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Caroline Reed 21067840 EDP243 Children as Mathematical Learners A1 – Investigation Report A. Relational / Conceptual In Chapter 2 of Van de Walle et al.'s (2019) book, the authors delve into the distinction between "relational" and "conceptual" understanding in the context of mathematics education. This differentiation is crucial for educators as it profoundly influences how students grasp mathematical concepts, particularly evident in the teaching and learning of place value (Van et al., 2019, p.22). This summary aims to clarify the distinctions between these two types of understanding and provide insight into their implications for place value instruction. Relational understanding refers to a fundamental comprehension of how mathematical concepts relate to each other and to real-world situations (Van et al., 2019, p.22). In a relational understanding, students grasp the interconnectedness of mathematical ideas rather than memorising isolated rules or procedures (Van et al., 2019, p.22). It involves the ability to flexibly apply mathematical principles across different contexts (Van et al., 2019, p.22). For example, in the context of place value, students with a relational understanding recognise that the value of a digit in a number depends on its position, enabling them to perform addition and subtraction with ease (Van et al., 2019, p.23). Conceptual understanding, on the other hand, goes deeper by focusing on the underlying structures and principles of mathematical concepts. It involves making sense of why mathematical procedures work and how they are derived (Van et al., 2019, p.25). In the context of place value, a student with a conceptual understanding not only knows that the value of a digit is determined by its position but also comprehends the mathematical reasoning behind it, such as powers of ten (Van et al., 2019, p.25). This deeper understanding fosters mathematical fluency and problem-solving abilities. One crucial term related to learning and understanding is scaffolding. Scaffolding involves providing students with support and guidance to help them learn and understand new concepts (Van et al., 2019, p.31). In the context of teaching place value, a teacher might scaffold a student's learning by using concrete manipulatives like base-ten blocks to demonstrate the relationship between digits and place value. As the student becomes more proficient, the scaffolding can gradually be reduced, allowing for independent understanding (Van et al., 2019, p.32).

Caroline Reed 21067840 EDP243 Children as Mathematical Learners Another important concept is productive struggle. This refers to the idea that learning often involves grappling with challenging problems or concepts. It is through this struggle that students deepen their understanding and develop problem-solving skills (Van et al., 2019, p.25). In place value instruction, educators should create opportunities for students to engage in productive struggle by presenting them with tasks that require thinking, reasoning, and experimentation. The process of learning and understanding in mathematics can also be associated with accommodation and assimilation, terms derived from Jean Piaget's cognitive development theory (Van et al., 2019, p.27). Accommodation occurs when students modify their existing mental structures to incorporate new information or experiences. Assimilation, on the other hand, involves incorporating new information into existing mental structures without significant modification (Van et al., 2019, p.27). In the context of place value, students might assimilate the idea of "borrowing" in subtraction into their existing knowledge of addition, while accommodation might involve reorganising their understanding of numbers to accommodate place value principles. In conclusion, Van de Walle et al.'s (2019) distinction between relational and conceptual understanding is pivotal in mathematics education. Place value instruction serves as a prime example of how these two types of understanding manifest in the classroom. Educators must employ scaffolding, encourage productive struggle, and differentiate their instruction to help students develop both relational and conceptual understandings, thus fostering true mathematical proficiency and problem-solving skills. Accommodation and assimilation also play a role in how students adapt and integrate new mathematical concepts into their existing mental frameworks, contributing to their overall mathematical development. B. Resources Year 2: Place Value Ice Cream Cones Resource: ACARA Year 2 Mathematics Workbook by Pascal Press. Rationale: This resource offers an engaging and fun way for Year 2 students to grasp the concept of place value (Year 2 Maths - Excel Basic Skills Maths | Pascal Press, n.d.). It aligns with the Australian Curriculum: Mathematics Year 2 content description code ACMNA027. This code states, "Recognise, model, represent and order numbers to at least 1000," (ACARA, n.d.) which is an essential skill in developing a strong foundation in mathematics.

Caroline Reed 21067840 EDP243 Children as Mathematical Learners The "Place Value Ice Cream Cones" activity involves printable worksheets with images of ice cream cones. Each cone has three scoops, with numbers written on them. Students are required to cut out the scoops and place them on the cones to form a 3-digit number. This hands-on activity allows students to physically manipulate the digits, promoting a deeper understanding of place value. It also incorporates visual aids, making it an excellent resource for visual learners. Year 2 Ice Cream Cones Activity (Twinkl, n.d.) Year 3: Place Value Bingo Resource: Australian Curriculum: Mathematics Year 3 – Bingo Cards by Australian Curriculum Lessons. Rationale: Place Value Bingo is a fantastic resource for Year 3 students, aligning with the Australian Curriculum: Mathematics Year 3 content description code ACMNA057. This code states, "Represent and solve problems involving multiplication using efficient mental and written strategies and appropriate digital technologies" (ACARA, n.d.). Bingo helps students develop fluency in recognising place value patterns in numbers, a crucial skill for solving problems involving multiplication (v_sheehan, 2013). The Place Value Bingo cards feature various 3-digit numbers in expanded form (e.g., 300 + 40 + 5) or as numerals (e.g., 345). Students mark the numbers on their cards as they are called out. This game promotes

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Caroline Reed 21067840 EDP243 Children as Mathematical Learners engagement, reinforces place value concepts, and enhances mental math skills. It also encourages healthy competition, making learning more enjoyable. (Twinkl, n.d.) Year 4: Place Value Flip Books Resource: Teaching Resource: Place Value Flip Books by Teach Starter. Rationale: Place Value Flip Books are a hands-on resource that caters to the Year 4 curriculum's content description code ACMNA073, which states, " apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems." Understanding place value is essential for building this connection. The flip books consist of printable templates where students create flip cards for each place value position (ones, tens, hundreds, thousands). They write numbers, words, and expanded forms on the flip cards and then assemble them. This interactive activity reinforces the concept of place value and its relationship to arithmetic operations (Starter), 2017). It helps students see how changing one digit impacts the overall value of a number. The visual and tactile aspects of flip books provide a multi-modal learning experience, catering to diverse learning styles. Year 4 Place Value Flip Books (Starter), 2017)

Caroline Reed 21067840 EDP243 Children as Mathematical Learners In conclusion, these three Australian-sourced resources are excellent for teaching place value at different year levels. The Place Value Ice Cream Cones activity is a hands-on introduction for Year 2 students. Place Value Bingo is an engaging game for Year 3, promoting fluency in recognising place value patterns. Place Value Flip Books offer Year 4 students a multi-modal approach to understand the connection between place value and arithmetic operations. These activities align with the Australian Curriculum: Mathematics and provide a well-rounded foundation in place value for students. C. Multiplicative Thinking Multiplicative thinking is a fundamental concept in mathematics, focusing on understanding and using multiplication as a primary operation to solve problems (Van et al., 2019, p.173). It goes beyond mere memorisation of times tables and instead involves a deep comprehension of the principles and relationships involved in multiplication (Van et al., 2019, p.173). This conceptual shift allows students to solve a wide range of mathematical problems more efficiently and accurately. In essence, it's about recognising the multiplicative relationships between numbers and using this understanding to make sense of and solve problems (Van et al., 2019, p.173). Diagnostic Questions for Year 5: "If you have 4 boxes, and each box contains 5 apples, how many apples do you have in total?" This question assesses a Year 5 student's ability to recognise the multiplicative relationship between the number of boxes and the number of apples in each box. A student using multiplicative thinking will correctly respond with "4 x 5 = 20 apples." "If there are 15 students in a class, and each student has 3 pencils, how many pencils are there in total for the entire class?" This question evaluates the student's comprehension of multiplication as repeated addition. A Year 5 student employing multiplicative thinking will calculate "15 x 3 = 45 pencils" by recognising that 15 groups of 3 pencils must be added together. Diagnostic Questions for Year 6:

Caroline Reed 21067840 EDP243 Children as Mathematical Learners "You want to buy 8 packs of trading cards, and each pack costs $5. How much will you spend in total?" This question assesses a Year 6 student's ability to use multiplicative thinking to solve real-life problems involving multiplication. A student employing multiplicative thinking will correctly determine "8 x $5 = $40." "If there are 24 students in a classroom, and each student needs 4 markers for an art project, how many markers are needed in total for the whole class?" This question evaluates the student's ability to understand multiplication as a way of finding the total in a group. A Year 6 student using multiplicative thinking will correctly calculate "24 x 4 = 96 markers," recognising that each student contributes 4 markers. For the Year 5 questions, if a student answers them correctly, it indicates that they possess basic multiplicative thinking skills. They can recognise and apply multiplication to solve problems involving equal groups or repeated addition (Van et al., 2019, p.185). An incorrect response suggests a need for further development in understanding the concept. In the Year 6 questions, a correct response shows that the student has a more advanced grasp of multiplicative thinking, successfully applying it to real-life situations (Van et al., 2019, p.185). An incorrect answer may signal that the student needs more practice in using multiplication for more complex problem- solving, indicating a need for further instruction. These diagnostic questions are effective tools for assessing a student's level of multiplicative thinking. Understanding and applying this concept is essential for a solid foundation in mathematics, as it forms the basis for more advanced mathematical concepts in the later years of education. D. Learning Activities Activity 1: NCTM Illuminations - "Factorise" The "Factorise" activity from NCTM Illuminations, noted by Van de Walle (2019, p.74), is an excellent choice for promoting multiplicative thinking. This online interactive tool helps students explore the concept of

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Caroline Reed 21067840 EDP243 Children as Mathematical Learners prime factorisation by breaking numbers into their prime factors. It encourages students to think multiplicatively because they must recognise how numbers can be decomposed into prime factors, which is a fundamental concept in multiplication (Illuminations, 2019). This activity also allows students to see the connection between multiplication and division, further enhancing their understanding. Follow-up Questions: 1. What did you learn about prime factorisation from this activity? 2. Can you explain how you determined the prime factors of the given numbers? 3. How do prime factors relate to multiplication and division? 4. Can you find other numbers that have the same prime factorisation? Activity 2: "Virtual Supermarket Multiplication" (Online Resource) Description: "Virtual Supermarket Multiplication" is an online activity accessible through Problemo. It engages students in a virtual supermarket scenario, helping them understand multiplication in a real-world context ( Problemo - Powered by Australian Maths Trust , n.d.). Instructions: 1. Present students with a virtual supermarket environment where they can "shop" for various items with different quantities and prices. 2. Students need to calculate the total cost of their shopping by multiplying the quantity of each item by its price. 3. Encourage them to use multiplication as a practical tool for solving a real-life problem. Online resources like "Virtual Supermarket Multiplication" offer an interactive and authentic context for learning multiplication. It connects mathematical concepts to real-world applications, which is essential for developing multiplicative thinking ( Problemo - Powered by Australian Maths Trust , n.d.). Such activities are engaging and suitable for digital-age learners. Follow-up Questions:

Caroline Reed 21067840 EDP243 Children as Mathematical Learners 1. How did you apply multiplication to calculate the total cost of your shopping? 2. Can you think of other real-life situations where multiplication is useful? 3. Was this online activity helpful in understanding the practical uses of multiplication? Why or why not? Incorporating these activities in the classroom fosters multiplicative thinking in a social constructivist manner by combining hands-on, creative, and digital approaches. Follow-up questions aim to assess students' comprehension and the effectiveness of these activities in achieving the intended learning gains. E. Investigation "Multiplication is just repeated addition." The claim that multiplication is merely repeated addition is a common oversimplification of a fundamental mathematical concept (Hurst, 2015, p.10). While multiplication does involve adding numbers, it is a more complex operation with distinct characteristics. Multiplication essentially represents the scaling or resizing of numbers, which is different from the notion of addition, where you combine values to find the sum (Hurst, 2015, p.10). Multiplication is fundamentally tied to the concept of groups or sets (Hurst, 2015, p.10). For example, when we multiply 3 by 4 (3 x 4), we are essentially finding the result of adding 3 four times. However, this is not the same as saying that 3 x 4 is the same as 3 + 3 + 3 + 3. This interpretation of multiplication may work for small integer values but becomes problematic when dealing with fractions, decimals, or negative numbers. Multiplication exhibits unique properties that addition lacks (Hurst, 2015, p.12). For instance, multiplication is commutative (a x b = b x a) and associative (a x (b x c) = (a x b) x c), while addition follows these properties as well. Still, it is essential to distinguish between these operations for a more comprehensive understanding of mathematics(Hurst, 2015, p.12). "All children learn mathematics in the same way." The claim that all children learn mathematics in the same way is not supported by empirical evidence or educational theory Burns. (n.d.). Children have diverse learning styles, cognitive abilities, and educational

Caroline Reed 21067840 EDP243 Children as Mathematical Learners backgrounds, which influence how they grasp mathematical concepts Burns. (n.d.). Mathematical learning is a complex process influenced by several factors, including cognitive development, socio-economic status, cultural background, and individual learning preferences Burns. (n.d.). Vygotsky's sociocultural theory highlights the significance of social and cultural context in learning (Gowrie, 2022). It suggests that children from different backgrounds may approach mathematical concepts differently (Gowrie, 2022). Moreover, Gardner's theory of multiple intelligences underscores that individuals possess various types of intelligence, and their mathematical abilities may vary based on these intelligence types (Brualdi, 1998). Additionally, research in mathematics education has shown that effective teaching methods need to cater to diverse learning styles, including visual, auditory, kinesthetic, and tactile learners (Van et al., 2019, p.68). Tailoring instruction to suit these different learning styles can enhance students' understanding and retention of mathematical concepts. Furthermore, special educational needs and disabilities also impact how children learn mathematics. Some students may require individualised approaches and additional support to master mathematical skills.

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Caroline Reed 21067840 EDP243 Children as Mathematical Learners References Australian Curriculum, Assessment and Reporting Authority [ACARA], n. d. a Brualdi, A. (1998). Gardner’s theory. Multiple Intelligences. Burns. (n.d.). Snapshots of student misunderstandings. Chris Hurst. (2015). The multiplicative situation. Australian Primary Mathematics Classroom, 20(3), 10–16. Gowrie. (2022). Lev Vygotsky’s Theory of Child Development - Gowrie NSW. Www.gowriensw.com.au. https://www.gowriensw.com.au/thought-leadership/vygotsky-theory Illuminations. (2019). Nctm.org. https://illuminations.nctm.org/ Starter), S. (Teach. (2017, May 18). 4-Digit Place Value Card Game - Flip It! Teach Starter. https://www.teachstarter.com/au/teaching-resource/4-digit-place-value-card-game-flip/? queryID=7e6f19e1e961c71b0b68b2a0272d82e2&objectID=487940 Twinkl. (n.d.). Ice Cream Cone Number and Word Matching Activity. https://www.twinkl.com.au/resource/t-n-4531-new-ice-cream-cone-number-and-word-matching-activity Van, D. W. J., Karp, K., Bay-Williams, J., Brass, A., Bentley, B., Ferguson, S., Goff, W., Livy, S., Marshman, M., & Martin, D. (2019). Primary and middle years mathematics : Teaching developmentally ebook. Pearson Education Australia v_sheehan. (2013, April 7). Multiplication Bingo - A lesson in practising timestables. Australian Curriculum Lessons. https://www.australiancurriculumlessons.com.au/2013/04/08/multiplication-bingo-a-lesson-in- practising-timestables/ Year 2 Maths - Excel Basic Skills Maths | Pascal Press. (n.d.). Pascalpress.com.au. Retrieved October 12, 2023, from https://pascalpress.com.au/mathematics-year-2-old/