EDC243 Children as Mathematic Learners

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Curtin University *

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Jan 9, 2024

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EDC243 Children as Mathematic Learners Assessment 1: Report Curtin University Introduction This report will analyse and discuss the results of a Place Value Assessment Interview (PVAI) conducted with a Year 5 student (Toni). This report will describe the aspects of place value this student is experiencing difficulties with and look to suggest a series of appropriate teaching activities that could be used to assist this student. This report will then discuss the planning of future interventions for the student to continue past her present stage while building on her previous skills, knowledge and understanding.

Place value (PV) is the basis of our number system, it is the system in which the position of a digit in a number determines its value. Place value understanding develops from a variety of experiences, such as mental computation and counting, usually in base tens to begin with (Reys et al., 2020). The four properties of PV as discussed by Ross (2002) are Additive property; where the quantity that represents the whole numeral is the sum of the values represented by the individual digits, Positional property; where quantities that represent the individual digits are determined by the positions that they hold in the whole numeral, Base-ten property; in which the values of the positions increase in powers of ten from right to left, and Multiplicative property; where the value of an individual numeral is found by multiplying the value of the digit by the value assigned to its position (Ross, 2002, p. 419). There are set stages of development for PV that align with the First Steps in Mathematics (FSiM) key understandings (KUs) (DoEWA, 2013), these stages help educators to understand a student's progress and mathematical development. To further assess a student's progress through these stages, diagnostic interviews such as a PVAI (Place Value Assessment Interview) are conducted, in which the stages of development structure a sequence of mathematic skills students should be able to achieve unassisted at various points academically. These interviews then help teachers identify misconceptions or spots of difficulty students may be experiencing, so they can then create learning interventions and corrections to ensure students are aligned to the average level of skill for their age group. Diagnostic interviews are a tool that is underpinned by a constructivist approach to learning, in which the student is asked to actively construct or make their own knowledge based off their own experiences and understandings of the materials taught to them (Elliott et al., 2000, p. 256). Tools such as a PVAI, use Vygotsky’s theory of the Zone of Proximal Development (ZPD) to identify gaps in a student’s knowledge and play on Vygotsky's concept that when a student is in the zone of proximal development for a particular task, providing the appropriate assistance will give the student enough of a boost to achieve the task (Vygotsky, 1978, p. 86). To assist the student through the ZPD teachers are asked to focus on three main components as explored for Toni in this report; the presence of a teacher or someone who has a higher level of skills than the learner, social interactions with a skilled tutor that allow the student to practice and observe their skills (such as a PVAI), and the use of scaffolding or supportive activities to support the student

through the ZPD as explored in this report in the intervention, contingency and future learning sections. Analysis of the Diagnostic Interview Through the PVAI conducted, it is clear Toni has several misconceptions and difficulties where correction is required. The following section will identify which questions from this interview she has issues. Questions 4 and 6 highlighted Toni’s issues with FSiM (DoEWA, 2013) KU2, with failing to be able to correctly look at groups and quantification and has been unable to correctly represent numbers with the material (icy pole sticks) as well as KU6 where she has been unable to correctly use partitioning of numbers by place value. In terms of place value development, here it is clear Toni sits at stage 2, with a slight development into stage 3, but is not ready to progress (DoEWA, 2013). Here it is identified that Toni is currently at a year 2 level, as she meets the Australian curriculum content descriptor ACMNA031, recognise and represent multiplication as repeated addition, groups, and arrays (ACARA, 2017), which is well below the appropriate skill level for a year 5 student. In question 10, it is shown that Toni is unable to successfully model FSiM KU8, failing to compare and order the numbers themselves. This is particularly clear in the second part of the question where Toni states she is unable to attempt ordering the numbers on a line as the second line starts at ‘39’ and not ‘0’. Here Toni demonstrates she is below a year 2 level in her place value development as she is unable to meet ACMNA027 by failing to order numbers to at least 1000 (ACARA, 2017) Question 12 is also a problem area for Toni, with her failing to correctly partition three-digit numbers, and therefore is unable to meet FSiM KU6, looking at flexible partitioning by place value. This places Toni at a stage 3 for her place value development and year 3 level according to the ACARA (2017) content descriptors, as she currently is unable to meet year 4 level ACMNA073, applying place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems (ACARA, 2017). It can be identified that Toni is at a year 2 level, as she is able to identify ACMNA028 as she has use partitioning to

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group 1000 in hundreds, tens, and ones to facilitate more efficient counting, and split the three- digit number into its base-ten properties. Question 13 demonstrates Toni’s inability to use multiplicative properties within place value. Here she has demonstrated in her answer that she is an additive thinker (Reys et al., 2020). She has failed to meet KU4 and KU5 of FSiM, and has shown a misconception with the term ‘times bigger’ and having interpreted that as addition and not a multiplication phrasing. Toni has been unbale to meet stage 5 of her place value development as her misconceptions around additive versus multiplicative properties requires correction. Due to this misconception, this means Toni’s skill level is at a year 2, aligning with ACMNA031 (ACARA, 2017). Through the answering of questions 14 and 15, it is identified that Toni has achieved up to level 2 of the FSiM development stages aligning with FSiM KU2 in being able to write the 'ten group' as the basis for building numbers beyond single digits and is working on progressing to level three. It is clear Toni is an additive thinker (Reys et al., 2020) and she has not yet mastered the skill of being able to use horizontal patterns to recognise the ‘second group’ of numbers in a 100- 10-1 pattern. In this question we can see that Toni is unable to meet the requirements of reading and writing four-digit numbers and therefore does not meet ACMNA052, recognising, modelling, and ordering numbers to at least 10,000, which is a year 3 level skill, and fails to successfully master FSiM KU4, with being unable to recognise the pattern in which number are said and KU5, with being able to write numbers correctly. Intervention and Contingencies To correct Toni’s inability to successfully model question 10 and KU8, failing to compare and order the numbers themselves, the following task can be used to scaffold and correct her misconception that place value in a line and ordering numbers is not dependent on the entire sequence from 0 – maximum number being there, for example, where she states she is unable to attempt ordering the numbers on a line as the second line starts at ‘39’ and not ‘0’. An intervention activity to correct this would be the ‘three in a line’ game (see Figure 1), as it looks at number ordering and place value This activity could be amended slightly so the line did not start with ‘0’ to assist Toni with being able to successfully identify where on a line the number

she rolls goes. This activity also is supported by the ZPD (Vygotsky, 1978, p. 86), as if Toni were to play with a higher skilled peer or teacher, she would use the social interaction with a skilled tutor allowing her to practice and observe her own skills. Figure 1. (Curtin University n.d.). If Toni continues to struggle with KU8 and fails to be able to complete this activity, the contingency would be to revert to using ‘0’ as a placeholder on the line and have her attempt this task with two-digit numbers only to bring her back to stage 3 of her place value development. This is a constructivist approach, as it looks to the student to highlight their level of knowledge, so the teacher can have them attempt a reasonable level of skill and reverts the student's skill level in attempt to discover where the misconception has occurred and then look to correct this.

FSiM KU6 is also an area of concern for Toni as she is at a year 3 level where she should be closer to year 5 and is unable to correctly use flexible partitioning to find the place value. The use of materials such as MAB blocks could assist in realigning Toni’s misconception here. The below activity (figure 2) is an example of how problems could be presented to Toni, where she would then use MAB blocks to visually assist her in being able to partition three-digit numbers, then having Toni break down how the number is represented by the blocks in written multiplicative thinking to correct her misconception that it is only additive. Using Toni’s prior knowledge, the number can be split by the base-ten property, this activity can be supported by the ZPD as smaller two-digit numbers, to begin with, then scaffolded up to three-digit numbers to support her progress with a skilled tutor to assist her. Figure 2. (Curtin University n.d.) Failing this, an ‘adding ten’ activity could be used to revert Toni’s place value development to a lower level and prepare her to build back up to a stage 4 or 5 of PV development. This activity would see the student start with MAB blocks of ten or one hundred and then build upon each ten to practice more structured partitioning in an additive property way. The activity would then ask the student to begin to build combinations of two-digit numbers with the MAB blocks until it is clear they understand the concept of flexible partitioning at a two-digit or stage 3 place value

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development level. This contingency is based on constructivism, asking the student to revert to where their own unassisted or prior knowledge and skill level is, to then help the teacher identify when the misconception could have occurred and correct it. Within Toni’s answers to question 13, it is clear due to her additive thinking, that there is a misconception about the term ‘times bigger’, demonstrating Toni is unable to use multiplicative properties (Ross, 2002, p. 419) to address the place value of numbers. To correct this, the multiplicative relationship must be taught to Toni to have her progress from her year 1 level place value development. Using the ‘powers of ten’ activity as shown in figure 3, Toni will be supported in developing her understanding of multiplicative relationships of numbers, by correcting her understanding of the ‘times bigger’ phrasing. The below activity highlights to students that when multiplied, numbers can fit into one another when they share a multiplicative relationship. This activity also gives Toni a visual representation of the ‘times bigger than’ function and is simplified by bringing it back to the base-ten properties. Figure 3 (Curtin University n.d.)

Failing this, a secondary activity to assist Toni would be ‘counting cards.’ This activity focuses on the base-ten properties of place value, helping Toni to develop the concept of multiplicative thinking by bringing her progress back to skip counting. The teacher uses a set of cards with 1, 10 and 100 printed on them and asks students to skip count according to the number printed on the card. Toni would be asked to consider why they reach the same total when the cards are presented in a different order. She would then explore the multiplicative relationship between the cards and the place-value property of the final number in the count, being prompted to identify how many ‘times bigger’ each skip count is (Australian Academy of Science, 2020). Future learning In order for Toni to progress her Zone of Proximal Development needs to be assessed to understand where her misconceptions are developing. The contingency activities listed above provide a framework for the teacher to work backwards with Toni to find the point at which she is developing misunderstandings of place value. Vygotsky (1986) suggests that student is in the zone of proximal development for a particular task, providing the appropriate assistance will give the student enough of a boost to achieve the task. This concept leans on social constructivism, with the belief that collaborative learning methods require students to develop teamwork skills and that an individual student's progress is essentially related to the success of group learning (Berkeley Graduate Division, 2019). In question 10, Toni shows her misconceptions around ordering numbers, and is currently at a year 2 level aligning with content descriptor ACMNA027 (ACARA, 2017). For Toni to progress, she will need to achieve a year 3 level first with the intention of building upon her skills up to a year 5 level. As outlined by FSiM Toni should have developed KU8 at year 5 level which states that at a level 5 achievement students can place whole number and decimal numbers on a number line in instances where calibration is not marked or when the number of places is unequal (DOeWA, 2013a, p.75). An activity to assist Toni in this progression would be to model the ‘number on a line’ task in a social constructivist lesson, in which the teacher runs through placing numbers on a line in front of the class to practice modelling and the lesson being a whole class social interaction with a

skilled tutor that allow the student to practice and observe their skills, as supported by Vygotsky’s (1978) theory of the zone of proximal development. Conclusion A strong understanding of place value underpins a student’s ability to successfully progress and continue to develop throughout their academic career. Without it, a student will form misconceptions around PV from an early age and begin to struggle to comprehend later concepts in mathematics as they progress through year levels. Through diagnostic interviews and development maps, teachers can successfully observe a student’s skill level and place value development stage, identify gaps in knowledge, and then implement teaching strategies and activities that are suited to the student’s skill level to progress them. References ACARA. (2017). The Australian Curriculum . Australiancurriculum.edu.au. https://www.australiancurriculum.edu.au Australian Academy of Science. (2020, June 12). reSolve website . Www.resolve.edu.au . https://www.resolve.edu.au/place-value-cards Berkeley Graduate Division. (2019). Social Constructivism | GSI Teaching & Resource Center . Berkeley.edu. https://gsi.berkeley.edu/gsi-guide-contents/learning-theory-research/social- constructivism/ Curtin University. (n.d.). EDC243 Children As Mathematical Learners: Topic 2, Place Value and Task Choice .

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DoEWA (Department of Education Western Australia). (2013). First Steps in Mathematics Overview . https://myresources.education.wa.edu.au/docs/default-source/resources/first- steps-mathematics/mathematics-overview.pdf?sfvrsn=3a1fc7d7_1 Elliott, S.N., Kratochwill, T.R., Littlefield Cook, J. & Travers, J. (2000). Educational psychology: Effective teaching, effective learning (3rd ed.). Boston, MA: McGraw-Hill College. Reys, R. E., Rogers, A., Bennett, S., Cooke, A., Robson, K., Ewing, B., & West, J. (2020). Helping Children Learn Mathematics (3rd ed.). John Wiley and Sons Australia, Ltd. Ross, S.R. (2002). Place value: Problem solving and written assessment. Teaching Children Mathematics, 8 (7), 419-423. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes . Cambridge, MA: Harvard University Press.

EDC243 Children as Mathematic Learners

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