Abstract
Classroom instruction focused on discussion-based learning opportunities can provide productive and inclusive learning experiences for all students, including students with learning disabilities in mathematics and those without learning disabilities. Mathematical discourse allows students to share their ideas, justify their thinking, critique the reasoning of others, and refine their thought processes. While one might typically envision mathematical discourse happening during face-to-face instruction, meaningful discourse can also occur in online learning environments. This article presents a blended format of both synchronous and asynchronous learning opportunities, coupled with Smith and Stein’s (2018) “5 Practices” for productive mathematical discourse, to support teachers in designing and facilitating lessons in which all students are actively engaged in the learning processes both for themselves and their classmates.
Discourse is a powerful pedagogical tool that allows students to gain a better understanding of mathematical ideas when meaningfully and appropriately designed interactions occur (Legesse et al., 2020). Mathematical discourse can be defined as “the purposeful exchange of ideas through classroom discussion, as well as other forms of verbal, visual, and written communication” (National Council of Teachers of Mathematics [NCTM], 2014, p. 24). Creating discussion-based opportunities has many benefits wherein students across all grade-bands (i.e., elementary, middle, and high school) can (a) engage in learning complex mathematical concepts that allow productive struggle and active learning; and (b) share their thinking, justify their thinking, critique the thinking of others, and make connections to others thinking (NCTM, 2014; Smith & Stein, 2018). Discussion-based learning helps support co-learning from all classmates, as well as allowing students to engage in learning where autonomy and agency are at the forefront of the experience. A classroom where mathematical discourse is a prominent teaching practice can also create a more equitable and inclusive learning environment where all ideas are valued and shared (Aguirre et al., 2013; Walshaw & Anthony, 2008).
The benefits of discussion-based learning extend to all students, including students with learning disabilities (LD) in mathematics. Indeed, research indicates instructional approaches that encourage students with LD in mathematics and students who show signs of struggle in mathematics to verbalize their thinking and reasoning significantly and positively influences their mathematics performance (Gersten et al., 2009). However, students with LD are often provided mathematics instruction in which the primary focus is on procedural learning rather than conceptual learning (Lambert & Tan, 2017). While procedural learning is certainly important, research shows that procedural fluency should be built from conceptual understanding and not taught as isolated skills to be memorized (National Mathematics Advisory Panel, 2008; National Research Council, 2001). Engaging students with LD in mathematics in learning experiences where rich, mathematical discourse is at the forefront (a) allows students to receive meaningful and timely feedback from their teacher(s) and peers, (b) provides students with more opportunities to respond and engage in their learning, and (c) creates an environment where modeling of making one’s thinking visible occurs regularly, all of which are practices supported in both mathematics and special education (Gersten et al., 2009; NCTM, 2014). In addition, discussion-based instruction allows for easier recognition of students who may need additional support, interventions, or enrichment on certain mathematical concepts as all students regularly expose their thinking.
While the benefits of mathematical discourse for all learners is well-established, facilitating quality discourse can be a challenging task for even seasoned teachers; this is especially true when trying to do so within an online learning environment (Hewitt, 2005; Kim, 2013). This article provides practical recommendations on how to facilitate productive mathematical discourse in general, and, specifically, in an online learning environment. The recommendations provided are situated within a context of an inclusive learning environment wherein both students with LD in mathematics and their typical peers learn collaboratively and are co-taught by a general elementary education teacher and a special education teacher. As noted earlier, a discussion-based classroom supports the learning of all students (NCTM, 2014), including students with LD in mathematics and students who have shown signs of struggle in mathematics (Gersten et al., 2009). This article offers recommendations that support this inclusive instructional approach, while also drawing attention to specific supports for students with LD in mathematics that may be needed or are appropriate.
Discourse in the Mathematics Classroom
Mathematical discourse provides a vehicle for students to share their ideas to better make sense of their mathematical understandings. As noted earlier, mathematical discourse can occur in different forms, including the most common verbal communication, as well as visual, gestural, and written forms (NCTM, 2014). When considering the setting in which discourse happens, partner work, group work, and whole-class discussions are often the most common settings. By working collaboratively with peers, students exchange, share, and alter their thinking through an iterative process of learning. Mathematical discourse provides the teacher with a window into students’ thought processes where they can then probe and clarify students’ thinking in ways that are accessible to all students (Herbel-Eisenmann et al., 2013).
Although many benefits of a discourse-based classroom exist, a classroom filled with rich mathematical discourse does not happen by accident or chance. Teachers are the key to creating opportunities for students to engage in discourse (Walshaw & Anthony, 2008) and supporting the use of language as a tool for reasoning (Mercer & Sams, 2006). Taken together, this means planning and preparation are essential. A well-known, research-informed process for generating productive, whole-class mathematical discussions comes from Smith and Stein (2018). While their instructional moves are characterized as “The Five Practices,” the authors note there are actually six practices a teacher should engage in as they plan and facilitate mathematical discourse: (a) Practice 0: setting goals and selecting tasks, (b) Practice 1: anticipating students’ thinking, (c) Practice 2: monitoring students’ thinking and responses, (d) Practice 3: selecting strategies to share, (e) Practice 4: sequencing responses to be shared, and (f) Practice 5: connecting shared responses (Smith & Stein, 2018, pp. 9–10). Although each practice, or instructional move, is important to facilitate productive classroom discourse, when considering the shift from traditional face-to-face teaching settings to an online learning environment, an emphasis on Practice 0 is of the utmost importance. Teachers must be conscious of how discourse traditionally takes place in online learning environments to increase the likelihood that the task they designed will result in quality mathematical discourse.
Discourse in Online Learning Environments
Online learning has become increasingly popular as it allows for learning to take place without the constraints of time and place. Within online learning environments, the tool most typically used for facilitating discourse is an online discussion board (Wise et al., 2013). These discussion boards are “a text-based human-to-human communication via computer networks that provides a platform for the participants to interact with one another to exchange ideas, insights and personal experiences” (Hew & Cheung, 2003, p. 249).
Discussion boards are typically organized hierarchically, with each post being a reply to the original post. So by their nature, discussion boards are asynchronous, meaning conversations are not taking place in real time. It is possible to conduct real-time synchronous discussions in online learning environments, but this would require the use of different tools. Online discussion boards are thought to be beneficial for learning as they require students to express their thoughts in writing while practicing higher-level learning skills such as reflection, analysis, synthesis, and evaluation (Hew & Cheung, 2008; Newman et al., 1997). Furthermore, online discussion boards provide a way for students to participate equally in the conversation (Zhu, 2006).
However, the affordances of online discussion boards can only be obtained if students participate in the conversation. This can be challenging as research has found that when facilitating discourse in online courses via asynchronous discussion boards, low engagement and participation is a common problem (Hewitt, 2005; Wise et al., 2013). A lack of clarity on the purpose of class discussion and what is expected of students also makes increasing participation difficult (Kim, 2013). Moreover, online teachers oftentimes struggle with creating quality initial questions and or prompts, regulating how involved they should be in the discussion, and creating an environment where students feel connected with their classmates and teacher (Aloni & Harrington, 2018). Within a face-to-face setting, students can see their classmates and teacher(s). This provides them with immediate responses and feedback, making it easier to understand meaning and intent because classmates’ facial and body expressions are visible (Ebner & Gegenfurtner, 2019; Wang & Woo, 2007). This is why some students might feel that face-to-face discussions are more authentic than online discussions.
Fostering quality mathematical discourse in online courses brings to bear an additional set of challenges. For some students, it can be challenging to translate the abstract symbology common to mathematics to the textual format of online discussion boards. For example, students may find it challenging to represent a fraction or a division problem using a text editor and not have the vocabulary needed to represent these operations with words. Fortunately, research on online teaching and learning provides a means of combating the issues related to online discussion boards. Just as with facilitating mathematical discourse in face-to-face classrooms, most solutions focus on the teacher taking on the role of a facilitator, the design of the discussion assignment itself, and leveraging the pedagogical affordances of modern learning technologies.
Facilitating Productive Mathematical Discourse
Setting the Stage
To facilitate a productive discussion, teachers first and foremost must engage in goal setting and planning (i.e., Practice 0; Smith & Stein, 2018), providing the foundation for how discourse can later occur. Determining the mathematical goal of the lesson is followed by selecting a task that provides students the opportunity to meet the mathematical goal of the lesson. Not all tasks are created equal, meaning that different types of tasks provide different opportunities for students to engage with the task (Boston & Smith, 2009; Stein et al., 2000). High-quality tasks (i.e., high cognitive demand tasks) provide students opportunities to grapple with the mathematics rather than producing a quick memorized procedural or answer (Smith & Stein, 2018). For example, high-quality tasks provide all students with multiple entry and exit points, thus focusing on equitable learning opportunities and inviting participation from all learners (Herbel-Eisenmann et al., 2013). How one student begins the solving process often looks different from that of another student, allowing them to build on their strengths and understanding (i.e., multiple entry points). Students are then able to solve the task in a way that makes sense to them (i.e., multiple exit points), creating opportunities for diverse solution paths (Herbel-Eisenmann et al., 2013; Smith & Stein, 2018). Because of these various entry and exit points, high-quality tasks are inherently differentiated as they provide multiple avenues for students with LD in mathematics and their typical peers to engage in the same task. This produces learning opportunities where various solving processes and strategies are modeled and shared (Jayanthi et al., 2008).
Once a task is selected, the teacher should then take time to anticipate students’ thinking (i.e., Practice 1; Smith & Stein, 2018). Anticipating student thinking is critical as it helps to prepare the teacher for possible strategy selection and potential roadblocks students may encounter. When structuring the lesson, students can work independently, in pairs, or in small groups to solve the task, allowing them time to grapple and engage with the mathematics in purposeful settings and formats.
Engaging in Productive Discourse
High-quality tasks take time to solve, allowing teachers ample opportunity to monitor students’ progress and thinking (i.e., Practice 2; Smith & Stein, 2018). Through monitoring, teachers can collect formative data to aid in their decision-making for the next steps within the lesson and next steps for future instruction. While monitoring, teachers can assist students with finding their entry point into the problem. Posing purposeful questions to guide students is more impactful on student learning as opposed to telling the student how to solve the problem. Researchers indicate a strong relationship between monitoring students’ thinking and posing questions to move student thinking forward (e.g., Martino & Maher, 1999). Potential questions may include the following: What strategy might you use? What information is important in the problem? What might be your first step? Monitoring can also include providing students with LD in mathematics or students who are struggling with more systematic and explicit instruction as needed (Gersten et al., 2009). Here, teachers can provide more concrete and specific instructional support such as, “Maybe you could try drawing a picture to help you,” thereby leveraging the effective teaching practices of explicit instruction and visual representations of problems (Gersten et al., 2009; Jayanthi et al., 2008). Monitoring also allows teachers the time and space to select strategies to share in the whole-group discussion and determine the most appropriate sequencing of said strategies (i.e., Practices 3 and 4; Smith & Stein, 2018). Selecting and sequencing provide the opportunity to ensure a variety of strategies and voices are valued and shared with others. Teachers and students then connect the shared responses to culminate the learning and relate shared strategies and the mathematical goals of the lesson (i.e., Practice 5: Smith & Stein, 2018).
Facilitating Productive Discourse in Online Environments
Here, an example of how teachers can facilitate productive discourse around a high-quality mathematics task in an online environment through a blend of synchronous and asynchronous learning opportunities is provided. A description of how each consideration the teacher engages in maps onto the 5 Practices (Smith & Stein, 2018) is highlighted.
Blending the Five Practices With Online Environments
Practice 0: Setting goals and selecting tasks
First, the mathematical goal of the lesson should be determined. In this example, the mathematical goal of the lesson is to allow students to develop an initial understanding of factors and multiples and specifically common factors. When selecting a task, it is important to keep in mind that the aim is to have students explore mathematics, not be told the mathematics. During task selection, consider: Will this task provide opportunities for productive discourse to happen? Will this task allow students to show their thinking and learning in varied ways? For this example, occurring in a fourth-grade inclusive mathematics classroom that serves students with and without LD in mathematics, the Counting Cogs task (NRICH, 2020; see Figure 1 for task description) was selected to address the mathematical goal and fourth-grade mathematics standard, “gain familiarity with factors and multiples.” Specifically, the task’s goal is to allow students to “recognize that a whole number is a multiple of each of its factors” (Common Core State Standards Initiative, 2010, p. 29).
The Counting Cogs task, a high-cognitive demand task, supports students in developing an understanding of factors and multiples.
Next, the structure of the lesson should be established. Since this example is situated in an online environment, careful attention to the pace and format (e.g., small group, whole group) is critical for successful implementation. Given the complexity of the Counting Cogs task (NRICH, 2020), this activity will likely span several days (e.g., 2–3 days) to allow for meaningful discourse and learning to occur. Below, a sample format sequence and specifically planned questions to pose to students during the Counting Cogs task (NRICH, 2020) are provided.
Day 1
To begin the lesson, the teacher introduces the task and sets the stage for learning in a whole-group, synchronous setting. The teacher can use a document camera to show the task, or may prepare slides (e.g., Google Slides) to share with the students. In the example, the Counting Cogs task can easily be shared with students via a website link (NRICH, https://nrich.maths.org/6966). Next, expectations are established and the initial task question is posed to students (e.g., “If you colored one tooth on each cog, which pairs of cogs let the colored tooth go into every gap on the other cogs?”; NRICH, 2020). Next, students are provided individual work time (e.g., offline; individual, asynchronous setting) to problem-solve and document their thinking to share with their teachers and classmates the following day.
Day 2
Initially, the teacher and students meet in a whole-group, synchronous setting to reflect on their independent work and prepare to share thinking in small groups. Individual sharing can certainly be accomplished verbally; however, students also can take a picture of their work and use a platform such as Google Docs to share their thinking with their peers. Similar to the first whole group, synchronous session, the teacher provides expectations for the small group work that will soon follow. This can also be a time in which some students share their problem-solving approach with others who may have felt uncertain in their own approach (e.g., How did you start? What strategy did you use to begin determining which cogs worked?) The teacher should then pose a purposeful question for students to discuss in groups and set expectations for the small group discussions. (e.g., “Which cogs have you found that work so far? Which pairs didn’t work? Can you explain why?”; NRICH, 2020). Next, in small-group, synchronous settings, groups or pairs of students discuss their findings and ideas, while the teacher moves among the groups to monitor discussions, understandings, and misconceptions; during this time, the teacher may need to provide clarification, review expectations, or pose additional questions to facilitate discussions. It is recommended that the student groups document their thinking, creating a product of their collective thinking to share with the teacher(s) and classmates. Suggestions for ways in which collective thinking can be documented include a shared Google Doc, Google Slides, Jamboard, or Padlet. These platforms all provide a way to capture text images while making it visible to others for review, reflection, and sharing.
Day 3
During the final whole-group, synchronous session, small groups share their thinking (e.g., “Can you explain how to determine which pairs work, and why?”; NRICH, 2020). Students continue to refine their thinking as the teacher poses questions to prompt additional thinking and move students toward a more precise and accurate response. The introduction of mathematical terms and vocabulary to generalize thinking (e.g., greatest common factor, multiple; Walshaw & Anthony, 2008) may also occur. Again, platforms that allow slide sharing (e.g., Google Slides) can be used to show these terms visually to students, along with verbal explanations. Collectively, teachers and students make connections to the ideas presented, noting similarities and differences among the strategies shared. After the Counting Cogs task, the teacher can then provide a similar task and follow a similar format for implementation.
Practice 1: Anticipate student thinking
Before meeting with students, teachers should take time to anticipate student thinking on the Counting Cog task (NRICH, 2020; see Figure 1); important questions to consider include: What strategies might students use? What roadblocks might students encounter? What misconceptions may arise? After contemplating these questions, teachers should then think about what information can or should be shared to support students, without “telling” them how to enter (i.e., begin) the problem. For instance, specific to the Counting Cogs task, (a) students may use certain colors when shading in cogs to help them process; (b) students may not understand why they are coloring one cog or connecting this to the mathematical purpose of the lesson, and (c) teachers might suggest the use of paper or electronic version of cogs to support students’ learning and help overcome potential barriers.
Practice 2: Monitor student thinking
Because different formats are being used in the Counting Cogs task example, there are multiple settings to monitor students’ thinking, including whole group, individual, and small group or pair settings. Knowing this, teachers should pay careful attention to students’ thinking, strategies, and discourse happening in each format. Suggestions for monitoring students’ thinking in different online formats include the following.
Individual work: Have students record or write their thinking down on paper and then capture their thinking in a picture to upload to a shareable platform, including the LMS system, Google Slides, or Padlet. The teacher may also prefer a verbal explanation of thinking wherein Flipgrap would be an appropriate platform to consider. The teacher can then monitor this work before the next class meeting.
Small group work: Move virtually among the groups, noting students’ understanding and providing more explicit instruction or different representations as needed (Gersten et al., 2009). Have groups record or write their thinking down in shareable platforms for documentation (e.g., Google Docs or Google Slides).
Whole group work: Note students’ understanding and providing support as needed. In addition to monitoring student thinking, it is important to consider how to keep track of their thinking and collect formative data to use for future instructional decisions (Gersten et al, 2008; Jayanthi et al., 2008).
One suggestion for data collection is to create a way to easily document student thinking during monitoring. For instance, a document with students’ names and space for notes provides a quick way to jot down notes. A checklist of students’ names and a quick system of recording progress (e.g., got it, almost there, needs support) or strategies students use can also be helpful. Additional examples of ways in which to collect formative data include (a) a chart to help identify student discourse that addresses the mathematical goals of the lesson (Smith & Stein, 2018), and (b) the products students create independently and in small groups (i.e., written formative assessment). These pieces of formative data can be collected for individual students, or during small group work (e.g., Group 1, Group 2). Moreover, teachers can explore their online learning management system for data collection tools to support this process, as well as consider some of the suggested resources described above (e.g., Google Docs, Google Slides, Flipgrid). While these are only a few methods for collecting formative evidence during discussion-based lessons, they highlight the frequency in which formative data can be collected in a blended class format and the importance of such practices.
Practice 3: Selecting strategies to share
Based on the monitoring data, teachers now select students’ strategies to share during the whole group discussion. How do you decide? A few suggestions include (a) select strategies that are frequently being used, (b) select unique strategies, (c) select strategies that hold common misconceptions. For instance, during Day 2 of the Counting Cogs task (NRICH, 2020), based on the monitoring of students’ independent work, teachers should select a few students to share their thinking/strategies. This sharing is beneficial for all students but is explicitly supported by research for students with LD in mathematics as it provides students who may have been unsure of how to enter the problem with opportunities to experience multiple modeled examples of how to approach the problem (Jayanthi et al., 2008). On Day 3 of the Counting Cogs Task, each group will share; therefore, there is no need to select certain strategies to share during this portion of the lesson. However, the selection of strategies may be important depending on available time, even with the Cognitive Cog task. If time is limited or simply running out, select approximately a few groups to share rather than all groups.
Practice 4: Sequencing responses
Following the selection of strategies, the sequencing of responses closely follows. How does one decide the order in which selected strategies are shared? Options include (a) strategies that grow in complexity or (b) clumping similar strategies together—meaning have groups share back-to-back who approach the problem in similar ways. Sequencing becomes even more important when not all groups share. Make careful decisions on what strategies to share with the whole group to (a) provide launching points for others, (b) clarify any misconceptions, or (c) move students thinking forward.
Practice 5: Connecting responses
Making connections among students’ thinking is a critical point for closing any discussion-based lesson. Following the whole group strategy share, teachers should pose purposeful and open-ended questions to students, such as: What did you notice? What similarities and differences did you notice across the groups? Specific to the Counting Cogs task: Can you explain how to determine which pairs will work, and why? (NRICH, 2020). While connecting responses, teachers can choose if students respond verbally in the whole group setting or write their responses in an available chat format. This is also a time for clarifying and/or introducing mathematical terminology (Walshaw & Anthony, 2008), such as factor and multiple in the Counting Cogs task (NRICH, 2020), and make generalizations when appropriate.
Additional Strategies
While the blend of synchronous and asynchronous learning environments is a recommended format as it allows for opportunities where rich and productive mathematical discourse can occur, other formats for lesson design may be needed depending on the individual teacher’s circumstances. Specifically, additional teaching strategies to consider can be placed into one of four best practices: (a) communicate purpose and expectations, (b) structure discussions, (c) create quality discussion prompts, and (d) provide effective discussion facilitation (Aloni & Harrington, 2018). For discussion boards, teachers should resist the natural urge to assume that students know what is being expected of them. Instead, teachers need to stress the importance and intent of the discussion board, make sure that students know that their participation board will count toward the course grade, and provide students with a rubric that lays out how students earn points for posting and participating in the discussion (Aloni & Harrington, 2018).
When it comes to structuring online discussions, teachers should leverage successful teaching strategies. For example, separating the class into small groups so that students will have a better opportunity to participate in discourse (Rovai, 2007). Students can be placed into groups based on the use of similar or dissimilar strategies on their initial post, or they can be placed into heterogeneous groups from the beginning of the task based on mathematical learning needs. Teachers can also use online tools that facilitate unthreaded discussion (e.g., Padlet, Jamboard). Finally, to help students manage their participation in discussions there should be separate deadlines for initial posts and responses to the postings of peers. Having the same due date for initial postings and responses usually results in students posting their initial response close to or on the due date, leaving little time for the discourse to take place.
It is worth noting again that online learning environments provide the opportunity to utilize different approaches for delivering and framing or prompts. The use of video prompts, word clouds, WebQuests, debates, and role-playing have been found to improve online discussions (Aloni & Harrington, 2018; Hou, 2012). Similar results have been found in the use of questions. Specifically, instead of encouraging students to close in on the same solution, research supports (a) encouraging students to diverge from their classmates, (b) asking Socratic questions, (c) helping students identify themes, (d) using assigned roles (e.g., moderator, skeptic, devil’s advocate), and (e) providing summaries of discussions (Aloni & Harrington, 2018). In addition, there is an increasing number of online applications designed specifically to help facilitate discourse in online environments. Online collaborative tools, including those mentioned previously, as well as others such as VoiceThread and NowComment, provide teachers with options beyond the discussion board found within a typical learning management system to engage students in both synchronous and asynchronous mathematical discourse.
Conclusion
Productive mathematical discourse does not happen by chance; the teacher(s) take on the role of designer and facilitator to create an environment where students are encouraged to engage in high-quality mathematical tasks and discuss, critique, and refine their thinking through classroom-based discussions. Through a collective and collaborative learning environment, rich mathematical discourse can occur, providing equitable and inclusive learning opportunities for all students. When considering an online environment, task selection, pedagogical affordances of various educational technologies, and design format in which discourse occurs require careful attention. By using the 5 Practices (Smith & Stein, 2018) as a guiding framework, mathematical discourse in an online environment can not only come to fruition but produce meaningful learning opportunities for all involved.
Footnotes
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.