Zayed_Dina_MTP501_Assessment_1

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Dina. Zayed 20432010 MTP501 - Teaching Number, Algebra and Probability in The Primary Years Assessment: (One) Date: 03/01/2022 1

Introduction “A good teacher can inspire hope, ignite the imagination, and instil a love of learning” (Henry, 2018). The following report analyses Bonnie, Christobel, and Charlie’s work samples on early number, multiplicative thinking, and place values. The students work samples are observed to identify mathematical errors and thus understand mathematical concepts that each student struggles with. Student errors are linked to relevant content descriptors from the Australian Curriculum Assessment and Reporting Authority (ACARA). The report further explores misconceptions around mathematical concepts that students portray whilst suggesting relevant hands-on and learning activities that assist in enhancing student knowledge and understanding. Subsequently, incorporated activities are scaffolded to explicitly demonstrate, engage, and actively motivate students (Henry, 2018; Reys et al., 2020). 2

Bonnie Lack of knowledge Based on Bonnie’s work sample analysis, one can identify that Bonnie lacks knowledge and understanding as she struggles to create a link between numerals and words. Evidently, Bonnie failed to reproduce numbers from their word representation. Furthermore, Bonnie lacks the understanding regarding place values, particularly place values of millions, thousands, and hundreds. It is evident that Bonnie does not understand the scaling of numbers and fails to understand the idea of place values based on division and multiplication of tens and hundreds that assist one with problem-solving ( Hurrel & Hurst, 2016; Reys et al., 2020). Relevant content descriptors: Year 4:  Recognise, represent, and order numbers to at least tens of thousands (ACMNA072) (ACARA, 2016). Elaboration: reproduce five-digit numbers in words utilising numerical representations and vice versa (ACARA, 2016).  Applying place value to partition, rearranging and regrouping numbers to at least tens of thousands assists calculation and solves problems (ACMNA073) (ACARA, 2016). Elaboration: Recognise and demonstrate that place value patterns are founded on the operations of division and multiplying by tens (ACARA, 2016).  Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for the division where there is no remainder (ACMNA076) (ACARA, 2016) Student misconception: Hurrel & Hurst (2016) confirm that multiplicative thinking is measured as the big idea of mathematics underpinned by mathematical knowledge and understanding (Hurrel & Hurst, 2016). Evidently, Bonnie seemed to struggle with associating ten times multiplication and the connect amongst places that extend to parts of 3

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wholes, which is referred to as how many times ‘smaller’ or ‘larger’ and ‘times as many’ (Hurrel & Hurst, 2016; Reys et al., 2020) .). Subsequently, students must develop the knowledge and understanding amongst the numerical names with the concept of the grouping of tens, hundreds, and thousands ( Reys et al., 2020) . Furthermore, Bonnie must be made aware that the way numbers are written must correspond with the idea of groupings. Furthermore, upon analysing Bonnie’s work sample, it was evident that she had troubles understanding where to place the digits within the numeral correctly. Evidently, Bonnie failed to grasp the concept of multiplication and instead misinterpreted it for subtraction (Bonnie has not grasped the understanding of how many times bigger means multiplication). Consequently, students commonly make the mistake of writing numbers as they are read out; an example of this includes one hundred and four thousand, five hundred and sixty as 100,4000560 (Mix & Partner, 2015; Reys et al., 2020 ). Activities: Activity 1: Match the numbers with the correct place value The activity has been selected to allow Bonnie to transform larger numbers from alphanumerical into numerical form. In turn, a place value chart is provided to assist Bonnie to establish the place value of each digit within the numeral. This activity aids Bonnie to write numbers out in an expanded form (Mix & Partner, 2015). The activity further reinforces knowledge in place values up to thousands in blocks of ten (Mix & Partner, 2015). Teachers can introduce the topic and confirm that they will be utilising place values to assist in reading numerals to assist write numbers of more significant value ( Reys et al., 2020). Consequently, teachers will review place values from smallest to largest as they begin with small values and slowly increase to larger ones (Mix & Partner, 2015). Example: Together, students will be asked to use their place value chart to relate numbers to their correct value as a class . Bonnie will be instructed to fill in the place value chart from right to left depending on the numeral given. Once Bonnie completes the activity, the answer will be presented as a class and individually checked by the teacher. Students who require further assistance will have a couple of examples scaffolded and demonstrated to allow them to grasp the concept (Mix & Partner, 2015; 4

Reys et al., 2020). Students will then complete a place value matching chart as part of a formative assessment to check for student understanding. Activity 2: Changing Places derived from: (DETWA, 2004) Bonnie will utilise Multi–Base Arithmetic Blocks (MAB) to explicitly demonstrate the association between places to enhance understanding of values and differences ( Reys et al., 2020). The teacher will begin by scaffolding the smallest MAB block and asking Bonnie, “what number does this look like?” Bonnie is expected to respond with the numeral one. Subsequently, students will be grouped in pairs of two and given a large base ten cubes to share and follow through with the teacher. Students will then be asked to count in hundreds with the teacher to confirm how many blocks there are. This demonstration engages and motivates students to effectively enhance their knowledge and understanding of counting large numbers ( Hurrel & Hurst, 2016; Reys et al., 2020). The teacher will then test students by showcasing the most extensive base ten cube and prompt students with “what number is this?” depending on Bonnie’s attentiveness within the lesson, she may or may not be able to respond with the correct answer of 1000 times larger. The teacher will then explain the answer and how it is 1000 times bigger to clarify and ensure student understanding by all. Bonnie will then be asked to write down the value of each demonstrated cube within their maths books to allow students to refer back to at any given time. 5

Both activities involve the explicit strategy of scaffolding to enhance student perception, knowledge, and understanding and thus, allow students to successfully and independently complete activities in future (Pfister & Opitz, 2015; Reys et al., 2020). Through such an explicit strategy, the teacher ensures a practical interpretation of student learning difficulty and can intervene by providing students with explicit guidance and instructions on independently completing allocated tasks (Pfister & Opitz, 2015). Furthermore, utilising tangible resources such as Multi–Base Arithmetic Blocks (MAB) and place value charts and activities explicitly and purposefully assist Bonnie with the ‘times bigger’ activity and develop enhanced levels of understanding (Reys et al., 2020). Christobel Lack of knowledge Upon analysing Christobel’s work sample, one can identify Christobel struggle with the concept of subtraction and addition and the representation of a digit when regrouping values. Christobel demonstrates her inability to align the equation in the correct sequence, which resulted in subtracting rather than adding the numbers. Furthermore, it is evident that Christobel lacks the conceptual knowledge and understanding of decimal place values. In turn, causing difficulty to distinguish and rationalise which decimal sets of value is greater (Reys et al., 2020). Relevant content descriptors: Year 3:  Apply place value to partition, rearrange, and regroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053) (ACARA, 2016). Elaboration: Recognise the value of 10,000 equals ten thousand, 100 equals one hundred, and 10,000 one’s plus justified choices regarding 6

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regrouping and partitioning based on usefulness for specific calculations (ACARA, 2016). Year 5:  Compare, order, and represent decimals with the elaboration of locating decimals on a number line sentence (ACMNA105) (ACARA, 2016). Elaboration: Locating decimals on number line sentences (ACARA, 2016). Student misconception: Commonly, students often make the mistake of mixing the concept of whole numbers with decimals. Students frequently believe numbers containing the most significant number of digits have a larger value. They assume numbers with the most significant number of digits after the decimal point has the larger value (Steinle, 2004). Nonetheless, this is not applicable to decimals. Furthermore, various students may select lengthier decimals as more significant numbers in other circumstances and often still select the correct answer if the first decimal digit is a 0 (Reys et al., 2020; Steinle, 2004). For example, students can often be under the impression that 0.43 is significant than 0.5; however, they will recognise that 0.042 is less in value than 0.5. these students are often referred to as ‘column overflow thinkers’ (Beckmann, 2006), whereby they recognise the relationship between decimal fractions; however, they experience difficulties understanding the fundamentals of place values (Clarke et al., 2008). Consequently, students who lack understanding of decimal points and what they signify have great misconceptions as they may recognise the decimal point as a divider amongst numbers (Jong & Fisher, n.d; Reys et al., 2020). Subsequently, this is evidently portrayed within Chirstabel’s work sample, whereby she has written that twenty-eight is greater than three. Alternatively, Christobel misconceived subtraction whereby she viewed each column as separate subtractions instead of looking at the entirety of 7

the calculation. Students who do this often misconceive subtraction from zero will most likely occur. Thus, Christobel must enhance her understanding and knowledge of zero as a placeholder as she fails to understand the concept of place value which causes difficulties to identify and justify which decimal is larger (Mathematics Mastery, 2014; Reys et al., 2020). Activities: Activity 1 : Colour the Decimals. Activity derived from (Roche, 2010) Christobel is provided with two dice, one regular six-sided dice and another six-sided dice labelled with 1/10, 1/100,1/100,1/1000,1/1000, 1/1000. Christobel will also be provided with a deciMAT game board along with coloured pencils and a table to record results. Importantly, students are grouped in pairs and will be provided with explicit instructions on what they are required to do. Christobel is instructed to begin by rolling the two dice simultaneously and then record and shade the displayed number on the table using the pencils provided. For example, rolling a three and 1/1000 forms 3/1000. Christobel and peers were then required to record the result as a decimal and fraction. Students will take turns to roll both dies and are required to record a decimal of the total amount shaded in the last column. As Christobel continues to play, she will be able to see that the last column will resemble the total shaded. Consequently, Christobel will be instructed to colour each result with a different colour to allow the teacher and student to observe and match the recorded result to check for any errors ( Mathematics Mastery, 2014; Reys et al., 2020). Activity 2: Regroup popsicle stick lesson derived from (Popsicle Stick Regrouping Fun –n.d) Christobel is introduced to the activity and advised that they will be practising how to regroup values. Teacher to provide Christobel with explicit instructions on what is required for the task to ensure involvement and engagement. Firstly, Christobel will create a number using cards and then with popsicle sticks. Doing this allows students to learn and practice regrouping numbers to assist with 8

subtracting numbers with ease through a hands-on approach ( Mathematics Mastery, 2014; Reys et al., 2020). Christobel is then instructed to place down the card with the numeral one and place all numbers with one digit under this card. Christobel will then be asked, “what is the total amount of popsicle sticks that fall under this?”. Furthermore, once students gather ten popsicle sticks, they must regroup them by placing an elastic band around to allow moving the group of sticks from one group to another. Subsequently, Christobel will be prompted to take the ‘ten’ card and lay it to the right of the ‘one’ card. Christobel advised only nine groups of popsicle sticks could remain. Once Christobel reaches up to ten groups of ten popsicle sticks, they are required to start a new column. Christobel is prompted with “take a few moments to think of what we will label this column?” Christobel may or may not know the answer. Teacher to confirm the correct answer of “the column will be labelled hundreds column” ( Mathematics Mastery, 2014; Reys et al., 2020). Christobel is then prompted with a subtraction question: What is fifty-one minus twenty-nine? And asked where they believe the five groups of ten should be placed. Students who understand the activity will be able to direct focus to the five-column and the single one on the one’s column. While the two groups of ten on the two and similarly the nine singles under the nine. Christobel will be asked, “can you take away the nine from the one?”, Christobel should be able to respond with no. Christobel is then asked to regroup the popsicle sticks to allow them to subtract the nine. Christobel will be asked to take one of the groups from the numeral five, undo the elastic, and place the ten popsicle sticks next to the one so they end up with eleven sticks. Christobel will then be asked, “can we take nine away from eleven after you have regrouped?” Christobel should be able to respond with yes and confirm that the answer is three. The above steps will be repeated numerous times; however, numbers will be changed to check for student understanding and ensure that Christobel has 9

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effectively grasped the concept of regrouping ( Mathematics Mastery, 2014; Reys et al., 2020). Charlie Lack of knowledge Upon analysing Charlie’s work sample, one can identify Charlie’s struggles with multiplying by utilising the partition rule appropriately (long multiplications and carrying numbers over). Furthermore, it is evident that Charlie demonstrates her inability to understand the concept of extending the place value system beyond the tens value as the numbers increase. Relevant content descriptors: Year 4:  Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for the division where there is no remainder (ACMNA076) (ACARA, 2016). Elaboration: Utilising known strategies and facts, including commutativity, by halving and doubling multiplication, along with the connection of division to multiplication when there are no remainders (ACARA, 2016). Year 5:  Solve problems involving multiplication of large numbers by one – or – two-digit numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100) (ACARA, 2016). Elaboration: Discovering methods to multiply, including partitioning of numbers, area model, and the Italian lattice method (ACARA, 2016). Student misconception: Evidently, it is clear that Charlie experiences great confusion with looking at each number separately instead of considering the value when multiplying two-digit factors. Students commonly misconceive multiplying numbers as they take the value they see and disregard the place value (Mix & Prather, 2015). It is apparent that Charlie understands that thirty times twenty is 10

equal to 600, and four times six is twenty-four, signifying that he understands the concept of regrouping numbers. However, Charlie fails to understand the split strategy also, known as partitioning in a multiplicative question, and that each number is required to be multiplied by every number within the question (Hurrel & Hurst, 2016). Charlie has wrongly completed her working out and thus resulting in the wrong answer. Charlie was required to break down the algorithm and utilise expanded notation. Student misconception of the split strategy commonly arises as they fail to grasp the basic foundational concepts of multiplication, such as expanded notations and place values (Reys et al., 2020). Activities: Activity 1 : The Box/Window Method The box/window multiplication method will effectively enhance Charlie’s knowledge and understanding of multi-digit multiplication (Reys et al., 2020). The teaching strategy emphasises on number sense knowledge and understanding to allow students to utilise the expanded form of each factor (Reys et al., 2020). Essentially, this will help Charlie to comprehend what each digit in every factor actually means. Charlie will be provided with explicit instructions beginning with drawing a box within their mathematics book; the total number of rows and columns drawn will depend on each factor's number of digits (Mix & Prather, 2015). Charlie will be asked to decide how to partition the box and break numbers into an expanded form. Such as, thirty-four is thirty and twenty- six is twenty and six. The expanded form will then be written along the grid. Once completed, Charlie will be asked to “multiply each number that corresponds”. Once Charlie multiplies all factors, she will be asked to add all the numbers within the boxes to find the answer (Mix & Prather, 2015; Reys et al., 2020). 11

Activity 2: Ten times smaller activity derived from (First Steps Mathematics – The Department of Education) Charlie is involved in a hands-on activity by using a calculator to solve mathematical questions. Charlie will use the calculator to identify what ten times, one hundred times bigger and smaller than the provided set of numbers. Charlie will be given the following set of numbers: 10, 100, 1000, 10000. The answer will be scaffolded to enhance Charlie’s knowledge and understanding (Mix & Prather, 2015; Reys et al., 2020). Charlie is then asked to record the answers to reflect and assist when completing the task independently. Charlie will be prompted with “how did we come up with the answer?” and “are we required to complete the same steps for each number?”. Charlie will then be given another set of numbers including 20, 200, 2000, 20000 and 30, 300, 3000, 30000, whereby they will be required to complete independently, and record answers as the teacher will check them over to ensure student understanding. Upon completing the activity, Charlie will then be asked to reflect on what they learned to check Charlie’s understanding whilst confirming whether it was effective. Based on Charlie’s answer, the teacher is given the opportunity to further elaborate on portrayed weaknesses and acknowledge portrayed strengths (Mix & Prather, 2015; Reys et al., 2020). 12

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Conclusion: Upon analysing Bonnie, Christobel, and Charlie's work samples, it is evident that the students share mutual mathematical misconceptions. As a result of the misconceptions, teachers are provided with opportunities to implement various explicit strategies and learning styles to enhance student knowledge and understanding within their learning journey (Mix & Prather, 2015; Reys et al., 2020). Implementing a student-centred learning style with explicit teaching strategies such as hands-on activities and scaffolding creatively and innovatively engages students enthusiastically and invitingly instead of the traditional teaching style (Reys et al., 2020). Studies suggest that to acknowledge and understand critical mathematical concepts, including multiplicative thinking and place values, one must ensure students understand the foundations to solve problems accurately (Mix & Prather, 2015). 13

References: Australian Curriculum Assessment and Reporting Authority. (2016). The Australian Curriculum, All curriculum elements. http://www.australiancurriculum.edu.au/download/f10 . Australian Curriculum and Assessment Reporting Authority. (2016). Year 3, Mathematics. Australian Curriculum and Assessment Reporting Authority. (2016). Year 4, Mathematics. Australian Curriculum and Assessment Reporting Authority. (2016). Year 5, Mathematics. Australian Institute for Teaching and School Leadership. (2017). Australian Professional Standards for Teachers . https://www.aitsl.edu.au/teach/standards. Baker, C., & Ward, J. (2015). Base Ten and Place Value . National Council of Teachers of Mathematics. Beckman, S. (2006). Mathematics for Elementary Teachers (2nd ed., pp. 7-12). Sydney: Pearson. 14

Clarke, D., Roche, A., & Mitchell, A. (2008). Ten practical tips for making fractions come alive and make sense. Teaching Children Mathematics (13th ed., pp. 372-380). Department of Education and Training in Western Australia [DETWA], (2004). First Steps in Mathematics : Book number 1. Perth, Western Australia: Roberts, J. Hurrel, D., & Hurst, C. (2016). Multiplicative thinking quiz (v.11). [Lecture notes]. https://lms.curtin.edu.au/webapps/blackboard/content/listContentEdi table.jsp? content_id=_9441894_1&course_id=_112507_1&mode=reset Jong, C., & Fisher, M. [n.d] Decimal Dilemmas: Interpreting and Addressing Misconceptions . Ohio Journal of School Mathematics, 14-18. Mathematics Mastery., Common Errors and Misconceptions in Column Addition and Subtraction. . Toolkit.mathematicsmastery.org. (2014). http://toolkit.mathematicsmastery.org/app/webroot/js/tiny_mce/plug ins/moxiemanager/data/files/Common%20errors%20and %20misconceptions%20in%20column%20addition%20and %20subtraction.pdf . Mix, K., & Prather, R. (2015). Young Children's Interpretation of Multi-digit Number Names: Emerging Competence to Mastery. Child Development, 85(3). Numbers and Operations in Base 10 and Numbers and operations – Fractions . Pdo.ascd.org. (2011). https://pdo.ascd.org/lmscourses/PD11OC129/media/CCSS_Math_K- 5_M05_Reading_NBT-FractionsII.pdf Pfister, M., & Opitz, E. (2015). Scaffolding for Mathematics teaching in inclusive primary classrooms. ZDM Mathematics Edition, 47(7). Popsicle Stick Regrouping Fun – Lesson This. Lessonthis.com. http://www.lessonthis.com/math/popsicle-stick-regrouping-fun/ Reys, R., Lindquist, M., Lambdin, D., Smith, M., Rogers, A., & Cooke, A. et al. (2020). Helping children learn mathematics (3rd ed.). Milton, QLD: Wiley. 15

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Roche, A. (2010). Decimats: Helping Students To Make Sense of Decimal Place Value. https://files.eric.ed.gov/fulltext/EJ891799.pdf. Steinle, V. (2004). Changes with Age in Students Misconceptions of Decimal Numbers. Department of Science and Mathematics Education – The University of Melbourne. 16

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