Chapter 4: Digital Learning in Mathematics - California Digital Learning Integration and Standards Guidance

Chapter 1 provided a wide range of general recommendations associated with effective teaching in a digital learning environment. This chapter includes additional strategies for a focused subset of topics most relevant to mathematics instruction and aligned to the ISTE Standards for Educators and National Standards for Quality Online Teaching (which were introduced in Chapter 1).

Additionally, the draft Mathematics Framework with an adoption target date of May 2022, devotes a chapter to the intersection of mathematics, technology, and distance learning. The Mathematics Framework emphasizes how technology “supports both the learning of meaningful mathematical content and the fostering of the productive habits of mind and habits of interaction embodied by the Standards for Mathematical Practice (SMPs).” The chapter expands upon three principles for incorporating technology in mathematics instruction, each of which are introduced and explained briefly below:

Principle 1: Strategic Use of Technology in a Learning Environment Can Facilitate Powerful Learning of Mathematics: The first principle focuses on the intentionality behind the use of technology for mathematics to contribute to students’ “learn[ing], experienc[ing], communicat[ing], and do[ing] mathematics. It includes access (technology availability for students and educators), usage (technology’s use in the learning process), and skills (the students’ and educators’ ability to use the technology in meaningful ways).
Principle 2: Support for Teachers of Mathematics Accompanies Use of Learning Technologies: The second principle concentrates on the need for robust and continual professional learning and support during a technology adoption process. Additionally, the school and/or district needs to provide time for teachers to learn how to use the technology, including ongoing and content-specific training, such as for mathematics.
Principle 3: Learning Technologies Are Accessible for All Students: The third and final principle emphasizes the need for access and equity to learning technologies, including devices and digital tools specific for mathematics, for all students, including foster youth, youth experiencing homelessness, ELs, and students with IEPs and 504 plans. This access can include devices and the internet.

The topics included within each of these principles are covered briefly in this chapter within the context of mathematics and more generally in Chapter 1.

When implementing the strategies in this chapter, educators are further encouraged to create ongoing partnerships with family members and caregivers who help their students with their learning. This cultivates a robust support system for students as they work through assignments and problems that may be challenging. Educators might invite family and caregivers to online office hours and/or one-on-one meetings with students to identify interventions and resources and further strengthen the support system.

Preparing and Supporting Teachers for Digital Teaching

Professional Responsibilities

As referenced in Chapter 1, both the ISTE Standards for Educators and National Standards for Quality Online Teaching emphasize the vital importance of teachers collaborating with colleagues to support one another as they create authentic learning experiences that leverage technology. As is true for any subject area, this pedagogical training should also be content specific, as mentioned above, so that teachers can leverage those technologies to support their teaching of mathematics and their students’ learning (National Council of Teachers of Mathematics, 2015).

One way to encourage collaboration in mathematics might be to develop a supportive cadre/group of teachers, forming a formal professional learning community (PLC), at the regional, district, school, or grade level. By leveraging various digital tools, such as collaborative documents, social media, and video conferencing software, the PLC might be responsible for dividing content based on areas of expertise, creating activities, and sharing ideas for integrating technology.^1,2

Teacher Presence

In Chapter 1, both the ISTE Standards for Educators and National Standards for Quality Online Teaching emphasize the use of digital tools to foster teacher-student relationships that build students’ sense of belonging to the school community. This focus on relationships is especially important in distance learning, where teacher presence is critical to helping students feel best supported for their success. This does not suggest that teachers have to be connecting to students synchronously all of the time. Instead, it can be achieved through a personalized note, quick feedback on an assignment, a private message of encouragement during group time, or email messages.

In the mathematics learning environment specifically, digital tools that allow teachers and peers to communicate feedback through video and/or audio help make the learning experience much more personable than purely text-based feedback. Additionally, videos allow students to stop and replay the content if they missed information the first time they heard it.

Digital Citizenship

Digital citizenship is one of the core components of both the ISTE Standards for Educators and National Standards for Quality Online Teaching, and it reminds teachers to model, guide, and encourage legal, ethical, and safe behavior related to students’ technology use. Chapter 1 presented the DigCitCommit competencies as a framework that allows teachers to consider strategies for teaching and reinforcing a comprehensive set of digital citizenship skills.³

Addressing digital citizenship through the lens of mathematics specifically can provide students an opportunity to reinforce the “Inclusive” competency of the DigCitCommit framework (“I am open to hearing and respectfully recognizing multiple viewpoints, and I engage with others online with respect and empathy.”). For example, collaborative tasks related to mathematical ideas that necessitate investigation provide an opportunity for students to learn how to interact in respectful ways with each other and provide productive feedback to peers.

Furthermore, mathematics can be an ideal content area for practicing the “Engaged” competency of the DigCitCommit framework (“I use technology and digital channels for civic engagement, to solve problems, and be a force for good in both physical and virtual communities.”). For example, teachers can guide students in discussing and reflecting on how investigating mathematical ideas, asking questions, and making conjectures in mathematics may help in solving local and global issues around them.⁴ This activity may be further facilitated by using video conferencing tools to connect with experts in the field who are using mathematics to solve these local and global issues.

Teachers will find additional ideas for teaching digital citizenship through the lens of mathematics from resources like Tech InCtrl that provide lesson materials and ideas.⁵ Refer to Digital Citizenship in Chapter 1 to learn about more strategies.

Data-Informed Instruction

Both the ISTE Standards for Educators and National Standards for Quality Online Teaching emphasize the importance of teachers’ ongoing use of data to inform their instruction. In the context of mathematics, there are many ways digital tools can be used for formative assessment in online and blended learning environments to determine pedagogical effectiveness, understand support needs for students, inform and individualize instruction, and accelerate learning. Chapter 2 shares many of these approaches under Assessment for Learning, which explores frequent, formative assessments that may be used to inform instruction. Some examples for mathematics include, but are not limited to, the following:

Students can meet in online work sessions where the teacher might give students just-in-time support, such as small lessons using Edpuzzle [a video-based lesson software], that provides data for the teacher as students collaboratively work through problems. As the students are working through problems, teachers can check in on students’ learning goals to identify students in need of additional support.
Students might also meet with their teachers one-on-one to discuss their progress related to a specific concept, engage in a discussion-based assessment via video conferencing, and/or discuss their next steps.
Students might write out their mathematical explanation as they solve selected problems in an online shared document. This enables teachers to closely monitor student progress and provide ongoing, supportive feedback and notes to bolster students’ motivation as they continue their problem-solving process. It also provides a space for students to share if they are asked to present.
Teachers can create a quick check-in survey using an online survey tool, such as Google Forms or Zoom polls, to ask students where they are with their understanding of concepts. Based on that assessment, the teacher can adjust their instructional approaches (e.g., pace of instruction).

Sample rubrics to assess and give feedback to students around their strengths and areas for growth in mathematics are included in Appendix D: Mathematics Rubric Examples. The rubrics connect the Drivers of Investigation to both the big ideas and the standards for mathematical practice (SMP).

Refer to Data-Informed Instruction in Chapter 1, as well as Chapter 2, to learn about more strategies.

Designing Meaningful Digital Learning Experiences

Aggregating Quality Synchronous vs. Asynchronous Instructional Time

Both the ISTE Standards for Educators and National Standards for Quality Online Teaching call on educators to design learning experiences that are best-suited for the specific learning environment. Teachers determine which information is better conveyed through real time, synchronous instruction with direct teacher-student interaction, and which information is appropriate for asynchronous instruction without direct teacher guidance or interaction.

As described in Chapter 1, when teaching synchronously, teachers are advised to present critical content information as concisely as possible after students engage actively in a task, reserving the remaining time for active learning activities that reinforce the content presented. In the context of mathematics, these might include the following:

Students can practice solving an authentic problem independently during asynchronous time and then join a live Number Talk and discuss ways to solve the problem (synchronously), allowing them to share a variety of perspectives on approaches. To motivate and increase engagement, consider using breakout rooms in Zoom for students to collaborate in small groups and then transition back to the whole class to share and compare strategies.
Teachers can facilitate a live discussion with an expert, such as a mathematician who works at NASA, around a math topic to elicit curiosity and provide student-centered, mathematical experiences. Students can ask the expert questions and engage in a discussion about something tied to math that they are passionate about.⁶
Teachers can build math activities using digital tools, such as Desmos [a mathematics lesson building software], NearPod [interactive learning platform], and Pear Deck [formative assessment platform], in which students develop conceptual understandings and reflect with their peers on what they are learning together.

It is important to note that educators can be mindful of how groups are formed. Catalyzing Change in Early Childhood and Elementary Mathematics states, “Challenge ability grouping and ensure all children have access to mathematics learning environments where each child interacts with, learns from, and contributes to shared and deep mathematical understanding within a classroom community” (p. 125). While asynchronous learning activities can include tasks and exercises students review in order to prepare for synchronous time, it can also leverage active learning opportunities, including, but not limited to, the following:

Students can record themselves using Screencast-O-Matic [screencasting tool] as they work with hands-on or virtual math manipulatives or simulation. While using the screencasting tool, they can speak about their understanding of what actions they are taking, what they are learning, and why it is important.
Students can create a digital infographic based on data they have analyzed for a project. The process of creating the infographic can help students decipher what information is the most critical to share in a presentation they give to the class. Tools, such as Google Slides [online presentation tool], help students create infographics. Activities such as this provide students with an outlet to creatively visualize data and apply data literacy and data science skills.
Students can create a graph using Google Sheets [online spreadsheet] to interpret and visualize data they have analyzed for a group project. Students can then share that sheet with peers who are working collaboratively on the project. Students can also write about what the visualization suggests within the context of the problem being solved.
To assist teachers in monitoring student learning, students can take a daily self-assessment using Google Forms that will help the teacher know what concepts need to be covered for future learning activities.
Students can use Geogebra [a modeling software for algebra and geometry] to visually model a problem to help provide an alternative representation of what they are learning.
Teachers can use Desmos [a mathematics lesson building software] to provide interactive math activities for students. See sample activities focused on how to land a plane, where students can “plot the linear equation of a plane so that it lands on a runway” in Chapter 10 of the Mathematics Framework.

Universal Design for Learning

The ISTE Standards for Educators and National Standards for Quality Online Teaching emphasize that educators must design digital learning experiences that take individual learner differences into careful consideration. This includes leveraging the Universal Design for Learning (UDL) framework, introduced in Chapter 1, to help support all learners with accessible learning experience design.

LD OnLine, a national education service organization working in partnership with the National Joint Committee on Learning Disabilities (NJCLD), shares a number of key technology-empowered approaches grounded in the UDL framework that teachers can use. For instance, digital tools allow mathematics teachers to provide multiple means of representing concepts, which are especially helpful for students with difficulty processing language, navigating spatial concepts, or retaining mathematics-related facts. Such tools may include, but are not limited to, digital manipulatives, videos, pictures, simulations, and other graphic representations.

Other suggested strategies for integrating the UDL framework in mathematics contexts include:⁷

building computational fluency, such as counting with objects rather than using drill and skill approaches (e.g., using physical objects at home that students can then take video of as they count);
converting symbols, notations, and text using text-to-speech software, which is typically built into platforms;⁸
building conceptual understanding by collaborating with others through video conferencing tools and digital whiteboards;
making calculations⁹ and creating visual mathematical representations through graphing technologies; and
using graphic organizers to help students depict and connect different mathematics concepts.

Infusing Opportunities for Creativity

The ISTE Standards for Educators call on educators to nurture creativity and creative expression to communicate ideas, knowledge, or connections. The Mathematics Framework encourages teachers to help students “view mathematics as a vibrant, inter-connected, beautiful, relevant, and creative set of ideas” (Chapter 2). An authentic activity or problem elicits students to wonder, ask questions, investigate, and be creative. Strategies for infusing mathematics instruction with imaginative and creative activities may include, but are not limited to, the following:

Focus on investigations around the big ideas of mathematics. Have students apply the Drivers of Investigations and explore patterns to solve authentic problems that include an open-ended, complex issue with multiple solutions. With video conferencing tools, students can work together to engage in discussions around creative ways to solve problems, play various leadership roles, ask reflective questions, consider multiple perspectives, and arrive collaboratively at possible solutions. Students can then use apps, such as Notability [application for documenting data and sharing learning with others], to take notes on what they have found in their deliberations.
Invite mathematicians and other professionals in the field to talk about the importance of mathematics in various career pathways and connections to real-world problems. Invite professionals who reflect the ethnic and gender diversity of the school community. Such opportunities to connect with experts can allow students to see direct connections between concepts and possibilities for what they may encounter in future career opportunities. To foster further engagement, invite students to collaboratively compose a set of questions relevant to their curiosities and interests for the guest speaker. To support these efforts via digital tools, many organizations, such as National Geographic, NASA, and local zoos, have created content to support educational programming, which includes interviews and presentations by professionals.¹⁰ Other examples of guest speakers might include pilots, who can share with students the connection of aerospace dynamics and applied mathematics with physics, or a construction worker or manufacturing engineer, who can talk about how fractions are part of their day-to-day work.
Invite students to create informative and explanatory tutorials focused on teaching a math concept to other students. One of the best ways to learn something is to teach someone else (Koh, Lee, & Lim, 2018). To promote motivation and engagement, consider offering students a choice in how to develop their tutorial. Options may include video, oral presentation, digital brochure, or poster. Teachers can curate a collection of student-produced tutorials (with permission from students and parents), cultivating an ever-expanding library of tools that other students can use into the future. This activity also provides students a chance to boost their confidence and take control of and have an empowered voice in their learning.
Ask students to generate their own problems or tasks for the class to solve that require their peers to recall previous concepts learned. Student-generated problems can help students “connect math concepts to their background knowledge and lived experiences…promot[ing] creative reflection, sense-making, and application of students’ procedural and conceptual knowledge.”¹¹
Use inquiry-based learning to provide students opportunities to devise their own questions to launch an investigation around a given topic and share them with their peers in a collaborative online space, such as a discussion forum. This allows students to have control and choice, as well as get ideas and questions from peers to expand their learning. When students are younger, there might be a need for more teacher guidance in this process, and the collaborative activity might be better done in a synchronous session in small groups with peers who are working on similar inquiries.¹²
Using virtual gallery walks with tools, such as Google Slides or VoiceThread, students can visit stations illustrating a variety of representations of manipulatives (hands-on and virtual) focused on a particular math concept. As a follow-up, students can use spatial skills to externally visualize a concept to process how they think about it (e.g., graphs, simulations, coding, infographics). Students can video conference in breakout rooms while they are working on their visualizations to share ideas to deepen their understanding.¹³
For younger students, it is essential to create daily themes to further engage students and provide them a variety of ways to represent their learning (Hege, n.d.). Mathematics learning can include items from their personal lives so that students can make a direct connection. Students can use digital choice boards to decide, based on the theme, how they are going to represent what they have learned.¹⁴

Refer to Infusing Opportunities for Creativity in Chapter 1 to learn about more strategies.

Encouraging Authentic Collaboration

The ISTE Standards for Educators call on educators to collaborate and co-learn with students to discover, use, and create new digital resources. This type of collaboration in online learning environments is critical to establishing meaningful relationships, cultivating a supportive community, deepening student learning, providing a foundation to grow students’ sense of belonging, and giving every student a chance to develop their own mathematical understandings. Teaching mathematics in a relevant and coherent way can be supported by multiple instructional approaches, including inquiry-based learning, problem-based learning, and project-based learning. Investigations, open-ended tasks, and meaningful problems in mathematics can provide a variety of opportunities for authentic collaboration among students.

Key considerations when building project-based learning opportunities into virtual mathematics instruction may include the following:¹⁵

When possible, allow students to choose the topic of their project so that they have more control and buy-in for what they are working on. Invite students to identify an interdisciplinary issue in their community and think through how mathematics can be part of the solution to the problem.
Invite students to create digital portfolios in Seesaw [portfolio-based learning application] to empower them to share their thought processes when approaching the problem, which can also help students see the relevance of mathematics for everyday life. Encouraging students to share their thought processes also provides peers with varying perspectives on how to approach the problem and solution.
Invite students to create a response to a mathematics problem using a variety of technologies of their choice. For instance, students can create tools, such as infographics (with Canva [a graphic design tool]), digital comic strips (with Google Slides), games (with Scratch [an online programming tool]¹⁶) and videos (with Flipgrid [a video-based discussion software]). This provides an opportunity for students to choose how to best express themselves and represent their learning. It also provides a unique opportunity for students to understand how their peers work through complex problems based on varying cultural and contextual perspectives and experiences.
Allow students to connect with and share their project with a mathematics professional and/or mathematics-focused organization so that they can expand their connections and receive feedback from a global network of experts (Drexler, 2018). Allowing students to network can provide greater balance between teacher control and student autonomy.

Additional examples of strategies to encourage authentic collaboration in mathematics learning in online and blended settings include the following:

After providing students with protocols and guidance related to productive and positive online communication, such as those shared in Chapter 1, invite them to engage in peer editing each other’s mathematics project work. This can be done using collaborative tools, such as Google Workspace [suite of online, shared tools]. This also provides students with feedback and evaluation experiences to further reinforce digital collaboration skills.¹⁷
Use and reference visuals that allow students to make direct links to materials and spaces students have immediate access to. For instance, students could find varying angles using the walls in their homes or learning spaces or use different objects that create angles.¹⁸
Invite students to relate their mathematics concepts to their home environments or their communities. For example, educators can ask students to identify a problem that directly relates to concepts learned. Students can document the problem as well as show how they may solve that problem. A variety of digital tools can help with this documenting process, including shared documents and digital spreadsheets to record data and develop visualizations. Students can use cameras, cell phones, as well as tablets to additionally document the process visually.
Structure a virtual Number Talk, which are explained in Chapter 5, through a video conferencing tool or breakout groups. Give students a problem to mentally solve and ask students to defend their answers using mathematical reasoning. Through discussion, students have the opportunity to explore, compare, and develop strategies. Number Talks provide students an opportunity to have their voice heard as well as to build new understandings as they talk through their process.

Refer to Encouraging Authentic Collaboration in Chapter 1 to learn about more strategies.