Remembering basic number facts: But I can't do division!

Author(s):

Catherine Pearn(presenting / submitting)

Conference:

ECER 2014

Network:

24. Mathematics Education Research

Format:

Paper

Session Information

24 SES 09 B, Paper Session

Paper Session

Time:

2014-09-04

11:00-12:30

Room:

B116 Sala de Aulas

Chair:

Javier Diez-Palomar

Contribution

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One Minute Tests of Basic Number Facts (2000) and a paper and pencil Number Screening Test (Pearn, Doig, & Hunting, unpublished manuscript).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.

Method

There were 122 Year 4 students, in five classes, from a large metropolitan primary state school in the outer northern suburbs of Melbourne. The classes ranged in size from 19 to 27 students. The students ranged in age from 8 years 11 months to 10 years 3 months with an average age of 9 years 7 months. Two paper and pencil assessment instruments were administered by the author to ensure consistency. They were the One Minute Basic Number Facts Tests(Westwood, 2000)and the Number Screening Tests 2A and 2B (Pearn, Doig, & Hunting, unpublished). There are four One Minute Basic Number Facts Tests. Each test has 33 items that focus on one of the four processes: addition, subtraction, multiplication and division. Items are ordered randomly and not by difficulty. Addition test items include one-digit addends with either one- or two-digit sums. Subtraction test items include one-digit minuends and subtrahends, and two-digit minuends and one-digit subtrahends, with a one-digit difference. Multiplication test items have one-digit multipliers and one-digit multiplicands while the division test has six one-digit dividends and 27 two-digit dividends divided by a one-digit divisor with one-digit quotients. Students are expected to complete as many items as possible in one minute. Students were given each test as per the administration instructions (Westwood, 2000). There were breaks between the tests and the author ensured students were aware of the process being used. For example, the author said something like: “Don’t forget this is subtraction. You are taking the number away this time (p.107).” The Number Screening Tests 2A and 2B (Pearn, Doig, & Hunting, unpublished) were designed to identify students mathematically ‘at risk’ in Years 3 and 4 (Pearn, 1999). Both versions of the test were deemed to be of similar difficulty which allowed students to be seated side by side for the test but unable to copy from the student beside them. The Number Screening Tests contains 34 items that focus on number. There are seven word problems including two quotitive division tasks. The One Minute Basic Number Facts Tests were analysed to determine the range, mean, and critically low score for each process. According to Westwood (2000) the “critically low score” is one standard deviation below the mean for the group. Student work samples were analysed for each of the tasks on the Number Screening Tests 2A and 2B (Pearn, Doig, & Hunting, unpublished) to determine the strategies being used.

Expected Outcomes

Scores for the four One Minute Basic Number Fact Tests (Westwood, 2000) ranged from 0 to 33. Some students were unable to answer any items correctly while others completed all items for each process in less than one minute. For the whole group, the means for addition, subtraction and multiplication were 22, 16, and 13 respectively. Three students correctly answered 33 division number facts in less than one minute while 12 students correctly answered one or less. The mean for division was 9, the median was 6 and the critically low score was one. When results were compared for each class there was considerable differences in the results. The means for division for the five classes varied from 5.6 to 13. All students completed one version of the Number Screening Test 2. Many students struggled with the word problems. Twelve students (8%) did not attempt any word problems. While 78% of students successfully answered the addition word problem, 49% were successful with the multiplication problem and more than half (57%) succeeded with one division problem but only 23% were successful with the second division problem. Both division problems were quotitive word problems based on real life contexts but the second problem was a two-step problem where students were expected to interpret their answer. Students had been randomly placed in classes so classes were not based on the mathematical ability of the students. Results from both assessment instruments highlight the large diversity of the ability to recall basic division facts within each class and between classes. The results from the Number Screening Tests highlighted the range of primitive strategies used by students completing short answer division word problems. These results had implications for the provision of professional development for this school but also the wider international mathematics education community.

References

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)

Author Information

Catherine Pearn (presenting / submitting)

Australian Council for Educational Research

ACER Institute

Melbourne

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