24 SES 09 B, Paper Session
Recent newspaper headlines highlighted difficulties caused by poor numeracy skills. For example, "School leavers 'lacking basic skills', say business leaders" (The Telegraph, 14/8/2013 ) and "Millions behind on basic skills, threatens Australia's international competitiveness" (The Australian, 4/4/2011).
There appear to be two main reasons for failure to recall basic number facts. Some researchers attribute difficulties to limitations of short-term memory. That is, students do not retain several pieces of information in working memory long enough to make use of them. Some students, however, have difficulty with the basic number processes because they simply have not had enough practice and the responses have not become automatic. Garnett (1998) suggested that students had difficulty remembering and recalling basic number facts because they "continue laboriously over years to count fingers, pencil marks or scribbled circles and seem unable to develop efficient memory strategies on their own” (Garnett, 1998). In previous work the author found that Year 3 and 4 students struggling with mathematics relied on rules and procedures even when these were inefficient and unreliable ( Pearn, 2009; Pearn & Merrifield, 1996). More recent observation has revealed that many students completed mathematical tasks using a ‘counting by ones’ strategy evidenced by tapping of fingers, nodding of heads and the drawing of tally marks on paper.
Australia has recently released a new national curriculum (ACARA, 2012). In part, the Year 4 Achievement Standard states that by the end of Year 4 students should "recall multiplication facts to 10 x 10 and related division facts". To determine which students needed additional mathematical support Year 4 students were tested using Peter Westwood’s One Minute Tests of Basic Number Facts (2000) and a paper and pencil Number Screening Test (Pearn, Doig, & Hunting, unpublished manuscript). The information from this testing was also to inform the professional development for teachers at the school.
During the tests several students commented: “But I can’t do division!” Observations revealed that many students had difficulties recalling any number facts particularly division facts. As students completed the Number Screening Test they struggled with the word problems. They could read the problems but had difficulty deciding which process to use.
Many students experience considerable difficulties with division and it is perceived to be the most difficult of the four operations to learn. When given a task such as 10 ÷ 2 = students can solve this in either of two ways. They could make two groups and share ten objects equally so there are five objects in each group (partition division) or could make groups of two five times (quotition division) (see for example, Fischbein, Deri, Nello & Merino 1985). However when this task is presented as part of a paper and pencil test where only an answer is recorded it would be difficult to ascertain whether students were sharing into two groups or into groups of two.
Research studies about students' solution strategies to one step division word problems suggest that they generally begin with direct modelling and unitary counting, move to skip counting, double counting, repeated addition or subtraction, then to the use of known multiplication or division facts, commutativity and derived facts (Mulligan, 1992; Mulligan & Mitchelmore, 1997). Kouba (1989) found children used two intuitive strategies when solving quotition problems: either repeated subtraction or repeatedly building (double counting and counting in multiples). For partitive division, they drew on three intuitive strategies: sharing by dealing out by ones until the dividend was exhausted; sharing by repeatedly taking away; and sharing by repeatedly building up.
This paper will focus on student responses to the division tasks from both assessment instruments.
Australian Curriculum, Assessment and Reporting Authority (2012). Australian Curriculum: Mathematics. Last accessed 30th January 2014 from http://www.australiancurriculum.edu.au/Mathematics/Curriculum/F-10 Downton A (2009). A study of comparative performance on partitive and quotitive division in solving division word problems. In M. Tzekaki; M. Kaldrimidou and H. Sakonidis (Eds). Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Vol 2, pp. 465-472. Thessaloniki, Greece: PME Fischbein, E., Deri, M., Nello, M. S., Marino, M.S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal of Research in Mathematics Education.16 (1), 3 -17. Garnett, K. (1998) Math Learning Disabilities http://www.ldonline.org/article/5896 last accessed 30/1/2014 Kouba, V. L. (1989). Children’s solution strategies for equivalent set multiplication and division problems. Journal for Research in Mathematics Education, 20 (2),147 -158. Pearn, C. (2009) Highlighting the similarities and differences of the mathematical knowledge and strategies of year 4 students. In R. Hunter, B. Bicknell, T. Burgess (Eds). Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia. (Vol 2, pp. 443 - 450) Wellington: MERGA Pearn, C. & Merrifield, M. (1996). Strategies for classroom teachers: A lesson from Mathematics Intervention. In H. Forgasz, A. Jones, G. Leder, J. Lynch, K. Maguire, & C. Pearn (Eds.), Mathematics: Making connections. Brunswick: Mathematical Association of Victoria. Pearn, C., Doig, B., & Hunting, R. (unpublished manuscript) Number Screening Test 2A, 2B Mulligan, J. (1992). Children's solutions to partition problems. In B. Southwell, R. Perry, & K. Owens (Eds.), Proceedings of the 15th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 410 - 420). Sydney: MERGA. Mulligan, J. T., & Mitchelmore, M.C. (1997). Identification of multiplicative thinking in children in Grades 1 - 5. Journal for Research in Mathematics Education, 28 (3), 309 - 331. The Age (4/4/2011) Last accessed 30/1/2014 from http://www.theaustralian.com.au/national-affairs/millions-behind-on-basic-skills-threatens-australias-international-competitiveness/story-fn59niix-1226032957469 The Telegraph (14/8/2013). Last accessed 30/1/2014 from http://www.telegraph.co.uk/education/universityeducation/clearing/10239980/School-leavers-lacking-basic-skills-say-business-leaders.html Westwood, P. (2000). Numeracy and learning difficulties: Approaches to teaching and assessment. Victoria: Australian Council for Educational Research (ACER)
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