Promoting Mathematical Practices in Online Learning
By Andrew Senkowski
Mathematical Practice 1 (MP1): Make sense of problems and persevere in solving them.
Making Sense of Problems
One of the greatest benefits of online education is the ability to easily provide meaningful synchronous and asynchronous opportunities to promote students’ abilities to make sense of problems and persevere in solving them.
Allowing for Think Time
One particularly beneficial way to promote this standard of mathematical practice is to use the Flipped Classroom Model:
Direct instruction occurs in the form of a pre-recorded instructional video.
Live classes (such as in Zoom) would focus on practice or application of the skills learned in the video.
In pre-recorded videos, teachers can build in cues for students to pause and practice individual think time before they see what comes next. This scripted pausing allows students time to do three things:
Understand the context of the problem
Determine what prior knowledge gives them an entrypoint to solving the problem
Determine a possible solution path
Pre-recorded videos can also be used to preview a larger scale scenario or problem that will be covered further in a live class or in an online group discussion. Prompting the students to consider an open-ended problem in such a way allows them to have ample individual think time to make sense of a problem and determine a possible solution path prior to engaging in discussion with others.
Research Tools
In an online environment, students also have the immediate tools to research a new context to better support their understanding of an open-ended problem. Search tools give them the ability to research which assumptions they will need to make in order to even begin to make sense of and solve a more open-ended problem.
For example, say the problem involves determining which commission-based sales job one should take based on little given information. Students can research the average price of the goods sold, average hourly wages for salespeople for various goods, typical sales per week/month for those goods, etc.
Visualizing and Modeling Tools
Online learning also supports perseverance in problem solving through its wealth of easily accessible modeling tools to help students “conceptualize and solve a problem.”
From Google to SketchToy, students have the ability to create their own models.
Some websites offer pre-built interactive models. For example, this simulation from PhET can support integer operations problems and this Fraction Wall from Visnos can support problems involving equivalent fractions.
Others offer a mix of pre-built interactives, such as this Quadratic Equation interactive from Desmos, as well as the option to create your own tools to share out with your students.
Online Graphic Organizers
Math-specific graphic organizers are excellent tools for making sense of a problem and planning out a specific approach for solving. Though not graphic organizers are not unique to online education, creating online versions of graphic organizers may increase the ease with which students can utilize them. All graphic organizers such as Using 4 Block and others can be converted to Google Docs or Google Slides for easy editing and duplication as needed by students.
Step-by-Step Graphic Organizer (Google Doc)
Using 4 Block - Frayer Model (Google Slides)
Feedback for Self-Evaluation
Online learning also provides great opportunities to “monitor and evaluate their progress and change course if necessary.” This standard of mathematical practices is all about thinking about your thinking - i.e. metacognition.
Peer and Teacher Feedback
Feedback from peers and the teacher is a crucial part of students being able to monitor and evaluate progress. In an online environment, there are almost limitless opportunities for sharing feedback that can push students to think critically about their progress on a problem. A few examples include:
Student collaboration on more complex problems can be easily built into the online course via discussion boards.
Live classes such as in Zoom can utilize breakout rooms for group discussions on solution paths.
Tools such as Nearpod that allow for live, anonymous responses can give students a low-stakes opportunity to get feedback on their ideas.
Tools such as Whiteboard.fi allow teachers to actually see students working out a problem during live class on a digital “whiteboard.” This allows them to provide immediate feedback either individually or as a whole group.
Practice Tools and Feedback
Beyond teacher-created opportunities for feedback, there are many practice-focused Open Educational Resources (OERs) that help students to make sense of problems and push them to “plan a solution pathway rather than simply jumping into a solution attempt.” One such resource is IXL.
To see an example in action, try out one problem from Multi-Step Problems with Unit Conversions (Algebra 1) and answer it incorrectly.
You will find that the feedback you receive provides valuable step-by-step insights for planning your approach to the next problem.
Regular exposure to such detailed, immediate feedback while practicing skills greatly promotes sense-making thought processes.
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.