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Start here!

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Rich mathematical tasks are essential for engaging learners and creating dynamic classrooms. In this post-adapted from her book, Modeling with Mathematics-Nancy Butler Wolf looks at six characteristics shared by the best math tasks.

How do we know a good task from a bad one? As teachers make the foray into mathematical modeling in their classrooms, it is important that they use rich and rewarding tasks. Unfortunately, too few textbooks contain such tasks, so teachers must scramble to find or develop their own. Here is a brief look at six important characteristics of rich modeling tasks:

1. Accessibility to All Learners The first characteristic should be obvious to almost any teacher in almost any classroom: the students in the desks in front of us come with a wide range of foundational knowledge, mathematics background, and experience with mathematics. If a task is truly rich, it will provide interest, motivation, and challenge for all our learners. It will not encourage our most struggling students to opt out and allow the more proficient students to take over, provide direction, and supply solutions. By the same token, it will not be quickly and easily solved by more advanced math students, but will provide challenge and extensions for these learners as well. A good task will provide opportunities for all learners to contribute with confidence.

2. Real-Life Tasks
The second criterion for a rich task has been the subject of extensive discussion among teachers, mathematicians, and researchers for many years. What, exactly, is a task with some basis in real life? But the question remains: What is an authentic task? I would suggest that an authentic task is one with which students have some real-life experience. I have used many problem-solving and modeling tasks that are, admittedly, not "really real," but are, nonetheless, interesting and engaging to students. For instance, mathematics with a literature connection can be very interesting and engaging to students despite the fact that the scenarios investigated would not be called "really real." Remember, the objective is to present tasks that are interesting and can help students understand that the concepts they are learning in their math classrooms are not completely separate from their lives outside the classroom. Do not stifle your creativity by imposing false limits regarding what is a "real world" application. Students will be engaged by problems that spring from literature, from other subject areas, and from their imaginations, as well as from their "really real" lives!

3. Multiple Approaches and Representation The third criterion of a rich math task is that it lends itself to a variety of approaches and representations. Our students approach mathematics from very different perspectives, and when we can recognize and respect that fact, we can provide more opportunities for students' success. When a variety of approaches can lead to an appropriate solution, students can become confident in their abilities and learn from one another.

4. Collaboration and Disscussion The fourth characteristic of a rich modeling task is that it encourages collaboration and discussion. The teaching strategies for modeling problems vary, but in every case should involve some degree of collaboration and discussion. Many tasks require students to first work independently, with guidance and encouragement from the teacher as needed. When students have reached an independent conclusion, they may share their results, approaches, and representations within collaborative groups and discuss the similarities and differences. Students gain new insights and perspectives by hearing from their peers; even struggling students begin to gain confidence in their ability to problem solve when given the opportunity to explain their reasoning on a problem.

5. Engagement, Curiosity, and Creativity The next characteristics of a rich task are fairly self-explanatory and go hand in hand with each other as well as with other characteristics. When a task is interesting and engaging, and gives a problem that interests students, they will persevere in the problem solving. A rich task presents students with a variety of possible approaches and representations, allowing access for all learners. When students problem solve, they must first decide which tools and what approach to try. As the problem unfolds, students may run into obstacles and roadblocks that will require them to make more decisions both individually and collaboratively, and, often, will demand creativity and variety in the way they apply their knowledge.

6. Opportunities for Extension The final characteristic of a rich modeling task is often the missing piece in problem-solving experiences in the classroom. We can probably all relate to the times we have presented students with an engaging task, only to find that one group finishes quickly while another group struggles with the problem. This can result in one of two scenarios: a frantic attempt to keep the more advanced group busy (or quiet) while the other group continues their work, or an attempt to hurry along the slower-paced group in order to keep everyone on task. A rich task keeps all students engaged by including challenges and extensions for more advanced learners to work out while other students continue to work without the pressure to "hurry up and finish."

There are many sources for rich math tasks, including the MARS tasks, the 3-Act Math tasks, and the Illuminations problems. As you work with problems like this, it will become easier to recognize a good mathematical modeling task and develop your own rich mathematical experiences for your students, and you will begin to see the return on your investment of time and energy in the enthusiasm, motivation, and depth of understanding of your students.

Want to learn more about getting started or going deeper with mathematical modeling? Watch Nancy talk about why modeling is so effective or preview the first chapter of Modeling with Mathematics.

♦ ♦ ♦ ♦

Nancy Butler Wolf is a national math consultant and curriculum developer who has served as a member of NCTM's Illuminations Writing Group. She has over 30 years of classroom teaching experience and has presented at various conferences on mathematical modeling and algebraic reasoning. Follow her on Twitter @drnanbut.

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Rich mathematical tasks are essential for engaging learners and creating dynamic classrooms. In this post-adapted from her book, Modeling with Mathematics -Nancy Butler Wolf looks at six characteristics shared by the best math tasks. How do we know a good task from a bad one?

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Formative Assessment Tasks

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Please Note: Balanced Assessment printed materials are available from this site, except for those indicated below. Please use the order form and follow the ordering directions carefully as they have changed. Current prices can be found on our order form.

Please also note that the Balanced Assessment Primary & Elementary Tasks have been published by Corwin Press. The Balanced Assessment Transition & Middle School Tasks have been published by Teachers' College Press. These tasks may still be viewed in.pdf format on this website but they may not be copied or printed.

About the program

From 1993 to 2003, the Balanced Assessment in Mathematics Program existed at the Harvard Graduate School of Education. The project group developed a large collection of innovative mathematics assessment tasks for grades K to 12, and trained teachers to use these assessments in their classrooms.

The library of over 300 mathematics assessment tasks developed during the project remains freely available through this web site. Teachers may use these materials in their own classrooms at no cost.

This site also offers reports on our approach to mathematics assessment and our distinctive objects x actions scoring system. A more detailed overview of the project is also available.

One of the most satisfying results of this ten-year project was the enthusiastic response of teachers to our assessment tasks and publications. We hope that you will continue to find the Balanced Assessment tasks to be a valuable resource.

Ordering Balanced Assessment publications

It is possible to purchase printed Balanced Assessment publications using our order form.

Our library of assessment tasks

If you are looking for a specific task, you can find it by name or by number.

If you're just getting to know our materials, we'd like to suggest a few of our favorite tasks:

Our approach to assessment and scoring

These reports and materials were developed while the Balanced Assessment program operated at the Harvard Graduate School of Education, between 1993 and 2003.

Assessing Mathematical Understanding and Skills Effectively (AMUSE)
Also available as a Word document.
This 1995 report describes our project group's approach to mathematics assessment.

Mathematics Content and Process Scoring System (MCAPS)
Also available as a Word document.
This report describes our objects x actions approach to scoring assessment tasks.

MCAPS Scoring Tool
Requires Microsoft Excel or compatible spreadsheet.
This spreadsheet implementation of the MCAPS scoring system enables teachers to efficiently use the system in their classes.

Judah L. Schwartz,
Principal Investigator/Director
Joan Kenney,
Project Coordinator/Director
Susan Ilias
Kevin Kelly
Teresa Sienkiewicz
Yesha Sivan
Victor Steinbok
Amy Warren
Michal Yerushalmy

For information on using BA tasks in the classroom, professional development for teachers and administrators, and issues regarding orders, please contact:
Joan Kenney
Joan_Kenney@post.harvard.edu

For technical web site issues, contact:
dan@mathshell.org

The materials on this web site were developed under National Science Foundation grants MDR-9252902 and ESI-9726403 (subcontracts from the University of California at Berkeley and Michigan State University), with additional funding from the American Academy of Arts and Sciences, the Boston Public Schools, the Cambridge Public Schools, and the Noyce Foundation..

All documents are Copyright © 1995-2003, President and Fellows of Harvard College.With the exceptions listed above in the heading to this page you are hereby granted the right to duplicate or print these materials for use in your classroom or school. Otherwise, these materials are protected by copyright.

This web site was originally designed by Kevin Kelly.

Last Update: 10/17/2015
Copyright © 2023, The Concord Consortium. All rights reserved.

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Click on a Grade Level and then a Unit.

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Tools 4 NC Teachers | Math Science Partnership Grant Website

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Tasks are found within each unit

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Click on the grade level & unit.

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Rich Learning Tasks

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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.

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Math PD in a blog by Mark Chubb. It includes some really great tasks.

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Do you have a great site for rich mathematical tasks you'd like to share?

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Do you have a great site for rich mathematical tasks you'd like to share?

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Email Leigh at belfordl14@ecu.edu

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Blog with rich tasks and a coaches story of how they were used in a classroom.

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Recently, I have been helping a third grade teacher learn how to use Number Talks to develop computational fluency and deepen student understanding of multiplication and division. The first time we met, we planned a Number Talk using MiniLessons for Early Multiplication and Division by Willem Uttenbogaard and Catherine Twomey Fosnot. We rehearsed the Number Talk. As we discussed the math, we anticipated what students might say, how Rachel might record student thinking, and what questions she might ask to connect student thinking. We read the teacher's guide and noticed some of our ideas in the description. We discussed how Rachel could follow the same structure we just used to plan subsequent Number Talks.

Last Thursday, Rachel and I met again to discuss how the number talk went. She told me she was so glad I showed her how to use her Smartboard for Number Talks. She described how helpful it was for her students to be able to describe their own thinking and see each other's thinking. She pulled up the Smartboard Notebook she had created for the last Number Talk so we could discuss her student's thinking. Unfortunately, during this Number Talk, the markers on her Smartboard weren't working so she had to record student thinking with numbers, symbols, and words.

At this point, Rachel asked me an awesome question. She wondered, "my students started doing this thing where they find a fact that doesn't work evenly and then add the extra squares. Should I encourage them to do that? Will it distract them for what they should be doing?" I asked her to tell me more. She showed me some examples. She said the students were thinking about division. They were noticing that you couldn't divide 25 in half. They tried to divide it into three groups. One of the students said, "You can do it any way you want. You just have to add the extras."

Rachel looked at me, waiting for me to tell her what to do next.

She said, "Should I let them do this?"

I responded, "I don't know. Let's find out."

Then, I got really excited. "Oh my gosh, are they thinking about division? Are they thinking about remainders, but they don't know it. This is so cool. Do they wonder if this will always work? Do they wonder how many ways there are? I'm wondering how many ways there are! Are there a certain number of ways? Does it depend on the array? What does this have to do with factors? There must be a pattern. Is there a pattern? Can we do the math? Do you want to explore this right now? I feel like we have to explore this right now to answer your question? Are you okay if we try this right now?"

Rachel smiled. I started scribbling things down on scrap paper.

It didn't take long before we realized we needed some kind of system for organizing our thinking.

At this point, we started talking about the commutative property. We discussed the difference between 3 groups of 8 and 8 groups of 3. We wondered, "are we only using arrays or can we use "groups"?" If we are only using arrays than we can't see any 3 by 8 arrays in a 5×5 array. We can see seven 3 by 1 arrays, but then we have 4 leftovers. We can't make anymore 3 by 1 arrays. If we are using groups, we can see 8 groups of 3 plus one leftover and we can also see 3 groups of 8 plus one left over. We were in this funny place. We were mixing array language with "equal groups of" language, trying to figure out how the constraints of contextualizing the community property impacted our problem. I thought about this some more this weekend. I tried to anticipate what students might do when they confront this situation.

There is the potential for this conversation to get messy fast. We could all end up on the fast track down a major rabbit hole. I think it is worth the risk. As I explored this problem, I was forced to articulate the difference between an area representation of multiplication and a set representation of multiplication. These third grade students are about to start their unit on area and perimeter. This seems like a meaningful mess. It might be worthwhile to pause during the exploration, examine a few of the decompositions and discuss; What is the same? What is different?

When Rachel and I were exploring this problem, we didn't define the constraints. We were in the initial stages of our problem solving. We were messing around with mathematics. We were thinking about arrays, but using "groups of" language. Towards the end of our exploration, Rachel stumbled upon a conjecture. She said, "I think we have found all the ways, but I can't really explain why." I asked her to tell me what she was thinking, even if it was still fuzzy. I wrote down what she said:

She said she wasn't sure how to describe what needed to be bigger, but she just knew that we could make more groups with the leftovers. I knew what she meant, but I couldn't find the precise words for it, either. I tried to use numbers and symbols to record what was happening:

Then, Rachel said, "I think I can explain why we can't find any more ways. If the number of leftovers is larger than the group size, you will always be able to make more groups."

"Yes. That makes sense to me."

While Rachel and I were working, every so often, we would zoom out and discuss the implications of Rachel exploring this problem with her students. She wondered, would it confuse them? Would it dissuade them from using efficient strategies? I kept trying to find connections to the third grade standards. I was reminded of an excerpt from Bill McCallum's progressions document:

I saw a lot of connections to fourth grade standards, particularly the one about interpreting remainders. Rachel and I both worried, "is it okay if these students start talking about remainders if they are only in third grade?"

I have been thinking about this 4th grade standard a lot this year. Marilyn Burns, Kristin Gray, and Jody Guarino have all pushed my thinking on this standard. I started a post about a lesson I did with a fourth grade class earlier in the year. I haven't finished it, but I'm posting it in draft form because it is really connected to this post. How is this fourth grade standard connected to work in third grade? It should be, right? Mathematics is a system of interconnected ideas. Remainders don't just drop out of the sky in fourth grade? They shouldn't, right? What might we explore in third grade that would connect to a deeper exploration of remainders in fourth grade? I'm thinking about this cluster:

I'm not totally sure if/how these standards might connect to the problem that Rachel is going to explore with her students, but I'm wondering about it.

I shared some of these thoughts with Rachel. Together, we decided this problem would be an opportunity to discuss the difference between arrays and groups, connect multiplication and division, look for structure, persevere, and organize thinking. Should Rachel create a space for her students to explore this problem?

Of course she should! They created it! It is their problem!

We talked about how she would present the problem and support the student exploration. We revisited how the question originally came about and how she would phrase the question so it was clear and still captured the student's voice.

Rachel took all of our work with her so she could share it with her students after they tackled the problem. She thought they would be really excited to see that we worked so hard on a problem that they created. I'm still thinking about ways that Rachel can use this problem as a spring board to deepen understanding of grade level standards. She plans on using it with her students this coming Monday. I would love to hear your thoughts about questions Rachel could ask, connections she might make, next steps for her students. Try this problem out! Get messy! Let us know what you find out.

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Recently, I have been helping a third grade teacher learn how to use Number Talks to develop computational fluency and deepen student understanding of multiplication and division. The first time we met, we planned a Number Talk using MiniLessons for Early Multiplication and Division by Willem Uttenbogaard and Catherine Twomey Fosnot.

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K-2 and 3-5 Tasks that promote mathematical justification.

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Hey, you have something to add to this list! Send me an email and I will add it to the site:

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180 days of estimation tasks to build number sense

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ESTIMATION 180 is a place for both students and teachers to enjoy making sense of math and develop their number sense together.

Pick from over 200 visual estimation tasks that invite all your students to be part of rich math conversations and learning.

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How do I get started? 1 USE THE FREE MATH LESSONS Estimation 180 Tasks 3-Act Tasks 2 SUBSCRIBE Email Newsletter Podcast

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Promote mathematical reasoning through "Which One Doesn't Belong" tasks

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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Problem based math tasks using videos and pictures.

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Tasks from Jo Boaler (often known for her work with growth mindset) and friends

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Posts related to elementary mathematics

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Fraction Tasks from pictures.