“There is always something new after nine”

“There is always something new after nine” Action research as a mode for teachers to develop a

tool for analysing their pupils numeracy skills

Dóróþea Reimarsdó-r, Dalvíkurskóli Hafdís Guðjónsdó-r and Jónína Vala Kris=nsdó-r

University of Iceland – School of Educa=on

Overview of the presentation Coopera=ve school-‐based ac=on research, conducted by a special-‐Ed teacher, a mathema=cs educator and a special educator. •  Background – the first cycle •  This study – the second cycle •  Purpose of the study •  Methodology and methods •  Findings from the second cycle •  The research con=nues •  Conclusions

Dóróþea Reimarsdó-r, Hafdís Guðjónsdó-r & Jónína Vala Kris=nsdó-r 15.11.2013 2

Background – the first cycle •  The beacon that guides this work is that all children should be supported in learning mathema=cs in a way that facilitates their understanding.

•  The work is based on Dóróþea’s experience as a mathema=cs and special-‐Ed teacher for almost 30 years.

•  She developed a framework through which teachers could learn how to understand children’s thinking about numbers and calcula=ons based mainly on the CGI-‐project (Carpenter, Fennema, Franke, Levi & Empson, 1999).

•  This framework was used in interviews with children that scored low on na=onal tests in grade 4.


This study – the second cycle •  Children that have difficul=es with mathema=cs need support from the

beginning of school. •  The Early Numeracy: Assessment for Teaching and Interven:on (the

Mathema:cs Recovery Project, MRP) (Wright, Martland & Stafford, 2006) was chosen as a support for developing and modifying a tool for assessment.

•  Interview is conducted as an assessment tool. The teacher uses a flexible approach; poses addi=onal tasks and ques=ons on the basis of the child’s ini=al responses to the tasks.

•  Following the assessments, a profile of the child’s knowledge is created.

Dóróþea Reimarsdó-r, Hafdís Guðjónsdó-r & Jónína Vala Kris=nsdó-r 15.11.2013

MRP aims to provide extensive and detailed informa=on about a child’s numerical knowledge. This includes obtaining detailed informa=on about the child’s current numerical strategies and knowledge of number words and numerals. 4

The goal with this research project is to improve prac3ce and develop assessment tools

The research ques3ons that guided the ac3on research:


•  How can a special-‐Ed teacher develop and use an assessment tool to create an individual educa=onal plan?

•  How can teachers create a learning environment that supports all children in learning mathema=cs?

•  How can teachers involve parents in the support of their children as they develop numeracy?

Methodology and methods •  Data consists of documents from the teacher in the form of her

assessments tools, individual plans, case wri=ngs and minutes from mee=ngs.

•  The findings from the data are incorporated into on-‐going cycles of the research.

•  The special-‐Ed teacher modified and developed the assessment tools and tried them out, collected data on the process, and interpreted and evaluated it during the con=nuing development of the assessment tool.

•  The co-‐researchers analysed the data extracted from the process, unfolding special events required in order to understand what was taking place, and bringing the data together again into narra=ves.


The narra=ves reported here give a picture of how the special-‐Ed teacher, reflec=ng in collabora=on with others, builds on her past

experience as she shapes her present prac=ce.

Findings To begin with, the special-‐Ed teacher used the assessment tools as she was developing and adap=ng them to the condi=ons in her school.


I tell Guðrún that I have seven red cubes and place them under a sheet of paper. She turns around and I add some yellow cubes and tell her that there are ten cubes under the sheet and ask if she can tell me how many cubes I added. She responds: Three. I ask her how she knew. Because I know that seven plus three is ten. I place 12 red cubes and all together they are 15. I ask Guðrún how many yellow cubes I placed under the sheet. She responds: Twelve (limle hesita=on), thirteen, fourteen, fi8een (stretches her fingers for 13, 14, and 15). Looks at her fingers and says: You put three. Next I show her the number sentence 16–12 and say: Can you read this? Guðrún nods and reads: Sixteen minus twelve. Then I ask if she can find the solu=on for me. She stretches up all her fingers and for each number she men=ons she bends one finger at =me. Fi8een, fourteen… six, then she stretches up two more fingers, five, four. It is four.

The teacher’s analysis of Guðrún’s numeracy •  Guðrún’s response was in coherence with what the special-‐Ed teacher

was familiar with from her earlier work and also correlated with the early numeracy project. –  Guðrún knew that seven plus three is ten and could use her knowledge to find out

how many cubes added to seven would make 10. –  When she does not know the facts about the numbers used, she counts her fingers

to help her keep track of her coun=ng. –  She is also capable of reading the number sentence 16–12 and since coun=ng

backwards is difficult the fingers again support her in keeping track of her coun=ng.

•  Guðrún solved this prac=cal task with the help of her speech, as well as her eyes and hands.

•  The teacher’s knowledge of children’s development, helped her analyse Guðrún’s knowledge of numbers.


“There is always something new after nine”


I was working with Siggi, a seven year old boy. He mastered coun=ng; from whatever number un=l he reached 109. Instead of saying 110 he said: 1000. I asked him to count from 96 and he said: 109, 1000. I asked if he remembered that he counted: 7, 8, 9, 10 and he nodded and said: Yes. I told him that we do the same with 107, 108, 109, 110. Siggi responded: No, there is always something new a8er nine. Reflec=ng on his response reminded me that this is a common response, and many children have difficul=es naming the right number aper 109, most open they men=on 200 or 1000. His response, that there is always something new aper nine, got me wondering if children at this age have figured out a system as they count but as they come to 110, their system doesn’t work because then they have to con=nue to say: hundred and …

The teacher’s analysis of Siggi’s understanding


•  Siggi was assimila=ng his coun=ng to his schema of the place value system, but was s=ll not able to filter in the teacher’s explana=on, to be able to modify his understanding of the place value.

•  The teachers’ knowledge of Piaget’s theory of assimila=on and accommoda=on helped her interpret Siggi’s explana=on and helped her relate to former experiences of similar explana=ons offered by other children.

Assisting other teachers •  The classroom teachers were hesitant to use the assessment tool and relied on the special-‐Ed teacher.

•  In the second cycle, one classroom teacher interviewed her first grade students.

•  This teacher was surprised how much she learned from interviewing them with the support of the assessment framework.


This teacher is slowly adop=ng the systema=c reflec=on on mathema=cal interac=ons that focus on student’s learning and understanding of processes, as well as on one’s own interac=on behaviour, that according to Mason (2002), represents an essen=al professional competence of teachers.

Collaboration with parents


I was working with Pétur who is an ac=ve six year old boy. I tell him that there are five red cubes and four yellow cubes under my sheet of paper. Then I ask him if he knows how many there are in total. Nine, responds Pétur quickly. When I ask him how he knows that he replies: Because five plus five is ten and minus one is nine. I then tell him that there are nine red cubes under my sheet and six yellow and he responds quickly: Fi8een. I ask how he knows that. If they were 16 altogether and there were 10 red cubes instead of nine it is just minus one. I show Peter the number sentence 17–14 and he reads: Seventeen minus fourteen are three. I ask him how he found that out and he replies: I just took four out of seven …… Why do you know mathema:cs so well: I ask him and he responds: Because my mom and dad are always making problems for my brother and me.

The research continues Par=cipa=on in the collabora=ve research has enhanced us to reflect more on our understanding of children’s development of mathema=cal thinking. •  Young children’s development in making sense of the base-‐ten system. •  In what way can teachers support children who have difficul=es with

understanding our number system? •  How can we support classroom teachers that may not have strong

background in teaching mathema=cs in learning to no=ce children’s abili=es in mathema=cs and respect their differences?

•  How can parents be supported in helping their children to develop their mathema=cal thinking ?

•  The third cycle is evolving – planning the interven=on


Conclusions •  The main goal of school is to provide students with a learning environment that is respecsul, caring and safe (OECD, 2009).

•  The special-‐Ed teacher is concerned about her student’s understandings of numeracy and how to respond to their individual differences.

•  She realises that if children’s everyday life is separated from what they learn in school, they might not make the connec=on needed between schoolwork and everyday work.

•  By invi=ng the parents into a partnership, they can in collabora=on help the children make the connec=on needed to make mathema=cs prac=cal.


Conclusions •  Prac==oners are deliberate intellectuals who constantly theorize prac=ce as a part of prac=ce itself (Cochran-‐Smith and Lytle, 2009)

•  The goal of teacher learning ini=a=ves is the joint construc=on of local knowledge both inside and outside contexts of prac=ce.

•  This special-‐Ed teacher’s collabora=on with her colleagues and with her pupil’s parents has resulted in improved prac=ce within the school and a suppor=ve learning environment.


Conclusions •  Narra=ves can be one way to collect data for professional

development pushing thinking about the work into an unforeseen direc=on and toward new learning (Amarda, 2012).

•  Through ac=on research methodology, prac==oners, in collabora=on with researchers can analyse, evaluate and make meaning of their authen=c narra=ves and support them in presen=ng their work.

•  Collabora=ng with teacher educators in theorizing and wri=ng about her prac=ce, this teacher’s developmental work has resulted in this ongoing ac=on research.


Our goal with collabora=ve research is to build a bridge between theories concerning mathema=cs teaching and learning and the prac=ce within schools where teachers are engaged in working with children and may neither have =me nor experience in researching their prac=ce and wri=ng about their work.

References


Attarda, K. (2012). The role of narrative writing in improving professional practice. Educational Action Research 20(1), 161-175.

Cochran-Smith, M. & Lytle, S. L. (2009). Inquiry as stance. Practitioner research for the next generation. New York/ London: Teachers College Press.

Mason, J. (2002). Researching your own practice. The discipline of noticing. London: Routledge Falmer.

OECD. (2009). Creating effective teaching and learning environments: First results fromTALIS. Retrieved from www.oecd.org/publishing/corrigenda

Wright, R. J., Martland, J. & Stafford, A. K. (2006). Early numeracy: Assessment for teaching and intervention (2. Ed.). London: Paul Chapman.

“There is always something new after nine”

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