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 Task

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).

#### Three-Act Task

“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

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

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 on students’ prior knowledge and experiences” (National Council of Teachers of Mathematics, 2014. p. 17). They need to make sure students have access to the problem, but that they do not lower the cognitive demand of the task by over-scaffolding or showing examples of how to solve the task prior to students solving. Here are a few frameworks for effectively facilitating tasks.

#### Launch, Explore, Summarize/Discuss

A basic structure for implementing tasks includes launching the problem, providing students time and structures to explore independently and with peers, and then bringing the class back together to discuss or summarize the strategies used along with the underlying mathematics concepts. This teaching model has been published by Michigan State University and is included in the Connected Mathematics Project. It is also an integral part of the Comprehensive Mathematics Instruction framework.

(Michigan State University, Accessed 2023)

Successful inquiry-based teaching moves through phases of a *Teaching Cycle* that begins by engaging students in a worthwhile mathematical task (*Launch), *allows students time to grapple with the mathematics of the task *(Explore), *and concludes with a class discussion in which students thinking is examine and exploited for its potential learning opportunities *(Discuss)*.

(Hendrickson, Hilton, & Bahr, 2008)

#### 5 Practices for Orchestrating Productive Mathematics Discussions

“The five practices were designed to help teachers to use students’ responses to advance the mathematical understanding of the class as a whole by providing teachers with some control over what is likely to happen in the discussion… The five practices are -

__anti__likely student responses to challenging mathematical tasks and questions to ask students who produce them;__cipating__students’ actual responses to the tasks (while students work o the task in pairs or small groups);__monitoring__particular students to present their mathematical work during the whole-class discussion;__selecting__the student responses that will be displayed in a specific order; and__sequencing__different students; responses and connecting the responses to key mathematical ideas” (Smith & Stein, 2018).__connecting__

Implementing the five practices can incredibly enhance the implementation of mathematical tasks. Teachers are more prepared when they anticipate student thinking and more connections are made when strategies and solutions are intentionally selected and sequenced in a whole-class discussion.

## Summing it up

Implementing mathematics tasks provides students with opportunities to think critically and solve problems in engaging and meaningful ways. It is important that students connect to the context and content of the tasks and that teachers do not remove the rigor by providing too much support upfront. Discourse is the most important part for students to make connections. Tasks introduced early on when building understanding of concepts should include discourse that allows students to develop and share strategies with classmates. Discourse should also be included with tasks that build fluency. For example, after students play a game or engage in a routine they can be asked about the patterns they notice and the skills they are refining. There are many nuances and resources related to math tasks. Look for additional future posts on implementing math tasks!

#### References:

Degarcia. (2008). *Mathematical Routines. *Strategies for Teaching Elementary

Mathematics. __https://mathteachingstrategies.wordpress.com/__

Hendrickson, S., Hilton, S. C., & Bahr, D. (2008).* The comprehensive mathematics *

* instruction (CMI) framework: A new lens for examining teaching and *

* learning in the mathematics classroom. *Utah Mathematics Teacher, 44-52*.*

Huinker, D. A. (2020). *Catalyzing change in early childhood and elementary *

*mathematics: initiating critical conversations*. National Council of

Teachers of Mathematics, Inc.

Meyer, D. (2011). *The Three Acts Of A Mathematical Story. *Three-Act Math.

Michigan State University. (Accessed 2023). *Connected Mathematics Project - *

* Curriculum Design - Philosophy. *College of Natural Science.

Open Middle Partnership. (2016-2022). *What’s Open Middle?* Open Middle.

San Francisco Unified School District. (Accessed 2022). * 3 Act Tasks. *Mathematics

Department. __https://www.sfusdmath.org/3-act-tasks.html__

Smith, M. S. & Stein, M. K. (2018). *5 Practices for Orchestrating Productive *

*Mathematics Discussions. Second Edition. *National Council of Teachers of

Mathematics.

University of Cambridge. (2005 Revised 2017). *Problem Solving: Opening up *

*Problems. NRICH. *__https://nrich.maths.org/2471__

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