It’s human nature to categorize items by common characteristics. This instinct provides an opportunity to build students' understanding of underlying mathematical structures more deeply by, say, having students classify shapes in geometry or identify the number of solutions in algebra.

But there are many ways to incorporate card sort activities into your mathematics classroom. Here are a few:

1. Snip and sort

There is tactile engagement in physically handling pieces of paper and sorting them into different categories. One example would be to have a piece of paper divided into three columns with the labels one solution, no solution and infinite many solutions. Then students cut out slips of paper with linear equations in one variable to sort into the three columns. You can find various concepts appropriate for a card sort activity if you think about your curriculum. What is one mathematical concept that comes to mind?

2. Digital options

A digital card sort can help students spend more time on tasks. Learners spend less time gathering materials, physically cutting or searching for missing pieces of paper on the floor. There are a variety of tools that teachers can use to create or implement digital card sorts into their daily lessons.

• Flippity manipulatives provide a Google Sheet template to create an online published manipulative. The Flippity manipulatives have a basic yet versatile design. The customizable template generates draggable terms over a background field. You can change the background color of the terms and the background of the sorting field.
  
  When I created a Flippity to sort systems of linear equations by the number of solutions, I gave the categories names (no solution, infinitely many solutions, and one solution) a background color to distinguish them from the systems. I also chose the three-column background. With this design, students can drag one category name into each column and then sort the systems of equations into the appropriate columns. It is easy to get started using Flippity's step-by-step directions.
   
• Desmos is an open educational resource that pairs an online graphing calculator with interactivity activities designed to build students' mathematical thinking. Among the engaging activities that Desmos provides are several card sort activities. In the Desmos activities, the card sorts are often a part of a larger activity meant to help students think mathematically and critically about the concept. You can search for card sort activities on the site and assign them to your class using Google Classroom or a shareable link.
   
• Classflow, free online collaborative software, provides a tool to create a sorting activity called Categorize, which is ideal for one-to-one device classrooms because students use a class code to access the activity and complete it on their devices. Categorize has the added functionality of using images to be sorted, which can be helpful in geometry card sorts. In addition, the categorized activity can tell students if they have sorted an item correctly or not. You can sign up for a free account to access tutorials and create your own Categorize activity.

3. Small group or pair work

Card sorts provide structure for meaningful math talk. Assigning card sorts in a group of two or three provides students an opportunity to explain their reasoning and assess the reasoning of others. I suggest students take turns placing (or dragging) a card into the appropriate category. As they place the card, they should explain why they place it in the chosen column. When finished, they should have a chance to compare with another group engaging in healthy dialogue about any cards they sorted differently.

4. Student-created examples

Expand the card-sorting activity by including blank cards that students can us to create examples to represent each category. Student-created examples allow students to engage in higher-order thinking and demonstrate their understanding of a skill. Want to take it further? Have students create a card sorting activity to demonstrate mastery of an objective.

5. Vocabulary building

By sorting cards into examples and nonexamples, students can build their definitions for a mathematical term instead of a teacher just presenting students with vocabulary and asking them to memorize it. In the article "Sorting out definitions," Erin Baldiger and colleagues detail a process in which students can define the term linear function by working in small groups to sort cards of examples and nonexamples while discussing their reasoning. After sorting the cards, students participate in a whole-class discussion. During this teacher-facilitated discussion, students share the reasoning for sorting the example and nonexample cards. Once all the cards are sorted, students construct and revise a linear function's definition. This approach demonstrates the potential of using a card sort to help students build vocabulary through an inquiry-based approach.

6. Open card sorts

In an open card sort, students create their own concepts and categories allowing for an exploratory mode of learning. Instead of giving students the categories, students develop their own by examining the cards and looking for patterns. These tasks have the added advantage of supporting computational thinking (CT) by helping students practice pattern recognition. In step two of the CT process, students use pattern recognition to look for similarities between the smaller pieces of the problem they found during the first step, decomposition. When you ask students to examine cards and create categories, students must engage in pattern recognition. When developing an open card-sort activity from your classroom, consider a card-sort activity with more than one way to sort the cards. This can lead to a class discussion about pattern recognition.