Using experiments to measure irregular objects in Year 5

Using technology and data to teach the volume of irregular objects: a Grade 5 mathematics case study  

Han Lei  

Aletenxire No. 5 Primary School, Yijinholuo Banner, Ordos, Inner Mongolia  

Abstract:This case focuses on the topic “Volume of Irregular Objects” in the fifth-grade mathematics textbook of the People’s Education Press. It utilizes technology to enhance experimental teaching and establishes a data-driven intelligent teaching model. Based on analysis of the textbook content and students’ capabilities, objectives for knowledge, skills, and competencies are set. Through technical methods such as a digital homework system for pre-testing, hierarchical previewing on the National Smart Education Platform for Primary and Secondary Schools, and in-depth interaction via Xibo Whiteboard, teaching is carried out throughout the entire process – before, during, and after class. By integrating teaching and assessment, and using various evaluation tools for real-time feedback, a closed loop of “data collection – precise analysis – precise intervention” is achieved. Practice shows that students’ ability to understand and apply the concept of transformation improves significantly, and classroom interaction increases. At the same time, this provides data support for teachers’ professional development, highlighting the important role of technology in transforming mathematical experiments from verification to exploration.    

Keywords:Technology empowerment; data-driven approach; integration of teaching and assessment    

1. Course background and overall design concept  

(1) Analysis of teaching materials and curriculum standards  

This lesson is a practical activity class for Unit 3 of the fifth-grade mathematics textbook of the People’s Education Press, which deals with “Cubes and Cuboids”. It falls under the “Geometries and Geometry” category. The curriculum requirements state that, through experimental operations, students should explore methods for measuring the volume of irregular objects, thereby developing spatial concepts and application skills.  

Developing a concept of space is one of the core competencies required in this lesson. Students need to break through the mechanical understanding that “volume equals a calculation formula” and understand the true nature of volume as “the amount of space that an object occupies”. They also need to be able to imagine and reason about the abstract process of transforming an irregular object into a measurable volume of water (or sand). Measuring the volume of an irregular object requires students to dynamically construct a mental model of “physical object → water level change → volume transformation”. For example, when a stone sinks into water, students must imagine a “virtual cuboid” formed by the rise in water level; its length and width correspond to the base area of the container, while its height represents the change in water level. In this way, the thinking from a three-dimensional entity to a two-dimensional representation can be achieved.  

In terms of developing the awareness of application, the curriculum standards require students to be able to apply mathematical knowledge and methods to solve practical problems. This lesson directly relates to real-life situations, such as measuring the volume of fruits, stones, and toys. It helps students understand that mathematics is a powerful tool for solving real-world problems and highlights its practical value. The process involves questions like “Why is it necessary to measure the volume of irregular objects?” “How to measure it?” and “Solving practical problems.” This awareness of application is present throughout the entire lesson.  

Recasting an unfamiliar problem in a form students can measure is the central mathematical idea of this lesson. Instead of measuring an irregular object directly, students determine its volume from the amount of water displaced or spilled, or from the amount of sand that overflows. This is an example of reduction: turning an unknown quantity into a known one and an irregular shape into a regular measurement. This strategy helps solve complex mathematical and scientific problems and prepares students to find areas by cutting and rearranging shapes and to think in terms of equations.  

(2) Analysis of students’ learning situation  

1. Knowledge foundation: Through preliminary tests using the digital and intelligent assignment system, it was found that 95% of students can correctly calculate the volume of regular objects such as cuboids and cubes. This indicates that students have mastered the basic formulas and methods for calculating the volume of regular shapes. However, their performance is not satisfactory when it comes to problems that involve the application of the “transformation concept”. For example, when asked to use the displacement method to measure the volume of irregular objects, only 32% of students could correctly explain the measurement principle. Most students’ understanding of the “transformation concept” is merely mechanical; they know that “when an object is placed in water, the volume of water that rises equals the volume of the object itself”, but they are unable to explain its essence in detail – that is, how to transform the unknown quantity (the volume of the irregular object) into a measurable quantity (the change in the volume of water). It is even more difficult for them to apply this concept flexibly in new situations. For instance, when dealing with complex situations such as objects floating or partially submerged, they often lack the necessary knowledge to handle such cases. Additionally, students make many mistakes in handling details such as unit conversions and analysis of data errors, which indicates a lack of proficiency and rigor in applying their knowledge.  

2. Technical competence: 90% of the students in the class are proficient in using the National Smart Education Platform for schools to record and upload homework. They can also use the micro-lectures available on the platform for pre-study, thus having a good foundation for digital learning. The students are able to watch videos on their own and submit online homework; they show high enthusiasm and adaptability in technical operations. However, there are still limitations when it comes to using technology for in-depth exploration. For example, most students only passively receive the knowledge presented in the micro-lectures, and they lack the awareness and ability to explore actively on their own. Additionally, in group cooperation scenarios that involve the use of technology, some students have problems with unclear division of tasks and low collaboration efficiency. For instance, when using the Xibo whiteboard for interaction, some students are not skilled in using its functions, which affects the overall progress of the group. This indicates that there is a need to improve the students’ technical application skills as well as their collaborative abilities.  

(3) Key points of technological integration and innovation  

1. The teacher used the digital homework system to distribute pre-test questions and basic post-test questions related to irregular objects. Subsequently, the system collected and analyzed students’ responses to the questions regarding the volume of irregular objects, thereby providing a basis for the teacher’s teaching activities.  

2. The National Primary and Secondary School Smart Education Platform provides support throughout the entire process: it offers comprehensive support for “data collection during lessons, resource utilization during lessons, and additional activities after lessons”.  

3. Deep applications of Xiwu whiteboard: Using the masking, mind mapping, and classroom activity tools provided by Xiwu Whiteboard 5, along with features such as mobile screen casting, photo uploading, random selection, and real-time evaluation, an immersive learning environment can be created.  

4. Professional development of data-driven teachers: After this lesson, teachers used “AI Great Teacher” to generate teaching analysis reports. By analyzing the generated experimental reports, they could understand the quality of teaching. Comprehensive and thorough analyses were carried out in various aspects such as teacher speaking speed, classroom language, teaching scenarios, teacher questions, S-T teaching methods, and teacher observations. This enabled teachers to improve their teaching practices based on the data, thereby facilitating their professional development.  

II. Goal setting based on student conditions and textbook analysis  

Students are able to fully understand the concept of the volume of irregular objects, and clearly grasp the principle of converting the volume of irregular objects into measurable volumes of regular objects (such as the volume of water or sand). They are proficient in using methods like the drainage method, overflow method, and sand overflow method to measure the volume of irregular objects, and can accurately apply these methods for actual measurement and calculation, ensuring the accuracy of the results.  

By participating in experimental activities and group-based collaborative investigations, students can gradually improve their practical skills and problem-solving abilities. They learn to identify problems, analyze them, and attempt to solve them in practice.  

In the process of exploring methods for measuring the volume of irregular objects, it is important to fully understand the significance of transformation concepts in mathematics learning. This helps to develop students’ mathematical thinking skills and their ability to apply knowledge flexibly to solve other mathematical problems.  

III. Deep integration of teaching implementation processes and technology  

(1) Evaluation of the overall framework construction  

Based on the concept of “integration of teaching and assessment”, a closed-loop system is established that is goal-oriented, with teaching as the pathway and assessment as the feedback mechanism. Teaching objectives are broken down into specific, observable and evaluable indicators, which are applied throughout the entire process – before, during, and after class. Multiple assessment tools are used to ensure the coordinated progress of “teaching – learning – assessment”. Specifically, this involves three dimensions: process evaluation (focusing on participation and cognitive development during learning), performance evaluation (focusing on experimental skills and problem-solving abilities), and result evaluation (focusing on knowledge acquisition and practical outcomes). Information technology is utilized to collect and analyze assessment data in real time.  

(II) Integration of teaching implementation and evaluation  

1. Pre-class preparation: The intelligent platform helps with pre-experiment preparation. (1) The digital and intelligent homework system enables monitoring of students’ understanding of concepts such as the volume of cuboids and cubes. (2) Teachers post differentiated pre-class tasks on the National Smart Education Platform for schools. Basic tasks (required): Watch the video on measuring the volume of irregular objects, and think about how the volume would be measured if potatoes were replaced by oranges, which would float on the water surface. Challenge tasks (optional): Use the overflow method to measure the volume of potatoes, and record a 2-minute video of the process. (3) After collecting the experiment videos, evaluations are conducted promptly: 25 student videos are collected, and teachers provide feedback on the students’ work, as well as collect examples of correct measurement methods. The objects were not completely submerged 7 times, and measurements were taken using calibrated containers such as measuring cups 18 times (with the line of sight tilted 3 times during reading), and the overflow method was used 16 times (with 4 instances where the water volume was not controlled properly).  

2. In-class implementation: Various information methods support teaching.

Step 1: Introduction to cultural context – from historical wisdom to mathematical concepts. Utilize online resources to find and play the 3D animation video “Cao Chong Weighing the Elephant”. Select a segment that lasts about one minute. Before playing the video, ask the following key questions: How did Cao Chong measure the weight of the elephant? Cao Chong converted the weight of the elephant to the weight of stones, and the weight of the stones was equal to the weight of the elephant.  

Step 2: Experimental exploration – Clear instructions for technical support. (1) Creation of a problem scenario on the Smart Whiteboard: The Smart Whiteboard displays various objects such as Rubik’s cubes, clay, potatoes, oranges, and small stones. Students are required to first classify these objects, then consider how to measure the volume of these irregular objects that can be deformed, such as Rubik’s cubes and clay. Subsequently, they use experimental equipment to determine the volume of non-deformable objects like potatoes, oranges, and small stones. (2) Time-bound experiment to calculate the volume of irregular objects using the displacement method: The Smart Whiteboard provides instructions for group-based exploration. Students are required to complete this experiment within 4 minutes. Students use one of the three irregular objects—potatoes, oranges, or small stones—along with measuring cups and sinks for the experiment. The procedure is as follows: pour an appropriate amount of water into the measuring cup and record the data; slowly add the irregular object to the measuring cup and observe the change in the water level; when the water level no longer changes, record the level of the water in the measuring cup; then think about the relationship between the volume of the irregular object and the difference in volume. (3) Deep reflection on the experimental process: First, consider what data needs to be recorded when using the displacement method to calculate the volume of irregular objects; then summarize what should be noted during the experiment; finally, analyze whether it is possible to first add potatoes and water to record the total volume, then remove the potatoes and record the volume of water, and the difference in volume will be the volume of the potatoes.  

Step 3: Presentation of Research Results – Interaction between students with the support of information technology. Photo upload: Teachers take photos of the experimental data and research records submitted by each group. The data from the experiments conducted by the groups on irregular objects such as potatoes, oranges, and small stones are also uploaded. Mobile phone projection: Teachers use the “Mobile Phone Projection” function on the Smartboard to record the typical mistakes made by students during their operations, and then display these recordings so that students can analyze the reasons for those mistakes and correct them. Class management evaluation function: When students present their results, teachers and students from other groups evaluate them based on three criteria: whether the experimental procedures were carried out properly and the process was recorded accurately; the accuracy of the data recorded in the records; and whether the explanation clearly explained the volume differences and the relationships between objects. Teachers use the class management evaluation function on the Smartboard to record and evaluate these evaluations promptly.  

Step 4: Enhanced Understanding and Deepening of Knowledge – Real-time interaction between school and home with the support of information technology. In this step, the teacher first poses a question: If there are only some containers and small measuring cups, can the volume of a larger object be measured? Then, the teacher displays the homework assigned by the National Smart Education Platform for primary and secondary schools in the classroom. The teacher also shows videos of students in the class performing the experiment of measuring the volume of potatoes using the overflow method at home before class, and asks the students to analyze whether this method is reasonable.  

Step 5: Overcoming cognitive conflicts – a progression in thinking from the drainage method to the sand overflow method. Question: Can the above methods be used to measure the volume of a table tennis ball or ice cubes? Why? Through discussion, students realize that these methods are not applicable, because the table tennis ball does not sink into the water, while the ice cubes mix with the water. Using information technology again, it is shown that when using the drainage method to measure the volume of a table tennis ball, the ball floats on the surface of the water, so this method cannot be used for measuring volume; instead, the sand overflow method must be used.  

3. Post-class activities: Personalized learning supported by the platform. (1) Display and timely feedback on classroom assignments. (2) Hierarchical assignment delivery supported by information technology: Basic tasks – calculating the volume of the large circle in the figure; Advanced tasks – after learning about interesting measurement methods, Zhang Qi and Li Beibei want to try measuring the volumes of several irregular objects. They conducted the following experiments: Which has a larger volume – a potato or a sweet potato? (Unit: centimeters) The digital homework system is used to analyze and assess students’ understanding of the concepts; Challenge tasks – students choose three irregular objects from their homes (such as pebbles, toy cars, teacups, etc.) and use the immersion method, overflow method, or sand overflow method to measure their volumes. They record videos and upload them to the platform, along with descriptions of any problems encountered during the measurement and the solutions they found. (3) For students who have issues with “objects not fully submerged” or “inappropriate water level control” during experiments, micro-videos and dynamic demonstration animations explaining the proper “drainage method” are provided.  

IV. Teaching effectiveness and improvement strategies  

(1) Teaching effectiveness  

Through the closed-loop design of “video recording before class – visual analysis of errors during class – application of the concepts learned in real-life scenarios after class”, 92% of students were able to accurately describe the application of “transformed ideas” in volume measurement. 85% of students were able to design measurement methods for two or more irregular objects on their own, which is a 40% improvement compared to traditional classrooms.  

The integration of Smartboard and Class Optimization Master creates an immediate interactive ecosystem based on “operation – presentation – evaluation”. The generative resources in the classroom, such as videos of students’ incorrect operations and creative measurement methods, are promptly transformed into teaching materials. As a result, students can better understand typical errors, and interaction between students, teachers, and students is facilitated.  

There are new pathways for teachers’ professional development. The teaching analysis report by “AI Great Teacher” provides multi-dimensional data support. Through the S-T teaching analysis method (analysis of teacher and student behavior codes), it is found that the time spent on “interactive communication” in the classroom accounts for 38.46%, while the time spent on “group collaboration” accounts for 15.98%. This is a significant increase compared to traditional laboratory classes, indicating a transformation towards a “student-centered” inquiry-based classroom. The data shows that comprehension-based questions occur most frequently (72.6%), followed by application-based questions (13.7%). The number of evaluation and innovation-based questions is limited, which hinders students’ critical thinking and innovative abilities. Teachers can encourage students to think critically and innovatively about mathematical problems by posing open-ended questions.  

(2) Improvement strategies  

In this lesson, the connection between science concepts such as “buoyancy” and information technology concepts like “data collection” is limited to the stage of problem creation. In the future, “interdisciplinary project-based learning” can be implemented. For example, students can use programming to record the changes in data during drainage experiments, or they can analyze the applicability of different measurement methods for objects made of various materials, thereby further enhancing the transfer of methods and the integration of ideas across different disciplines.  

Although technologies such as mobile screen casting and AI analysis have improved classroom efficiency, there is still a problem of transmission delay. Therefore, it is necessary to adjust the network environment before class, and use a wired screen-casting method as a backup option. Alternatively, typical error videos can be cached in the Xunuo whiteboard in advance, thereby reducing the pressure of real-time transmission.  

V. Deep consideration on the empowerment of technology  

In this teaching practice, the deep application of information technology has not only transformed the teaching methods, but also redefined the educational value of mathematical experiments. Through the coordinated use of digital homework systems, the National Smart Education Platform for primary and secondary schools, XEVO Whiteboard, and AI analysis tools, technology has evolved from a tool to assist teaching into a core element that drives educational reforms. Its enabling role is reflected in three aspects:

Firstly, technology enables a dynamic and precise teaching cycle. With the pre-test analysis provided by the digital learning system and the classroom diagnostics offered by AI, teachers can accurately identify the difficulties in teaching based on data. The National Smart Education Platform for Primary and Secondary Schools ensures the full process of teaching resources, supporting students’ preparation before class, exploration during class, and further learning after class. The real-time interactive features of the Xibo whiteboard allow for the integration of evaluation into the teaching process, creating a complete chain of “data collection – precise analysis – teaching intervention – feedback on outcomes”. This closed-loop design breaks the limitations of “experience-driven” teaching in traditional education, transforming teaching decisions from subjective judgments into data-driven ones.

Secondly, technology facilitates the visualization and deeper development of students’ thinking. Through features such as mobile phone projection and photo uploads, the process of students’ experimental operations and their thought processes can be presented in a clear manner. Errors can be turned into teaching opportunities. Mind mapping tools help students organize their knowledge logically, and 3D animations make the abstract process of “transformation” more tangible. The use of technology not only reduces the difficulty of learning, but also encourages students to think actively through visual representations, helping them move from mere imitation to creative exploration.

Finally, technology has promoted the diversification and intelligence of teaching evaluations. AI analysis tools enable quantitative assessment of classroom behavior, and multi-party evaluations through platforms (e.g., mutual evaluations between teachers and students, as well as evaluations involving parents and schools). This allows evaluations to transcend time and space constraints, covering all aspects and involving everyone throughout the process. In particular, the inclusion of experimental operations and thought processes in the evaluation system changes the traditional approach, which focuses on outcomes rather than the process, thereby truly achieving the educational goals centered on developing skills.

However, practice also reveals the deeper challenges associated with the application of technology: first, how to balance the abundance of technical tools with the focus on teaching objectives, so that the essence of the subject is not lost due to an excessive emphasis on technical forms; second, how to use technology to reduce the individual differences among students and provide more precise personalized learning support; third, how to deepen the integration of interdisciplinary technologies, so that information technology truly becomes a catalyst for the integrated development of academic competencies. In future teaching, it is necessary to continue exploring innovative ways to integrate technology with teaching, so that information technology can better serve the overall improvement of students’ core competencies.

The true essence of technology-driven education lies not in the advanced nature of the devices, but in whether they serve as a “lens for seeing what is invisible and thinking about what is not considered”. In this lesson, through the deep integration of the National Primary and Secondary School Smart Education Platform and the XEVO ecosystem, a closed-loop system of smart teaching—“data collection – precise analysis – precise intervention – diverse evaluation”—has been established. Technology is no longer just a collection of simple tools; it becomes a cognitive framework that supports deep learning, enabling mathematical experiments to transform from “verification-based operations” into “creative explorations”.