Tasks That Inspire Reasoning - What Are They and How Do I Implement Them?
- Shannon Olson
- 5 days ago
- 9 min read
In Principles to Actions (NCTM, 2014), one of the cornerstone practices is Implement Tasks That Promote Reasoning and Problem Solving. Great math learning grows from tasks - they invite students to wonder, explore, and explain their thinking. In this post, we’ll unpack what we mean by “tasks,” the key features that make them high-quality, and practical ways to launch and facilitate them so every student can engage in meaningful problem solving.
What is a Math Task?
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? Math tasks are problems students engage in without the teacher providing direct instruction or telling the student how to solve the problem. These types of math tasks can take many forms including Open Tasks, 3-Act Tasks, Open-Middle Tasks, Games, and Number Routines (more info on Types of Tasks can be found in my blogpost “What is a Math Task and How Do I Use One?”). There are also "curricular tasks” that clearly align to grade level standards and “non-curricular tasks” that engage students in problem solving, reasoning, and collaboration but may not have specific content goals. Our math classrooms should consistently utilize the power of math tasks.
What makes a math task high-quality?
Not all tasks provide the same opportunities for student thinking and learning. 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). In other words, a high-quality task should be relevant to students and of high-cognitive demand with multiple solution pathways.
1. Relevant
When tasks feel like they belong in students’ worlds—not just the math world—engagement and reasoning take off. Students bring rich knowledge from their families, communities, cultures, and interests. When we tap into that, we make space for math to feel more relevant, more inclusive, and more joyful.
Here are some ways this can look in today’s classrooms:
Make it about students in the class:
Use their names and interests in word problems and tasks.
Screen time and app usage:
Analyze and graph weekly screen-time data from apps they actually use.
Local celebrations and traditions:
Design seating or food portions for events like Día de los Muertos or Lunar New Year.
Sports stats & drafts:
Analyze player performance data from the NBA, WNBA, NFL, or other emerging leagues.
2. High-Cognitive Demand with Multiple Solution Pathways
High-cognitive-demand tasks give students the chance to truly do mathematics—not just follow steps. In these tasks, students:
Use complex, non-algorithmic thinking
There isn’t a single “right” method or a familiar set of steps to follow — students have to figure out their own approach.
Understand the math, not just do the math
They explore how ideas connect and why they work.
Keep track of their own thinking
They have to reflect, adjust strategies, and make decisions about their cognitive processes as they go.
Build on what they already know
They draw on experiences and prior learning, and choose what knowledge is useful for the problem in front of them.
Encourage them to examine the problem itself
They look closely at the information given, the limits, and the possibilities to decide how they might solve it.
Use considerable cognitive effort — and that can feel a little uncomfortable
Students may struggle at first, but that struggle is part of the learning. The path to a solution isn’t obvious, and that’s okay.
Examples:
Elementary - Addition within any number of digits:
Diva has______stickers. She then goes to the store and gets_____ more. How many stickers does Diva have now?
Middle School - Ratios and Proportional Reasoning:
A candy jar contains 5 Jolly Ranchers and 13 jawbreakers. Suppose that you had a new candy jar with the same ratio of Jolly Ranchers to jawbreakers, but it contains 100 Jolly Ranchers. How many jawbreakers would you have? Explain how you know.
High School - Exponential Functions:
In the movie Pay It Forward, a student, Trevor, comes up with an idea that he thinks could change the world. He decides to do a good deed for three people, and then each of the three people would do a good deed for three more people and so on. He believes that before long there would be good things happening to billions of people. At stage 1 of the process, Trevor completes three good deeds. How does the number of good deeds grow from stage to stage? How many good deeds would be completed at stage 5? Describe a function that would model the Pay It Forward process at any stage.
Practical tips for tailoring curriculum tasks to increase cognitive demand:
Remove step-by-step directions that give away the solution.
Ask for multiple strategies or representations.
Pose extension questions: “Does this always work?” or “Can you generalize this?”
Replace convenient numbers with less obvious ones that require reasoning.
Structuring Tasks
Although it’s important to select tasks of high-cognitive demand, tasks with high cognitive demand are the most difficult to implement well and are often transformed into less demanding tasks during instruction (National Council for Teachers of Mathematics, 2014.) How can we set up our classrooms so the task stays challenging and students stay engaged? Using the structures below can be a good place to start.
Structure 1: Random Groups of Three
Many teachers are turning to Peter Liljedahl’s ideas when looking for fresh, engaging ways to teach math. A practice Liljedahl recommends is to consider "How We Form Collaborative Groups in a Thinking Classroom." He recommends frequently forming visibly random groups. This means that the groups are random and students can visibly see they are random (a.k.a. the teacher is not choosing groupings and telling students they are random). He recommends forming groups of two in grades K-2 and groups of three in grades 3-12.
Liljedahl states, "We know from research that student collaboration is an important aspect of classroom practice, because when it functions as intended, it has a powerful impact on learning (Edwards & Jones, 2003; Hattie, 2009; Slavin, 1996).
How we have traditionally been forming groups, however, makes it very difficult to achieve the powerful learning we know is possible. Whether we grouped students strategically (Dweck & Leggett, 1988; Hatano, 1988; Jansen, 2006) or we let students form their own groups (Urdan & Maehr, 1995), we found that 80% of students entered these groups with the mindset that, within this group, their job is not to think. However, when we frequently formed visibly random groups, within six weeks, 100% of students entered their groups with the mindset that they were not only going to think, but that they were going to contribute. In addition, the use of frequent and visibly random groupings was shown to break down social barriers within the room, increase knowledge mobility, reduce stress, and increase enthusiasm for mathematics" (Liljedahl, 2024).
As students are getting trained to work in groups of three, it might be helpful to start with pairs and build up to three. This keeps everyone accountable, helps students get used to collaborating, and may limit classroom management challenges. You could also distribute roles to students in groups. Roles could include scribe, materials collector, quality control, or facilitator.
Structure 2: Vertical Non-Permanent Surface (VNPS)
Another practice from Peter Liljedahl that has gained momentum in math classrooms is the use of vertical, non-permanent surfaces—think whiteboards, glass, windows, or dry-erase boards. The goal is to get students standing, thinking, and sharing their ideas where everyone can see them.
Liljedahl’s research shows that when students work vertically and visibly, several important things happen:
Students get started faster—there’s no hiding behind a desk.
Thinking becomes more public, which encourages collaboration and communication.
Mistakes feel low-stakes because the work can be easily erased and revised.
Movement keeps energy high and helps sustain focus.
Teachers can quickly scan the room to see strategies, misconceptions, and progress.
In his words, “Vertical non-permanent surfaces support the visibility of thinking and the mobility of knowledge, allowing ideas to move easily around the room and between students.” (Liljedahl, 2024)
This practice shifts students away from individual “sit-and-wait” thinking and turns the classroom into a community of problem solvers. Students begin to look at each other’s work for inspiration, ask questions, and adjust their strategies—without the teacher being the sole source of help.
“Nothing we have tried has had such a positive and profound effect on student thinking as having them work in random groups at vertical whiteboards. Students were thinking longer, discussing more mathematics, and persisting when the tasks were hard.”
- Liljedahl, P., 2020, p. 58-59
Structure 3: Clear Classroom Routines
Having clear classroom routines for tasks, such as using a ‘launch, explore, discuss’ routine as well as a ‘5 Practices for Discourse’ routine can be helpful. More information about these can be found in my previous blogpost, “What is a Math Task and How Do I Use It?”
Structure 4: Notice/Wonder
When you introduce a task, try starting with an image that gives a glimpse into the story. For instance, if the problem is about boxing up cookies at a cookie shop, you could show a picture of a bakery with cookies being packed before looking at the task. As students look at the image, ask questions like, “What do you notice?” and “What do you wonder?” This simple step helps students connect to the situation, sparks their curiosity, and gets them thinking about the problem even before they see the full task.
Structure 5: Teaching Student Collaboration
Students need explicit routines and language for collaboration that they can use during tasks. Teachers can help by setting clear expectations for collaboration and giving students sentence frames to use while discussing.
Sentence stems and phrases:
1. Sharing Ideas
“I think ___ because ___.”
“My strategy is ___.”
“One way to solve this is ___.”
“Another approach could be ___.”
“I noticed that ___.”
2. Asking for Help
“Can you help me understand ___?”
“I’m stuck on ___, can you show me your thinking?”
“I’m not sure if ___ is correct; what do you think?”
3. Building on Others’ Thinking
“I agree/disagree because ___.”
“That makes me think ___.”
“Can you explain why ___ works?”
“I’d like to add ___ to your idea.”
4. Explaining Reasoning
“I know this works because ___.”
“I solved it this way ___, and it shows ___.”
“My answer makes sense because ___.”
“I used ___ strategy to solve this problem.”
5. Making Predictions or Generalizations
“I predict that ___ will happen if ___.”
“I think this always works because ___.”
“A pattern I notice is ___.”
“This can be generalized to ___.”
6. Problem-Solving and Reflection
“The problem asks me to ___.”
“A challenge I faced was ___.”
“One way to check my work is ___.”
“Next time I would ___ differently.”
“This reminds me of ___.”
7. Collaborative Roles and Task Management
“I will take the role of ___ (facilitator, scribe, materials manager).”
“Let’s decide who will ___.”
“I will listen while you explain ___.”
“Can we combine our ideas to ___?”
Small Shifts You Can Try Tomorrow
Show an image or object from the task to spark curiosity. Ask students, “What do you notice? What do you wonder?”
Use random groups of 3 (or 2) during a math task.
Give students vertical whiteboards to solve a task on.
Try not to over scaffold or provide too much instruction upfront.
Even small adjustments like these can transform a classroom from “doing math” to thinking and reasoning mathematically, helping students build confidence, persistence, and a sense of ownership over their learning. I’ve been implementing tasks for over 20 years but the facilitation and engagement has increased in the last four years after regularly implementing some of these structures, specifically the random groups of three and vertical non-permanent surfaces. However, not ALL tasks must follow these practices. Sometimes tasks require tables, sometimes groups of 2-4 work better, and sometimes working individually is necessary to allow the teacher to assess each student's understanding and promote confidence in all students. The key is to be intentional—try different structures based on the learning goals of the task and how students are responding.
Conclusion
Implementing high-quality math tasks is more than just giving students problems to solve—it’s about creating a classroom where thinking, reasoning, and collaboration are at the center of learning. By selecting relevant, high-cognitive-demand tasks and pairing them with thoughtful structures—like random groups, vertical surfaces, clear routines, and notice/wonder prompts—we set students up to engage deeply, share ideas, and build confidence in their mathematical abilities. Even small shifts in how we launch, facilitate, and support tasks can make a big difference. The key is to experiment, observe, and adjust so that every student has the opportunity to do meaningful math every day.
References:
Huinker, D., & Bill, V. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in K–Grade 5. Reston, VA: National Council of Teachers of Mathematics.
Smith, M., Steele, M., & Raith, M. L. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 6–8. Reston, VA: National Council of Teachers of Mathematics
Boston, M., Dillon, F., Smith, M., & Miller, S. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 9–12. Reston, VA: National Council of Teachers of Mathematics
National Council of Teachers of Mathematics. (2014). Principles to Actions: Ensuring Mathematical Success for All. Reston, VA: National Council of Teachers of Mathematics
Liljedahl, P. (2021). Building Thinking Classrooms in Mathematics, Grades K–12: 14 Teaching Practices for Enhancing Learning
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