Technological Resource Development

This website will Use a variety of digital tools and interactive resources can help fourth-grade students learn number operations such as addition, subtraction, multiplication, and division without remainders.

This initiative aims to create technical materials that emphasise mathematical language, communication, and problem solving using open-ended questions. Students can better understand and apply this knowledge to real-world settings by taking use of online courses, teaching apps, and engaging tasks.

The task is to design a series of lessons that use digital resources( PhET/GeoGebra) and interactive tools to teach number operations effectively. The conversation shows that mathematics textbooks might be produced as educational tools that place more importance on numerous communication resources, as varied communication resources are required to represent mathematics. (Norberg 2021, p. 24) The architect was tasked with creating a set of courses that would transform the way basic mathematical operations like addition, subtraction, multiplication, and division are taught. It will focus on how children were asked to memorise tables of multiplication and try to solve a huge number of mathematical arithmetic problems. Yet, these methods are already out of date.

Our present technological progress no longer demands people to have these quick calculating skills but rather requires children to innovate and discover creative ways to address real-life issues, as well as encouraging children's problem-solving abilities. Developing effective procedures for addition, subtraction, multiplication, and division while utilising appropriate digital tools for problem solving. Encourages flexible mathematical communication and collaborative learning settings. Addressing open-ended challenges and determining outcomes in real-world circumstances.

Lesson 1: Introduction to Multiplication Strategies (AC9M4N06)

In this lesson, students participate in an innovative examination of multiplication, focusing on doubling and halving procedures utilising interactive PCs, as well as digital simulation tools like as GeoGebra. Students not only understand this technique but also improve their mathematical fluency by engaging in group work and hands-on experimentation using GeoGebra, supported by reinforcement through math game applications. In also, children learn to estimate and calculate the products of whole numbers with multi-digit components, as well as use partial product and area models to find the product of two-digit numbers in engaging instructional games designed for preschool children.

This full method allows students to face multiplication issues with confidence while also developing a better comprehension of mathematical ideas through engaging learning experiences. https://www.geogebra.org/m/qd3s5nqs

Begin the lesson with a discussion about the students' existing approach to tackling multiplication problems. For example, provide a problem like 5×18 and ask students about shortcuts to answer it effectively. Introduce the doubling and halving approach by demonstrating how doubling one number and halving another simplifies multiplication. According to the double change-in-units concept, multiplication involves counting a larger unit with a pre-existing link to a smaller unit. When constructing composite units, the unit of measurement (the multiplicand) must be defined before the unit amount (the multiplier). （Bakos 2022，p53）By doubling 5 to get 10 and halving 18 to reach 9, the equation 5 × 18,5×18 becomes 10 × 9,10×9, which is easier to answer.

Next, use an interactive whiteboard to visually demonstrate the procedure. Write out the methods to translate 5×18=10×18/2=180/2=90. Use diagrams and pictures to demonstrate how breaking down the problem into smaller steps simplifies its understanding and solution. Give an open-ended question to the students, such as, "You have 5 packs of 18 stickers apiece. How do you use the doubling and halving approach to quickly figure out the total number of stickers?" Students must work together in small groups to experiment with alternative problem-solving strategies. Encourage them to apply the doubling technique while also thinking creatively about additional ways. Following group work, have each group present their techniques and solutions to the class and analyse the various approaches and their efficacy. GeoGebra software allows teachers to geometrically and successfully teach the ideas and methods of math operations in primary school. (Kllogjeri 2015, P2) Help students use GeoGebra on their PCs to visually understand the concept of doubling and halving. Demonstrate how to set up the problem and alter the numbers to observe the impact of doubling and halving.

Allow students to practice GeoGebra by experimenting with different numbers and seeing how the method works with different problems. Use apps like Mathletics to get extra practice. These apps provide active puzzles and fast feedback, allowing students to reinforce their mastery of the method. Distribute printed worksheets including a variety of math questions designed to practice the doubling and halving approach. Include an area for the students to display their work and explain their reasoning. Assign homework that contains both simple and difficult problems requiring the use of the doubling and halving technique. Conclude the lecture with a class discussion about what was learned. Ask students to discuss the doubling and halving approach and how it helped them solve multiplication issues. Provide feedback on their strategies and solutions, emphasising successful approaches and correcting any misunderstandings. By the end of this lesson, students will know how to apply the doubling and halving approach to simplify and solve multiplication problems. They will also have gained familiarity with digital tools for studying and developing mathematical topics.

Lesson 2: Exploring Doubling and Halving with Apps

The lesson concludes with students sharing their findings and discussing how the app contributed to their comprehension of the strategy, addressing any remaining misconceptions and enhancing their learning. To shift from traditional mathematics instruction to digital technology, teachers must learn how to successfully use technology as a pedagogical tool. (Bakos 2022，p32）This lesson effectively utilizes technology tools to model and demonstrate standards-referenced mathematics, providing students with an excellent exploration of standard mathematical concepts through the integration of technology. Students will enhance their learning experience by exploring multiple approaches and receiving fast feedback with apps such as mathematical games.

Start the session with a quick review of the doubling and halving approach. Remind students of the last lesson, in which they learned how to simplify multiplication problems by doubling one number and halving another. For example, doubling 5 to reach 10 and halving 18 to get 9 simplifies 5 × 18 to 10 × 9. Support this concept by providing additional examples and allowing students to ask questions and clarify any misconceptions. Pose an open-ended question to the class, such as, "Imagine you're organising 5 gift bags, each with 18 items. How can you use the doubling and halving approach to quickly determine the total number of items?" Encourage students to consider other approaches to this challenge. Encourage them to use the Mathletics app on their PCs to answer the challenge. Mathletics provides an environment where students may enter numbers, use the doubling and halving approach, and see the answers right away. This software also provides a variety of practice problems and fast feedback, allowing students to compare numerous methods and determine which way is the most effective.

As students work through the problems, they can experiment with different numbers and observe the effects of doubling and halving in various contexts. To reinforce their understanding of the doubling and halving strategy, have students use Mathletics independently. This will encourage individual practice. At the end of the lesson, have students share their findings and discuss how the app helped them understand the strategy better. This will address any remaining misconceptions and strengthen their learning.

Lesson 3: Real-World Applications(AC9M4N06)

In this lesson, students harness multiplication techniques, particularly doubling and halving, to tackle practical scenarios, leveraging digital tools like digital laptops to enhance their learning experience. Experiential learning theory (ELT) emphasises focusing learners' attention on their real-life experiences. Experiential learning involves four steps: concrete experience, reflective observation, abstract conceptualisation, and active experimentation. (Wei 2011, p11) Commencing with a discussion on the real-world applicability of multiplication strategies, students confront a concrete problem

Encourage students to consider how multiplication strategies can be used outside of the classroom by starting the lesson with a discussion and presenting a real-world problem such as, "A gardener needs to plant 8 rows of 15 plants each. How can we use multiplication to find the total number of plants?" Stress the usefulness of doubling and halving in simplifying such calculations, reinforcing earlier learning, and connecting it to real-world applications.

The students should then be divided into smaller groups and given the assignment to work together to solve the problem. Give each group access to digital notebooks so they may record their plans and solutions, along with PCs. In addition to encouraging students to apply the doubling and halving approach, encourage them to investigate other multiplication strategies they have studied. To simplify the computation, they might tackle the problem by halving 15 to obtain 7.5 and doubling 8 to get 16. This educational exercise encourages group conversations among students regarding different approaches. Let each group give a presentation to wrap up the class. Ask them to describe the steps they took to solve the problems and the techniques they employed in their digital notes. In addition to strengthening their comprehension, this gives students a chance to learn from one another as they examine different methods for solving the same issue. By allowing students to both visually and textually record their work, using digital notebooks improves this experience and facilitates sharing and revision. Students will be able to express their ideas clearly and comprehend how multiplication strategies are used in real-world situations thanks to this reflection exercise.

Lesson 4: Simulations and Visualizations (AC9M4N06)

Goal: Gain a deeper grasp of multiplication techniques with the use of simulation software. Materials: PhET Interactive Simulations; computers or tablets. This lesson's main goal is to improve students' understanding of multiplication techniques by using modelling software. Students may modify and understand mathematical concepts through the use of interactive technologies such as PhET Interactive Modelling, which help translate abstract notions.  for example：use  the Area Model Introduction in PhET（https://phet.colorado.edu/en/simulations/area-model-introduction）

Introduce the features of the PhET Interactive Models programme at the beginning of the course. Describe the ways in which students will represent and visualise different mathematical issues with this tool. Emphasise important features that can assist students in understanding the immediate effects of doubling and halving multiplication procedures, such as changing variables and observing changes in real time. This introduction ensures that students are comfortable with the technology and sets the stage for the following hands-on engagement. Next, give the students an open-ended assignment to complete, like figuring out how many things a company produced in total during a given time period. Motivate them to model this problem using the PhET simulations. For instance, students may study how increasing the quantity of boxes and increasing the number of things in each box (or vice versa) impacts the overall production if the factory produces 12 boxes with 20 items in each box.

Through this exploration, students may experiment with different methods in a visual and interactive way while also learning about the practical application of multiplication strategies. End the class with a discussion in which the students present their research and consider the effectiveness of the various tactics they employed. Invite them to talk about how the pictures improved their understanding of the ideas. By helping students witness the instant results of their mathematical manipulations, PhET simulations not only promote deeper knowledge but also reinforce learning through interactive enquiry. Students that engage in this reflective activity will be able to express their ideas clearly and understand how useful simulation software may be while working through challenging multiplication issues.

The courses' teaching techniques include an increased focus on collaborative learning and real-world applications, which greatly enhances student learning results. Students are encouraged to actively engage in the learning process through group projects, conversations, and demonstrations, which fosters a sense of ownership and independence in mathematical research.  These courses encourage students to apply multiplication skills to real-world challenges, reinforcing learning and fostering deeper comprehension. The use of interactive digital tools, collaborative learning approaches, and real-world applications enhances student learning and engagement in multiplication strategy courses. By utilising technology and developing an interactive and inquiry-based learning environment, these courses successfully give students with the required skills and knowledge to confidently answer multiplication problems.

A written quiz might be given, with multiplication questions that need the doubling and halving approach presented in the session. This exam might include both number and word questions to test students' ability to use the method in various settings. There are several ways to measure student understanding and competency with the doubling and halving approach. One way may be to use digital assessment platforms such as Mathletics, in which students are given multiplication problems to answer using the doubling and halving strategy. This would offer quick feedback on both students' and teachers' performance and comprehension.

In this Multiplication Strategies (AC9M4N06) course, students learn about the principles of multiplication, starting with an introduction to doubling and halving procedures and utilising interactive PCs and simulation tools like GeoGebra to see and explore with math ideas. Students learn doubling and halving strategies via group projects and hands-on exercises, while also improving their math fluency and problem-solving abilities. Furthermore, including math game applications and platforms like Mathletics gives kids more practice chances and fast feedback, which strengthens their comprehension and confidence in applying these concepts.

As the course advances, students improve their understanding by investigating real-world applications of multiplication methods and delving further into models with tools like PhET Interactive Models. By relating math ideas to real-world settings and utilising interactive technology, students develop a better understanding of the relevance and use of multiplication in everyday life. Students strengthen their comprehension of multiplication procedures via collaborative problem solving, demonstrations, and reflective conversations, as well as critical thinking abilities and a feeling of mathematical enquiry. Overall, this set of courses provides students with the information, skills, and confidence they need to handle multiplication problems creatively, well, and with a greater understanding of mathematics.

Reference

• Norberg, M. (2021). Exercise design in mathematics textbooks: The case of subtraction. Nordic Studies in Mathematics Education, 26(1), 5-30.

• Wei, C. W., Hung, I., Lee, L., & Chen, N. S. (2011). A joyful classroom learning system with robot learning companion for children to learn mathematics multiplication. Turkish Online Journal of Educational Technology-TOJET, 10(2), 11-23.

• Bakos, C. (2022). The teacher–tool–mathematics ensemble: Before, during and after implementing TouchTimes.

• Pierce, R., & Ball, L. (2009). Perceptions that may affect teachers’ intention to use technology in secondary mathematics classes. Educational studies in mathematics, 71, 299-317.

• Kllogjeri, L., & Kllogjeri, P. (2015). Dynamic models for multiplication and division offered by GeoGebra. American Journal of Software Engineering and Applications, 4(2-1), 1-6.

SplashLearn: Provides numerous resources and interactive activities for fourth-grade math, including multiplication and division problems​ ​. GeoGebra: Offers powerful tools for visualizing and exploring mathematical concepts. Mathletics: A comprehensive app for practicing various math strategies with built-in feedback.

The following set of classes intends to improve students' problem-solving abilities and learn of multiplication strategies through the effective use of technology. Using technology in math classrooms involves a significant shift in attitude for teachers who have previously taught with pen and paper. （Pierce 2009, p300) Teachers will have more challenges to find math resources online, or create games themselves, and prepare more so that children can engage in learning. Encourage students to talk and reflect on their learning on a regular schedule, creating a classroom environment which encourages interactive and collaborative learning.