There is a wealth of literature that outlines the benefits of the social construction of learning. Researchers have found that learning is optimized when students share learning opportunities with their peers. Incorporating teaching strategies like Cooperative Learning and Think/Pair/Share give students a chance to do just that.

In the image shown, my students are working together to measure the circumference and diameter of a table. After doing this for several circular objects (also shown in the picture), they created a graph and found that the constant of proportionality between the two dimensions is approximately Pi. This activity demonstrates Cooperative Learning because each student in the group was asked to measure a different sized circle. Without all students' participation, the group would not have enough points to draw a line of best fit on the graph.

Given the diversity in our schools, it is vital to incorporate research-backed strategies for supporting students who are learning English as a second language. In my classroom, I frequently facilitated card sorts in order to help students make sense of concepts and practice using new vocabulary. I also provide sentence frames to aid with language production. In doing so, students' cognitive load is shifted from forming sentences to correctly applying concepts. I like these strategies because they are useful for multilingual learners, but they may also be used by others in the classroom.

In the slide above, I include sentence frames with the card sort instructions in the spirit of Universal Design for Learning. In this particular activity, students sorted problem statements involving circular objects (e.g., "How much fabric is needed for a round table cloth?") that could be solved by finding either circumference or area.

In order to give students a strong foundation for procedural fluency in mathematics, teachers must first provide an opportunity to develop conceptual understanding. That is, before students learn how to apply a procedure, they should understand why they are doing it. Research has shown that traditional mathematics pedagogy has often failed at this task, and is in part the reason for students' poor performance in the subject. Once students have conceptual understanding and procedural fluency, they are able to reason with their new learning in "real-world" applications.

Sometimes, however, inviting students to consider contexts with which they are already familiar can help them develop conceptual understanding. I used the thermometer chart, shown above, to introduce our unit on rational number arithmetic. Before students ever added negative and positive numbers, we explored real-world situations in which negative numbers make sense—namely, elevation and temperature.

The term "low-floor, high-ceiling" in the context of mathematics pedagogy describes activities in which all students have an opportunity to learn. When an activity has a low-floor and high-ceiling, it is highly conducive to partner and class share-outs because every student will have something to say. Two popular prompts that demonstrate this trait are "What do you notice and wonder?" and "Which of these does not belong?" These are excellent because there is no incorrect answer; students can share whatever ideas they want.

In the image above, students explore one representation of adding rational numbers. I asked them which pair of arrows does not belong, and why. Their reasoning ranged from the direction of the arrows to the fact that the arrows represented addition/subtraction. Clearly, the latter is more mathematically sophisticated, but all students were able to come up with a reason.