What is a Math Task and How Do I Use One?
The concept of "math task" is found in professional books, in educational posts on social media, and used in schools regularly. What does it mean?
What is a math task?
Math tasks are problems students engage in without the teacher providing direct instruction or telling the student how to solve the problem. High quality math “tasks encourage reasoning and access to … mathematics through multiple entry points, including the use of different representations and tools, and they foster the solving of problems through varied solution strategies.” (National Council of Teachers of Mathematics, 2014. p. 17)
What types of math tasks are there?
“Mathematical tasks can range from a set of routine exercises to a complex and challenging problem that focuses students’ attention on a particular mathematical idea” (National Council of Teachers of Mathematics, 2014. p. 17). There are several types of mathematical tasks that vary in complexity and level of cognitive demand. Some math tasks are rich in context and some may involve deep thinking with mathematics without a context. Here are a few types of tasks commonly used.
Open tasks usually refer to tasks that have multiple solutions or multiple solution paths. Tasks with multiple solutions are known as open-ended tasks or problems. A task is also open if it allows for a variety of strategies to be used. “In general, the more open-ended types of problems have greater potential for stimulating higher order mathematical thinking. This is partly because they usually involve a search for patterns and relationships between elements in the problem” (University of Cambridge, 2005 Revised 2017).
Open Middle Problem
“Most (Open Middle problems) have:
a “closed beginning” meaning that they all start with the same initial problem,
a “closed end” meaning that they all end with the same answer,
an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.
Some additional characteristics of open middle problems include:
They generally have multiple ways of solving them as opposed to a problem where you are told to solve it using a specific method.
They may involve optimization such that it is easy to get an answer but more challenging to get the best or optimal answer.
They may appear to be simple and procedural in nature but turn out to be more challenging and complex when you start to solve it.
They are generally not as complex as a performance task which may require significant background context to complete” (Open Middle Partnership, 2016-2022).
“A Three-Act Task is a whole-group mathematics task consisting of three distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three” (San Francisco Unified School District, Accessed 2023).
For more detailed information see: https://www.sfusdmath.org/3-act-tasks.html and https://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/.
Mathematics routines are purposefully structured activities that help children develop procedural fluency, as well as reasoning and problem-solving skills, through meaningful practice. … Where math lessons tend to be more exploratory, routines tend to be more practical. Instead of isolated events, purposeful routines are most successful if they are done in a series, which allows the students to focus on certain mathematical ideas. Initially, the routine may take a little longer, because the goal is to establish the classroom norms in its process and discourse. Once these norms are in place, the focus shifts from the process to the content” (Degarcia, 2008). Routines may include counting activities, quick images that are shown to help students develop subitizing and spatial reasoning skills, number talks, and other activities that support mathematical development by repeating activities regularly over time.
Games can include card games, board games, and other games that involve elements of fun and/or competition. “Honoring the resources that children and families bring to a classroom allows mathematics to be a bridge to their cultures and communities. This may include the beauty of mathematics in both the process and product of braiding hair, beading and weaving patterns, playing games, and creating art” (Huinker, 2020). Game-based tasks can be used as resources to build fluency as well as to involve families.
When to use various types of tasks
Typically problem solving tasks including, but not limited to open-ended word problems and Three-Act tasks are great to introduce concepts as students build conceptual understanding by solving problems with rich contexts in ways that make sense to them. Open Middle problems are great to engage students in critical thinking as they are building procedural fluency from conceptual understanding. Games are most often more effective when used after students have had experience with concepts as a way to practice and build fluency. Routines are activities that are done regularly so students make connections and become familiar with patterns.
How do I use a math task?
As teachers implement tasks they should carefully plan “how contexts, culture, conditions, and language can be used to create mathematical tasks that draw o