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Children as Mathematical Learners
Through conducting the diagnostic interview process, the child, Sally, exhibited an extensive understanding in regards to place value and its appropriate mathematically application. The 10 year olds mathematically ability was apparent as a result of a sound foundational understanding of the base ten number system and its relationships between the tens, hundreds, and thousands places. Sally was also extremely capable when tasked with reading writing and saying numbers being able to successfully do so well into the the hundreds of thousands whilst also demonstrating a profound conceptual awareness of whole numbers and their connection with one another. Additionally, though the one on one diagnostic interview process Sally revealed a rudimentary understanding of the decimal system by being able to successfully read the decimals posed to her.
One of Sally’s greatest assets was her capacity to read, write and order whole numbers.
Through revisiting the interview transcript, it was identified that Sally possessed a profound comprehension around the concept of place value and the effect it has on how numbers can be spoken, written and ordered. The Department of Education WA (2013) detail eight key understandings of mathematics that teachers need to effectively scaffold into their students learning experiences. Through the process of the diagnostic interview Sally was able to successfully display an understanding of each of the eight key understanding of mathematics. For example, during question 21 Sally was asked to write a series of large numbers such as ‘one million and twenty-four’, not only was Sally able to successfully write the number but she was able to detail how the zero can be used as a “marker” for each place value and thus demonstrating a sound conceptual understanding of place value through her explanation (Appendix B). Sally’s strength for ordering whole numbers was evident throughout the entirety of the interview process but in particular during question 9 when asked to order the numbers 156, 403 and 813. Sally immediately without hesitation successfully ordered the three numbers and then when probed as to how she arrived at the answer she articulated that the number 813 was the greatest number as the number 8 in the hundreds column was larger then the 4 in the number 403 and 1 in the number 156 (Appendix C). Sally’s rationale for her answer exhibits her conceptual understanding of place value and in particular key understanding 5 in the First Step in Mathematics text that “there are patterns in the way we write whole number that help us remember their order” (Department of Education WA, 2013).
Despite Sally being able to correctly answer the majority of the questions asked of her during the diagnostic interview, there was still however a number of question posed that challenged her mathematical knowledge and tested her conceptual understanding of place value. One of the areas that generated the most concern for Sally was those questions that related to partitioning. Initially when posed questions regarding partitioning Sally required continuous reminding as to what was meant by the term and found it challenging to recognise the relationship between separating a whole number into a sum of whole numbers. For example, during question 19 Sally was tasked with partitioning the number 2791 and was unable to formulate an answer demonstrating a potential gap in her mathematical learning (Appendix D). Sally also commonly found that the questions that involved number lines, ordering of decimals, and fractions difficult. Through undertaking the diagnostic interview process, it provided the opportunity for a deeper analysis of Sally’s mathematical understandings. Significantly, although a great deal of Sally’s responses to the interview questions where successfully, the opportunity of listening to her responses in a one on one situation allowed for her mathematical misconceptions and red lights to be highlighted. Through the analysis of the results attained in the diagnostic interview process the following series of tutoring lesson where planned in an attempt to target Sally’s misconceptions and effectively develop her overarching mathematical understanding.
Sally also exhibited difficulty when tasked with questions related to the use of numbers lines and in particular uneven number lines. This difficulty surrounding number lines was first apparent during question ten of the diagnostic interview when presented with a number line ranging from 39 to 172. Before attempting this question Sally had previously demonstrated the capacity to successfully answer and provide an appropriate explanation to an even number line question ranging from 0 to 100 in question 5. The justification she provided made reference to the key mathematical understandings of fractional awareness and subitising. Although when tasked with an uneven number line as was the case in question 10, which did not start with zero or end with a multiple of ten, Sally’s methodology that aided her during question 5 was not applicable in this scenario. As Sally’s process for answering even number lines was not transferable to answer uneven lines this illustrated a fault in her understanding of the relationship between how many numbers are in a number line and what the exact number marked is. In question 5 Sally was able to identify through visual awareness that the number marked was 50 as it was directly half way between the 0 and the 100 points, however, with a number line ranging between 39 and 172 she was unable to implement this same process. This gap in Sally’s conceptual understanding identified within the diagnostic interview process allowed for a series of tutoring which where tailored to resolve this void in knowledge. By successfully identifying this problem area within Sally’s mathematical processing the tutoring sessions could efficiently work towards rectifying these misconceptions.
In order to correct Sally’s misconceptions surrounding number lines a blend of visually based hands on activities and written worksheets where devised to be undertaken through the course of the tutoring sessions. During the initial session Sally worked through a series of worksheets and upon completion engaged in discussion as to why she believed she was finding these challenging. Through observation of Sally’s workings and actively listening to her elaborations it was decided that in order to effectively progress her understandings of number lines a concrete hands on activity such as the clothesline activity would be more beneficial where numbers are pegged on a string which stretches the length of the room. This preliminary approach was valuable as through the initial discourse the teacher was able recognise Sally’s perspective which influenced the teacher’s instruction for the remainder of the tutoring sessions. This form of exercise was advantageous for Sally as it provided a physical representation of a number line for her to efficiently visualise how each of the numbers would align with one another. During this activity two numbers are placed upon respective ends of the number line and the student is given the third to add to an appropriate space. For example, during the second session the number line given to Sally began at 34 and ended in 90, Sally was given the number 51 and asked to place the marker where she believed it would sit along the line. Through her explanation she correctly articulated that there was “56 numbers between 90 and 34” and from this was able place the number 51 accurately along the physical representation of a number line. Another method Sally found beneficial was finding the middle of the number line and using it as a point of reference when placing any number. Through the completion of these tutoring sessions and attainment of new strategies Sally strengthened her conceptual understanding around the use of number lines.
Rationale: One on One Interviews
A one on one diagnostic interview is an effective method of evaluating a student’s mathematical aptitude. The use of alternative forms of assessment in mathematics, such as interviewing, was first promoted by Piaget and has grown in reputation as a result of reform in mathematics education (Moyer & Milewicz, 2002, p. 4). Through the student interview process, the classroom teacher can guide and inform their mathematics instruction by asking students questions which help the students make sense of mathematics and develop their mathematical reasoning (Moyer & Milewicz, 2002, p.5). As outlined by Burns (2004) students often see the need to remember an assortment of rules and steps when partaking in mathematics and it is common that students may implement these incorrectly when attempting to answer questions. Notably, the diagnostic interview process has developed from more then a means of identifying a student’s misconceptions and red lights, with its intention to also discover “the nature and extent of an individual’s knowledge about a particular domain by identifying the relevant conceptions he or she holds and the perceived relationships among those conceptions” (Moyer & Milewicz, 2002, p.5). This form of assessment is invaluable in the modern classroom as it allows the classroom teacher the opportunity to provide the students their undivided attention in a one on one scenario which intern allows the educator the chance to comprehensively observe and analyse student’s mathematical knowledge (Burns, 2004). Furthermore, the classroom teacher can use this form of assessment to identify potential areas of weaknesses or problem areas in their own explanation of mathematical concepts. This will be apparent when alike problems arise across a series of interviews with students within the class. For example, a teacher through interviewing may identify that a number of students when asked to create a number one hundred times more then itself just add two zeroes onto the original number, the teacher then may identify that they have not described the conceptual understanding of place value sufficiently and revisit this with a follow up lesson. In this regard, the diagnostic interview as a means of assessment is valuable for both the classroom teacher and the students.
Central to the effectiveness of a diagnostic assessment is the role of the educator. As outlined by Reys et al. the diagnostic interview is most successful when the teacher takes care to develop a rapport with the student, practice meaningful listening to the student answers, ensures that all student responses are accepted and valued, and that the nature of questioning is contusive for the student to actively want to engage and elaborate have they arrived at the answer (Rey et al., 2017, p. 93). Furthermore, it is important that teachers establish effective communication throughout the diagnostic interview as one on one interviews by definition rely on verbal communication as the primary means for successfully obtaining of information between the interviewer and the interviewee (Moyer & Milewiz, 2002, p. 5).
The Department of Education WA in the First Step in Mathematic (2013) texts details the professional judgement cycle as a dynamic process where the teacher uses their professional judgment based on their knowledge, experience and evidence to scaffold classroom experiences so that students have the greatest prospect of achieving the required mathematics outcomes. The fluid nature of the process of using professional judgements to design classroom learning experiences is dependent on the context of the class and is driven by the teacher’s knowledge of their students and their needs, the mathematics, and the mathematics pedagogy (Department of Education, 2013, p. 32). These three components require teachers to make decisions on the mathematics needed to “move students on”, careful observation of students to deduce what they “do and say”, and the choosing of the appropriate learning activities and focus questions (Department of Education, 2013, p. 32). Therefore, when considering a one on one diagnostic interview as a part of this dynamic process it is beneficial as it allows for careful observation which is needed to identify the mathematical misconceptions that need to be attended to in order for students to progress and continually develop their understanding of mathematical concepts. Furthermore, it is also valuable as it assists teachers when making the decision in regards to the appropriate pedagogy to be applied. Critical to the application of the professional judgment cycle is ensuring that the teachers endeavour to develop a profound conceptual understanding of the mathematics they are teaching rather then relying on simple procedures. The important concept for teachers to understand during this process is that their role is not solely the transfer of knowledge to students but rather as facilitators of students to develop their own mathematical knowledge “in their own way and at a pace that enables them to make sense of the mathematical situations and ideas they encounter” (Department of Education WA, 2013, p .4). Students are more inclined to genuinely engage in mathematics through “robust learning” where mathematical outcomes are understood fully and in depth as against “short term” procedural knowledge (Department of Education WA, 2013, p. 5).
Key Mathematics involved in Place Value
The numeration system has continually evolved and developed over time and the number system that is commonly used today is the Hindu-Arabic system which consists of four characteristics; place value, base ten, use of zero and additive and multiplicative property (Reys et al., 2017). In order to develop a comprehensive mathematical understanding it is essential that students develop an understanding of each of these four characteristics. The concept of place value in the numeration system indicates that the position of a digit within the number represents its place value (Reys et al., 2017). For example, the 3 in $32 labels three tens or thirty and has a different meaning from the 2 in $32 which labels two ones. The concept of base ten in the numeration system determines that only ten digits are used, 0 through 9, and that the numbers are considered a collection rather than the individual itself (Reys et al., 2017). The use of zero is an important symbol to consider in the numeration system as it is considered a place holder to represent that there is something absent (Reys et al., 2017). For example, in the the number 607, the symbol of zero suggests that there are no tens in the given number. The final characteristic within the numeration system is that numbers have additive and multiplicative properties which means numbers can be written in notation form to represent place value (Reys et al., 2017). A sound understanding of each of these characteristics within the numeration system leads itself to developing a comprehensive number sense.
Developing a profound understanding of place value is integral as it serves as one of the key underpinning ideas of all mathematical understandings. As well as this trusting the count, multiplicative thinking, multiplicative partitioning, proportional reasoning and generalising algebraic reasoning, are significant as they are considered the big ideas (Appendix A) of mathematics that are essential for further mathematical development (Hurst, 2017). Furthermore, the concept of place value is underpinned by a number of fundamental understandings. Sorting and grouping, subitising, skip counting, the ten group and partitioning all function as essential building blocks of place value, along with an understanding of whole and counting numbers (Hurst, 2017). Through developing this thorough understanding of place value students acquire the necessary mathematical knowledge required for computational algorithms for addition, subtraction, multiplication and division to be learned and applied in a successful way (Rey et al., 2017, p.220). In order for children to successfully learn about place value they must first develop an understanding of counting and whole numbers. When students commence their mathematical learning journey they begin by learning to read, write, say, interpret and use whole numbers (Department of Education WA, 2013). As outlined in the First Steps in Mathematics (Department of Education WA, 2013) students develop a basic understanding of whole numbers and realise the key understandings (1-8) and specifically key understanding 4 and 5, which prescribe that whole number are organised in a certain order and that there are particular patterns in the way say them which aid us to remember their order. Grasping of these key understandings is important as they form these foundational understanding of early number sense and counting which leads to a natural progression of understanding place values as an organisation structure for numbers (Reys et al., 2017).
Common Place Value Resources
There are a numerous resources and activities that can be used by the classroom teacher to successfully reinforce and develop students conceptual understanding of the place value system. (Reys et al., 2017). This plethora of resources can include calculators, place value charts, virtual resources, symbols, abacuses, and base ten blocks (Rey et al., 2017). When determining the appropriate resource for a chosen place value exercise it is significant for the classroom teacher to consider whether to use grouped or un-grouped materials, and proportional or non-proportional materials (Rey et al., 2017). Grouped resources are commonly formed into collections by the classroom teacher prior to the students using them whereas ungrouped materials such as individual counters and beads are seen as beneficial as the students can use these materials to form their own groups of 10 for the particular exercise (Reys et al., 2017). Proportional place value models such as base ten block can be beneficial for students’ to establish an association through measurement of height whereas with non proportional models such as currency, the size relationship in not maintained (Reys et al., 2017). Students are more inclined to develop a conceptual understanding of place value by actively participating in learning activities which focus on concrete models partnered with detailed oral descriptions and symbolic representations of models (Reys et al., 2017). As all students learn differently offering them with a number of different activities and resources to learn and understand the concept of place value is important. Through providing these opportunities the classroom teacher allows students to make mistake that can be simply changed then learned from in order to gain stronger understanding of how place value works.
As outlined by Reys et al. (2017) there are a number of common misconceptions that can potentially arise with some students’ mathematical understandings when learning place. As counting forms, the bases of mathematical understanding if students have difficulties in this area then this can have detrimental effect on their understanding of place value. For example, if students are unfamiliar with the pattern of the base ten number system, they will find challenging counting above ten and therefore will have difficulties with conceptualising the idea of a decade (Rey et al., 2017). Another misconception that can often arise during the place value learning process relates to students’ ability to recognise place parts, for example, 4 hundreds, 5 tens and 7 ones in 457 is considered as collections of one i.e. 400 ones, 50 ones and 7 ones (Education and Training Victoria State Government, n.d.). This misconception arises as students do not understand the structural basis of two or more-digit number and do not recognise the significance of 5 in the number 57 as a count of 5 tens. This means that although students can implement place value conventions they do not truly understand them. Another common misconception surrounding place value is around students ordering of numbers. Often students order numbers based on the value of the digits rather then the place, for example, when asked which is this larger number 68 or 202 students will answer with 68 because 6 and 8 are greater then 2 and 0. In order to help students over these misconceptions it is essential that the classroom teacher can firstly identify these errors and then chose the appropriate resources and activities to remedy these misunderstandings.
- Burns, M. (1993). The twelve most important things you can do to be a better maths teacher. Retrieved from: https://lms.curtin.edu.au/bbcswebdav/pid-4709411-dt-content-rid-26027084_1/xid-26027084_1
- Burns, M (2001). Snapshots of student misunderstandings. Educational leadership Vol 67. No5. Pp16-22. Retrieved from: https://link.library.curtin.edu.au/ereserve/DC60272306/0?display=1
- Burns, M. (2004). Ten Big Math Ideas. Retrieved from: https://lms.curtin.edu.au/bbcswebdav/pid-4709412-dt-content-rid-26041777_1/courses/EDPR2004-FacHum-254043853/EDPR2006-DVCEducatio-1459099305_ImportedContent_20150508113435/BURNS%202004%20Ten%20Big%20Ideas.pdf
- Department of Education WA. (2013) First Steps in Mathematics Number Books 1. Perth: DETWA. (http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/) (ISBN/ISSN: 9780730744832)
- Department of Education WA. (2013) First Steps in Mathematics Number Books 2. Perth: DETWA. (http://det.wa.edu.au/stepsresources/detcms/navigation/first-steps-mathematics/) (ISBN/ISSN: 9780730744832)
- Education and Training Victoria State Government. Common Misunderstandings Place Value. Melbourne: Education and Training Victoria State Government.
- Mason, C. (2013). Red Lights. The common denominator, 1(13), pp. 12-13.
- Moyer, P. S., & Milewicz, E. (2002). Learning to question: Categories of questioning used by preservice teachers during diagnostic mathematics interviews. Journal of Mathematics Teacher Education, 5(4), 293-315.
- Reys, R., Lindquist, M., Lambdin, D. V., Smith, N. L., Rogers, A., Cooke, A… Bennett, S. (2017) Helping Children Learn Mathematics. (2nd Australiasian Ed). Brisbane: Wiley.
Visual representation of the big ideas of mathematics that are essential for further mathematical development (Hurst, 2017)
This section of transcript from the diagnostic interview illustrates Sally’s ability to successfully write large numbers and explain using appropriate mathematical terminology how she arrived at her answer during question 21.
This section of transcript from the diagnostic interview illustrates Sally’s ability to successfully order whole numbers and explain how she worked at her answer during question 9.
This section of the transcript from the diagnostic interview illustrates Sally’s difficulty in the partitioning of numbers.
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