Engagement in the Mathematics Classroom

By: E. Keith Harrison

Mathematics education helps to develop students’ ability to analyze complex situations and improve their problem-solving skills (Aguilar, 2021).  The ability to think critically and engage in problem-solving is valuable not only in the academic setting, but also in the workforce. Despite its importance, many students struggle with mathematics.  This struggle can be compounded by a lack of engagement in the classroom, which can lead to disinterest, poor performance, and a negative attitude towards the subject matter. In the context of mathematics, engagement is particularly essential, as it not only enhances the understanding of complex curricular concepts, but also fosters a positive learning environment that can transform students' attitudes and abilities about mathematics and learning. While engagement has been defined in many different ways, including but not limited to, solely an internal attitude, a reciprocal relationship between students and institutions, or categorized into multiple strands (Alba & Fraumeni, 2019), for the purposes of this paper, we will consider engagement to be student perseverance in what Liljedahl (2021) refers to as a productive struggle.

Engagement

Jumping into engaging lessons takes planning and preparation. Foundational information that is necessary to be successful on a performance task must be presented before it can be implemented, so scaffolding is an essential part of planning engaging lessons. Barkley and Major (2020) write, “although some college teachers resist providing this kind of assistance and criticize it for coddling students, scaffolding can provide the support that students require to persist with a difficult task that might otherwise become overwhelming” (p. 128).  

For example, a teacher might want to show students the dangers of credit card debt through a project in which they take on the role of a debt consolidation specialist helping customers with different amounts of credit card debt at various interest rates.  The students might be tasked with finding basic information such as the number of months to repay the credit card making only minimum payments, the number of months making double or triple the minimum, and perhaps even the number of months making larger payments, but still using the credit card.  This work could be supported through graphing software such as the online Desmos calculator.  The students could then research current interest rates for debt consolidation loans using the internet, propose a loan and terms, and show how much money the customer could save by using this service.  

In order for all students to be successful on such a project, the teacher might choose to scaffold.  This process has already been started by the teacher through the selection of the tasks and their progressively more difficult nature.  However, by providing some guidance during the first section in identifying and selecting variables, providing the required equations, or even engaging in a Text Dependent Question (TDQ) or Depth of Knoweldge (DoK) activity prior to starting the activity in which the students are able to generate these needs for themselves can help ensure all students are able to participate and remain engaged in the project.  

Peter Liljedahl (2021) writes about the importance of helping to keep students in flow, or engaged in productive struggle. He found that students naturally transition between boredom and frustration even when working on a highly engaging thinking task. The teacher’s role, Liljedahl defines, is to provide added challenge or hints when needed and to know when those challenges or hints are required.  If students find a task too daunting or challenging, no matter the level of interest, engagement may be negatively impacted. 

Barkley and Major (2020) explain that one of the reasons teachers find engagement so difficult to accomplish in the classroom is that the definition of engagement is often disputed.  While the definition and underlying motivations behind classroom engagement are muddled, “what is clear,” Barkley and Major write, “is that it is exceedingly difficult to be against student engagement” (p. 4). Sutton (2024) explores how motivational theories for workplace also apply to education, focusing on Victor Vroom’s notion of expectancy theory. Expectancy theory is “based on the suggestion that an individual’s behavior is motivated by anticipated results and potential success” (para. 2) and can be applied when considering what engagement might mean for students. Essentially, “the effort that people are willing to expend on a task is the product of the degree to which they expect to be able to perform the task successfully (expectancy) and the degree to which they value the rewards” (Barkley & Major, 2020, p. 20).  When students see value and relevance in what they are doing, they are more likely to be engaged in the content. 

Kuh et al. (2010) explain that there are two components when it comes to increasing student engagement: student effort and institutional resource allocation. However, teachers are often expected to increase student engagement without additional resources or funding to design lessons that may be more engaging for students. How might that be accomplished, then? Barkley and Major (2020) argue that active learning and real-world application are at the heart of designing lessons that students view as meaningful, and when those learning experiences are authentic and intentional, true engagement flourishes. When students are provided with active learning opportunities, they don’t just passively absorb, or more likely, ignore, the course content.  Instead, they are given a chance to explore the topic in a way that allows them to make connections to previous learning, the real world, and their lives.  Furthermore, critical thinking and strategic thinking (problem solving) are two soft skills that rank amongst the highest demand in business settings (Wells, 2024).  Providing students with active learning opportunities is a great way to help students develop these skills.    

Critical Thinking and Problem Solving

Mathematics plays a critical role in the development of critical thinking and problem-solving skills by providing students with structured opportunities to engage in analytical reasoning and logical deduction. The problem-solving processes inherent in mathematics require students to identify patterns, formulate conjectures, and test hypotheses, thereby honing their ability to think critically. For instance, Jonassen (2000) asserts that math problems, particularly those that are ill-structured and open-ended, promote deep cognitive engagement, as students must navigate ambiguity and apply various strategies to arrive at solutions. This non-traditional approach to selecting problems encourages an iterative process of hypothesizing, testing, and refining solutions that enhances the students’ cognitive flexibility and ability to approach complex problems systematically. Mathematical problem-solving encourages the development of metacognitive skills, as students reflect on their thinking processes and evaluate the effectiveness of different strategies. By engaging in these activities, students not only improve their mathematical proficiency, but also transfer these critical thinking and problem-solving skills to other academic and real-world contexts.

Teaching for Transfer

Melzer (2014) explains that there are two types of learning for transfer: high road and low road. In low road transfer situations, students practice a skill until rote practice leads to automatic recall; in high road transfer, however, students make connections between prior learning and new experiences. Considering that high road transfer expects students to be able to apply their understanding across a broad range of contexts (Perkins & Salomon, 1989), designing lessons with metacognitive activities and real-world application can help them see how mathematics goes beyond the classroom setting. By designing lessons that present foundational knowledge while offering strategies and opportunities to apply that knowledge in meaningful and relevant ways, teachers can help students both internalize their learning and develop a more flexible, adaptive understanding of the course concepts. When teachers are supported by administration, cross-curricular lessons allow students to directly see how their learning is relevant across two or more disciplines. This, in turn, may help to drive engagement in multiple classrooms (Barkley & Major, 2020). 

Most teachers are familiar with Bloom’s taxonomy. However, a contemporary of Bloom, Krathwohl, developed the taxonomy of the Affective Domain (along with Bloom) in 1964, which “addresses the manner in which we deal with things emotionally, such as feelings, values, appreciation, enthusiasms, motivation, and attitudes” (Barkley & Major, 2020, p. 136). When viewed in the context of the cognitive domain set by Bloom, when students’ affective domain is considered, educational outcomes can become more attainable for students. As Eyler (2018) writes, “most of the time, emotion and cognition cooperate quite nicely” (p. 115). This reconciles with the notion that when students are engaged cognitively, affectively, and kinesthetically, a holistic approach has the potential to foster engagement across a diverse population of learners (Barkley & Major, 2020).  In short, “student engagement means students are thinking – not [simply] being entertained” (p. 129). 

Gregory (2013) explains that “because active learning is a balance between student learning centered activities and direct instruction, sometimes the results or solutions generated are unexpected” (p. 120). As with any new tool or strategy, there are going to be challenges to the process. Teachers will likely require additional training or planning time to be effective in implementing new strategies that activate student interest, are relevant to the real-world, take social-emotional needs into account, and encourage students to think. However, when teachers practice these strategies with fidelity, engagement and buy-in can become easier to attain.

The Real-World Application of Mathematics

When students see how mathematics can solve real-world problems, they are more likely to find the subject interesting and worthwhile. According to Boaler (2015), integrating real-world contexts into mathematics instruction can enhance students' motivation and engagement by making learning more meaningful. For instance, project-based learning that involves real-life scenarios, such as designing and building a tiny house to make use of geometric and budgetary concepts, allows students to apply mathematical concepts in practical ways, fostering a deeper understanding and appreciation of the subject. Swan (2006) further emphasizes that real-world applications of mathematics can improve students' problem-solving skills and critical thinking abilities, as they learn to approach and solve complex, authentic problems. Real-world applications can also be constructed in such a way that they spiral back in topics from previous math classes. This can help show students how the math they learn in different years connects and builds on itself. This not only enhances their engagement but also prepares them for future careers in fields that require strong mathematical skills.

Creating Lessons with Fun in Mind

When authentic learning tasks are developed, active learning should be considered. Barkley and Major (2020) argue that “just engaging students in a given task is not sufficient” (p. 42). In fact, the level of activity does not equal the rigor of the assignment or the appropriateness of the task in relation to engagement. For example, Rawson et al. (2014) explore the concept that “providing students with illustrative examples can significantly enhance conceptual learning for declarative concepts” (p. 502). In short, students should be given examples or samples of what the expectations are for the outcomes they will be producing and enough room for their own creativity to shine through when completing engaging tasks.

Lecture has its place in lessons.  In fact, lecture can be a great way to begin the scaffolding process! Gregory (2013) explains that like any instructional practice, there are effective and ineffective ways to use lecture: “for lecture to be most beneficial, students need to have familiarity with the information on which it is based” (p. 117). When a teacher only uses lecture as an instructional medium, however, engagement can be negatively impacted. Barkley & Major (2020) found that lecture plus active learning techniques improves the learning process for students, which causes information retention to improve, as well. Furthermore, they found that:

Research generally shows that the amount of retention corresponds with the degree to which a student is dynamically participating in the learning activity.  It also intimates that students may retain more if they are using multiple senses to process information and are given opportunities at regular intervals to participate in a variety of rehearsal activities that help them to make sense and meaning of the information. (p. 135)

That leads to the natural conclusion that balancing information distribution by the teacher with interactive learning opportunities can improve both retention and engagement. This is also alluded to by Liljedahl (2021), when he encourages teachers to keep lectures to no more than 5 minutes at a time, immediately follow lectures with opportunities to solve problems, and frequently pause to discuss the mathematics occurring around the room.

Environmental Learning

By “setting up [the] conditions for active learning” (Barkley & Major, 2020, p. 43), teachers can intentionally construct activities that challenge students and increase engagement. This builds off Hillocks’ (1986) notion of environmental learning, who argues that environmental learning requires three components to be successful: authentic performance tasks, specific skills that can transfer from one discipline to another, and opportunities for collaboration. For example, an environmental lesson on algebraic expressions might begin with a demonstration on forming and solving expressions, then guiding students through similar problems before being presented with a real-world scenario (perhaps budgeting for a school event). In their groups, students may be asked to collaborate to solve problems related to the scenario, using algebra tiles or Desmos to visualize and manipulate expressions. This would create a rich learning environment that supports peer interaction, real-world application, and engagement, causing students to be more likely to be able to transfer their learning to new and different contexts. 

Building Thinking Classrooms

A new framework for teaching that has recently started gaining traction world-wide is Building Thinking Classrooms. When using this framework, students are placed in random groups of three and do all work at vertical non-permanent surfaces (Liljedahl, 2021). It is important to note that this “…research means exploring important, testable questions with more than four hundred teachers and their thousands of students over 15 years. Success means getting more of these students thinking in math class, for longer amounts of time” (Zager, n.d., as cited in Liljedahl, 2021, p. xiv).  The groups are visibly random so that students know they aren’t being singled out for any reason, and the work spaces are non-permanent surfaces so that students know they can erase wrong work at any time.  The surfaces are vertical because after hundreds of tests, vertical surfaces had the shortest wait time prior to beginning work. Liljedahl (2021) explains that once students have been grouped and sent to their non-permanent vertical surface, they should be presented with a series of progressive problems that activate prior knowledge and lead students to discover new learning. For example, when teaching a lesson on factoring, a teacher might begin by having students multiply binomials together using the box method (a 2x2 grid) as a scaffold (see Figure1). Students would then receive a series of problems with the grid scaffold removed. 

Figure 1

A Completed “Box” Problem

Note.  This figure was created by E. Keith Harrison using SMART Notebook software.

After the teacher has visually verified that all groups understand the prior learning, students work on a series of polynomials that need to be “un-multiplied”.  Again, the 2x2 grid would initially be provided as a scaffold. The teacher uses the fact that the learning is visible to ensure all students are able to remain in flow, and encourages struggling groups to acquire knowledge from groups that have discovered it. Once students have mastered the new content, the teacher might provide a rich task: one that is ambiguous and open-ended for student groups to apply their new knowledge (Liljedahl, 2021).   

Gamification

Teaching with gaming is another option that can help provide students with an interactive opportunity to learn. Prisms is a virtual reality (VR) company that has produced a variety of modules for middle and high school math and science classes. In each module, students take on the role of a professional such as a small business owner, city planner, or climate scientist. They then learn about a new math topic through literal hands-on activities to help solve a problem.  Teachers have long used Kahoot! to help liven up a test review, but Gimkit (2024) offers teachers a way to turn the review into a full fledge video game.  Students have the opportunity to play tag, have snowball fights, or try to climb an endless height.  The students often find the gimmick behind the game entertaining, and teachers don’t mind the gamification since students have to frequently stop playing to answer questions to earn energy if they want to continue playing.  However, “even without using full-scale electronic games, teachers can take advantage of game characteristics (goals, rules, challenge, interaction) and types (puzzles, role-playing, strategy, board games, and so forth)” (Barkley & Major, 2020, p. 133) in order to present lessons in a way that reinforces environmental learning concepts. 

Practical classroom experience reflects that students are often much more willing to solve problems when presented as a Jeopardy game rather than a worksheet. They also tend to enjoy web-based or real-life scavenger hunts in which they need to solve a problem to find the next problem in the chain. Unfortunately, many online games are riddled with ads and either prioritize gameplay over learning or are nothing more than a worksheet. However, new and emerging technologies can do so much more than just let students practice problems. The online Desmos (2024) calculator, for example, is an amazing, free piece of web-based software. The ability to add sliders to any equation in place of constants means that students can deeply explore how changes to an equation transform a graph (see Figure 2). Furthermore, this kind of exploratory actiGrvity tends to drive engagement in students (Barkley & Major, 2020).

Figure 2

Exploration Using Sliders in Desmos.  

Note. This figure was created by E. Keith Harrison using SMART Notebook Software and Desmos.

According to Flinn et al. (2024), these types of rich experiences create deep connections in the brain, which aids in retention of the knowledge. This means that in later classes, the teacher can reference the activity rather than the outcome when a student has questions. 

Recall that Hillocks (1986) stated that environmental learning requires authentic performance tasks, specific skills that can transfer from one discipline to another, and opportunities for collaboration. These ideas can be easily enhanced with effective technology use. Consider the algebra teacher who has just finished teaching systems of linear equations. Students might then be asked to design a small business plan complete with projections for revenue and expenses. Those equations can be graphed on the Desmos (2024) calculator and incorporated into a Google Slide or other presentation software for a business pitch. Students can either be allowed to work in small teams or given time to regularly consult with one another during the project duration. Technology would play a large role in this project outside of merely graphing equations and presenting results. Students would also need to research the cost of raw materials, any one time purchases they need to make, and the cost of using a third-party website to sell their products. They will also need to research the current market rate for their products, how many vendors they are competing with, and how they will stand out from the competition.  By providing reflection questions for students to answer at the end, teachers can help guide students into seeing how this process could be applied to other parts of their life.  Furthermore, by supporting soft skills like collaboration, critical thinking, and strategic thinking, the teacher is providing more than just a project for a portfolio that the student can use when applying to colleges or entering the workforce. 

Conclusion

Mathematics instruction, especially when grounded in the principles of student engagement, has the potential to significantly improve student achievement. Educators can equip students with critical thinking and problem-solving skills when designing lessons that intentionally teach for transfer.  Hillocks’ notion of environmental learning and the real-world application of curricular tasks can help create a more effective educational experience. When students are able to apply mathematical concepts to new and diverse situations, their ability to think critically and problem solve is enhanced; by integrating real-world applications into the curriculum, the learning experience becomes more engaging and meaningful.  

This can be achieved by teachers who purposefully plan instruction. They must dive into pacing guides and the school calendar to find times for rich learning experiences. They also need to devote the time required to plan them. However, when these practices of integrating technology, activating student interest, providing rich educational opportunities, and working on soft skills like collaboration, communication, and problem-solving teachers will often find that student engagement increases. As engagement improves, learning often becomes easier for students and retention almost always improves (James et al., 2024). This can then create a positive feedback loop in which the teacher is able to explore content more deeply with students and the students are excited to learn about topics relevant to their lives.

References

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