Summary of student understanding at the beginning of the process.

After completing the diagnostic place value interview (Australian Catholic University, 2011) and based on her responses, it was concluded that Polly needed to begin with consolidating her knowledge of 1-9 number concepts. The diagnostic interview provided valuable insight into her current level of understandings and her mathematical way of thinking. Her responses placed her level of mathematical knowledge as growth point two (Australian Catholic University, 2011), which focuses on two digit numbers. The lessons were planned with a view to progress her to growth point three, three digit numbers (Australian Catholic University).

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Booker, Bond, Sparrow & Swan (2014) note that using the tens frames is beneficial as a mental representation as it helps children develop an ability of viewing numbers without counting through the building patterns, by seeing ten when the frame is full and giving zero a meaning when the frame is empty. Reys, Lindquist, Lambdin, Smith, Rogers, Falle, Frid & Bennett (2012) view learning opportunities with the tens frame as promoting the development addition, subtraction, multiplication and importantly place value. In addition, according to Reys et al, (2012) it is beneficial to enter into discussions about different patterns. Therefore, Polly was engaged in verbal questions such as “give me one more” or “one less”. The reason for this is that it aids in conceptual understanding of ten as exploring these relationships promotes flexibility of numbers and overall number sense. (Reys et al, 2012). The concept of ten ones, make one ten was modelled and incorporated into asking questions like “one more than five”, “one less than 5” and “how many more to make one ten”. Polly initially struggled with the language used as when she was asked to show one more than five she did not understand what was required. Further elaboration was required such as put “five in your head and then one more”. The lesson also incorporated writing numbers in numeric and written form. Polly showed understanding, so the lesson was extended by introducing another tens frame to challenge her understanding (Department of Education, 2013). For example she was asked” how many counters would fit into these two frames”. Polly answered twenty correctly without counting each box. She confidently made numbers in the teens and removed the counters correctly when asked to show various lower numbers. Two tens frames were used (see appendix 1) to represent two different groups such as “ten and two make