children's views about mathematics learning after participation in a numeracy initiative

7/28/2019 Children's views about mathematics learning after participation in a numeracy initiative

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Children's views about mathematics learningafter participation in a numeracy initiativeJennifer Young-Loveridge and Merilyn Taylor University of Waikato

L ike many other Western countries. New Zealand responded to its poorresults on international comparisons of mathematics achievement byfocusing on numeracy learning (Ministry of Education, 2001a). In 2000 amajor initiative in numeracy was launched by the Ministry of Education,designed to improve student performance in mathematics through improv- S-ing the professional ability of teachers. Support for teaching and learning inmathematics included the development of a number framework and dia- ^gnostic interview for assessing number knowledge and mental strategies. (For |the most recent versions see Ministry of Education, 2005a, 2005b.) The |-teaching approach encourages students to use knowledge of number proper- |-ties to partition (break apart) numbers and recombine them in ways that 3.make calculation easier. Although New Zealand's numeracy initiative has


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schools for two years and tw o for one year). There were approximately equalnumbers of girls and boys. The majority (two-thirds) of students were ofEuropean descent, approximately one-fifth were Maori, the indigenouspeople of New Zealand, and the others consisted of Asian, Pacific Island andother ethnic groups. Three of the schools were from low socio-economicareas, and the fourth was from a medium socio-economic area.All participants were interviewed individually by one of the researchers ina quiet room outside the classroom at a time the teacher though t would causeleast disruption of their schoolwork. The purpose of the study was explainedto the child, and their consent obtained. (Parental/caregiver consent hadalready been given for the child to participate in the study.) Each interviewwas audio-taped for later transcription and analysis. We were interested in

exploring children's views about strategies for solving addition problems,their suggestions for alternative strategies, and their responses to a questionabout the importance of getting an answer correct.ResultsAn addition task (27 + 54) was presented orally and on paper, and studentswere asked to solve it in whatever way they wanted. A pencil was offered sothey could record their strategies. If a student did not w ant to record his/heranswer, the researcher wrote down the process as it was talked through. Toour surprise, the great majority of students (three-quarters) used a traditionalalgorithmic approach to solve the problem (see Table 1). Only a small pro-portion of students (~10 per cent) spontaneously used a part-w hole strategy(e.g. 27-1-3 makes 30, plus 50 makes 80 , plus 1 makes 81) to solve the prob-lem. A small group of students used a combination of part-whole strategyand written algorithm. Another small group worked from left to right, usingstandard place-value partitioning, adding the 20 and 50, then the 7 and 4,combining 70 and 11 to m ake 81.Table I Students who responded in particular ways to the multi-digit additionproblem: 27 + 54

Overall Girls BoysResponseVertical written algorithmPart-whole strategyMixture of algorithm and part-wholestrategyStandard place-value partitioning, left to

o/o74.410.3

2.6

n588

2

o/o67.411.6

2.3

n19

51

o/o82.98.6

2.9

n193

1


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After children had given their responses, they were asked if they had S'another way of working out the answer' (see Table 2). More than half thestudents were unable (or unwilling) to suggest an alternative strategy forsolving the problem. Of those who gave another strategy, the most frequentresponse (20 per cent) was to work from left to right, using standard place- ^value partitioning, as described above. Slightly more students gave a |part-whole strategy in response to the request for an alternative strategy thanhad done so initially. A few students gave an incorrect response on theirsecond attempt.

Tab l e 2 Students who responded in particular ways when asked to show an alternativestrategy for solving the wr i t ten problem

ResponsePart-whole strategyMixture of algorithm and part-wholestrategyStandard place-value partitioning, left torightCould not show an alternative strategyGave incorrect response - miscellaneousstrategiesNo.

Overallo/o13.75.519.256.25.5

n104

1441473

Girls%17.17.3

22.043.99.7

n739184

41

Boyso/o9.43.1

15.671.9

0.0

n315230

32

Twice as many girls as boys (half the girls and a quarter of the boys) offeredan alternative strategy for solving the problem. Almost a quarter of the girlsworked from left to right, using standard place-value partitioning, andanother substantial group used a part-whole strategy. Although boys seemedreluctant to show an alternative strategy, none gave a wrong answer, whereasa few girls did.

The children were also asked, 'How important is it to get the right answer?'Approximately a quarter of all students thought that getting the right answerwas important (see Table 3). The largest group (over a third) considered thatlearning from the process was more important than a correct solution. Amuch smaller group thought that what mattered more than getting theanswer right was the effort put into solving the problem and 'doing yourbest'. Quite a sizeable group thought that getting the answer right was notimportant, but they couldn't explain why or said they didn't care. Two chil-


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Table 3 Students who responded init to get the right answer?'

Response

particular ways to the

Overallo/o n

question HowGirls

% n

important isBoys

% n

36.014.718.75.3

2 711144

42.99.514.34.8

18462

27.321.224.26.1

9782

Yes, getting the right answer is important 25.3 19 28.6 12 21.2 7N o, learning from the process is moreimportantN o, trying your best is more importantN o, but I don't know why or don 't careOtherNo. 7S 42 33

disappointment when an answer was incorrect. For example, Henry's view-point was:

Well, you get a bit disappointed, like if it's a really, really hard problem andyou really want to solve it and you get really just a bit disappointed if youget the answer wrong, and the same with the little ones.Others saw a correct response as an indicator of progress. Georgia put it thisway:

Quite important, 'cause then I know that I'm doing well.Another group saw a right answer as being important so they could giveadvice to other children who hadn't been able to solve a problem. James'sview was that there was a social purpose in being right:

Quite important, because, like, then you know, you already know theanswer, then you could help other people who don't know it.An interesting group responded that they needed to get answers correct nowto ensure their future success in later life.

It is important because I want to be brainy at intermediate and high school.[Karen]A few children were conc erned abo ut ho w they wo uld be seen by their pee rs.Alana's though ts were expressed like this:

Because I'm scared that people will laugh at me if I get it wrong.Although a common response to the question about the importance of being


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Not that important, 'cause you learn by your mistakes. [Nita] i;Sometimes people get itwrong. There's this guy in our group and he alwayshas this saying, 'If you do n't make mistakes you don 't make anything.' . . .Yea, we always learn from our mistakes and stuff, and you d on 't really careif somebody gets it wrong. [Briar]Not that important, because if I get it wrong I can learn new ways to get itright. [Alan]Not very important because if you get it wrong, then you're just there towork out another way to work it out properly. [Ruby]

Another common response to the 'right answer' question was that trying yourbest is more important. More boys than girls responded that putting effortinto working out the solution to the problem was more important than get-ting the answer right. The majority of the children who responded that effortwas important were from the upper stages of the framework. Children saidthings like:

It doesn't matter - at least you try - try your best. [Mary]Because it doesn't m atter if you get it wrong because you can just keep try-ing and trying and getting the right answer for it. [Bob]Well it's not that important. The main thing is that you try. [Douglas]

A sizeable group thought that gett ing an answer right was not impor tant .More boys than girls responded that gett ing an answer correct was notimpor tant , but they couldn' t say why or didn' t seem to care. M ore chi ldrenat the upper stages of the number f ramework responded in this way. Chil-dren said things like:Well, I don 't really care, it's not really important, because you don 't alwayshave to get things right and it doesn't matter if you get something w rong.Yeah and it will help. [Tim]Not that much. I can be wrong or right. I don't really care. I don't changemy answer when she says i t . . . what it is . . . keep it as it is. [Steven]Not at all - because just getting the right answer is OK, but if you get itwrong there 's always next time - it doesn't really matter if you get itwrongor right. [Karen]I don't really care - so long as I get an answer down. [Sally]

Discussion


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^ numeracy initiative had been in place only for a relatively short time, and" many of the children had probably been taught the traditional verticalalgorithm before their involvement in the numeracy project. The childrenused the language associated with an algorithmic approach (4 and 7 are 11 ,

put down 1 and carry one, etc.). Perhaps, after several years of working withalgorithms, children are reluctant to let go of this approach and begin usingthe part-w hole strategies introduced to them m ore recently. In our study thenumbers were presented on paper in columns, and that may have acted as aparticular stimulus to choosing an algorithmic process, although the childrenhad been told that they could solve the problem any way they wished. Onreflection we decided that, in future research, presenting tasks in the form ofword problems might avoid the potentially confounding effect of format.Some of the Year 5 and 6 students in the present study had participated inthe project for more than two years (when they were in years 3 and 4), andsome had been involved in the project for only a year and that had been inthe year prior to our study (when they were in year 5). It is possible that theimpact of the project on students varied according to the year level, with chil-dren at the senior primary levels more strongly wedded to vertical writtenalgorithms and inclined to prefer a process or method that was familiar.It was interesting to note that many of the children who had been assessedby their teachers as being at higher stages on the framework were not so con-

cerned about getting a correct answer to a problem. Children assessed asbeing at lower stages on the framework tended to think that the correctanswer was the most importan t goal of mathematics. More capable childrenseemed able to discern and articulate the difference between knowingand using a process and attaining a specific solution. Although Swan andcolleagues (2000) found that older (Year 8) children shared their teachers'orientation about learning mathematics, we found that only our more capa-ble Year 5-6 students were able to change and reflect upon the new ways thatwere being introduced.Half the children were able to work out the answer in a different way whenasked to, but it was noticeable that more girls than boys offered an alterna-tive strategy. We wonder if this means that, once boys have one method theyare satisfied with, they choose to ignore any other possibilities. This is con-sistent with Boaler's (1997) finding that girls were more concerned with theconnectedness of their knowing and the links between ideas, whereas boystended to take a more 'separate' approach which valued certainty, absolutetruth, procedures and algorithms.More girls than boys thought that learning from the process of solving aproblem was more important than obtaining a correct answer. Forgasz and


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Q .might have on the expectation that all children articulate their thinkingprocesses. SMore boys than girls thought that trying their best (i.e. putting in effort)was more important than achieving a particular answer. This fits with theliterature on causal attributions, which shows that boys are more likely than g.girls to explain their lack of success as the result of putting in less effort(Forgasz and Leder, 1996; Walkerdine, 1998). IConclusionAlthough New Zealand's numeracy initiative has been designed to improvechildren's understanding of number, this is clearly not a straightforwardprocess. Our study has highlighted the importance of asking children to artic-ulate their views about learning mathematics. It is evident that some childrenhave deep-seated beliefs about learning mathematics, and these influence theway they respond to an initiative such as the numeracy project. Although itwas not our original intention to analyse the data in terms of gender and abil-ity (as reflected in teachers' assessments of students' framework stages), wefound that these factors were influential in shaping the children 's ideas abouttheir learning of mathematics. Further research is needed to better under-stand children's perspectives on mathematics learning.ReferencesBoaler, J. (1997), 'Equity empowerment and different ways of knowing', Mathemat-ics Education Research Journal 9 (3), 32542.Carr, K. (2003), 'What mathematics is important to learn? School students' views',Research in Education 70, 1-8.(2000), 'Seeking children's perspectives about their learning', in A. B. Smith,N . J. Taylor and M. M. GoUop (eds). Children's Voices: Research, Policy and Prac-tice, Auckland N Z : Longman.Duffield, J., Allan, J., Turner, E., and Morris, B. (2000), 'Pupils' voices on achieve-

ment: an alternative to the standards agenda', Cambridge Journal of Education 3 0(2), 2 6 3 - 7 4 .Fielding, M., Fuller, A., and Loose, T. (1999), 'Taking pupil perspectives seriously:the central place of pupil voice in primary school im prov em ent' , in G. S outhw orthand P. Lincoln (eds). Supporting Improving Primary Schools: the Role of Heads andLEAs in raising Standards, London: Falmer Press.Forgasz, H. J. , and Leder, G. G. (1996), 'Mathe ma tics classrooms, gender and affect '.Mathematics Education Research Journal 8 (1) , 15 3-7 3 .Freeman, J. G., McPhail, J. G., and Berndt, J. A. (2002), 'Sixth-graders' views ofactivities that do and do not help them learn' . Elementary School Journal 102 (4),3 3 5 - 4 7 .Kershner, R., and Pointon, P. (2000), 'Ghildren's views of the primary classroom as


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(2005h), Book 2, The Diagnostic Interview, Wellington NZ: Ministry of Educa-tion; available on-line: www.nzmaths.co.nz/numeracy/project_material.htm.Paley, V. G. (1986), 'On listening to what the children say'. Harvard EducationalReview 56(1), 112-31.Phelan, P., Davidson, L., and Gao, H. T . (1992), 'Speaking up: studen ts' perspectiveson schoo l'. Phi Delta Kappan 73 (9), 695 -704.Pollard, A., Thiessen, D., and Filer, A. (1997), 'Introduction: new challenges in takingchildren's curricular perspectives seriously', in A. Pollard, D. Thiessen and A. Filer(eds). Children and their Curriculum: the Perspectives of Primary and ElementaSchool Children, London: Falmer Press.Rudduck, J., and Flutter, J. (2000), 'Pupil participation and pupil perspective: "carv-ing a new order of experience'", Cambridge Journal of Education 30 (1), 75 -89 .Swan, M., Bell, A., Phillips, R., and Shannon, A. (2000), 'The purposes of mathe-matical activities and pupils' perceptions of them'. Research in Education 63,11-20.Walkerdine, V. (1998), Counting Girls out: Girls and Mathematics, new edn, Lon-don : Falmer Press.Young-Loveridge, J., and Taylor, M. (2003), 'The perspectives of two childrenwho participated in the Advanced Numeracy Project', in L. Bragg, G. Gampbell, G.Herbert and J. Mousley (eds), MERINO: Mathematics Education Research: Innovation, Networking, Opportunity, proceedings of the twenty-sixth annual conferenceof the Mathematics Education Research Group of Australasia, Deakin University,6-10 July.AcknowledgementsSincere thanks are extended to all the students and the ir teachers. We are grateful alsofor funding from the University of Waikato School of Education Research Gommit-tee.

Address for correspondenceDr J. M. Young-Loveridge, School of Education, University of Waikato, Private hag3105, Hamilton, New Zealand. E-mail [email protected]


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children's views about mathematics learning after participation in a numeracy initiative

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