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.

Part B

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

## Comparison between American and Belgian Education

American and Belgian Education: The Learning Journey

Schools in America work diligently to increase their graduation rates. Many schools have a mission statement for students to be college and career ready. In efforts to get students to that point, districts and teachers look at ways to achieve success. Retention is highly frowned upon and thought to be ineffective. There is lots of research that discuss the effects of retaining students. With that stigma of retention many Americans believe the emotional and social effects highly out way the lack of academic success. As America continues to push students along, more and more students are not proficient in skills and struggle. As these students struggle they often become disciplinary issues and fall further and further behind. This academic struggle sometimes is a factor in students dropping out. Where is the happy medium? We want students to graduate, be successful, and be college and career ready, BUT we don’t want to retain students that are not proficient or academically ready to move on. In Belgium, doubling (repeating a year) is common. There is very little stigma with doubling because this is the norm if students are not ready to move up. Children are tested with different methods (assessment/supervised test for preschool

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