Promoting Mathematical Practices in Online Learning
By Andrew Senkowski
Mathematical Practice 2: Reason abstractly and quantitatively. (Common Core State Standards Initiative)
Abstract reasoning and quantitative reasoning are both hugely important abilities that must be developed in every student. The emphasis of mathematical practice 2 (MP2) is not to develop these separately but to develop them in tandem. Students must be able to toggle between abstract and quantitative reasoning and do so with ease. They must decontextualize and contextualize mathematical information regularly in problem solving and even do this multiple times in the course of solving a single problem.
As with any mathematical practice, you would take the same basic approach in an online setting as you would for in-person learning. You would look for opportunities to guide students between abstract and quantitative representations and to make connections between these representations. This could be as simple as students completing a problem in which they write an equation to represent a situation, solve that equation, and then interpret the solution in the context of that situation.
For many math teachers this is a natural part of their pedagogical approach, however when thinking about this from an online learning perspective, you want to ask the following questions:
In what ways is this mathematical practice already a part of my pedagogical practice?
Could I incorporate this practice into my live classes, online assessments, and asynchronous online activities?
Can I adapt in-person learning activities or assessments to fit within a synchronous or asynchronous online learning situation?
How could I leverage online educational tools and platforms to promote this practice in a new and engaging way for my students?
The following examples will jump start your thinking about incorporating mathematical practice 2 in an online learning environment. Keep in mind that these activities may seem familiar. Maybe you have even done one of them with students already. The intent is to provide a glimpse into how effective teachers leverage online learning tools and platforms to create equally, and maybe even more, meaningful and engaging opportunities for students to grow in this practice.
Elementary Example
Although reasoning abstractly and quantitatively may sound like a lofty goal for elementary students, it’s an essential part of mathematical thinking and completely achievable at the lower levels. This could be as simple as having students translate a visual diagram into numerical expressions (or vice versa) and explain the connection.
One simple yet powerful example of an activity for promoting mathematical practice 2 was shared by the Education Development Center (Elementary Education at EDC). In this example, students are given a diagram involving tiles and challenged to write numerical expressions to describe the number of tiles. The open-endedness of this activity allows students to move between the concrete diagram of tiles to abstract representations.
Just as the activity itself is open-ended, how you approach this in online education is also open-ended. Here are two examples to consider.
Example 1
Whole Class Discussion
During a live online class, display a tile diagram and prompt students to share their numerical expression representations. Chat features could be used, but tools that collect responses anonymously may be best. For example, students can be given editing access to a Google Slide document (such as this one) to add their responses as a textbox on the appropriate slide. Once this is completed, the teacher can guide a discussion on how the numerical expressions connect to the diagram.
To incorporate a different skill set, students could examine expressions for a tile diagram and sort them based on if they are a true representation of the number of tiles (such as on the second slide of this Google Slides document).
Example 2
Create-Your-Own with Collaboration
Have students create their own diagram of color tiles using free online manipulatives such as Didax Unifix Cubes or GeoGebra Color Tiles. Then ask them to share their diagrams via discussion boards so that other students can try to create numerical expressions for their peers' creations.
This could also be reversed where each student shares a numerical expression and their peers create tile diagrams that could represent the expression in a discussion board.
Secondary Example
A commonly used activity that promotes Mathematical Practice 2 is the Banquet Table Problem (CollectEdNY). This activity can be adapted for multiple levels to cover multiple standards while allowing for great opportunities to toggle between abstract and quantitative reasoning.
The basic premise of this problem is that four people can sit around one table. When two tables are placed next to each other, then a total of six people can sit together. And the pattern continues on.
There are many great ways to approach this activity from live whole class discussions to asynchronous, self-guided exploration via course assignments.
These Google Slides show one approach. Some of the benefits of Google Slides are:
They can be created to be more interactive
Copies can be easily created for individual students or shared among groups of students
The format of slides can help break down this activity of exploring multiple representations into smaller chunks.
Through these slides, students get to use online tools, such as Desmos, and interactive elements of Google Slides, such as shapes and tables, to create multiple representations and connect them back to the original context. They can contextualize and decontextualize as they consider questions that arise such as:
How does the equation connect back to the real-world pattern I noticed with the number of tables and seats?
Does a solution to my equation make sense if I plug in an odd number for y?
Do odd values and negative values make sense in this context? Why or why not?
Although this activity may just look like an online worksheet, its application can vary greatly based on one’s desired methodology:
Students could work asynchronously on these Slides on their own individual copy and submit their completed Slides via a learning management system or via an emailed link.
Slides can be displayed by the teacher in a live class and students can work alongside the teacher on their own copy and share out their responses using a chat feature (or interactive slides such as Nearpod) or by volunteering to share their own screens.
Breakout rooms in a live class can be utilized to have students collaborate on the activity and create a team copy of the Slides.
In the learning management system, teachers could assign specific groups of students to complete a specific slide so that students focus on just one representation. Then a discussion board can be utilized to have students share their experiences with their assigned representation and how it connected to the original problem.
Andrew is a former Math educator who has worked in both brick & mortar and online schools over the past decade. He currently lives and works from Pennsylvania as a Content Editor Manager for a national curriculum company, but stays connected to online teaching as an Instructional Coach for SYS. He is passionate about decreasing “math anxiety” for students and figuring out ways to take math education to new heights in the virtual setting.