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High-Order Tasks for Deep Thinking and Understanding
Think back to the last time you got a new phone. You probably didn’t get it up and running by reading the user’s manual. Instead, you used your prior knowledge of how other phones have worked. Sure, maybe the browser was a little different, or the camera had more features, but you could troubleshoot based on your previous experiences.
Your students do something similar when you assign high-order math tasks in your classroom. They think about what they already know and test out different ways to complete the task until they identify one that works. In the process, they build problem-solving skills, collaborate successfully with each other, and exercise their creativity.
Margaret Schwan Smith and Mary Kay Stein, authors of the influential "Mathematical Tasks as a Framework for Reflection," explain that these high-order tasks ask students to connect the “what?” to the “why?” They also prioritize productive struggle and create the kind of deep mathematical understanding that can only come from applying knowledge instead of just recalling or memorizing it.
MATHbook, the write-in, consumable textbook component of the our Middle School and High School Math Solutions, helps you bring these high-order tasks to your classroom so you can help your students become thinkers who don’t just do the math–but understand it.
Here are five ways MATHbook promotes high-order tasks for math classrooms.
1. MATHbook Has Compiled Tasks, so You Don’t Have To
Teachers committed to consistently providing their students with rich tasks spend a lot of time gathering and reworking them for their particular groups of students. Even before factoring in the million other things teachers do, it’s a lot.
MATHbook has already gathered high-order tasks that fit seamlessly with the concepts and skills of every lesson. These tasks spiral in students’ knowledge from past units while subtly preparing them for what’s to come. You’ll have a clear sense of how tasks fit in with learning goals and will be able to articulate this to your students.
MATHbook’s Engage, Develop, Demonstrate instructional model, built from both cognitive science research and decades of classroom experience, incorporates several types of tasks into single lessons, emphasizing high-order tasks.
In this example, students are initially engaged in the open-ended activity of creating a variety of rectangles based on their perspectives. This high-order task supports the development of conceptual and procedural understanding, with subsequent tasks to allow them to formalize their thinking. When tasks are complete, students demonstrate what they have learned through summary and discussion.
This example is just one of hundreds. Every MATHbook lesson has students actively completing tasks to learn math.
2. Tasks in MATHbook Are Consistently High-Order
Landing on the right mix of tasks to address all levels of student need can be a daunting task. That’s where MATHbook comes in. It contains a carefully selected mix of tasks, with the majority being high-order, so you can focus on planning and preparing for engaging students with the tasks.
Moreover, you can use the “Learning Goals” section in every lesson to remind your students why they are completing a particular task. Understanding the goal for learning is a great motivator and encourages students to reflect on what they know and need to keep working on. With this metacognition in play, all tasks will require students to think deeply.
3. MATHbook Helps You Discern What Tasks Are Most Appropriate
Let’s say you have oodles of tasks at hand. It can still be unclear when in the lesson to incorporate them for maximum effectiveness or even which kind of task to use at a given time. Tasks are just one part of your toolbox, and it can be hard to know when a task is the best tool for the job versus direct instruction, pair-and-share, online skill practice, or even journaling.
MATHbook uses instructional design research to determine when, where, and which tasks are most appropriate. Some lessons start with a task, and some begin with direct instruction and incorporate tasks later. Some lessons have three connected mini-tasks, while others revolve around one large task.
The point is that tasks aren’t selected randomly. MATHbook considers the objectives and flow of every lesson and asks questions like, “What can we do to energize or consolidate learning here?” “Is there a different way to have students explore the same concept?” or even “What can we do to get students on their feet and talking to each other?” When you use MATHbook, you can rest assured that every piece of your lesson, tasks included, works logically with all the other parts.
4. MATHbook Supports Don’t End With Tasks
As much as we love them, just having students do tasks is insufficient math instruction. The scaffolds that surround tasks matter too. Cognitive research shows that consolidation is essential for learning, specifically for ensuring that concepts enter long-term memory where they can be retrieved and built upon at will.
MATHbook’s “Questions to Support Discourse” sections ask students to explore challenging concepts with each other, ask questions about what they don’t yet understand, and explain what they have learned. All these actions help them consolidate new knowledge.
In MATHbook’s “Talk the Talk” sections, students apply what they have learned to solve real-world problems. This application lets students see how they can use math in their daily lives and enables them to demonstrate that they understand new skills and concepts well enough to use them in a different context. When students can do this, it’s a safe bet that they have consolidated knowledge and will be able to use it again in future tasks.
5. MATHbook Makes High-Order Tasks More Accessible
While every teacher wants to bring challenging mathematical tasks to their classroom, doing this without underlying support can be tricky. Sometimes, even the best tasks get watered down and become less rigorous during instruction.
MATHbook makes it easier for teachers to implement high-order tasks by building in all the support you need to introduce, guide, and debrief. The “Questions to Support Discourse” sections provide opportunities to ask students additional questions about their reasoning or thought process. These questions further students’ exploration of complex concepts with each other, ask questions about what they don’t yet understand, and explain what they have learned. All these actions help them consolidate new knowledge.
MATHbook’s many discussion, writing, and reflection opportunities also help create a safe learning environment where students feel comfortable pushing themselves, taking risks, and supporting each other.
MATHbook Puts High-Order Tasks Within Your Reach
Varied and thoughtful high-order tasks can help teachers create the active, safe, and student-centered spaces we all strive for. But knowing how, when, and where to implement relevant and effective tasks, day in and day out, can be daunting.
MATHbook was designed with the philosophy that tasks shouldn’t supplement math learning but anchor it. In this text, you’ll find high-order tasks inviting all your students to approach math with curiosity and the conviction that they, too, can be math people.
Before joining Carnegie Learning's marketing team in 2022, Karen spent 16 years teaching mathematics and social studies in Ohio classrooms. She has a passion for inclusive education and believes that all learners can be meaningfully included in academic settings from day one. As a former math and special education teacher, she is excited to provide educators with the latest in best-practices content so that they can set all students on the path to becoming confident "math people."
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Before joining Carnegie Learning’s marketing team in 2021, Emily Anderson spent 16 years teaching middle school, high school, and college English in classrooms throughout Virginia, Pennsylvania, California, and Minnesota. During these years, Emily developed a passion for designing exciting, relatable curricula and developing transformative teaching strategies. She holds master's degrees in English and Women’s Studies and a doctorate in American literature and lives for those classroom moments when students learn something that will forever change them. She loves helping amazing teachers achieve more of these moments in their classrooms.
Explore more related to this authorMATHbook, the write-in, consumable textbook component of the Carnegie Learning Math Solution, helps you bring high-order tasks to your classroom so you can help your students become the kind of thinkers who don’t just do the math–but understand it.
Karen Sloan, Math Teacher and Intervention Specialist of 16 Years
