### Rich Problems – Part 1

by Marvin Cohen and Karen Rothschild

One of the underlying beliefs that guides Math for All is that in order to learn mathematics well, students must engage with rich problems. Rich problems allow ALL students, with a variety of neurodevelopmental strengths and challenges, to engage in mathematical reasoning and become flexible and creative thinkers about mathematical ideas. In this Math for All Updates, we review **what** rich problems are, **why** they are important, and **where** to find some ready to use. In a later Math for All Updates we will discuss how to create your own rich problems customized for your curriculum.

**What are Rich Problems?**

At Math for All, we believe that all rich problems provide:

- opportunities to engage the problem solver in thinking about mathematical ideas in a variety of non-routine ways.
- an appropriate level of productive struggle.
- an opportunity for students to communicate their thinking about mathematical ideas.

Rich problems increase both the problem solver’s reasoning skills and the depth of their mathematical understanding. Rich problems are rich because they are not amenable to the application of a known algorithm, but require non-routine use of the student’s knowledge, skills, and ingenuity. They usually offer multiple entry pathways and methods of representation. This provides students with diverse abilities and challenges the opportunity to create solution strategies that leverage their particular strengths.

Rich problems usually have one or more of the following characteristics:

- Several correct answers. For example, “Find four numbers whose sum is 20.”
- A single answer but with many pathways to a solution. For example, “There are 10 animals in the barnyard, some chickens, some pigs. Altogether there are 24 legs. How many of the animals are chickens and how many are pigs?”
- A level of complexity that may require an entire class period or more to solve.
- An opportunity to look for patterns and make connections to previous problems, other students’ strategies, and other areas of mathematics. For example, see the staircase problem below.
- A “low floor and high ceiling,” meaning both that all your students will be able to engage with the mathematics of the problem in some way, and that the problem has sufficient complexity to challenge all your students. NRICH summarizes this approach as “everyone can get started, and everyone can get stuck” (2013). For example, a problem could have a variety of questions related to the following sequence, such as: How many squares are in the next staircase? How many in the 20th staircase? What is the rule for finding the number of squares in any staircase?

- An expectation that the student be able to communicate their ideas and defend their approach.
- An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
- An opportunity to learn some new mathematics (a mathematical residue) through working on the problem.
- An opportunity to practice routine skills in the service of engaging with a complex problem.
- An opportunity for a teacher to deepen their understanding of their students as learners and to build new lessons based on what students know, their developmental level, and their neurodevelopmental strengths and challenges.

**Why Rich Problems?**

All adults need mathematical understanding to solve problems in their daily lives. Most adults use calculators and computers to perform routine computation beyond what they can do mentally. They must, however, understand enough mathematics to know what to enter into the machines and how to evaluate what comes out. Our personal financial situations are deeply affected by our understanding of pricing schemes for the things we buy, the mortgages we hold, and fees we pay. As citizens, understanding mathematics can help us evaluate government policies, understand political polls, and make decisions. Building and designing our homes, and scaling up recipes for crowds also require math. Now especially, mathematical understanding is crucial for making sense of policies related to the pandemic. Decisions about shutdowns, medical treatments, and vaccines are all grounded in mathematics. For all these reasons, it is important students develop their capacities to reason about mathematics. Research has demonstrated that experience with rich problems improves children’s mathematical reasoning (Hattie, Fisher, & Frey, 2017).

**Where to Find Rich Problems**

Several types of rich problems are available online, ready to use or adapt. The sites below are some of many places where rich problems can be found:

- Which One Doesn’t Belong – These problems consist of squares divided into 4 quadrants with numbers, shapes, or graphs. In every problem there is at least one way that each of the quadrants “doesn’t belong.” Thus, any quadrant can be argued to be different from the others.
- “Open Middle” Problems – These are problems with a single answer but with many ways to reach the answer. They are organized by both topic and grade level.
- NRICH Maths – This is a multifaceted site from the University of Cambridge in Great Britain. It has both articles and ready-made problems. The site includes problems for grades 1–5 (scroll down to the “Collections” section) and problems for younger children. We encourage you to explore NRICH more fully as well. There are many informative articles and discussions on the site.
- Rich tasks from Virginia – These are tasks published by the Virginia Department of education. They come with complete lesson plans as well as example anticipated student responses.
- Rich tasks from Georgia – This site contains a complete framework of tasks designed to address all standards at all grades. They include 3-Act Tasks, YouCubed Tasks, and many other tasks that are open ended or feature an open middle approach.

The problems can be used “as is” or adapted to the specific neurodevelopmental strengths and challenges of your students. Carefully adapted, they can engage ALL your students in thinking about mathematical ideas in a variety of ways, thereby not only increasing their skills but also their abilities to think flexibly and deeply.

**References**

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.

NRICH Team. (2013). Low Threshold High Ceiling – an Introduction. Cambridge University, United Kingdom: NRICH Maths.https://nrich.maths.org/10345

Math for All is a professional development program that brings general and special education teachers together to enhance their skills in

planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.