| aMethodology
|
The
purpose of this investigation was to develop a conceptual model
to support teaching and learning about place value. My research
was informed firstly by a literature review and then by looking
at how children showed their understandings of place value. A tentative
conceptual model emerged from the literature and was refined by
the insights gained from observations of these children. The research,
aiming for descriptions of their knowledge and ways of knowing,
is situated within a qualitative research methodology. The qualitative
research methodology shifts the focus away from knowledge as objective
and recognises it as contextual and negotiated. Therefore I used
observations and interviews as my data gathering methods. These
methods have been described as the tools with which researchers
attempt to capture rich, thickly layered data that provide descriptions
and understandings about individuals or groups (Kvale, 1996). My
role as the researcher was to try to interpret and make sense of
the data by looking for the patterns and themes that emerged. |
| |
|
|
|
|
| aResearch
design |
In
this study I analysed interviews with two groups of children. The
first group was made up of children who had not participated in
the ENP. The second group was a made up of children from my school
who had participated in the ENP. I wanted the two groups to be as
similar as possible except for whether or not they had taken part
in the ENP.
For the first
group I approached the National Educational Monitoring Project (NEMP)
to see if they could provide me with suitable video recordings of
Year 4 children working on place value activities from their annual
surveys. I hoped that from these I would be able to find a group
of Year 4 children who had not yet participated in the ENP. NEMP
provided me with a list of schools that had been surveyed in 2001
that matched my ENP group as closely as possible. I refer to these
children as the NEMP group and my criteria for choosing them were:
|
| • |
Year 4 children |
| • |
similar decile to my school |
| • |
situated in the Christchurch area |
| • |
had not participated in the ENP. |
| |
|
|
|
|
NEMP did not
have information on this last criterion. Following a number of
telephone calls, however, I was able to find a school that had
not participated in the ENP professional development. After explaining
my project to the principal, I received permission to use the
NEMP data (see Appendix 1, p.40) collected from his school in
2001. This group was made up of twelve children who completed
several of the activities in smaller groups of four. These children
were identified by NEMP as A1, A2, A3, A4, B1, B2, B3, B4, C1,
C2, C3 and C4. One reason for looking at video data rather than
just checking the NEMP report was that I wanted to observe the
children closely and see cues, such as body language, that were
not included in the report. In choosing this school I assumed
that the children in the videos had not recently come from another
school that had participated in the ENP.
For the second
group of children, I decided that five would be a manageable number
of children to interview. They were in their second year of participation
in the ENP at my school and were selected using a purposive sampling
strategy. This meant that, rather than choosing them randomly,
I chose them according to pre-determined criteria. These were:
|
| • |
self-confidence in an interview situation |
| • |
representative of a range of mathematical knowledge (as identified
by the children's teacher). |
| • |
Year 4 level |
| • |
had participated in the ENP |
| • |
ability to articulate their thinking. |
| |
|
|
|
|
This
last criterion contrasts with views about intuitive learning and
presents a problem for assessment tools, such as the ones that I
used, that rely on children being able to articulate and explain
their thinking (Claxton, 1997). I called this group the ENP group.
In order to keep their names confidential I refer to these children
in this report as A, B, C, D and E. |
| |
|
|
|
|
| aData
gathering |
Data
gathering involved, firstlywatching the video recordings provided
for me by NEMP. This allowed me to become familiar with the types
of questions and activities that the children had been asked about
place value. The NEMP (2001) survey consisted of a series of interviews
conducted by a teacher / administrator with individual Year 4 children
from a group of twelve. Each child completed some of the eight place
value activities. As a group they covered them all. I made notes
of most of the conversations and actions of the individual children
where they seemed relevant to the research questions.
My next step
was to set up structured, face-to-face interviews with the ENP children
from my school. I recorded these on video. I wanted to make the
groups similar in as many areas as possible and this included the
structure of the interview. Therefore I decided to replicate the
NEMP (2001) activities relating to place value when I interviewed
the ENP group at my school. The following activities were used:
|
| • |
Girls and Boys: showing 26 and 32 with place value
blocks; dividing 26 and 32 by two using place value blocks. |
| • |
Population: reading 4-digit numbers and explaining
what the digits represent. |
| • |
Calculator ordering: putting 4010, 140, 41, 4100,
104 and 14 in order of size and reading these numbers. |
| • |
Number A: multiplying by 100 |
| • |
Number C: writing words for 3,4 and 6 digit numbers
in figures; multiplying by 10 |
| • |
Motorway: estimating and then multiplying by 100
and describing a strategy. |
| • |
36 + 29: mental calculation |
| • |
Speedo: adding and subtracting 1,10, 100, 1000 to
and from a 4-digit number. |
| |
(for more detail on these activities see Appendix 1, p40.) |
| |
|
|
|
|
It is possible
to question the validity of some of the activities by building
on observations from the literature. Kamii (1986), for example,
suggests that numbers such as those listed in Calculator ordering
(see above) could be arranged in order according to cues other
than the place values of the digits. The child could know that
larger numbers have more digits without recognising specifically
that place values are ordered according to ones, tens, hundreds
and thousands. Likewise the 14 and 41 could be ordered according
to the face value rather than the place value of the digit in
the tens place. In Girls & Boys, the second part of this activity
involved division with regrouping. One strategy for this would
have been to exchange one of the tens blocks for ten ones. The
choices for regrouping were limited by the provision of only tens
and ones blocks for the children. Cotter (2000) would suggest
the availability of fives as well as tens blocks. This would probably
fit in better with the ENP teaching that focuses on fives from
early stages of the Number Framework.
Claxton (1997)
reminds us that schools tend to focus on the cognitive modes of
thinking where being able to articulate and explain are valued
thinking skills. He suggests that, as a result, intuitive thinking
can receive less emphasis because assessment activities tend to
focus on explaining and articulating thinking. This could be a
criticism of the sorts of activities that I used in this project,
as they mostly required the child to explain their thinking. When
we ask a child to explain, we are making an assumption that thought
processes are in the conscious domain. The inability to articulate
reasoning may reflect the intuitive mode of thinking described
by Claxton, and needs to be taken into account when designing
assessment activities.
One of the
assumptions of any qualitative methodology is that during a research
interview, data is actively created that would not exist apart
from the interaction of the participants (Silverman, 2001). It
is important to allow the voice of the children to emerge and
shape the interview and indeed the research itself. Therefore
I tried to remain open to unanticipated answers that the children
might give to my questions and to keep the interview process as
informal, encouraging and comfortable as possible.
|
| |
|
|
|
|
| aAnalysing
the data |
As
I viewed each tape, I recorded my observations in a table so as
to make a written record of key points (see Appendix 3, pp.43 -
52). Reading through this table and my notes I looked for similarities
and differences, groupings, patterns and ideas of particular significance.
Some came up during the interview and before I had completed writing
up the notes. I initially read once through what I had recorded
so as to gain an overall picture and then start looking for data
that related to the research questions themselves while keeping
on open mind for unexpected responses. I found that I needed to
re-examine my notes and the video recordings several times during
the analysis process to check on some of the interpretations that
I was making.
For example, in an activity that required the children to explain
the place value of each digit in a four-digit number (see Appendix
3, Population, Child E, p.51) one of the ENP group seemed confused
by the question and did not identify the hundreds and tens digits
correctly. She did, however, at the end of the activity say that
the 2 in 2495 was in the thousands column. I interpreted this answer
to mean that she might have understood more about place value than
the question had initially revealed. In hindsight, it would have
been interesting to ask this child to go to the earlier parts of
this activity. She may simply have needed more time than I gave
her to think about the question. In the NEMP group, B2 and B4 also
seemed confused by the same question and were not able to furnish
an answer. The children appeared confused by the teacher prompt
(”What does this mean?”) which was ambiguous as it could imply face,
place or total value. It is not clear whether the NEMP children
had a lesser degree of place value knowledge or whether they were
simply confused by the question.
Kvale
(1996) likens the interview process to a snapshot, where the interviewer
records data at a given time. During the interviews I chose to reword
questions where I felt that they were ambiguous. My thinking was
influenced by the concept of reflexivity (Denzin and Lincoln, 1994)
that describes the interview as a dialogue that is created by both
of the participants in the interview. I would allow my own thoughts
about the questions and activities to be expressed where appropriate
in the interview rather than attempting to adhere rigidly to the
format followed by the NEMP teacher / administrators. |
| |
|
|
|
|
| aEthical
issues |
In a project
such as this, researchers are guided by the following ethical
principles: no harm is done, participation is voluntary, consent
is informed and the interviewee retains the right to withdraw
data at any stage (Tolich and Davidson, 1999). It is essential
that anything that the interviewee shares is kept confidential
to the researcher (myself), the interviewees (the individual teachers)
and the Christchurch College of Education supervisors and examiners.
I initially
contacted the principal of my school to explain the research that
I had in mind and gained his verbal and written consent. The next
step was to approach the teacher of the Year 4 children. After
gaining her verbal and written consent for the research we discussed
which children would be suitable for the interviews. The reason
for this was that I wanted children who would be confident enough
when faced by a video camera to provide data and attempt the place
value activities. I also wanted to have a range of mathematical
ability represented in the group. Once this was completed I introduced
myself to the group of children, briefly explained my research
and gave them an information letter and consent form for their
parents to read and sign. All of the children returned their forms
with parental approval to take part in the interviews.
I was aware
of the implications of conducting this research in my own workplace.
It was possible that a child might divulge personally sensitive
information or information that compromised another teacher or
child. I conducted the interviews at school in classrooms that
were temporarily vacated by classes who had gone to Manual training.
I tried to keep the room informal and comfortable for the children,
keeping doors open to encourage the normal interactions with other
children as far as possible and to make the interview environment
as close to “normal” as possible.
|
|
