the problem written on it and demanded they work in silence?  Some may have reached similar conclusions, but I suspect some would not.  So, the potential of a task to be rich is not enough in my opinion.  There are two other elements (at least!).

If we want children to get better at solving mathematical problems, then we need to encourage them to think in a mathematical way and to have a range of strategies at their fingertips, which they can draw upon.  Therefore, the questions and prompts we use, in conjunction with the tasks we provide, are crucial.  In this first session with the teachers, I showed them the ' Primary Questions and Prompts for Mathematical Thinking ' book, published by the ATM and give them a taster of its contents.  I am a huge fan of this book.  The authors define certain activities, which 'typify mathematical thinking', and suggest questions and prompts to encourage these.  These suggestions are entirely context-free, in other words they could be used when children are working on any topic, from number to calculating to shape to measuring to data handling.  So, the first element to consider in conjunction with using rich tasks is the way we question learners in the classroom.

The second element is what I term the classroom 'culture'.  Rich tasks and good questioning will thrive in a classroom where children are encouraged to talk to each other, where they are happy to offer ideas without the fear of being wrong, where their opinions are welcomed.  The culture of your classroom reflects your values so in all four schools, we discussed what we value in mathematics and how this affects the way we work in class. 

Reflecting on all three of these inter-related aspects of mathematics teaching (rich tasks, questioning and classroom culture) is a lot to cram into a staff meeting, let alone half a day.  And I threw in a quick tour of the NRICH website too!  Along the way, we talked about the benefits of such an approach. Many children who are currently 'high-attaining' may feel uncomfortable when presented with such tasks.  They may not be used to being challenged in mathematics.  They may be used to knowing immediately what to do when faced with a problem.  However, surely as teachers we have an obligation to equip our children with skills that will carry them in good stead as they get older?  Encouraging an ethic of perseverance and the idea of relishing a challenge is part and parcel of mathematics teaching, although it is something that perhaps feels rather daunting as a teacher. 

We arranged a date for me to return to each school so there was time for everyone to mull over the first session.  All the participating teachers agreed to try out at least one rich task with their children in the intervening weeks.  They will come to the second session prepared to talk about their experience:  the things that went well and those that didn't go well; the surprises and the lessons learned.  We hope then to find some ways forward for each school so that they can build on their achievements and continue to go from strength to strength.

Cristina Milos

Open-Ended Tasks and Questions in Mathematics

by CristinaM. | Sep 13, 2014 | inquiry , math , thinking | 5 comments

One way to differentiate in math class is creating open-ended tasks and questions (I talked about several differentiation strategies I use here – Mathematically Speaking ).

I think it is useful to clarify the scheme of mathematical problems – below I used Foong Pui’s research paper:

Problem types

“Problems in this classification scheme have their different roles in mathematics instruction as in teaching for problem solving, teaching about problem solving, or teaching via problem solving.”

1.   CLOSED problems   are well-structured problems in terms of clearly formulated tasks where the one correct answer can always be determined in some fixed ways from the necessary data given in the problem situation.

 A. Routine closed problems – are usually multi-step challenging problems that require the use of a specific procedure to arrive to the correct, unique, answer.

B. Non-routine closed problems – imply the use of heuristics strategies * in order to determine, again, a single correct answer.

*Problem-solving heuristics: work systematically, tabulate the data, try simpler examples, look for a pattern, generalize a rule etc.

Routine problem : Minah had a bag of rice. Her family ate an equal amount of rice each day. After 3 days, she had 1/3 of the rice left. After another 7 days, she had 24 kg of rice left. How much rice was in the bag at first?

Non-routine problem : How many squares are there in a chess board?

2. OPEN –ENDED problems – are often named “ill-structured” problems as they involve a higher degree of ambiguity and may allow for several correct solutions. Real-life mathematical problems or mathematical investigations are of this type – e.g. “How much water can our school save on a period of four months?” or “Design a better gym room considering the amount of money we can spend.”

FEATURES of open-ended problems :

  • There is no fixed answer (many possible answers)
  • Solved in different ways and on different levels (accessible to mixed abilities)
  • Empower students to make their own mathematical decisions and make room for own mathematical thinking
  • Develop reasoning and communication skills

HOW do you create open-ended tasks?

Usually, in order to create open-ended questions or problems, the teacher has to work backwards :

  • Indentify a mathematical topic or concept.
  • Think of a closed question and write down the answer.
  • Make up a new question that includes (or addresses) the answer.

STRATEGIES to convert closed problems/questions

  • Turning around a question

CLOSED: What is half of 20?

OPEN: 10 is the fraction of a number. What could the fraction and the number be? Explain.

CLOSED:  Find the difference between 23 and 7.

OPEN: The difference between two numbers is 16. What might the numbers be? Explain your thinking.

CLOSED: Round this decimal to the decimal place 5.7347

OPEN: A number has been rounded to 5.8. What might the number be?

CLOSED: There are 12 apples on the table and some in a basket. In all there are 50 apples. How many apples are in the basket?

OPEN: There are some apples on the table and some in a basket. In all there are 50 apples. How many apples might be on the table? Explain your thinking.

  • Asking for similarities and differences.

Choose two numbers, shapes, graphs, probabilities, measurements etc. and ask students how they are alike and how they are different.

Example: How are 95 and 100 alike? How are they different?

Possible answers:

They are alike because you can skip count by 5s, both are less than 200, both are greater than 90 etc.

They are different because one is a three-digit number, only one ends in 5, only one is greater than 99 etc.

Example: How are the numbers 6.001 and 1.006 alike? How are they different?

  • Asking for explanations.

Example: Compare two fractions with different denominators. Tell how you compare them.

Example: 4 is a factor for two different numbers. What else might be true about both numbers?

  • Creating a sentence

Students are asked to create a mathematical sentence that includes certain numbers and words.

Example: Create a sentence that includes numbers 3 and 4 along with the words “more” and “and”.

  • 3 and 4 are more than 2
  • 3 and 4 together are more than 6
  • 34 and 26 are more than 34 and 20 etc.

Example: Create a question involving multiplication or division of decimals where the digits 4, 9, and 2 appear somewhere.

Example: Create a sentence involving ½  and 64 and the words “less” and “twice as much”.

  • Using “soft” words.

Using the word “close” (or other equivalents) allows for a richer, more interesting mathematical discussion.

Example: You multiply two numbers and the product is almost 600. What could the numbers have been? Explain.

Example: Add two numbers whose sum is close to 750. What can the numbers be? Explain.

Example: Create two triangles with different but close areas. (*instead of, “Create a triangle with an area of 20 square inches.”)

……………………………………………………………………………………………………………………………………………………………………………………………………

A few important considerations are to be made when creating open-ended problems or questions.

  • Know your mathematical focus .
  • Develop questions with the right degree of ambiguity (vague enough to be interesting and to allow for different responses, but not too vague so as students get frustrated).
  • Plan for two types of prompts :
  • enabling prompts (for students who seem unable to start working)
  • extension prompts (for students who finish quickly)

High quality responses from students have the following features:

  • Are systematic (e.g. may record responses in a table or pattern).
  • If the solutions are finite, all solutions are found.
  • If patterns can be found, then they are evident in the response.
  • Where a student has challenged themselves and shown complex examples which satisfy the constraints.
  • Make connections to other content areas.

……………………………………………………………………………………………………………………………………………………………………………………………………………….

References:

Designing Quality Open-Ended Tasks in Mathematics , Louise Hodgson, 2012

Using Short Open-ended Mathematics Questions to Promote Thinking and Understanding , Foong Pui Yee, National Institute of Education, Singapore

Good Questions – Great Ways to Differentiate Mathematics Instruction , Marian Small, 2012

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Jessica

Thank you for posting this. I appreciate how there is a comparison between the two (closed and open ended) types of questions and the considerations that go along with each. Thanks!

CristinaM.

You are welcome!

Jen

Wow, well-written, thank you. I’m excited that my teaching is getting great, clear, and organized at the level that I’m at. But this article reminds me there are many higher levels I can get to, including this area of more open-endedness. Thank you!

I am happy to have helped even in a small way!

drmuneer

Thank you so much

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Open ended math questions and problems for elementary students.

Does your current math instruction involve only situations where there is one answer? Are students expected to solve problems following rigid procedures that do not require critical or creative thinking? These components are important; however, it’s time to take your math instruction to the next level! The answer: open ended math questions!

Open ended math questions, also known as open ended math problems, help learners grow into true mathematicians who use diverse problem solving strategies to explore mathematical situations where there isn’t necessarily one “right” answer. It equips them with the critical thinking skills they need to solve real world problems in the twenty-first century.

a girl looking up and thinking about what math strategy to use

This blog post will answer the following questions:

  • What is an open ended math question?
  • What are the differences between open-ended and closed-ended problems in math?
  • Why should I implement open ended questions in my classroom?
  • What are the disadvantages of using open-ended math problems?
  • How do I implement open ended math questions in my classroom?
  • How do I create open ended math questions?
  • What are some examples of open ended math problems?
  • How do I grade open ended math tasks?

math journal cover and data tracker on a teacher's desk

What is an Open Ended Math Question?

An open ended math question (which is known as an open ended math problem or open ended math task) is a real world math situation presented to students in a word problem format where there is more than one solution, approach, and representation. This instructional strategy is more than reciting a fact or repeating a procedure. It requires students to apply what they have learned while using their problem solving, reasoning, critical thinking, and communication skills to solve a given problem.

This strategy naturally allows for differentiation because of its open-ended nature. In addition, it is a valuable formative assessment tool that allows teachers to assess accuracy in computation and abilities to think of and flexibly apply more than one strategy. In addition to the teacher being able to learn about their students from this tool, the students can thoughtfully extend their learning and reflect on their own thinking through whole group discussions or partner talks.

math prompts for guided math workshop with a pencil

What are the Differences Between Open-Ended and Closed-Ended Problems in Math?

The major difference between open-ended math problems and closed-ended math problems is that close-ended ones have one answer and open-ended ones have more than one answer. This simple difference creates a very different learning experience for elementary students when they work on solving the problem.

elementary students solving open ended math questions

What are the Advantages and Disadvantages of Open Ended Math Problems?

Advantages of open ended math problems.

There are many benefits to using open ended math questions in your classroom. This list of advantages of open ended questions will help you understand their ability to transform your math block! Here are 8 advantages to using open ended math tasks:

  • Provides valuable and specific information to the teacher about student understanding and application of learning
  • Allows the teacher to assess accuracy in computation and abilities to think of and flexibly apply more than one strategy
  • Permits the teacher to see flexibility in student thinking
  • Gives students the opportunity to practice and fine tune their problems solving, reasoning, critical thinking, and communication skills
  • Creates opportunities for real-world application of math
  • Empowers students to extend their learning and reflect on their thinking
  • Fosters creativity, collaboration, and engagement in students
  • Facilitates a differentiated learning experience where all students can access the task

math printables on a clipboard and desk

Disadvantages of Open Ended Math Problems

Although there are tons of benefits to using open ended math problems in your classroom, it is important to note that there are some disadvantages. Here are 3 disadvantages using open ended math tasks:

  • Increases time in collecting data
  • Provides a higher complexity of data
  • Requires the implementation and practice of routines

3 Ways to Implement Open Ended Problems During Your Math Block

Here are 3 ways you can implement open ended problems in your elementary classroom:

  • Start a lesson with an open-ended math problems for students to solve independently. Invite them to share their work and reasoning with a partner. Ask a few students to share their ideas with the whole class.
  • Use the open-ended math problems for fast finishers . If a student or a group of students tend to finish independent work before the rest of the class, invite them to work on an open-ended math problem.
  • Utilize open-ended math problems as a center during math workshop . You will not have to worry about students finishing that math center before it is time to switch to the next center.

open ended math questions in an elementary classroom

3 Ways to Write Open Ended Math Questions with Examples

Here are 3 ways to create open ended math questions accompanied with easy-to-understand open ended math problems examples:

  • Start with a Closed-Ended Question. For example, a closed-ended question could be: What is the sum of 10 plus 10? The related open-ended question would be: The sum is 20. What could the addends be? There are an infinite number of responses because students could use negative numbers.
  • Ask Students to Explain, Prove, or Justify their Thinking. An example of this is, “Prove 5 + 6 = 11.” One possible student response could be that they know the sum is 11 because of the doubles + 1 rule. Another student may take out counters, while another draws a picture.
  • Invite Students to Compare 2 Concepts. For example, ask students to identify the similarities and differences between 2D and 3D shapes. Some possible responses for similarities are that they are both geometry concepts and classifications of shapes. A difference they could say is that 2D shapes are flat, while a 3D shape is solid.

3rd grader drawing quadrilaterals on a geometry math worksheet

How do you Grade Open-Ended Math Questions?

Grading open-ended math tasks is not as clear cut as closed-ended questions. If you are using the task as a formative assessment for your own planning purposes, then you have flexibility on how you choose to evaluate students’ work. However, I recommend you use a rubric if you plan to use it as a summative assessment. Remember to share the rubric with students so that the expectations are clear.

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We would love for you to try these open ended question resources with your students. They offer students daily opportunities to practice solving open ended problems. You can download worksheets specific to your grade level (along with lots of other math freebies) in our free printable math resources bundle using this link: free printable math worksheets for elementary teachers .

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Model Teaching

Opening Minds with Open-Ended Math Problems in the Primary Classroom

by Model Teaching | December 21, 2018.

Wait, so, what is the RIGHT answer?” “Sarah got a different answer than I did…how can we BOTH be right?” You will most likely hear all kinds of responses like this when you start to incorporate open-ended math activities into your classroom. At first, they’ll probably make your students look at you as if you have two heads. But, these kinds of reactions will begin to subside once your students have been exposed to the idea that there are many ways to solve problems, even math problems! Encouraging this kind of “endless possibility” thinking is an effective way to teach your students to challenge themselves and think outside of the “normal” problem solving thinking.

Open ended math stations

What are open-ended math problems?

Open-ended math problems are problems that have more than one possible answer. These problems might present an end result and then ask students to work backward to figure out how that end result might have been achieved or they might ask students to compare two concepts that can be compared in a variety of different ways. But whatever way they are presented, the purpose of open-ended math problems is always to encourage students to use higher order thinking skills to solve problems and understand that some problems can be solved in many ways, with many outcomes.

Examples of Open-ended Math Problems

If you teach pre-k or kindergarten, an open-ended math problem might be: “You have 2 shapes that have a different amount of sides. What 2 shapes could you have? Show and name the shapes.” You would provide them with crayons, paper, pattern blocks, or whatever other manipulative they might be used to using when discussing shapes and students would use these manipulatives to come up with as many answers as they can. Your little ones may answer with a variety of answers based on their current skill level. You may get answers like “triangle and square”, “hexagon and parallelogram”, or “a circle is a shape” depending on what each student knows about shapes. This is a great way to reinforce what students already know and to quickly assess where they are in their knowledge.

If you teach first grade, an open-ended math problem might be: “I’m thinking of the number 8. What two numbers work could work together to make the number 8?” Again, you would provide them with manipulatives they normally use for composing and decomposing numbers, like counters, small erasers, counting bears, unifix cubes, or even playdoh balls. The extra bonus about this kind of problem is that it’s extremely easy for students to show their math skills. Some might use addition, others will use subtraction, and you may even run into a kiddo or two who can use multiplication to find the number. However students choose to explore all the possibilities for answers, be sure to give them a few options for how to show their thinking. This might include simply writing equations, drawing pictures with the equations, or even building the number with a manipulative and then taking a picture of it with an iPad.

As students get older and move onto more abstract thinking in second and third grade, you might incorporate more word problems like: “The difference between the temperature on Monday and Tuesday was 13 degrees. What could the temperature have been on each day? Find and explain at least 5 different answers.” Or “Penelope sees 37 children playing in a corn maze. If those children split into four groups, how many children could be in each group? Find and explain at least 5 different answers.” As always, be sure to provide students with manipulatives, paper and pencils, dry erase markers and whiteboards, or whatever you normally use to help them solve problems and then let them go to work! By presenting these kinds of word problems, you’ll expose students to a variety of math concepts (such as division in this example) just by allowing them to think about how to solve the problem on their own. Then, when these concepts are formally introduced, they will hopefully feel more familiar to some students.

Why should I use open-ended math problems with my students?

There are many benefits to incorporating these kinds of problems into your students’ daily routine, but here are a few of the most obvious and effective ones:

  • Open-ended problems encourage higher order thinking skills. Students will not only be “recognizing”, “identifying”, or “describing” their thinking; they’ll be “justifying”, “defending”, and “evaluating” their problem solving skills and how they arrived at their answers.
  • Open-ended problems build confidence in your students. Once students recognize that there are many possibilities for correct answers and thinking, they begin to participate more readily because they can bring to the table. Students who normally struggle with math might solve the problem on a very basic level, using a basic strategy, but they’ll be correct! And your advanced students can solve it on their advanced level and be just as correct as the student who struggles. Simply knowing that the way that they specifically thought and solved the problem was considered correct builds confidence for students.
  • Open-ended problems are engaging! Students are immediately engaged in these kinds of problems because they recognize that there are so many different ways to solve it. Whether students are working in small groups or independently, there is possibility for so many different ideas and answers to be correct that everyone wants in on it. This engagement, in turn, encourages collaboration among students and soon, they’re sharing their thinking and learning from each other to solve problems.
  • Open-ended problems encourage creativity. Students are capable of using so many strategies that they’ve learned over the years to solve problems and, given the space and time, they can even come up with some of their own strategies for solving problems. Open-ended problems give students permission to be creative in their thinking and problem solving.
  • Open-ended problems make it easy for teachers to see what levels students are working at. Simply by walking around the room while students are working to solve an open-ended math problem, you’ll be able to informally assess what kind of level they are independently working on. This can be extremely beneficial as you are collecting data, forming groups, or simply getting a feel for what kind of skills each student is working with.

For more information about the benefits of using open-ended math problems, read:

https://nzmaths.co.nz/benefits-problem-solving

How do I incorporate open-ended math problems into my math instructional time?

Some of the simplest ways to incorporate open-ended math problems into your math instructional time is to include them in math stations, use them in small groups, and use them as a warm up.

  • Math Stations:  You can implement open-ended problems into your math stations a number of ways, including thinking mats, task cards, or interactive math journals. The simplest way to implement them into your math stations is by using task cards. Task cards are pre-made cards that you can create or purchase to cut and laminate for students to use repeatedly. Task cards usually include words, pictures, diagrams, or a combination of all to present a problem to students. To use task cards in a math station, simply create or purchase the cards you want with open-ended word problems or picture problems. Then, simply print them out and cut/laminate them to make them durable and easily reused. (TEACHER TIP: Most dry erase markers wipe off of lamination pretty easily if it’s wiped off within a reasonable amount of time. Your students may want to mark the important parts of the problem on the actual task card with dry erase marker if you want them to. Just wipe if off after use!) I would suggest storing cards in a labeled plastic container or ziplock bag to keep them organized. It is suggested that you always allow students to use manipulatives as needed, as this can help students feel allowed to express their creative problem-solving thoughts. So, be sure your task card station provides anything students might use to solve problems in their own way: whiteboards, markers, papers, crayons, counters, manipulatives, thinking mats, laminated task cards, etc. For example, if you give students a task card with this problem on it: “Marcy finds 47 apples on the ground. What 3 addends could create this sum? Find and explain at least 5 answers.”, I would provide them with small apple erasers or counters, a whiteboard and dry erase marker, and an iPad to take picture evidence of their five (or more!) answers when they’re finished.  Please refer to pages 10-16 in the resource provided to you below this article for some sample open-ended word problem task cards that you can use with your students immediately!
  • Small Groups: To implement open-ended problems in your small groups, using thinking mats, manipulatives, and prepared open-ended problems is a great way to ease students into working on open-ended problems independently. This is a great way to model your own thinking and problem-solving to allow students to see how they can begin their own ways of solving the problems. Take a moment to download and look at the thinking mat activity in the downloadable resource below. You can incorporate these mats into your small group activity by providing each student with a laminated copy of the mat you want to use and manipulatives for them to work with to follow the mat’s directions. For example, the thinking mat that says “Make patterns out of these shapes and name them.” would be an excellent open-ended activity for a group of kindergartners who are working on shapes OR patterns. Give each student a few of each of the pattern blocks shown on the mat and a dry erase marker. Explain and model how YOU would complete the activity by creating a pattern with the pattern blocks, tracing the blocks or drawing your pattern, and then naming it with letters (ie.: rhombus, rhombus, circle would be named an AAB pattern). This will give your students an idea of what’s expected and their little brains can get started coming up with their own patterns!
  • Warm Up: Using your warm-up time to practice with open-ended problems is a great way to model your own thinking to the whole. Modeling how to solve these problems step-by-step along with the whole class can help give reluctant participants the courage and understanding to participate and ready participants the reassurance that they’re on the right track. As an example, look at “Activity 3: Creating and Solving Problems” in the downloadable resource. You will notice a few thinking mats included, along with cards that correspond to the mats. For a warm-up activity before you begin your lesson for the day, you could give each student a laminated thinking mat and a corresponding manipulative (like, pass out the table and basket cards and give every student some small apple erasers). Then, project a corresponding task card so that everyone can see it. Read the card together, model one way you could solve the problem using your own thinking mat and manipulatives, and then allow students to solve it their own way to find one or two other answers. I would ask students to record their thinking and answers in a math journal or something similar so I could look back on their skills from early in the year and compare them to the end of the year. This is a quick, great way to collect data on student’s skills without a lot of involvement from you!

These are just a few ways to incorporate open-ended problems into your math time. I encourage you to try one way for a week or two and then experiment with another way once your students are showing they feel confident in the first implementation.

How do I make sure to provide students with open ended math problems during math each day?

In order to provide your students with activities and resources that encourage deep thinking and allow every student to participate, detailed planning is required. Deciding what standards and concept you want to focus on and choosing the best way to practice skills related to that concept before having students complete an activity is crucial to creating an effective learning time. If you peek at page 37 of the resource below, you’ll see a planning page that you can use to plan out the open-ended activities you want to use in your classroom. By editing this page with your own information, you can plan for a week of open-ended activities quickly and efficiently. This is also a great way to hold yourself accountable for how often you’re giving your students the opportunity to work on open-ended activities.

Any way you choose to implement open-ended problems in your classroom, your students are sure to grow in their problem-solving abilities and confidence. Creating a space that is safe for your students to take chances and risks with their learning is one of the greatest gifts you can give them. By incorporating ways for your students to express their individual ways of thinking, like open-ended math problems, you’ll foster a love of creative thinking and confidence in problem-solving skills.

Notes About the Included Resources:

The resources included in this blog post are for you to use in your classroom with your students. More detailed explanations for how to incorporate the activities are included in the resources themselves. There are also a few blank templates within the resource so that you can create your own task cards, thinking mats, and activity plans.

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open ended problem solving tasks

Open Ended Math Activities for the Primary Classroom

Use these templates and graphic organizers for students who may need additional support. Feel free to download and modify the editable version, including the Frayer model template and word bank template.

IMPLEMENTATION GOAL

Choose or create an open-ended math activity to incorporate into your math instructional time. Plan to introduce the activity to your students at the beginning of the week, model and practice how to complete the activity together, and then allow them to work on the activity for 10-15 minutes per day throughout the rest of the week. Take anecdotal notes about growth you notice and how your students react to these kinds of problems. Do they enjoy them? Dread them? Are you seeing improvement in their thinking and willingness to participate? Take note of these kinds of things as the week goes on. Then, decide what open-ended problems you’ll implement the following week.

  • What is Open-Ended Problem Solving? – https://mste.illinois.edu/users/aki/open_ended/WhatIsOpen-ended.html
  • The Effect of Open-ended Tasks- http://journals.yu.edu.jo/jjes/Issues/2013/Vol9No3/8.pdf
  • Clip art generated by Creative Clips Clipart by Krista Walden, http://www.teacherspayteachers/store/krista-walden

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open ended problem solving tasks

In maths classes teachers often approach differentiation by setting different levels of work for learners at different levels of ‘ability’. However, this approach may fail to develop the types of mathematical thinking needed most for students’  futures.

Here’s why, and how, ‘rich tasks’ can help.

Teachers are often being asked to differentiate teaching and learning to ensure they are catering for a broad range of student abilities, interests and readiness to learn.

In mathematics, this means simultaneously teaching new content to all students, providing ample opportunity for students to master content, supporting students experiencing difficulties with mathematical understanding and ensuring experiences in higher levels that extend and enrich more capable learners.

It can be exhausting to think about, especially when we know that in our classrooms we have such a broad range of students. As teachers, how can we manage this in real-world classrooms, without driving ourselves to the edge of exhaustion by micro-managing separate tasks for each and every student?

Back in 2016, the World Economic Forum predicted that within five years, over one-third of skills that were considered important in the workforce will have changed. They argued that by 2020, artificial intelligence, advanced robotics, autonomous transport, new materials, biotechnology and genomics will have transformed the kind of skills required in the workforce and in the way we live. The three skills that topped the WEF’s ‘list of 21st Century Thinking Skills’ were, in order, complex problem solving, critical thinking and creativity (Grey, A. 2016).

As educators we can’t ignore these realities.  We know we need to be preparing as many of our students as possible for the imperatives of life in the 21st century, developing their problem solving, reasoning and creativity proficiencies, especially in mathematics. These skills underpin so many roles in the new economy.  However, it’s also important that at the same time we are catering for students who are having difficulty understanding concepts at much more basic levels.

The standard approach to differentiation in mathematics has been to prepare multiple ‘levels’ of content set to different standards, and/or to make available a variety of learning resources, individualised for each learner in the classroom. When teachers are responsible for classes of around 30 students, often within the context of engagement or behavioural issues, this can be time consuming and professionally draining.

Some digital or online learning providers claim to have solved this by developing software that allows students to follow an ‘individualised’ program of learning in mathematics. Such programs use adaptive pathways which respond according to students’ progressive successes and failures against set tasks. Smart, huh?

However, the problem with many such online or digitally based mathematics learning solutions is the same as that facing the traditional ‘textbook only’ based approaches to mathematics that have plagued maths learning since the 1950s. Typical ‘drill and answer’ exercises and closed worded problems – such as those typically used in textbooks and on online mathematics websites – usually focus students’ learning only on developing understanding and fluency in mathematics.

These two math proficiencies – the skill of grasping a maths concept (understanding) and of using it flexibly and efficiently (fluency) are of course necessary – but by no means sufficient.  Tasks in maths that develop only understanding and fluency for students tend to suggest a single linear pathway to ‘working out’ and a single correct answer.

More importantly, they neglect the problem solving and reasoning proficiencies both prescribed within the Australian mathematics curriculum framework and required for humans to cope with life and work in the 21st century.

Problem solving, reasoning and creativity: ‘rich’ mathematical thinking

In mathematical learning, generally ‘closed’ (single answer/single method) tasks tend to stimulate basic conceptual understanding and develop concept fluency. However, ‘open-ended’ types of problems (more than one correct answer, multiple pathways for working out and justifying) have greater potential for stimulating higher order mathematical thinking, that is, creative problem solving and complex reasoning capacities.

This is partly because such tasks involve a search for patterns and relationships between elements in the problem. Students must ‘play around with’ different variables in order to generate different solution pathways, use trial and error (much the same as for problems faced in the real world) and explore a range of methods. They must compare the efficiency and accuracy of solution pathways and use reasoning to adapt and apply previously learned concepts to new situations.

Of course, students who have not understood and/or who are not fluent in math concepts will struggle to solve problems and to apply mathematical reasoning. The key issue for teachers is thus how we effectively and efficiently cater for these less fluent students without compromising opportunities for more fluent students to extend and enrich their mathematical thinking.

‘Rich’ or ‘Low Threshold, High Ceiling’ (LTHC) tasks

Above we mentioned the desirability of open ended maths tasks, or tasks which provide scope for more capable students to move toward more sophisticated thinking skills.

As an extension to this, ‘rich tasks’ – or ‘low threshold, high ceiling’ tasks in mathematics are structured so that all students can make a start to the problem, even if needing support. The ‘low threshold’ of such tasks reinforces understanding and fluency of a given concept and allow less confident learners to experience some level of success.

At the same time, however, further levels or iterations of the same tasks are designed to engage more independent learners in deep or complex problem solving and reasoning. The ‘high ceiling’ in these tasks provide plenty of opportunity for the participants to have a go at much more challenging maths, albeit within the same concept, topic or skill area.

According to Lynne McClure (2011), “A LTHC mathematical activity is one in which pretty well everyone in the group can begin, and then work on at their own level of engagement, but which has lots of possibilities for the participants to do much more challenging mathematics.”

She goes on to explain that rich tasks can go some way toward resolving the dilemma of more efficiently differentiating for a diverse group of learners; a single task designed with low thresholds and high ceilings can provide for the whole class.  Teachers often believe that the only way to challenge learners is to offer them different content at a higher grade level. However, in rich mathematical tasks the content itself remains quite simple but the level of thinking required – such as non-linear problem solving and mathematical reasoning – can become very complex (McClure 2011, p.2).

Rich or LTHC tasks allow teachers to set one problem for all students, provide some explicit whole-class instruction, and then respond to individual and small group strengths and needs as they arise during the problem. An important role for the teacher during such learning is to cultivate a problem solving ‘culture of iteration’ whereby students learn to push themselves into levels of the task that are difficult or ‘problematic’ for them.

Having most students persevering on harder levels of the task (for them) will also often free up the teacher to work more intensively with students struggling with basic concepts. Establishing collaborative problem solving guidelines such as time for quiet thinking and reflection, ‘ask three before you ask me’, pausing periodically to discuss the approaches of different students in various parts of the task ( not asking for ‘answers’!) and analysing errors as well as ‘correct answers’, will further facilitate the freeing of teacher time to be spent with less confident individuals.

Developing LTHC or ‘Rich’ tasks in Mathematics

Click to here to  d ownload an exemplar LTHC (rich) task: ‘ catering-canapes ’ (approximately Grade 5 or 6 level).

To write your own rich tasks in maths in a grade and topic suited to your own classroom (which is always much more fun than using someone else’s!), the following development guidelines might prove useful.

1. Start with a closed version of a problem within a given topic or concept. Grade level textbooks and standardised tests (such as past NAPLAN papers) are often a good source of closed mathematics problems.

2. ‘Open up’ the problem by removing or adapting parameters, to allow for a range of solutions. As an example, a closed problem might read:

‘ Sarah takes 4 hours and 55 minutes to complete a 200-page novel, while Derek takes 5 hours and 12 minutes to read the same novel. Assuming they are each reading at the same speeds as they were for the 200 page novel, how much faster is Sarah than Derek, in seconds, if they both read a 300-page novel?’

To open up this question up, we could remove the parameters in the second sentence of the question, such that it instead reads:

‘… Assuming they are each reading at the same speeds as they were for the 200 page novel, compare the times taken by Sarah and Derek to read one of the novels selected from a book shelf either in your classroom or your library.’

3. Push further by adding mathematical or procedural ‘problem solving’ complexity. For example, in the example above, we could add a third reader (with a different reading speed), select a range of different length novels to read and/or ask students to produce line graphs that compare the reading speeds of students against novels at different lengths.

They key concern here is to ensure a few different iterations of the same type of problem, each of which add increasingly complex levels of problem solving for students in the class.

4. Introduce a requirement for students to demonstrate reasoning and justification for a version or versions of the problem. Ways in which to do this might be:

  • have students challenge one another by setting their own versions of the task (they will need to have reasoned out their own version and to have ‘worked backwards’ in order to do this);
  • have students compare several different methods of working out and write reasons for which they think one is ‘better’ than another; or
  • have students design a model, experiment or product that applies the concept or topic in the real world. (An example of the above might be to design an experiment that compares any students’ reading speeds and draws conclusions from this experiment that could be useful to students when choosing the type and length of novel to read).

Designing and using rich tasks in mathematics can be a rewarding and motivating experience for teachers as they watch their students engage in problem solving and demonstrate thinking in ways that traditional ‘text book’ questions rarely allow.

They can also provide a new and rich source of assessment information as teachers gain new insights into how their students are working mathematically.

– Marcus Garrett

EduGains (2015), ‘Knowing and Responding to Learners in Mathematics’. Online resource available from http://www.edugains.ca/newsite/di/knowing_responding_to_learners.html. Accessed 17/01/18. Ontario Ministry of Education : Ontario, Canada.

Grey, A. (2016), ‘The 10 skills you need to thrive in the fourth industrial revolution’. Article published by World Economic Forum, 19 January, 2016. URL: https://www.weforum.org/agenda/2016/01/the-10-skills-you-need-to-thrive-in-the-fourth-industrial-revolution/ .

Herter, R. (2015), ‘Growth Mindset for Math – Mistakes’ (Youtube). URL: https://www.youtube.com/watch?v=LrgpKjiQbQw . Accessed 8/11/2017

McDonald, S. and Watson, A. (2012) What’s in a task? Generating mathematically rich activity. A report commissioned by the United Kingdom Qualifications and Curriculum Development Agency (now the United Kingdom  Standards and Testing Agency). London : United Kingdom. Online report – URL: http://xtec.cat/centres/a8005072/articles/rich. pdf . Accessed 17/05/18.

McLeod, S. A. (2012). Zone of proximal development . Retrieved from www.simplypsychology.org/Zone-of-Proximal-Development.html

McClure, L. (2011), ‘ Using Low Threshold High Ceiling Tasks in Ordinary Primary Classrooms’ . Online article on nrich.maths.org. URL: http://nrich.maths.org/ 7701 . Accessed 10/1/2018. NRich Maths, Cambridge University : United Kingdom.

Motter, A. (Date uncertain), ‘George Polya’. Online article, in ‘math.wichita.edu’. URL: http://www.math.wichita.edu/history/men/polya.html. Accessed 10/04/16. Wichita State University : Kansas, United States.

New Zealand Ministry of Education (2010 ? – 2017), ‘Problem Solving’ online resource page. URL https://nzmaths.co.nz/level-5-problems . Accessed 3/11/2017. New Zealand Ministry of Education : Dunedin, NZ.

NRICH Mathematics (2018), ‘What Was In the Box’. URL: https://nrich.maths.org/7819 . Accessed 17/01/18. Cambridge University, United Kingdom.

NSW Department of Education (2014), ‘Newman’s Error Analysis’. On Numeracy Skills Framework support website. URL: http://numeracyskills.com.au/newman-s-error-analysis . Accessed 23 October, 2016. Government of NSW : Sydney, Australia.

Piggott, J. (20018) ‘Rich Tasks and Contexts’ . Online article on nrich.maths.org. URL: https://nrich.maths.org/ 5662 . Accessed 10/1/2018. NRich Maths, Cambridge University : United Kingdom.

Piggott, J. (2011), ‘Integrating Rich Tasks’. Online article on nrich.maths.org. URL: https://nrich.maths.org/ 6089 . Accessed 10/1/2018. NRich Maths, Cambridge University : United Kingdom.

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open ended problem solving tasks

Open-Ended Math Tasks: The Benefits

Open-Ended Math Tasks!  Are you using enough rich math tasks in your class?  Check out this post with a ton of reasons (OK, TWELVE of them!) why we should be incorporating more of these open-ended problem-solving experiences.  Check out all the amazing things we can accomplish with them!

Reasons to use open ended math tasks

Gets students tackling problems that have more than one solution

So often we give students problems where there truly is one solution.  They get very accustomed to “filling in the box” and seeking the right answer rather than digging in and finding the complexity of the math.  Finding quality open-ended math tasks can help them see that math is far more intricate than merely finding a quick answer!

Helps students learn to make sense of a complex problem

Indeed, one of the Standards for Mathematical Practice involves students tackling a problem to determine what it’s asking.  If we consistently give students simple word problems that accompany our units, we take away all that thinking.  For example, when I teach a unit on multiplication if I only present word problems that involve multiplication–I’ve taken the thinking away.  Using more complex tasks forces students to think about what type of math they need to do and to dig in to get started.

Allows students to choose from multiple strategies or entry points

Similarly, these open-ended math tasks are perfect for giving students an entire arsenal of strategies–many of which can work in any given situation.  When there are multiple steps and lots of information, there are different places for students to dig in and get started.  Similarly, students can move back and forth between different parts of the task as they work and made progress toward understanding.  When given a task like designing a new library display, students can collect all the different pieces of information in the task and CHOOSE what makes sense to get them started.  They may start with a sketch or an estimate or any number of other strategies, but there is a LOT to grab onto as they begin.

Encourages math talk and discourse

This is pretty self-explanatory!  When you assign a rich math task, there is a lot to talk about.  When we ask students to tackle them, it only makes sense to give them “talk time”.  This is truly one of the best parts about using quality problems.  The opportunity to share thinking and help get classmates “unstuck” is invaluable.

Increasing math talk

Allows students to see real-world applications of math

We’ve all seen the memes making fun of word problems.

“If Lois buys 16 watermelons…”

“There are 693 kittens…”

(You get my drift.)

Quality open-ended tasks are MEANT to be real-world math.  Students are asked to solve real-world problems.  They may be working with a budget.  They may need to make decisions about “how many” or “which ones”.  Students may need to design something.  THIS is math.  When you find math problems that mimic students’ real lives, it’s priceless.  Our students eat lunch in the cafeteria.  They go on field trips.  They have sleepovers.  This is engaging math.

Helps students see the interconnectedness of math concepts

Like I mentioned before, teaching math in isolation takes away the thinking and deep understanding.  By using rich math tasks, students are forced to examine ALL the math they know as they look for ways to accomplish their goal.  They may need to add and subtract.  They may need to make a chart.  Sometimes they may see that drawing an array can help them solve a multiplication problem. In a great task, they may need to do all of these things for one problem!  So often, we fail to give students enough credit for being able to do this kind of work, and we do them a disservice when we don’t give them those opportunities.

Immerses students in the Standards for Mathematical Practice along with the math content

As we rush to work through the endless list of standards for our grade levels, we cannot forget to include the math practice standards in our plans.  When you read through these 8 standards, you can see the role open-ended math challenges play.  In fact, every single one of those standards can be met when we provide students with this type of math work.

Provides an excellent forum for partner and small group work

As I mentioned before, providing students with these rich tasks encourages accountable talk.  They are perfect for partner and small group work.  In fact, using these to explicitly teach students HOW to work together, how to coach each other, and how to encourage each other is invaluable.  Building relationships and adopting a growth mindset is a critical piece of learning productive struggle.  Learning to struggle and overcome challenges TOGETHER is a key part of our math coaching.

Naturally differentiate as students take “just right” approaches to solving them

Unlike more simplistic math problems, open-ended tasks can naturally differentiate.  Some students may want access to calculators.  Some may draw pictures.  There may be a time when groups of students choose to use manipulatives to model their math thinking.  As teachers coach, they can give math “hints” as needed to students based on their needs.  I have used countless math tasks where I have assigned the same task to ALL students but have provided differing levels of support as they tackle them.  Remember, students can still work on problem solving skills and strategies if they haven’t yet mastered algorithms.

A perfect example is a back-to-school shopping challenge where students have a budget and a set list of things to purchase.  There are multiple things to consider–from the number of items required to the different prices at different stores.  Should we save this rich task for students ONLY when they have mastered the additional algorithm with decimals?  Of course not.  We can have students work in teams, give calculators or other tools–or anything that will make the task more accessible.  This is an example of providing QUALITY education to all students and is a powerful way to build math confidence and self-esteem.

Allows teachers to serve as coaches as they watch and interact with students as they solve them

This is big.  Rather than teachers being the lecturer or “provider of knowledge”, these tasks naturally lend themselves to teachers serving as coaches.  So what?

  • We learn so much about our students’ thinking as we watch.
  • We have the power to differentiate on the fly as needed.
  • Some students benefit from having more direct teacher contact while others thrive on being more independent.
  • Learning to ask good questions to get students “unstuck” is so much more powerful than merely “showing them how”.
  • Teachers can truly maximize their time to reach WHO needs help, WHEN they need help.
  • We can help show students how to coach each other, making a room full of places for students to get coaching and guidance.
  • We can notice when “a ha” moments happen and recognize them.
  • This coaching time is perfect for finding work samples to share and discuss
  • And so much more!

Encourages creative thinking and strategies

I think this point is being made throughout this post, but I’ll reiterate it here.  When there is more than one solution, there are countless solution paths.  When there are multiple solution paths, students see the value in thinking differently and there is not one “right” way to do business.  What a powerful message!

To make it even more powerful, make sure you build in time to SHARE strategies.  Whether you have students showcase their work under a document camera, share their ideas in small groups, project their work, or hang it up–celebrate the amazing solutions students come up with!  Even more valuable?  Make sure students know that the final answer is less important than the process to get there.

Do we want students to realize that we value their thinking?  You bet.  Learning to think independently and creatively is another one of those life lessons that will extend far beyond your math class.  It’s how REAL problems get solved.

Encourages students (and teachers) to focus on the PROCESS over the solution

As I stated before, of course, we want students to do math accurately and get correct solutions.  The truth is, when students tackle these complex tasks, there are lots of places where errors can occur.  If we stress getting a correct answer over the complex processes, all value is lost.  The value in these tasks is the process of understanding the problem, choosing strategies, modifying strategies when needed, selecting models or tools to use, estimating, reviewing answers, organizing work, and so much more.

When students know that WE value their hard work as much as correct answers, it’s so freeing for them!  They see that we value their thinking and strategies and even their mistakes.  This takes away any compulsion to get the answer from others or to shut down due to frustration.

As you can see, the power of these tasks is obvious.  Unfortunately, very few math programs provide enough of them to really help students develop these skills.  Because of that, one role I have taken on as a curriculum designer is to try to fill this gap by creating tasks like this for different age levels.

Convinced?  Give some a try!

CLICK HERE to grab a free guide that summarizes this blog post AND tells exactly what skills some of my open-ended tasks cover.

CLICK HERE to see a bundle of tasks for grades 2/3 (or for beginning problem solvers in grade 4)

Looking for tasks for experienced grade 3 problem solvers and grades 4/5?  CLICK HERE!

Finally, looking for some even more in-depth tasks that are differentiated and have tons of extensions?  These tasks can last multiple days and can be used with your entire class or with small enrichment groups or for fast finishers.  CHECK THEM OUT HERE !

(NOTE:  All tasks are AVAILABLE IN PRINT AND DIGITAL FORMATS)

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Research Studies on Learning and Teaching of Mathematics pp 49–69 Cite as

To What Extent Are Open Problems Open? Interplay Between Problem Context and Structure

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  • First Online: 03 August 2023

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Part of the book series: Research in Mathematics Education ((RME))

The notion of open mathematical problems that appears in the mathematics education literature includes a variety of mathematical questions and tasks. Observation of the distinction between open-end and open-start problems leads us to draw a distinction between Multiple Solution Strategies Tasks (MSTs) and Multiple Solution Outcomes Tasks (MOTs). Whereas MSTs are inherently open, MOTs are not necessarily open ended. Their openness depends on the formulation of a question. MOTs can either be open or require completeness of solution sets. In this chapter, we discuss MOTs from the point of view of their openness and completeness and draw a connection between MOTs and “sense making” tasks, the complexity of which is thought to be rooted in the lack of realistic considerations applied by students and teachers. We argue that the complexity of these problems is linked to their multiple possible outcomes and the requirement to find a complete solution, both of which are unconventional, and to the fact that the solutions of these problems are insight based.

  • Multiple solution outcomes tasks (mots)
  • Sense making
  • Completeness of solution space
  • Unconventional tasks

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Acknowledgment

The research experiment was supported by Grant # 887/18 from the Israel Science Foundation. The opinions expressed in this publication are those of the author(s) and do not necessarily reflect the views of the Israel Science Foundation.

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This is the 22nd version of this paper, on which Roza Leikin had been working under the supervision of Ed Silver during postdoctorate studies (1997–1998) at LRDC. The paper is coauthored with Sigal Klein and Ilana Waisman who are currently working with Roza Leikin on different aspects of integration of context-rich and open problems in mathematics instruction, which are reflected in the paper.

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Leikin, R., Klein, S., Waisman, I. (2023). To What Extent Are Open Problems Open? Interplay Between Problem Context and Structure. In: Cai, J., Stylianides, G.J., Kenney, P.A. (eds) Research Studies on Learning and Teaching of Mathematics. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-35459-5_3

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