Math: An Evidence-Based Instructional Strategy

Paper Info
Page count 7
Word count 7517
Read time 29 min
Topic Science
Type Assessment
Language 🇺🇸 US


The selected lesson to complete the course activities and the final project is an introduction to calculus. The lesson is taught in higher education (undergraduate level), and it entails teaching students how to find the exact derivative of functions from their formula without utilizing graphical methods. The rules learned can use used to calculate derivatives of almost all functions one will encounter in mathematics. An undergraduate curriculum was chosen because as the world changes, technology is advancing unlimitedly, leading to complex challenges in politics, business, and the environment. Higher education prepares students to resolve these challenges effectively (Krause, 2022). This paper provides a rationale for choosing the aforementioned lesson and analyzes its components and technologies applied to achieve success in learning. In addition, it recommends an evidence-based instructional strategy that can be used to deliver the lesson effectively.

Rationale for Selected Lesson

The rationale for choosing the introduction to calculus lesson is that it teaches individuals how to determine a quantity’s rate of change. In the wake of technological advancement, climate variation, and the adoption of automation, differential calculus can be employed to determine the rate of change to predict future events (Tarasov, 2020). The class concepts offer a framework for molding systems that involve change and deducing future expectations from such models.

Components of the Lesson

The components of the introduction to calculus lesson include estimating derivatives of polynomials, interpreting and comparing curve slopes at different points, and applying the acquired knowledge in studying rates of change in real-world situations. The first module of the class introduces the phenomenon by teaching the various techniques used in differential calculus (Hashemi et al., 2020). The subsequent section of the lesson involves features and applications of derivatives. The instructor familiarizes students with the first and second-degree function derivatives. Sign diagrams are utilized in developing systematic curve sketching protocols. Additionally, individuals are taught to find derivatives of complex functions that emerge from simple ones. The product rule, chain rule, quotient rule, and data on the correct derivative to resolve complicated optimization problems are introduced in this module.

Another module of the lesson is the introduction to integral calculus; it considers tangent line slopes are areas under curves, building the fundamental theorem of calculus. Students explore how areas under velocity curves can determine displacement and approximation limits to estimate the formula for the areas under parabolas and circles. Moreover, definite integrals and Riemann sums are employed in capturing particular regions under curves (Hashemi et al., 2020). Lastly, instructors use this module to teach indefinite integrals, substitution as a method of integration, and reflectional and rotational symmetry. Differential calculus can be applied in evaluating the rate of change of atmospheric carbon (IV) oxide using polynomial differentiation. Subsequently, students will predict future CO2 levels based on current trends, thus informing the climate’s corresponding impact.

Technologies that Promote Lesson Success

A technology that can be incorporated into the lesson to enhance its success is the SimCalc software, which fosters visualization in the instruction process and fulfills both symbolization and generalization requests. SimCalc enables students to interact with calculus concepts and create tools utilized in studying change (Tatar et al., 2020). The digital environment enables students to create and organize content in such a way that students can intuitively work on critical facts embedded in the estimation of integrals and derivatives. Consequently, this builds their confidence when sharing results with others in the class. Additionally, they might develop mathematical theorems that explain the qualitative relationship between derivatives and their functions. As a result, this creates an ideal lesson where learners can think and produce conjectures, support conclusions, and challenge hypotheses.

Another technology that can be incorporated into the calculus lesson is a computer algebra system (CAS), such as Mathematica and Maple. The digital tool can be integrated with portable computing devices to allow symbolic and graphical computation (Bognar et al., 2018). Learning institutions can acquire site licenses for Mathematica to ensure students access it on their local machines. According to research, a CAS-oriented calculus lesson has a significant positive impact on students’ spatial visualization skills. Therefore, one implication of using CAS technology is improving spatial visualization skills through training using necessary content. In addition, using the software allows learners to construct solids, curves, and lines on the screen in 3D. Consequently, enhancing spatial visualization ability promotes calculus success.

Researched Instructional Strategy for Lesson Delivery

An evidence-based instructional strategy that can effectively support the delivery of the lesson is the use of visual representations. The practice is gaining popularity in mathematics education and involves establishing, analyzing, and reflecting images and pictures (David et al., 2020). In the introduction to calculus lesson, diagrams can be used to visualize, for instance, definite integral concepts to solve problems. Instructors can recommend using high percentage of imagination images and algebraic expressions for students with high visualization skills. Conjoining these two properties results in successful problem-solving. On the other hand, they should advise learners with low visualization abili3ty to utilize memory images to resolve issues effectively. The visualization strategy will help students articulate representations to produce other prototypes that promote solving problems.

To conclude, undergraduate education offers a holistic experience to explore one’s individuality, skill, and perseverance, thus regarded as a transformation from potential to realization. Introduction to calculus is a mathematics lesson taught at the undergraduate level. It examines change rates of curves and slopes; it involves the study of functions’ rates of change in relation to their variables by using differentials and derivatives. Through the use of SimCalc and CAS technologies, instructors can enhance learning and comprehension of the lesson by using visual representations.

Needs Analysis

In order to approach the process of shaping the framework that will help the learner achieve the expected outcomes, the learner’s needs must be identified. Needs and learner analysis involves the systematic assessment of a lesson to determine the actual problem for which instruction is the solution. Educators employ the approach to understand the problems faced by learners and design practical educational programs that solve these challenges. Ultimately, resolution programs that target particular documented needs are marketable and applicable. They enable educators to make informed decisions and plans about teaching content, thus extending the impact and reach of education. This research conducts a needs and learner analysis of the selected introduction to calculus lesson. It provides the different approaches used to ensure that the developed instructional plan is suitable for the students.

The significance of needs analysis for an introduction to calculus class is that it establishes the relevant requirements of the lesson by identifying the problems affecting performance. According to Mager (1989), a needs analysis should be conducted in twelve different steps, which focus on the critical needs with the most substantial implication to the students. The first step of Mager’s performance assessment is determining the group whose performance is being scrutinized. Specifically, first years doing mathematics-related courses are the targeted audience for this analysis. The second step is describing the negative behavior that results to the problem of failure. Although freshman learners seem prepared to handle the class initially, they often experience difficulties with the subject, which raises concerns. The students do not master all processes involved in finding derivatives of functions and utilize high-level monitoring systems to estimate these figures (Gordon, 2022). As a result, even though they know what is expected of them, they are still unable to complete tasks correctly, thus failing exams. Thirdly, since the calculus topic is introduced in some high schools, learners should excel in the first-year examination.

In the fourth step of the assessment, the cost of the discrepancy is evaluated. Failure to master the introduction to calculus concepts is likely to affect the performance of mathematics students because they are required to apply these processes in upcoming semesters. In the subsequent stage, Mager suggests that the process stops if the discrepancy’s cost is negligible. However, the implication of this problem is significant because not passing the unit disqualifies one from majoring in STEM sectors. Since calculus 1 is among the classes with high withdrawal and failure rates in universities and colleges, the consequences of this problem affect students, campus systems, and the nation. Since failure thwarts many promising engineers and scientists, the cost of this challenge is significant enough to warrant research and the establishment of possible resolutions (Gordon, 2022). Therefore, the evaluation proceeds to the next step, which involves establishing whether the learners know what is expected of them. Most first-year students do not understand why they follow specific algebraic processes in estimating derivatives of functions. Therefore, the primary cause of failure is not comprehending the algebra and arithmetic associated with the calculus lesson.

Next, since the learners do not know how to do what is expected of them, the seventh step of determining the consequences of underperforming is skipped. Students are not equipped with adequate skills from high school to understand and apply the mathematics that underlies algebraic manipulations. The following level eight discusses how the task can be simplified for learners to master it. To simplify the task and enhance mastery, educators should focus more time and resources on uncovering the misinformation that resulted in calculation errors and replacing them with accurate scientific understandings (Brezavšček et al., 2020). Algebraic skills are often used in a calculus 1 lesson because they are a prerequisite in finding derivatives of functions and, therefore, students should master it. Students have a great potential to gain algebraic skills through understanding their theoretical underpinnings, getting the step-by-step processes to be followed, and organizing their work well to avoid mistakes.

Success in the introduction to calculus lesson familiarizes students with concepts and formulas that underpin function derivatives, ensuring students pass subsequent related courses. As a result, there will be equitable science and engineering opportunities for coming generations, and the economy will thrive. Following the aforementioned responses, the subsequent step involves drafting potential solutions. One resolution to this problem is changing the K-12 curriculum framework so that students take similar mathematics classes in high school sophomore year. Secondly, introducing equal advanced placement math and calculus courses in high school reduces the gap between ill-prepared students and experienced learners in calculus. Lastly, educators can integrate active learning and technology in the present lecture and test strategy to enhance algebraic skill development (Hite et al., 2021).

In stage ten, educators are required to estimate the cost of implementing the various solutions. Introducing calculus in high school is challenging because it will mean rushing students into advanced mathematics, yet they need more foundational arithmetic. Further, equalizing learners’ paths will restrain some advanced students who want to go beyond. Finally, instructors compare the cost of the solutions to the problem’s impact and select the most suitable recommendation in phase eleven and twelve of the analysis. Since incorporating calculus lessons in high school has discrepancies and does not predict the success of the introduction class, integrating knowledge and software is the most practical and less costly recommendation.

Conclusively, conducting needs analysis establishes a method for exploring what has been covered in a lesson and what gaps exist in the learning process. The assessments offer insights on the most challenging concepts in the introduction to calculus lesson. Subsequently, an integrated instructional design is recommended to enhance mastery and understanding of the processes, formulas, and representations involved. Ultimately, understanding and passing the introduction to calculus exam guarantees a smooth application and transition to other complex calculus units.

Learner Analysis

In addition to needs analysis, assessing the learners themselves is crucial in determining the path that an educator should take in the process of managing a student’s needs. Educators should design lessons that best suit the needs of students. To do this, they must understand the learners and their needs through learner analysis. Critical aspects of the assessment include learner characteristics, prior knowledge, demographics, and access to technology. The general behavior exhibited by students in an introduction to calculus lesson is that they are anxious, quick, motivated, interested in the subject, and have a positive attitude towards the learning (Wintarti & Fardah, 2019). The students believe that calculus problems are easy to solve because they need straightforward applications of formulas or algorithms. The learners are first-year undergraduates who grasp concepts quickly and have good retention abilities. However, they do not recognize the significance of critical thinking in relating class principles to real-life situations.

Since pre-calculus was introduced in high school, most students are familiar with fundamental theorems and rules. Therefore, the entry skills of the lesson are trigonometry, algebra, and geometry skills. However, existing data reveals that most learners lack vital algebra skills, which is a weakness. The difference between students is that some have mastered basic concepts taught in high school pre-calculus while others have not. Students join the introductory class because it forms the foundation of subsequent science, mathematics, and engineering units. The learning styles employed by most learners enrolled in the lesson are active and student-centered. Lastly, introduction to calculus students can access institutions’ technology, including SimCalc and CAS technologies. Therefore, visualizing graphs and curves related to algebraic functions is simplified for them (Tatar et al., 2020) The iterative process that conjoins these critical learner aspects informs significant instructional design decisions. It enables educators to tailor teaching based on the existing student knowledge to control their learning and gain a deeper understanding of class content.

Performance analysis flowchart for introduction to calculus lesson
Figure 1: Performance analysis flowchart for introduction to calculus lesson

Conclusively, conducting needs and learner analysis establishes a method for exploring what has been covered in a lesson and what gaps exist in the learning process. The assessments offer insights on the most challenging concepts in the introduction to calculus lesson. Subsequently, an integrated instructional design is recommended to enhance mastery and understanding of the processes, formulas, and representations involved. Ultimately, understanding and passing the introduction to calculus exam guarantees a smooth application and transition to other complex calculus units.

Task Analysis

Furthermore, evaluating the task and its efficacy represents a crucial step in designing the strategy that will guide an educator toward the desired outcome. Partaking in an introduction to calculus lesson is essential because the emergence of technology has increased the demand for mathematically skilled people. As a result, a higher burden has been placed on mathematics departments to train a more diverse and extensive pool of students on campus. Many factors contribute to the success rate of the lesson, including the effectiveness of placement examinations, quality of teaching, and learners’ prior experiences with the subject. For instructors to ensure students have acquired all necessary skills in the class, it is required to conduct task analysis (Brown & Green, 2020). The process involves identifying and explicitly teaching students how to perform each step involved in calculating derivatives of functions and integrating. Task analysis helps educators select the most vital task that aligns with the lesson’s goal and most benefits learners (Khoshaim & Aiadi, 2018) This assessment highlights the instruction content needed for an introduction to calculus lesson and the order in which this information will be taught. Additionally, the essay discusses the depth of learning based on the previously conducted needs analysis.

Broad Level Task to be Completed

The needs analysis of the lesson revealed that many students fail introduction to calculus examinations because they do not understand the arithmetic and algebra associated with concepts of the class. Most learners know the calculus concepts and formulas, such as the chain rule, product rule, and quotient rule. However, they cannot critically think and apply the correct principle to some questions in an exam (Gordon, 2022). The primary reason for this problem is that students do not have adequate critical thinking skills to understand and apply concepts that underlie calculus algebraic expressions. Most learners often struggle with differentiating and integrating problems whose roots can be grounded in challenges related to critical thinking. Calculus has many formulas and concepts that students can apply when solving functions-choosing the wrong algorithm makes the final answer wrong (Hitt & Dufour, 2021). The most common mistake made by learners is copying procedures used in the examples in the textbook without considering other competencies used. Therefore, the broad-level task that should be completed through this lesson is equipping learners with the necessary cognitive skills to integrate and differentiate functions appropriately. As a result, students will overcome difficulties and comprehend conceptual knowledge better, performing well in subsequent calculus lessons.

Overview of Analysis of Task Aspects

The aspects of the aforementioned task will be analyzed through a series of steps, which ensure all critical components of the lesson are taught. The evaluation is based on Dick and Carey’s instructional model (Almazyad & Alqarawy, 2020). The initial step in breaking down this task is identifying the lesson’s goal. For example, the lesson’s objective in this case would be to equip students with necessary skills to compute the integrals, limits, and derivatives of functions. Another purpose would be to enable students to identify the correct principles and formulas to solve applied calculus problems. In defining the purpose of a class, educators should state what students will be able to do at the end of the class (Brown & Green, 2020). The introduction to the calculus course purposes to provide learners with basic integral and differential calculus knowledge. Consequently, this will enable them to solve application problems that need such knowledge.

The subsequent step in task analysis is carrying out instructional research to establish the current state of knowledge and skills among first-year undergraduate students. According to a literature review on students’ perspectives and previous test responses, many students have difficulty understanding and appropriately applying calculus concepts in solving problems. They do not have adequate algebraic skills to apply these manipulations when answering open-ended questions efficiently. However, most of them have mastered calculus concepts and formulas; the issue is using them correctly.

Another essential aspect of task analysis is writing specific learning objectives that students will accomplish. These SMART goals should portray the processes and tasks students should master and their assessment procedures. At the end of this lesson, learners will be able to think critically about open-ended and word questions and apply the correct formula to solve them. They will appropriately manipulate algebraic functions to find solutions to problems and interpret graphs correctly. Lastly, learners will attain at least sixty percent in the final examinations. Developing criterion tests is the fifth step of task analysis, and it involves monitoring the effectiveness and progress of the instruction for first-year students. In this case, students will be tested through open-ended questions that represent real-life situations, true or false questions, and multiple-choice queries.

Flowchart representing actual task analysis of the lesson
Figure 1: Flowchart representing actual task analysis of the lesson

Learner’s Entry and Subordinate Skills

Determining learners’ entry and subordinate behaviors is another critical aspect of task analysis. Understanding students’ behaviors, motivational levels, and traits will help educators design appropriate learning techniques. The entry skills needed in an introduction to calculus lesson are understanding trigonometry, algebra, and geometry concepts. Students are required to know the principles of manipulating algebraic functions, calculate the area and volume of simple and 3D shapes, and master essential trigonometric functions, such as sin(x), cos(x), and tan(x) (Nursyahidah & Albab, 2017). Subordinate skills may include manipulating linear functions, operations of numbers, and probability and statistics knowledge. Even though learners are never ready to handle the complex aspects of calculus, they are highly motivated and have a positive attitude towards the lesson.

Foundational reasoning and understanding abilities are critical in learning calculus. However, entry-level students have severe weaknesses in the aforementioned capabilities since they do not know the conceptual background of the subject. Learners cannot answer all function word and proportional reasoning problems. Most of them cannot create meaningful formulas from quantities presented in a dynamic word problem. Additionally, students have difficulties composing two functions since they cannot correctly define the area under a circle in terms of circumference.

Flowchart representing entry and subordinate skills required in the lesson
Figure 2: Flowchart representing entry and subordinate skills required in the lesson

Additional Tasks for Learners to Accomplish

Additional tasks that learners will accomplish within the lesson include interpreting and comparing slopes of curves at different points and using sign diagrams to sketch curves derived from functions systematically. An instructional strategy that can be employed in teaching the lesson is active learning using visual representations. The design will enable students to visualize functions, thus activating their cognitive skills when applying formulas to these expressions (Jaafar & Lin, 2017). Additionally, active learning will familiarize them with various aspects of the lesson, making them understand the principles behind different questions or problems. Instructional materials used for this lesson include the textbook, graphical and open randomized exercises, CAS tools, SimCalc software, many examples, and understandable explanations. The remaining steps of analyzing this task involve creating and conducting formative and summative evaluations to assess the effectiveness of the learning initiative and obtain feedback from students. Finally, educators are encouraged to continually review their instructional design and revise according to feedback obtained and students’ performance outcomes.

Thus, instructors need to plan lessons systematically by analyzing tasks to develop an effective, individualized instructional strategy. By focusing on the conceptual aspects and calculations involved in the introduction to calculus lesson, students will better understand the subject. They will sharpen their cognitive, arithmetic, and algebraic skills to ensure they apply the right concepts and formulas to the appropriate problem. As a result, significant improvement and success will be seen in their grades. Additionally, they will be well equipped to apply the acquired knowledge in subsequent calculus units.

Events of Instruction

Events of instruction of a lesson are constructed based on the learning outcomes and objectives of the class. Learning outcomes are the quantifiable statements that express what learners should value, know, or do at the end of the lesson. The significance of providing these accounts at the beginning of the instruction is that they form the cornerstone of assessments and task design (Hashemi et al., 2020). They ensure students pay attention to what is vital by clarifying expectations. On the other hand, learning objectives organize the specific activities or topics that should be covered to achieve the overall outcome of the lesson. This assessment discusses the learning outcomes and performance objectives of the selected introduction to calculus lesson. The events of instruction are based on the constructivism learning theory, which is characterized by active learners and a student-centered approach to teaching.

Learning Outcomes

The task analysis of the introduction to calculus lesson identified critical or cognitive thinking, technological skills, and visual and verbal communication as essential competencies in the class. The following are the core learning outcomes to be observed:

First learning outcome:

  • Occurring at the end of the course, when the students should be able to state the fundamental calculus theorem, the formal description of the derivative of a function, and the roles of all symbols in the explanation (Rahman et al., 2020).

Second learning outcome:

  • Students should be able to compute the graphical, algebraic, and numerical limits of a function and apply the differentiability and continuity notions to transcendental and algebraic functions.

Third leaning outcome:

  • The learners are expected to understand and apply the definite integral principles to calculate the rate of change, total change of a function, and areas below or above curves. In other words, they should be able to use integration and differentiation to resolve real-world challenges.

Performance Objectives

  1. Given a relevant assignment, students will demonstrate performance at the end of the lesson by explaining the relationship between integrals and derivatives using the fundamental calculus theorem, to demonstrate their understanding of the issue.
  2. Given a task involving limits, the students should be able to use limits to establish whether a function is differentiable or continuous at a particular point using the fundamental calculus theorem, to demonstrate their ability to apply core principles to the analysis (Rahman et al., 2020).
  3. Given a task including tangents and areas, the learners should be able to solve area and tangent problems using the principles of integrals, limits, and derivatives, to demonstrate their knowledge of differentiation concepts, as well as transcendental and algebraic functions. Additionally, they should be able to draw graphs of the aforementioned functions bearing in mind the rules of differentiability at a specified point, continuity, and limits.
  4. Given a task involving the relevant theorems, learners should be able to identify the appropriate calculus techniques and principles to obtain mathematical models to find answers to real-world problems.

Why the Selected Objectives Are Appropriate

Learning calculus is essential for science students because the topic can be hugely applied in various settings. Calculus principles are not limited to analysis and mathematics; they can be used in physics, economics, dynamic systems, and engineering. Therefore, the aforementioned learning and performance objectives are crucial because they help students master three essential tools: integrals, derivatives, and limits. The competencies can be utilized in solving application problems in many settings, including business environments. Through the performance objectives, learners will be able to compose detailed responses to functions using correct mathematical language. They will be able to identify and apply calculus to other subjects and fields where the course is significant.

Lastly, completing the learning objectives will enable students to generate answers to unfamiliar challenges in the world today. Applying critical thinking to determine solutions to advanced problems in the business world is the most crucial aspect of learning calculus. The learning outcomes can be achieved sequentially through describing fundamentals, synthesis, and mastery. Fundamentals are the building blocks of the lesson, and they often appear at the beginning of the class. They form the basis of more complex principles and are combined by breaking problems into small practical steps in the synthesis stage. Each division of the challenge is solved separately and conjoined to create one solution. A key component of synthesis is matching appropriate techniques to open-ended problems and considering the particular rules of the chosen principle. Eventually, students will master the content and be able to take a calculus question, sub-divide it into parts, craft a formula and execute and interpret results. Critical aspects of mastery include differentiating between competing principles, evaluating the effectiveness of functions in different contexts, and explaining results competently.

Events Mapped to Each Objective

Events of instruction mapped to performance objectives and learning outcomes
Figure 1: Events of instruction mapped to performance objectives and learning outcomes

How Events Are Effective and Aligned

To attain the performance objectives, events of instructions that ensure each goal is achieved should be formulated. Specifically, the following steps will be carried out:

  1. Educators will use lecture outlines covering the definition of terms, concepts, problems, and practice questions to teach students the fundamental calculus theorem and the relationship between integrals and derivatives (Stanberry, 2018);
  2. Handing out the outline in advance will guarantee that students will organize their notes well and correctly copy the problems and questions. The second objective of using limits to establish the differentiability of a function at a point can be instructed through group discussions.
  3. Putting students in groups depending on their capabilities and give them questions to solve;
  4. Discussing in teams to enable students to enhance their problem-solving and critical thinking skills and build the confidence of students, making them express themselves better (Mingus & Koelling, 2021);
  5. For area and tangent problems, teaching students to solve them using worked examples that show the step-by-step mechanism of solving a task. The method supports the initial acquisition of critical thinking skills by introducing a situation and demonstrating the steps to reaching a final solution.
  6. Using differentiation techniques to differentiate transcendental and algebraic functions. Inquiry-based learning will accomplish the goal by connecting learners to real-world problems through high-level questioning and exploration. Educators will supplement group activities with assignments and independent research to ensure this outcome is reached. In addition, instructors will use technological tools (SimCalc software and a computer algebra system) to teach learners how to draw and interpret graphs (Tatar et al., 2020). The techniques support visualization as they allow symbolic and graphical computations of functions. Using this method will ensure students understand the relationship between functions and their derivatives and interpret graphical representations (Jafar et al., 2020).
  7. Identifying the best technique to develop an arithmetic model for answering situational problems. Students will develop this competency through problem-based learning, a student-centered approach that involves working in groups to find solutions to open-ended questions. In this approach, students are motivated to learn through encountering problems and discussing with others to find the appropriate mechanism to solve them. The practice will ensure learners deepen their understanding of the calculus concepts and how and why they are applied in a particular manner.

To conclude, events of instruction are formulated based on performance objectives and their associated learning outcomes. Educators must include these aspects in their lesson plans to ensure the class content focuses on the expected learning outcomes. In the introduction to calculus lesson, the selected teaching techniques are mapped to the performance objectives to achieve all the learning goals. Subsequently, the learning objectives are matched to learning outcomes and the overall task to ensure students develop cognitive skills to know the appropriate principles to apply in specific open-ended and real-world problems. Through fulfilling the aforementioned objectives and outcomes, the overarching goal of the lesson will be achieved.

Performance Objectives and Learning Outcomes

Instructional events play a significant role in the learning process since the success of all activities depends on the correctness of their construction and execution. This concept refers to the activities that educators and students perform when learning new information. Furthermore, it is essential to apply such actions that will help students better assimilate knowledge and, most importantly, put it into practice. Therefore, this scientific paper aims to study the concept of events of instruction by analyzing performance objects. Moreover, they will be examined using Gagne’s events of instruction model for the most detailed understanding.

Task A

Each learning process involves drawing up a plan that includes instructional events. They are necessary to ensure smooth and productive assimilation of the information provided, especially if it concerns more complex sciences. Therefore, educators should know precisely how their tasks, as well as the objectives of each event, what is required of students, and in which cases it is possible to skip a particular step (Wong, 2018). Gagne’s instruction event model, which consists of nine essential steps to achieve the necessary learning tasks, can help in this process. The peculiarity of this approach is that it provides an opportunity to adjust the educational process for educators and students; that is, it gives it flexibility.

It is worth noting that this paper is based on previous work that provided performance objectives and learning objectives for the introduction to calculus lessons. Based on this information, Gagne’s events of instruction were selected, which are most suitable for the goals provided in work. Using this approach will help to better understand participants’ requirements at each stage and provide a basis for further formative and summative assessments.

Eliciting the Performance. Event #1 is a vital step in setting the stage for the student’s performance since it involves establishing core goals and introducing the task. Furthermore, the first event of the instructions becomes an elicitation of the performance. This stage is intermediate and is performed to strengthen the knowledge that students acquire in the learning process.

Objective of the Activity. Students should demonstrate performance at the end of the lesson by explaining the relationship between integrals and derivatives using the fundamental calculus theorem. They should be able to use limits to establish whether a function is differentiable or continuous at a particular point.

Teacher’s Performance. In this case, the teacher is assigned a mentoring role rather than a leadership one. This is due to the fact that at this stage, special importance is given to the independent work of students. Hence, the teacher must provide them with tasks that the students themselves must solve using the information and decision skills they have received. In addition, the educator in this process can play a motivating role to ensure the most productive work in groups and individually and reinforcement of knowledge strengthening. It is also possible to conduct a formative assessment, which will show the areas that need improvement and refinement. The way in which the teacher can elicit performance is through the application of tests, group projects, and quizzes. Specifically, a formative assessment could involve a quiz to which students should provide short answers. By asking the questions pertaining to the nature of the studied issue and the way in which it interact with other items, a teacher can test the learners’ understanding of the topic.

Student’s Performance. Students should apply the information received during the class to the investigated event of instructions. Thus, they should have activation of thought processes, and they should determine which of the new knowledge can be helpful for them. Students complete assignments or projects provided to them by the teacher, the primary purpose of which is to demonstrate the level of assimilation of class knowledge (Azizan et al., 2019). One of the main criteria is that students’ activity must be evaluated. This event is valuable as it provides an opportunity to track the success of training through the practical application of knowledge. In turn, the students’ actual performance is expected to be characterized by high levels of initiative, curiosity, and engagement. Specifically, students are expected to interact with one another during group discussions actively, demonstrating a proper understanding of the studied issue, and, possibly, go beyond the provided information to demonstrate their critical thinking faculties.

When/if the Event Can Be Skipped. It is worth emphasizing that this activity should not be skipped, as it provides better memorization of the information provided. Thus, it will be much more efficient and productive to achieve the performance objectives that were set by the educators at the very beginning of the process of studying the topic. The specified consideration applies to eliciting a student’s performance, in particular, since learners are expected to develop engagement and enthusiasm by this point.

Enhance Retention and Transfer to The Job. The second instruction event that was selected for this academic work, which was highlighted in the Gagne model, becomes the enhancement of retention and transfer to the job.

Objective of the Activity. The main objective on which this activity is based was also highlighted from previous work. Henceforth, at the end of the course, the students should be able to state the fundamental calculus theorem, the formal description of the derivative of a function, and the roles of all symbols in the explanation (Rahman et al., 2020). In other words, they should be able to use integration and differentiation to resolve real-world challenges.

Teacher’s Performance. At this stage, the teacher takes actions that will help consolidate the information received even more. It provides value since here, students must independently activate the available information and understand in what conditions what knowledge and skills are needed (Goode, 2018). An educator may face the main problem at this step is the lack of sufficient time resources. Hence, teachers may lack the opportunity and time to verify and strengthen all information.

Since the enhancement of retention and transfer to the job implies the internalization of the acquired knowledge, the teacher should be able to ask questions that awaken her. Moreover, it is possible to introduce exercises into the program related to the actual topic while repeating the one passed. Consequently, from the very beginning of training, students should have a complete understanding of what is expected of them at the end of the training, which means that the teacher should provide transparency in all aspects.

Student Performance. Based on the previously mentioned, the main task of students is to acquire skills that would allow them to apply the information received in practice without significant limitations and difficulties. The main task facing the students is to activate the processes of strengthening their knowledge during the course (Islam & Salam, 2019). In addition, this process includes the provision of reflection, which will help to understand how knowledge has become helpful for students and what can be changed when providing new information.

When/if the Event Can Be Skipped. This event can be skipped if the teacher, at the last stage of the enhancement of retention and transfer, ensures that the students have fully assimilated the new information and are ready to apply it freely and without difficulties and restrictions in practice.

Task B

Assessments as part of the Gagne model of events of instructions are necessary for evaluating the learner. It implies formative and summative evaluation. This section provides methods with which this check could be carried out. Thus, the formative evaluation of the learner aims to gain knowledge and results from students. Based on them, a decision is made on the need to change this or that aspect of the training plan. This kind of assessment method includes the provision of tasks to find out how they learned knowledge, for example, reflection at the end of the lesson. In the situation under analysis, the formative assessment will take the form of written quizzes during which students are expected to provide short answers to the questions that capture the core information. Summative assessment, in turn, implies the final examination of students’ knowledge. Summative assessments will be carried out as final projects for which students will have to write a report on the topic. A certain standard is taken as the basis for this kind of assessment, which will help determine the effectiveness of memorization. An example could be an exam, a test, or a final project.

Learner Evaluation and Instructional Design Evaluation

Evaluating an instructional design is crucial because it offers feedback that informs all other design process stages to ensure continual improvement of the teaching methodology. The introduction to calculus class trains on the concepts and skills, such as solving equations and manipulating functions, which are vital for learners to explore rates of change, limits, and integrals. I chose active learning using interactive technological tools as the best instructional design for this lesson.


First, I considered the strengths and weaknesses of the instructional design to determine what needs to be done before the lesson starts. The evaluation established the starting point of the task by finding out what the instructional design has already covered and what skills required more instruction (Jaafar & Lin, 2017). My initial examination of the lesson involved a criterion-referenced test, which identified mini-quizzes and surveys as items that test the overall comfortability with entry skills of the class (algebra, geometry, and trigonometry). The current active learning design is based on the cognitive constructivist learning theory, which dictates that my instructional design should work on modifying the existing algebraic and trigonometric structures to accommodate the new coaching material needed to solve advanced calculus problems. The team involved in the educational process includes proactive students and the instructor (me).

Formative Assessment

Secondly, formative assessment is used by educators throughout the teaching process to establish the effectiveness of the learning and modify it accordingly. I formatively analyzed my instructional design using alternative testing forms, such homework, informal tests, and exit tickets (Chigonga, 2020). Specifically, I gave learners daily written exercises, informal tests, exit tickets and plickers. The evaluation of the feedback obtained determined how well the instruction covered the various aspects of the topic. Additionally, these results enabled me to track learning, provide feedback, and adjust teaching to focus on enhancing cognitive skills, thus progress towards achieving the learning goals. As a result, I targeted a specific problem area within the instructional content—applying principles to open-ended and word problems. Therefore, I adapted the instruction to focus more on training cognitive skills rather than algebraic and trigonometric formulas.

Summative Assessment

Summative assessment is employed at the end of the lesson to determine the instructional design’s impact on students’ knowledge and skills. The evaluation informs and reports on the gaps and strengths of the instructional procedure, helping educators decide whether they should continue using it (Chytry & Kubiatko, 2021). The Kirkpatrick Model was the best evaluation technique for my instructional program as it accounted for the active learning with interactive technology style. The method has five levels, including reaction, learning, behavior, results, and, finally, return on investments (Kurt, 2018). In the first step, I asked learners whether they enjoyed the learning experiences. Most students loved the active teaching method that engaged them throughout the lesson through visualization techniques and other technologies. Besides, the learning material was valuable in representing the subject ideas visually to students, enhancing comprehension of the background principles and application of calculus concepts to practical problems. Secondly, the learning stage involved gauging the level of knowledge and skills that students have gained through informal and formal assessments. I administered mid-term examinations to measure how much learners have comprehended the class material. The results showed that most students appropriately applied concepts to situations and could explain the principles behind their calculations. There was an improvement from previous lessons that did not employ this instructional design.

The third phase, behavioral change, entailed analyzing how students’ attitudes, knowledge, and skills in calculus have changed following the instruction. The instructional design enabled learners to apply critical thinking and calculus principles in solving word and open-ended problems. The final level of the model was dedicated to measuring the direct results of the learning. I assessed whether the learning outcomes and objectives of the lesson had been achieved (Almazyad & Alqarawy, 2020). Examining the quizzes set for learners showed that they aligned with the lesson outcomes and goals. Additionally, the design improved performance in subsequent calculus units, making it effective over time. Conclusively, active learning provides the resources, assessments, and skills needed for educators to evaluate learning after the lesson. In addition, it coaches on higher-level thinking skills and the ability to synthesize data from word and open-ended situations.

Difference between Learner and Instructional Design Evaluation

Evaluating learners entails a systematic process of establishing whether students achieved the educational objectives of a program. On the other hand, assessing my instructional design involves reviewing the plan’s teaching components and resulting outcomes to establish if it attains the desired outcomes. Therefore, instructional design evaluation focuses on me and how I convey class material, while learner assessment considers the student, how they acquire knowledge, and their performance. Also, design evaluation tends to consider the issues such as the process of development and the choice of strategies for managing it. Thus, the framework for meeting the exact goals and learner-specific needs can be built. Furthermore, instructional design evaluation allows coordinating the performance of a team and the interactions within it, thus, minimizing instances of misunderstandings and promoting collaboration.

Steps to Ensure Success of an Instructional Design

Educators should take various steps for their instructional design to succeed. The first step is establishing the problem faced by students participating in the lesson. Secondly, they determine the broad level task to be accomplished and the specific learning objectives, goals, and outcomes. Then, learner analysis is done to understand learners’ entry behaviors, strengths, and weaknesses. Subsequently, instructors develop an engaging instructional design that embraces active learning and focuses on achieving the established goals and objectives. The teaching method should allow educators to obtain feedback throughout the process and accommodate necessary adjustments. Additionally, it should employ the use of graphics, clips, and other forms of technology.

Design Alignment to an Evaluation Theory

My active learning with interactive technology design aligns with the Kirkpatrick evaluation model because the paradigm effectively measures the effectiveness of the teaching method. It provides a simple approach to assessing aspects of my instructional design in four phases. Additionally, it is easy to implement because it enables the establishment of straightforward structures for getting information from me and the teaching framework without interfering with learning (Brown & Green, 2020). For example, it requires me to ask learners to examine the learning processes to know whether the expected outcomes have been met. Besides, it gauges the effectiveness of teaching through quizzes, tests, and examinations, which is part of the instructional process. The model used for the design appears to be appropriate, yet it lacks the nuance that the updated version provides.

Effectiveness of Instructional Designers

As a program designer, I evaluate my effectiveness through conducting an alignment mapping activity to ensure the instructional design meets the learning outcomes and objectives. Besides, I can undertake a lesson evaluation process after its completion to set quality standards that should be completed or measured. To assess the effectiveness of my team, I will consider the frequency to which teamwork skills, which include coordination, interpersonal skills, leadership, communication, and decision-making, are applied by every member in the development team of the instruction.


To conclude, evaluating the active learning instructional design will allow educators to ensure training offers the necessary skills to appropriately apply cognitive thinking and accurate principles to real-life business problems and subsequent calculus units. Therefore, assessing the instructional design will allow instructors to tailor their teaching throughout the class to ensure the design teaches the necessary calculus concepts and how cognitive thinking can be applied in different situations. As a result, the lesson will ensure knowledge and skills that align with the expected goals and outcomes are disseminated, thus lower the possibility of inappropriate instruction.


Almazyad, R., & Alqarawy, M. (2020). The design of Dick and Carey Model. In Society for information technology & teacher education international conference (pp. 544-547). Association for the Advancement of Computing in Education (AACE).

Azizan, U. H., Yatim, M. H. M., Ibharim, L. F. M., & Zain, N. Z. M. (2019). Analysis of game elements in digital educational game according to Gagne nine events of instruction. International Journal of Academic Research in Business and Social Sciences, 9(7), 131-135. Web.

Bognar, L., Fáncsikné, É., Horvath, P., Joos, A., Nagy, B., & Strauber, G. (2018). Improved learning environment for calculus courses. Journal of Applied Technical and Educational Sciences, 8(4), 35-43. Web.

Brezavšček, A., Jerebic, J., Rus, G., & Žnidaršič, A. (2020). Factors influencing mathematics achievement of university students of social sciences. Mathematics, 8(12), 2134. Web.

Brown, A. H., & Green, T. D. (2020). The essentials of instructional design: Connecting fundamental principles with process and practice. Routledge.

Chigonga, B. (2020). Formative assessment in mathematics education in the twenty-first century. In G. F. Kehdinga (Ed.), Theorizing STEM education in the 21st century. IntechOpen.

Chytry, V., & Kubiatko, M. (2021). Pupils’ summative assessments in mathematics as dependent on selected factors. EURASIA Journal of Mathematics, Science and Technology Education, 17(8), em1995. Web.

David, E., Hah Roh, K., & Sellers, M. (2020). Teaching the representations of concepts in Calculus: The case of the Intermediate Value Theorem. PRIMUS, 30(2), 191-210. Web.

Goode, P. (2018). Using the ASSURE model and Gagne’s 9 events of instruction as a teaching strategy. Nurse Educator, 43(4), 205.

Gordon, L. (2022). High calculus failure rates thwart students across CSU. EdSource. Web.

Hashemi, N., Kashefi, H., & Abu, M. (2020). The emphasis on generalization strategies in teaching integral: Calculus lesson plans. Sains Humanika, 12(3), 35-43. Web.

Hite, R., Johnson, L., Velasco, R., Williams, G., & Griffith, K. (2021). Supporting undergraduate STEM education: Perspectives from faculty mentors and learning assistants in calculus II. Education Sciences, 11(3), 143. Web.

Hitt, F., & Dufour, S. (2021). Introduction to calculus through an open-ended task in the context of speed: Representations and actions by students in action. ZDM–Mathematics Education, 53(3), 635-647. Web.

Islam, N., & Salam, A. (2019). Evaluation of training session applying Gagne’s events of instructions. Bangladesh Journal of Medical Science, 18(3), 552-556. Web.

Jaafar, R., & Lin, Y. (2017). Assessment for learning in the calculus classroom: A proactive approach to engage students in active learning. International Electronic Journal of Mathematics Education, 12(3), 503-520. Web.

Jafar, A. F., Rusli, R., Dinar, M., Irwan, I., & Hastuty, H. (2020). The effectiveness of video-assisted flipped classroom learning model implementation in integral calculus. Journal of Applied Science, Engineering, Technology, and Education, 2(1), 97-103. Web.

Khoshaim, H., & Aiadi, S. (2018). Learning calculus concepts through interactive real-life examples. African Journal of Educational Studies in Mathematics and Sciences, 14, 115-124.

Krause, K. (2022). Vectors of change in higher education curricula. Journal of Curriculum Studies, 54(1), 38-52. Web.

Kurt, S. (2018). Kirkpatrick Model: Four levels of learning evaluation. Educational Technology. Web.

Mingus, T., & Koelling, M. (2021). A collaborative approach to coordinating Calculus 1 to improve student outcomes. PRIMUS, 31(3-5), 393-412. Web.

Nursyahidah, F., & Albab, I. (2017). Investigating student difficulties on integral calculus based on critical thinking aspects. Jurnal Riset Pendidikan Matematika, 4(2), 211-218. Web.

Rahman, M., Ling, L., & Yin, O. (2020). Augmented reality for learning calculus: a research framework of interactive learning system. In Computational science and technology (pp. 491-499). Springer, Singapore.

Rahman, M., Ling, L., & Yin, O. (2020). Computational science and technology. Springer.

Stanberry, M. (2018). Active learning: A case study of student engagement in college calculus. International Journal of Mathematical Education in Science and Technology, 49(6), 959-969. Web.

Tarasov, V. (2020). Mathematical economics: application of fractional calculus. Mathematics, 8(5), 660. Web.

Tatar, D., Roschelle, J., & Hegedus, S. (2020). SimCalc: Democratizing access to advanced mathematics (1992–present). In Historical instructional design cases (pp. 283-314). Routledge.

Wintarti, A., & Fardah, D. (2019). The instructional design of blended learning on differential calculus using successive approximation model. Journal of Physics: Conference Series, 1417 (1), 012064. Web.

Wong, Y. L. (2018). Utilizing the principles of Gagne’s nine events of instruction in the teaching of Goldmann Applanation Tonometry. Advances in Medical education and Practice, 9, 45.

Cite this paper


NerdyHound. (2023, October 12). Math: An Evidence-Based Instructional Strategy. Retrieved from


NerdyHound. (2023, October 12). Math: An Evidence-Based Instructional Strategy.

Work Cited

"Math: An Evidence-Based Instructional Strategy." NerdyHound, 12 Oct. 2023,


NerdyHound. (2023) 'Math: An Evidence-Based Instructional Strategy'. 12 October.


NerdyHound. 2023. "Math: An Evidence-Based Instructional Strategy." October 12, 2023.

1. NerdyHound. "Math: An Evidence-Based Instructional Strategy." October 12, 2023.


NerdyHound. "Math: An Evidence-Based Instructional Strategy." October 12, 2023.


NerdyHound. 2023. "Math: An Evidence-Based Instructional Strategy." October 12, 2023.

1. NerdyHound. "Math: An Evidence-Based Instructional Strategy." October 12, 2023.


NerdyHound. "Math: An Evidence-Based Instructional Strategy." October 12, 2023.