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The Development of Children's Self-Assessment in Mathematics within the Framework of a Problem-Solving Lesson: A Participatory Action Research Project
\par }\pard \qc \li0\ri0\sb100\sa100\sbauto1\saauto1\sl-320\slmult0\nowidctlpar\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\faauto\rin0\lin0\itap0\pararsid9314395 {\cf1\up6\insrsid8128269 Rachel D}{
\cf1\up6\insrsid1123806 eitcher}{\cf1\up6\insrsid8128269
\par }{\cf1\up6\insrsid1123806 David Yellin Academic College of Education}{\cf1\up6\insrsid8128269
\par Email:}{\cf1\up6\insrsid1123806 lendeit@zahav.net.il}{\cf1\up6\insrsid8128269
\par }\pard \qj \li0\ri0\sb100\sa100\sbauto1\saauto1\sl-320\slmult0\nowidctlpar\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\faauto\rin0\lin0\itap0\pararsid9314395 {\ul\cf1\up6\insrsid8128269 Abstract
\par }{\cf1\up6\insrsid8128269 This article reports one aspect of a wider participatory action research project directed at the improvement of }{\cf1\up6\insrsid4274021 the mathematics education of a}{\cf1\up6\insrsid8128269 third-}{\cf1\up6\insrsid4274021
grade class}{\cf1\up6\insrsid8128269 through formative assessment and self-assessment}{\cf1\up6\insrsid4274021 . It looks at }{\cf1\up6\insrsid8128269 how problem-solving lessons in wh
ich children were asked to solve challenging mathematical problems using their own personal solution strategies, and subsequently to listen and comment on the solution strategies of others, were used to further children's ability to assess their own math
ematical understanding and to direct their future problem-solving efforts. }{\cf1\up6\insrsid8155612 It was found that there }{\cf1\up6\insrsid8155612\charrsid8155612 were }{\cf1\up6\insrsid8155612 t}{\cf1\up6\insrsid8155612\charrsid8155612
wo interrelated axes of development}{\cf1\up6\insrsid8155612 , }{\cf1\up6\insrsid8155612\charrsid8155612 one with regard to the children and the other with regard to the teachers. Notable progress was seen in the c
hildren's ability to use their judgment in choosing more advanced ways of solving and reporting solution}{\cf1\up6\insrsid8155612 strategies}{\cf1\up6\insrsid8155612\charrsid8155612 for challenging problems}{\cf1\up6\insrsid8155612 . }{
\cf1\up6\insrsid8155612\charrsid8155612 }{\cf1\up6\insrsid8155612 The teachers, on the other hand, developed a deeper understanding of ways to structure the lesson so as to better}{\cf1\up6\insrsid8155612\charrsid8155612 access the children's }{
\cf1\up6\insrsid8155612 thinking, a development which has helped them to better facilitate their students' learning}{\cf1\up6\insrsid9314395 .}{\cf1\up6\insrsid8128269
\par }{\cf1\up6\insrsid9314395
\par }{\ul\cf1\up6\insrsid8128269 Introduction}{\cf1\up6\insrsid8128269
\par The Experimental School in Jerusalem is a pre-k - 12 progressive school whose guiding principle is the centrality of t
he child, and his or her social, emotional and cognitive development. This educational perspective has led the school to view children's interests, prior knowledge and interactions with the environment as primary factors in the development of knowledge. T
h
is approach to education is in keeping with social-constructivist theory which sees learning as a process in which new understandings are built on previous knowledge and present interest (Cobb & Bowers, 1999), are holistic in nature, and must be socially
negotiated (Lave, 1988, 1996).
\par In support of this child-centred approach, assessment in the school is done informally, without the use of tests, grades, or traditional report cards. The assessment is done in the form of semi-annual written evaluations whi
ch, in order to take into account the children's point of view, are first completed by the children regarding their own learning and subsequently responded to by their teachers. The mid-year assessment is a relatively new addition, meant to offer teache
rs and students the opportunity to take advantage of its conclusions in order to inform subsequent educational decisions.
\par
\par Within the philosophical framework of the school, the way mathematics has been taught and assessed has been problematic for many ye
ars. As a result of a lack of professional guidance, mathematics teaching has been carried out in a traditional manner, where teaching is seen as a process of handing down information, and children's progress as increasing facility in the solving of de
c
ontextualized computation problems. Seven years ago, against this background, I suggested the implementation of a project to improve the mathematics education in the school, and, in the process, to better align its mathematics teaching with the school'
s general approach to education.
\par As dictated by a social constructivist view of learning, in order to increase the effectiveness of their teaching, teachers must be well-informed regarding their pupils' knowledge and understandings. The research presented
here is part of a wider participatory action research project which investigates ways to improve, expand and deepen the ongoing assessment of children's mathematical understandings. The goal of this part of the research is to find ways to modify the stru
c
ture of a problem-solving lesson that will improve its use as an effective assessment framework, both by the teacher, in her endeavour to understand the children's thinking processes, and by the children, to better evaluate their own progress and understa
nding.
\par
\par }{\ul\cf1\up6\insrsid8128269 Theoretical Considerations}{\cf1\up6\insrsid8128269
\par Stiggins (1995), Wiggins (1998) note three conditio
ns that are necessary in order to guarantee the effectiveness of assessment practices. The first is the collection of detailed and precise information regarding students' achievement; the second is the utilization of that information to improve the plann
ing of subsequent teaching; and the third is the involvement of the students in the evaluation of their own learning.
\par }{\ul\cf17\insrsid8128269 The collection of precise information and it}{\ul\cf17\insrsid12597546 s}{\ul\cf17\insrsid8128269 utilization in the teaching-learning process}{\cf1\insrsid8128269
\par According to Stiggins (2005) and Wiggins (1998)
, in order to arrive at detailed and precise information regarding children's learning it is necessary to determine explicit learning criteria according to which it will be possible to analyze and evaluate their achievements. They hold that in spite of
increased activity and attention to the subject of evaluation in recent years, little has been done to pinpoint important criteria of learning. Traditional evaluation looks at children's achievement at the end of a period of learning, and expresses asse
s
sment results in terms of one all-inclusive grade. In regard to the effects of this kind of summative evaluation on children's achievement, it is assumed that the focus on grades will increase students' motivation to learn and thus improve their performa
n
ce. In regard to its effects on teachers, as may be seen in the recent emphasis on teacher accountability, it is assumed that the focus on grades will increase teachers' motivation to work harder and to teach more effectively. In general there is no atte
m
pt to take advantage of the information acquired in order to assist students or teachers in improving their teaching and learning. The recording of the results of the assessment as one inclusive grade reinforces this phenomenon: even when the assessmen
t
items are scrupulously devised and theoretically able to provide important information regarding the children's understanding, as soon as the information is summarized in the form of a single grade, it loses its potential to direct further teaching and
learning.
\par }{\ul\cf1\insrsid8128269 The }{\ul\cf1\insrsid12597546 i}{\ul\cf1\insrsid8128269 nvolvement of }{\ul\cf1\insrsid12597546 s}{\ul\cf1\insrsid8128269 tudents in their }{\ul\cf1\insrsid12597546 o}{\ul\cf1\insrsid8128269 wn }{\ul\cf1\insrsid12597546 a}{
\ul\cf1\insrsid8128269 ssessment}{\cf1\insrsid8128269
\par The importance of the third criterion is crucial. Stiggins (2005) calls the understanding of importance of student self-assessment basic to formative
assessment. According to him, the decisions regarding the success or failure of their learning that are most influential are those of the children themselves. It is the children who decide if it is worth it for them to expend the effort needed in order to
succeed, and this decision is based, to a great extent, on their perception of their capabilities: are they "smart" enough in order to achieve their goals? Stiggins holds that the students' self-assessment, according to explicit criteria, is the way to he
lp children believe that they are indeed able to succeed.
\par In addition to this, Shepard (2005) holds that self-assessment allows children to understand and internalize the criteria that have been determined in that they think about them and try and put the
m into practice within the context of their own work. Black and Wiliams (1998) connect this argument to the research on cognition, and hold that if assessment is to be effective, children must learn how to assess their own work so that they will understa
nd the main goals of the learning task and will therefore understand what is expected of them in order to succeed.}{\cf1\up6\insrsid8128269
\par }{\cf1\up6\insrsid9314395
\par }{\ul\cf1\up6\insrsid8128269 Problem-Solving in Mathematics Education
\par }{\cf1\up6\insrsid8128269 The NCTM document }{\i\cf1\up6\insrsid8128269 Curriculum and Evaluation Standards for School Mathematics}{\cf1\up6\insrsid8128269 . (1989) discusses prob
lem-solving as one of the main strands in children's mathematics education. When presented with challenging problems that deal with subjects that are relevant and of interest to them, children are motivated to devise solution strategies based on their pre
vious knowledge and understanding. Solving challenging problems gives children the opportunity to tackle new mathematical content and concepts as they deal with different situations in new mathematical ways.
\par Cognitively Guided Instruction (Carpenter, Fen
nema, Franke & Empson, 1999) is a mathematics education program that centres around problem-solving. In it, teachers, provided with the background needed regarding children's solution strategies and an understanding of the factors that determine the compl
e
xity of different problem situations, use story problems in order to provide children with contexts within which they can build their own mathematical knowledge. The lessons are made up of the presentation of a single problem that is challenging for the
l
ion's share of the children in a heterogeneous learning group; time for the children to solve the problem, alone or in small groups, using their own devised problem-solving strategies; and the presentation of different solution strategies before the large
r
group during which children listen to each others strategies, compare them with their own, and discuss relevant mathematical concepts within the framework of these presentations. When carefully planned, these problem-solving lessons can provide a framewo
rk for the learning of the entire mathematics curriculum.
\par
\par }{\ul\cf1\up6\insrsid8128269 Mathematics Assessment
\par }{\cf1\up6\insrsid8128269 According to different educational organizations, among them the National Council for Teachers of Mathematics and the National Research Council in the US, current widespread
mathematics assessment practices undermine agreed-upon educational principles arrived at as a result of a large body of research and experience (NCTM, 1989; NRC, 2001). These traditional assessment practices, meant to compare children's achievement, oft
e
n carry with them the threat of sanctions, a situation that debilitates many children, particularly those who are most in need of support and a relaxed atmosphere in order to progress. The pressure on teachers to have their students perform well on the
se assessments has led them to neglect many significant and meaningful aspects of mathematics in favour of those subjects that will appear on the tests. In addition, the necessity of covering all the material to be tested prevents them from }{
\cf1\up6\insrsid10618063 even those subjects in a deep and meaningful way.}{\cf1\up6\insrsid8128269
\par }\pard \qr \li0\ri0\sb100\sa100\sbauto1\saauto1\sl-320\slmult0\nowidctlpar\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\faauto\rin0\lin0\itap0\pararsid9314395 {\cf1\up6\insrsid8128269
\par }\pard \ql \li0\ri0\sb100\sa100\sbauto1\saauto1\sl-320\slmult0\nowidctlpar\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\faauto\rin0\lin0\itap0\pararsid9314395 {\b\ul\cf1\up6\insrsid8128269\charrsid1123806 Methodology}
{\b\cf1\up6\insrsid8128269\charrsid1123806
\par }\pard \qj \li0\ri0\sb100\sa100\sbauto1\saauto1\sl-320\slmult0\nowidctlpar\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\faauto\rin0\lin0\itap0\pararsid9314395 {\cf1\up6\insrsid8128269
The research reported here, carried out as part of a participatory action research project involving myself, Shoshi, a third grade teacher}{\cf1\up6\insrsid10618063 ,}{\cf1\up6\insrsid8128269
and her class of students, investigates ways to improve, expand and deepen the assessment potential of children's mathematical understandings within the framework of a problem-solving lesson.
\par As a mathematics consultant and the originator of the mathemat
ics project in the school, I have had the opportunity to work with teachers with a variety of backgrounds in mathematics. Shoshi, who suffered in the past from low self-esteem in the area of mathematics, is today a successful and creative mathematics tea
c
hers. She has been involved in developing her mathematics teaching according to the spirit of the project for the past four years. The alternative approach to mathematics education that she has been exposed to has been both significant for her teaching a
n
d personally healing. She is convinced of its advantages and is continually involved in attempts to improve her practice. After an initial explanation on my part of action research, its advantages and its demands, she was willing and enthusiastic about t
aking part in this project. Nonetheless, because of the many demands on her time, she was unable to promise that she would perform all the activities of a full-fledged researcher. She agreed to take part in }{\cf1\up6\insrsid10618063
this action research }{\cf1\up6\insrsid8128269 project in the role of a cooperative teacher, and would participate as a researcher to the extent that that would prove possible. The children, too, are participants}{\cf1\up6\insrsid10618063 :}{
\cf1\up6\insrsid8128269 }{\cf1\up6\insrsid10618063 they }{\cf1\up6\insrsid8128269 contribute both in their role as students, and as interested informants}{\cf1\up6\insrsid10618063 ,}{\cf1\up6\insrsid8128269 providing Shoshi and me with in
sights and points of view regarding their own personal progress and the progress of the project.
\par Because this is the second year that Shoshi has taught this same class, both she and the children have developed their own agreed-upon and unique approach to
the study of mathematics and mathematical problem-solving. The children are active in their learning, and often cooperate with each other in the accomplishment of learning tasks. Different understandings and points of view are taken into consideration
in discussion among themselves and with Shoshi and me. Thus, many of the basic principles of social constructivism are put into practice in the teaching and learning that takes place in the class.
\par Although the goal of the project is to integrate assessme
nt into all parts of the class's mathematics program, the main arena of the research has been an hour-long lesson given once a week for which the class is divided into two sections, with 12 or 13 students in each section. Although the use of this time-s
l
ot has been expanded this year for research purposes, it is usually, and largely continues to be, dedicated to mathematical problem-solving, the central part of the mathematics program (Carpenter et al, 1999). Before each lesson, consideration is giv
e
n both to the mathematical content deemed important at that point in time, and to interesting occurrences in the lives of the children which provide a relevant context for the mathematical work. This lesson is often the context where the children are
e
xposed, or exposed at a higher level, to important mathematical content. The children know that they are expected to learn this content through their own efforts: through the use of their own chosen individual strategies, through explaining and listenin
g to the explanations of others regarding either their solution procedures and through discussion of relevant mathematical concepts.
\par }{\ul\cf1\up6\insrsid8128269 Methods}{\cf1\up6\insrsid8128269
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Data has been collected in the form of taped conversations between myself and the teachers, observation of the use o
f the assessment tools in the class, written and taped records of children's work on the assessments, taped structured interviews with the children, and my own research diary. Most of the data is qualitative, in keeping with the holistic view of reality c
h
aracteristic of qualitative - constructivist research, and categories have been induced through thematic analysis in which the texts were divided into meaning units and examined in terms of the whole. (Shkedi, 2003). Nonetheless, at this stage in the r
esearch, in order to attain a more general, overall picture of the preliminary findings, the emphasis has been on the more qu}{\cf1\up6\insrsid10618063 antitative aspects of the data}{\cf1\up6\insrsid8323278 :}{\cf1\up6\insrsid10618063 }{
\cf1\up6\insrsid8323278 }{\cf1\up6\insrsid10618063 when going through }{\cf1\up6\insrsid8323278 the }{\cf1\up6\insrsid10618063 work done by the children in the problem-solving lessons, I }{\cf1\up6\insrsid8323278 sorted their papers according to }{
\cf1\up6\insrsid10618063 }{\cf1\up6\insrsid8323278 the relevant aspects of their solution strategies and written presentations, and counted the number of papers in each categor}{\cf1\up6\insrsid4071793 y. I my talk I will present }{
\cf1\up6\insrsid4344279 a richer, more qualitative description of the data, as well }{\cf1\up6\insrsid4071793 examples of the children's work}{\cf1\up6\insrsid4344279 . }{\cf1\up6\insrsid4071793 }{\cf1\insrsid4071793 }{\cf1\insrsid8128269
\par }{\b\ul\cf1\insrsid8128269\charrsid1123806 Findings}{\b\cf1\insrsid8128269\charrsid1123806
\par }{\cf1\up6\insrsid8128269 At the outset the assessment project focused on the development of formative assessment methods meant to improve the quality of the mathematics education that took place in Shoshi's class. Self-assessment was co
nsidered part of that assessment. As time went on the central role of the children's self-assessment became increasingly clear. Almost from the beginning, Shoshi reported on the children's enthusiasm regarding being seen as full partners in the assessme
n
t of their understanding. From a preliminary analysis of the data, two main intersecting axes of development have become apparent. The first, relating to the children's work, is the increasing efficiency, flexibility and sophistication of the childr
e
n's solution strategies, and the other, relating to my own and Shoshi's work, is the developing format of the problem-solving worksheet on which the children were asked to report those solution strategies. Although it may be impossible to separate the
various influences on these developments, both seem to have been largely affected by the growing emphasis on self-assessment.
\par }{\ul\cf1\up6\insrsid8128269 The Children's Work}{\cf1\up6\insrsid8128269
\par Data collected from the children's problem-solving worksheets, from my own and Shoshi's observations of the ch
ildren's work during the lesson and from structured interviews that I have recently begun to hold with the children, point to the gradual development of their problem-solving abilities in a number of areas. One development is a progression from the us
e
of words or pictures in the documentation of their solution strategies, to the use of numbers or more abstract graphical representations. Of the six children who at the beginning of the year used mainly written sentences or pictures with which to report
their solution strategies, five of them have begun to use numbers and symbols almost exclusively. This development may be attributed at least partially to the fact that the advantages of the use of more abstract notation are often pointed out and discu
s
sed when children present their strategies to the rest of the class and show their solution processes on the blackboard. The children are thus presented with different models of more or less abstract notation, have the opportunity to consider their adva
ntages or disadvantages and to compare them with their own work.
\par Another development is the increased structure of the children's solution strategies. A large number of children seem to have a growing ability to identify the different stages of the solu
tion process, note the results of each stage and use these results to arrive at the final solution to the problem. This, one of Shoshi's main goals during the first part of the year, has been a major topic of discussion throughout the year. The importa
n
ce of approaching problems in an orderly and structured fashion, including orderly written representation of solution strategies, has been discussed, and apparently understood by many, as a way to greatly increase the possibility of arriving at correct
solutions. From examination of the problem-solving worksheets, and from Shoshi and my own observations, it may be seen that of the eight children who originally had difficulty organizing their thoughts when approaching a problem, three have begun to wor
k in a much more orderly fashion and another three have shown significant improvement. In addition, of the nine children who originally had problems in the clear written presentation of their strategies, five have shown significant improvement.
\par In addit
ion, in this regard, it is interesting to note that of the five children who have been interviewed to date regarding the progress they feel they have made during the year, regardless of their original level or mode of problem-solving, all felt that
the clear written presentation of the solution strategies on the blackboard has been one of the reasons for their progress.
\par A third development is the increasing efficiency and sophistication of their solution methods. At the beginning of the year four o
f the most advanced students used and presented highly sophisticated solution procedures, and at this point of the year another seven children have been observed using similar strategies. These strategies include thinking about the problem before they b
e
gin to solve it, using previous results to aid in the finding of others, using the attributes of numbers to help in their computation and looking for different angles from which to tackle the problem in the first place. Again, all of these different ways
of looking at the problem and solving it have been presented over the course of the year, and their advantages discussed. It seems that this evaluation of other children's strategies naturally leads to reflective self-assessment, allowing children to c
ompare other strategies with their own, to decide which of the methods make the most sense to them and which they might decide to adopt in the future.
\par A further and connected development in the children's work is the relative ease with which the great maj
ority of them are now able to begin to solve problems independently, immediately after hearing the problem presented. According to Shoshi's report and my own observation, at the beginning of the year twelve of the children, seven in one group and fiv
e
in the other, needed individual help before they were able to continue solving the problem on their own. This, needless to say, made the running of the lesson difficult, even with the help that I was able to provide as an additional adult in the class. B
y
the middle of the year, however, eleven of these children have begun to feel capable of beginning their thinking processes on their own, and, even if they still need help, are confident enough to work on their own, or with a friend, until a teacher beco
m
es available. This development may also be attributed to the self-assessment facet of the assessment, in that they are now aware of their abilities and feel that, if they put their minds to it, they may well be able to tackle the challenging problems
that are the general fare of this lesson.
\par }{\ul\cf1\up6\insrsid8128269 The Teachers' Work}{\cf1\up6\insrsid8128269
\par The development of the problem-solving worksheets was a process that at each point in time came in response to Shoshi's and my own felt needs in out attempts to improve the problem-solving less
on in its role as an evaluation opportunity. In this way the improvement of the format of the worksheet reflects well the action research nature of this project. In our attempts to better assess the children's thinking, and to have the children better ass
e
ss their own progress, we began to devise ways to use the worksheet to direct attention to those issues we deemed important, to ensure that this information be obtained for all the children, and to gather the information in an orderly way. The first ste
p
in this process was the initiation of the use of the worksheet in the first place: my own experience obtained from a pilot project carried out the previous year led me to understand the importance of having the children solve the problems on a worksheet
w
hich the teacher could easily take home, rather than in their notebooks as had previously been the case . This simple change allowed the teacher to collect the work at the end of the lesson, to review it later in the day in a more leisurely fashion, to m
a
ke notes regarding the significance of the information obtained and to plan subsequent steps in order to ensure further progress for each child. In addition to the direct influence of this simple change, it also has the effect of encouraging and facilit
ating teachers' taking of responsibility for their pupils' progress.
\par At the beginning of the year the worksheet consisted only of room to write the child's name and the date at the top of the page, the actual problem worded in the laconic language of tr
aditional story problems, and room for the children to show their solution strategies. This original format began to change once the children had become familiar with the expression "difficult but possible", a "motto" which they had begun to use it whe
n
judging the difficulty of computation exercises they solved in the framework of periodic self-tests At that point we added a question at the end of the worksheet: "Was this problem difficult but possible for you? Please explain." The addition of thi
s
metacognitive question directed the children's attention both to the attributes of the problem that determined its level of difficulty for them, and to the assessment of their understanding of the problem and their ability to solve it. The children's in
c
reasing self-confidence in their abilities seems to have been reinforced by this question, either by strengthening their sense of accomplishment at having succeeded in solving a difficult problem, or by making it clear to them that, for them, the prob
lem was actually not so difficult to begin with.
\par The next addition to the worksheet was the inclusion of a section at the bottom of the page which asked them to copy a solution strategy used by one of those children who had presented their work before t
he class: "Copy one strategy from the board that you understand and that you might use in the future." This was done as a way to have the children seriously consider the strategies presented and assess the extent to which they understood them. Also, the
instruction was added after Shoshi had begun to put more emphasis on the way the children recorded their strategies in writing in order to encourage greater organization and tidiness. The instruction to copy one of the strategies, was meant to direct th
eir attention and to reinforce the effect of their exposure to more structured written reporting .
\par At this point we had begun to understand the value of these added questions and began to devise additional ways to access the children's thinking that wo
uld be appropriate in different circumstances. If a particular problem was similar to a problem they had received in the past, we asked them whether they had chosen to use a strategy that they had learned from another child. When we began to place empha
s
is on estimating the solution before they did the actual computation, particularly when the solution to the problem was more complex such as in division problems or those involving two-digit multiplication, we added an instruction to estimate the solution
just under the problem itself, and allowed them two or three minutes in which to do so . And, particularly for the very advanced students who had received a second, much more difficult problem to solve, we asked "Did you feel that this problem was diffi
c
ult but possible for you? If it was too easy, change the numbers to make it more difficult". This last question will likely be modified in the future to make it more general: "Change the problem so that it will be sufficiently challenging for you." It is
also likely that as time goes on and different needs are felt, additional questions will be devised to enhance the children's learning and to help both them and the teacher better assess their thinking and progress.}{
\cf1\up6\insrsid8128269\charrsid4333579
\par }{\ul\cf1\up6\insrsid8128269 Conclusion}{\cf1\up6\insrsid8128269
\par From the data considered to date it seems legitimate to infer the important place of self-assessment in the children's mathematics progress. Because it is the children themselves that choose the ways in which they solve problems and report their solutio
n strategies, changes in these }{\cf1\up6\insrsid4344279 strategies }{\cf1\up6\insrsid8128269 over time would seem to be a result of their reflecting on different strategies, comparing }{\cf1\up6\insrsid2511394 those}{\cf1\up6\insrsid8128269
of others to their own, and coming to decisions regarding th}{\cf1\up6\insrsid2511394 e ones}{\cf1\up6\insrsid8128269 that they feel are most useful for them.
\par Both for research purposes and for educational ones, it is important to obtain the most explicit information possible. The }{\cf1\up6\insrsid2511394 gradual }{\cf1\up6\insrsid8128269 development of the problem-solving worksheet }{\cf1\up6\insrsid2511394
as an effective assessment tool is}{\cf1\up6\insrsid8128269 an ongoing attempt to do so. Each change or addition to the questions asked has been an attempt to answer a need, whether this be}{\cf1\up6\insrsid12017583 with }{\cf1\up6\insrsid8128269
regard to }{\cf1\up6\insrsid4274021 children's }{\cf1\up6\insrsid8128269 metacognitive abilities that will advance the}{\cf1\up6\insrsid4274021 ir}{\cf1\up6\insrsid8128269 mathematical understanding, }{\cf1\up6\insrsid12017583 or with }{
\cf1\up6\insrsid8128269 regard to}{\cf1\up6\insrsid4274021 ways}{\cf1\up6\insrsid8128269 the t}{\cf1\up6\insrsid4274021 eachers can gather better information about }{\cf1\up6\insrsid8128269 the
children's thinking. This study shows the interaction between these two }{\cf1\up6\insrsid4274021 processes}{\cf1\up6\insrsid8128269 : the }{\cf1\up6\insrsid4274021 process of the }{\cf1\up6\insrsid8128269 children's mathematical development, and}{
\cf1\up6\insrsid4274021 that of }{\cf1\up6\insrsid8128269 the teachers' }{\cf1\up6\insrsid4274021 growing }{\cf1\up6\insrsid8128269 ability to facilitate this development.
\par }{\fs20\ul\cf1\up6\insrsid8128269\charrsid8128269 Bibliography}{\fs20\cf1\up6\insrsid8128269
\par }{\fs20\cf1\up6\insrsid4333579\charrsid4333579 Black, P. & Wiliam, D. (1998). Inside the Black Box: Raising standards through classroom assessment. }{\i\fs20\cf1\up6\insrsid4333579\charrsid4333579 Phi Delta Kappan}{
\fs20\cf1\up6\insrsid4333579\charrsid4333579 , 80(2): 139-144, October.
\par Carpenter, T.P., Fennema, E., Franke, M.L., Empson, S.B. (1999). }{\i\fs20\cf1\up6\insrsid4333579\charrsid4333579 Children's Mathematics: Cognitively Guided Instruction}{\fs20\cf1\up6\insrsid4333579\charrsid4333579 . Portsmouth, N.H.: Heinemann.}{
\fs20\cf6\up6\insrsid4333579\charrsid4333579
\par }{\fs20\cf1\up6\insrsid4333579\charrsid4333579 Cobb, P. & Bowers, J. (1999). Cognitive and Situated Learning Perspectives in Theory and Practice. }{\i\fs20\cf1\up6\insrsid4333579\charrsid4333579 Educational Researcher }{
\fs20\cf1\up6\insrsid4333579\charrsid4333579 28(2) March: 4-15.
\par }{\fs20\cf1\up6\insrsid8128269\charrsid8128269 Lave, J. (1988). }{\i\fs20\cf1\up6\insrsid8128269\charrsid8128269 Cognition in Practice: Mind, mathematics and culture in everyday life}{\fs20\cf1\up6\insrsid8128269\charrsid8128269
. Cambridge: Cambridge University Press.
\par Lave, J. (1996). Teaching, As Learning, in Practice. }{\i\fs20\cf1\up6\insrsid8128269\charrsid8128269 Mind; Culture and Activity}{\fs20\cf1\up6\insrsid8128269\charrsid8128269 , 3(3): 149-164.
\par National Council of Teachers of Mathematics (1989). }{\i\fs20\cf1\up6\insrsid8128269\charrsid8128269 Curriculum and Evaluation Standards for School Mathematics}{\fs20\cf1\up6\insrsid8128269\charrsid8128269
. Reston, Va.: National Council of Teachers of Mathematics.}{\fs20\cf1\up6\insrsid8128269
\par }{\fs20\cf1\up6\insrsid4290163 Shepard, L. (2005). Linking Fo}{\fs20\cf1\up6\insrsid4396140 s}{\fs20\cf1\up6\insrsid4290163 rmative Assessment to Scaffolding, Educational Leadership 63 (3): 66-70. }{\cf6\up6\insrsid4290163
\par }{\fs20\cf1\up6\insrsid8128269\charrsid8128269 Shkedi, A. (2003) Words of Meaning: Qualitatitve Resarch - Theory and Practice. Tel Aviv: Ramot.
\par Stiggins, R.J. (2005). }{\i\fs20\cf1\up6\insrsid8128269\charrsid8128269 Student-Involved Assessment FOR Learning}{\fs20\cf1\up6\insrsid8128269\charrsid8128269 . Upper Saddle River, NJ: Pearson Education.
\par Wiggins, G. (1998). }{\i\fs20\cf1\up6\insrsid8128269\charrsid8128269 Educative Assessment: Designing Assessments to Inform and Improve Student Performance. }{\fs20\cf1\up6\insrsid8128269\charrsid8128269 San Francisco}{
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\par }}