The internet can be a great resource for finding math tasks at every level of cognitive demand. While elementary students need exposure to tasks at all levels—lower as well as higher—an emphasis should be placed on those at the higher levels. That means we need the skills to evaluate what is and isn’t cognitively demanding.
To determine the quality of online activities, my research partners and I used Margaret Schwan Smith and Mary Kay Stein’s 1998 Task Analysis Guide (TAG), which consists of four distinct levels of cognitive demand: memorization, procedures without connections, procedures with connections, and doing mathematics.
With memorization, critical thinking isn’t necessary, no connections are made to understanding why the answer works, and procedures are bypassed. This type of task can look like recalling facts. Procedures without connections are algorithmic; students come up with an answer without making connections to other math concepts and aren’t required to explain their work. Problems that follow simple procedures, like requiring the U.S. standard algorithm for addition, fall under this category. Memorization and procedures without connections are low cognitive demand tasks because they don’t require a lot of thinking.
Teachers often present visual diagrams or manipulatives like Unifix cubes or base 10 blocks to solve math tasks that are procedures with connections, which allow students to approach the problem from multiple angles. These problems use procedures, such as the partial product algorithm for multiplication, to help students understand why the answer works as opposed to only knowing how to find the answer.
The highest level problems, doing mathematics, require non-algorithmic thinking, demand self-monitoring, and allow for multiple strategies to be used—students at this point are exploring mathematical concepts.
Procedures with connections and doing mathematics are high cognitive demand tasks because students need to make connections, analyze information, and draw conclusions to solve them, according to Smith and Stein.
Math Tasks Need to Be Chosen Critically
In order to present elementary students with problems at every cognitive level, teachers must be critical consumers of the resources available. In our research, the following points helped my colleagues and me evaluate the cognitive demand and quality of online tasks.
Age matters. The level of cognitive demand can change depending on the age of the children a problem was created for. For example, completing a worksheet of basic one-digit addition problems would be coded as memorization for a fourth grader, who is expected to have them memorized (even more so if the student is being timed), but it would be considered doing procedures without connections for kindergarteners, who are just learning what it means to add two parts to make one whole.
If you’re looking for high cognitive demand tasks, a resource that meets any of the following criteria can be considered a procedure with connections; to be classified as doing mathematics, there must be multiple ways to solve the task:
- The problem usually involves manipulatives (e.g,. 10 frames, base 10 blocks, number lines, number grids).
- There are directions calling for students to provide explanations of how they found the answer (through models, words, or both).
- There is a high level of critical thinking required. For example, students decide how to tackle a problem that can be solved in more than one way, make real-world connections to the math, or explain their mathematical thinking.
When evaluating a math task, teachers should also evaluate any images that accompany it. Is an image included solely for decorative purposes, or does it have a functional role in solving the problem? Images with functional roles include clock faces, 10 frames, and graphs. If an activity has a decorative image, it is significantly more likely to be a low cognitive demand task; if it has a functional image, it is much more likely to be coded at a high level of cognitive demand. While an activity might be popular because of its decorative, cute images, visual appeal does not correlate with high levels of cognitive demand. It’s important to focus on the content rather than the art.
Where to Find Cognitively Demanding Math Tasks
You have a notably higher chance of finding math activities at a high level of cognitive demand on websites where resources are reviewed before publication as opposed to sites like Teachers Pay Teachers or Pinterest where anyone can post. The following websites publish reviewed resources:
- Illustrative Mathematics allows teachers to search for tasks based on content standards by domain or grade for K–12 (free).
- EngageNY is a set of pre-K to grade 8 English language arts and mathematics curricula created by the New York State Department of Education. It also has math curricula for higher grades—Algebra I and II, Geometry, Precalculus, and above (free).
- NRICH, run by the University of Cambridge in England, provides a library of resources and curriculum-mapping documents for students ages 3 to 18 (free).
- youcubed, founded by Stanford University mathematics education professor Jo Boaler, provides high-quality math tasks that can be searched for by grade (K–12) or topic. Some tasks have been created by the researchers who run youcubed, while others are drawn from a variety of sites, including NRICH (free).
- Illuminations is an online resource available through the National Council of Teachers of Mathematics (NCTM) that provides materials based on both NCTM standards and Common Core State Standards for grades pre-K to 12. Access requires an NCTM membership (cost: $49 to $139 a year).