In contrast to providing teachers with a deep understanding of higher-level mathematics, a framework that focuses on a clear understanding of key mathematical "growth points" in children’s learning may be a more critical need. Such a framework can offer an enhanced understanding of the mathematics which teachers need to teach, and a lens through which they can view their own children’s growth. Using the example of the Early Numeracy Research Project in Victoria, Australia, I will illustrate the power of a framework for mathematical learning in understanding, assessing and developing children’s mathematical thinking. At the beginning of the school year, teachers use a task-based interview with every child one-to-one for approximately thirty minutes, providing detailed information on individual children’s mathematical understanding. Teachers meet throughout the year at school, district and statewide professional development, during which they share insights and are presented with ideas for the classroom from research and the wisdom of other teachers. Implications of this approach for preservice and inservice teachers will be discussed, as will early data from the project.
There is widespread agreement among those involved in mathematics education that teacher knowledge is a major determinant of mathematics instruction and student learning (Fennema & Franke, 1992). But what forms of knowledge are most powerful, and how these might be developed and extended in preservice and inservice settings.
Various scholars have categorised the knowledge possessed by mathematics teachers in a variety of ways. Shulman (1987) identified the components of teachers’ knowledge (general and not specific to mathematics teaching) as:
Shulman’s contribution was to broaden the debate particularly in relation to pedagogical content knowledge which refers to how specific knowledge can be interpreted in teaching situations (Cooney, 1994).
Fennema and Franke (1992), in a review of teaching knowledge in mathematics, discussed four components: knowledge of mathematics, knowledge of mathematical representations, knowledge of students, and general knowledge of teaching and decision making, again articulating the importance of knowledge specific to the teaching of mathematics.
Mason and Spence (1999) reviewed a range of categorisations and focused particularly on teachers’ knowledge as dynamic and evolving and of the importance of knowing-to as it requires "relevant knowledge to come to the fore so it can be acted upon" (p.139). It is here that knowledge and practice intersect/interact and the knowledge can prove to be useful or otherwise. Much of the content knowledge that teachers have is not accessible. Brophy (1991) argues in relation to content knowledge that
where (teachers’) knowledge is more explicit, better connected, and more integrated, they will tend to teach the subject more dynamically, represent it in more varied ways and encourage and respond fully to students’ comments and questions. Where their knowledge is limited, they will tend to depend on the text for content, de-emphasize interactive discourse in favor of seatwork assignments, and in general, portray the subject as a collection of static, factual knowledge. (p. 352)
Clearly, the way knowledge is organised and accessed as well as the nature of that knowledge is important. It must also be acknowledged that in many countries (including Australia) there has been a shift in focus from a transmission model of teaching to an emphasis on teaching for understanding (Fennema & Romberg, 1999). It is no longer a case of the student "working out what is in the teacher’s head" but instead on teaching that aims to understand and build on what the student is thinking. It should be stated however that this shift is present in policy statements and curriculum documents more so than in the reality of classrooms, reflecting the challenge this poses for teachers. Teachers using innovative curriculum materials for the first time in a small project in Wisconsin, USA articulated this challenge.
Many of the strategies we had use to help our students master manipulative skills were useless in this new environment. . . . The possibility that several points of view and, consequently, several answers, were reasonable is difficult to accept, especially when you have spent an average of fifteen years rewarding thought processes that were identical to yours. . . . Our role was shifting from that of one who directs the thought processes of the students to one who reacts and guides their reasoning; it was not easy to resist telling students what to do or showing them how, but instead to ask leading questions. We had to listen to students, examine their work, and try to learn what they were thinking as they solved a problem. . . Communication became an integral part of the classroom dynamics. (deLange, van Reeuwijk, Burrill, & Romberg, 1993, pp.190-192)
Moving to a more learner-centred approach places greater demands on teacher knowledge, as the lesson can take many possible directions, given the more responsive nature of the teaching process, and students’ strategies and reasoning could well challenge the teacher’s mathematical "comfort zone".
For the purpose of this discussion, I will use a structure for
discussing teachers’ knowledge that has emerged from an
Australian project, titled Excellence in Teaching of Mathematics:
Professional Standards Project (see Bishop & Clarke, 1998;
Morony, 1999). This project is a collaboration between the Australian
Association of Mathematics Teachers (AAMT) and Monash University and
aims in part to determine consensual views on national professional
standards for excellence in teaching mathematics in Australian
schools. The following draft descriptors should not be viewed as
defining teachers’ knowledge but rather as a categorization of
the range of knowledge identified by Australian teachers as
describing the excellent teacher of mathematics.
1.1 Knowledge of students
Excellent teachers of mathematics have a thorough knowledge of the students they teach. This includes knowledge of students’ social and cultural contexts, the mathematics they know and use, their preferred ways of learning, and how confident they feel about learning mathematics.
1.2 Knowledge of mathematics
Excellent teachers of mathematics have a sound, coherent knowledge of the mathematics appropriate to the student level they teach, and which is situated in their knowledge and understanding of the broader mathematics curriculum. They understand how mathematics is represented and communicated, and why mathematics is taught. They are confident and competent users of mathematics who understand connections within mathematics, between mathematics and other subject areas, and how mathematics is related to society.
- 1.3 Knowledge of students’ learning of mathematics
Excellent teachers of mathematics have rich knowledge of how students learn mathematics. They have an understanding of current theories relevant to the learning of mathematics. They have knowledge of the mathematical development of students including learning sequences, appropriate representations, models and language. They are aware of a range of effective strategies and techniques for: teaching and learning mathematics; promoting enjoyment of learning and positive attitudes to mathematics; utilizing information and communication technologies; encouraging and enabling parental involvement; and for being an effective role model for students and the community in ways they deal with mathematics.
(AAMT, 2000, p. 2)
While there has been much written in this area, there is little agreement and even "less evidence about what knowledge will enable teachers to teach so that students learn mathematics with understanding" (Fennema, Carpenter, Franke, Levi, Jacobs, & Empson, 1996, p. 403). I will argue that in the context of developing children’s understanding in the early years of school, knowledge of students’ learning of mathematics and knowledge of students as it relates to their mathematical understandings are key.
The Cognitively Guided Instruction (CGI) project based at the University of Wisconsin &emdash; Madison in the USA is a research and professional development program that focused on helping teachers understand the development of children’s mathematical thinking by interaction with a specific research-based model. (Carpenter & Fennema, 1992; Fennema et al., 1996). Teachers’ detailed understanding of various types of addition and subtraction story problems, their relative difficulty for children, and common solution strategies enabled the teachers to carefully choose problems to pose to students, and resulted in enhanced student understanding of relevant concepts. The goal of CGI teacher development was to help teachers develop an understanding of their own students’ mathematical thinking and its development and how this knowledge can form the basis for the development of more advanced mathematical ideas. Their results support the value of providing such knowledge to teachers.
This study provides strong evidence that knowledge of children’s thinking is a powerful tool that enables teachers to transform this knowledge and use it to change instruction. These findings, when viewed in conjunction with those of other studies, provide a convincing argument that one major way to improve mathematics instruction and learning is to help teachers understand the mathematical thought processes of their students. (Fennema et al., 1996, p. 432)
In the following sections, another project which aims to support teachers in understanding, assessing and developing children’s thinking will be discussed. As with CGI, knowledge of children’s learning of mathematics and knowledge of individual children’s thinking is paramount.
The Early Numeracy Research Project (ENRP) is a collaborative venture between Australian Catholic University, Monash University, the Victorian Department of Employment, Education and Training, the Catholic Education Office (Melbourne), and the Association of Independent Schools Victoria. The ENRP aims to identify effective approaches to the teaching of mathematics in the first three years of school. The project began in January 1999 and is funded to early 2002 in 35 project ("trial") schools and 35 control ("reference") schools. The term "numeracy" has become used more frequently by politicians and school systems, but in the context of young children, for the purpose of this project, the terms numeracy and mathematics are used interchangeably.
The aims of the project include:
The Early Years Mathematics Coordinator and the teachers at Prep to Year 2 form the "professional learning team" at each school. Such teams meet weekly to share experiences and to plan programs together. Geographical clusters of learning teams from the 35 schools meet monthly, and all teachers in the project (around 230) meet as a large group several times a year. Each team and cluster is supported by one or more university researchers, who have regular involvement in classrooms and cluster meetings. (For more detail see Clarke, 1999).
The Framework
The research team identified the need for the development of a comprehensive and appropriate learning and assessment framework for early mathematics, as well as the need to address the personal confidence with and understanding of mathematics of many primary teachers.
The impetus for the project was a desire to improve mathematics learning and so it was necessary to quantify such improvement. It would not have been adequate to describe, for example, the effectiveness of the professional development in terms of teachers’ professional growth, or the children’s engagement, or even to produce some success stories. It was decided to create a framework of key "growth points" in mathematics learning. Students’ movement through these growth points in trial schools could then be compared to that of students in the reference schools. Although the initial impetus for developing the framework was to show growth, it will be argued that teacher understanding and "ownership" of the framework and its growth points has led to greatly enhanced understanding of children’s mathematical learning, more focused planning for teaching, and greater personal confidence with mathematics.
The project team studied available research on key "stages" or "levels" in young children’s mathematics learning (e.g., Clements, Swaminathan, Hannibal, & Sarama, 1999; Fuson, 1992; Lehrer & Chazan, 1998; Mulligan & Mitchelmore, 1995, 1996; Pearn & Merrifield, 1992; Steffe, Cobb, & von Glasersfeld, 1988), as well as frameworks developed by other authors and groups to describe learning.
It was intended that the framework would
At the time of writing, the domains addressed by the framework were Counting, Place Value, Addition and Subtraction Strategies, Multiplication and Division Strategies (within Number), Length, Mass and Time (within Measurement), Classification, and Visualisation and Orientation (within Geometry).
Within each mathematical domain, growth points were stated with brief descriptors in each case. There were typically five or six growth points in each domain. To illustrate the notion of a growth point, consider the child who is asked to find the total of two collections of objects (say nine objects and another four objects). Many young children will "count all" to find the total ("1, 2, 3, . . . , 11, 12, 13"), even once they are aware that there are nine objects in one set and four in the other. Other children will realise that by starting at 9 and counting on ("10, 11, 12, 13"), they can solve the problem in an easier way. Counting All and Counting On are therefore two important growth points in children’s developing understanding of Addition.
This paper presents only some results from the Addition and
Subtraction Strategies domain of the framework. The growth points
are shown in Figure 1.
0. Not apparentUnable to combine and count two collections of objects. 1. Count all (two collections)
Counts all to find the total of two collections.
2. Count on
Counts on from one number to find the total of two collections.
3. Count back/count down to/count up from
Given a subtraction situation, chooses appropriately from strategies including count back, count down to and count up from.
4. Basic strategies (doubles, commutativity, adding 10, tens facts, other known facts)
Given an addition or subtraction problem, strategies such as doubles, commutativity, adding 10, tens facts, and other known facts are evident.
5. Derived strategies (near doubles, adding 9, build to next ten, fact families, intuitive strategies)
Given an addition or subtraction problem, strategies such as near doubles, adding 9, build to next ten, fact families and intuitive strategies are evident.
6. Extending and applying addition and subtraction using basic, derived and intuitive strategies
Given a range of tasks (including multi-digit numbers), can solve them mentally, using the appropriate strategies and a clear understanding of key concepts.
Figure 1. ENRP growth points for addition and subtraction strategies.
In discussions with teachers, we have come to describe growth points as key "stepping stones" along paths to mathematical understanding. However, we do not claim that all growth points are passed by every student along the way. For example, Growth Point 3 in Addition and Subtraction involves "counting back", "counting down to" and "counting up from" in subtraction situations, as appropriate. There appears to be a number of children who view a subtraction situation (say, 12-9) as "what do I need to add to 9 to give 12?" and appear to make little use one of those three strategies.
Also, the growth points should not be regarded as necessarily discrete. As with Wright’s (1998) framework, the extent of the overlap is likely to vary widely across young children, and "it is insufficient to think that all children’s early arithmetical knowledge develops along a common developmental path" (p. 702).
The Assessment Interview
Once the early drafts of the framework were developed, assessment tasks were created to match the framework. A major feature of the project is a one-to-one interview with every child in trial schools and a random sample of around 40 children in each reference school at the beginning and end of the school year (February/March and November respectively), over a 30- to 40-minute period. It is a task-based interview conducted by the classroom teacher in trial schools and a team of trained interviewers in reference schools. Manipulative materials are provided for many tasks, with small plastic teddy bears a feature. These help to provide interest and motivation on the part of the children and have greatly increased the accessibility of the tasks.
Although the full text of the interview involves around 50 tasks (with several sub-tasks in many cases), no child moves through all of these. The interview is of the form "choose your own ending", in that the interviewer makes one of three decisions after each task. Given success with the task, the interviewer continues with the next task in the given mathematical domain as far as the child can go with success. Given difficulty with the task, the interviewer either abandons that section of the interview and moves on to the next domain or moves into a detour, designed to elaborate more clearly the difficulty a child might be having with a particular content area. The interviewer records the student responses on a four page record sheet.
To further illustrate let me describe one task from the Addition and Subtraction Strategies section of the interview. The interviewer asks "imagine you have 11 strawberries in your play lunch, and you eat 9. How many would you have left?" They then indicate the answer given and choose from the following options on the record sheet:
In order to categorise appropriately the child’s response, teachers and other interviewers are involved in detailed discussion during the professional development program, with the aim of providing a sound understanding of the relative sophistication and efficiency of each strategy.
The interview provides information about those growth points achieved by a child in each of the seven domains. Our aim in the interview is to gather information on the most sophisticated strategies that a child accesses in a particular domain. However, depending upon the context and the complexity of the numbers in a given task, a child (or an adult) may use a less sophisticated strategy than they actually understand, as the simpler strategy may do the job quite nicely in that situation.
Wright (1998) warns of the challenge of determining the actual strategy used by a child in solving a problem, as "a child may unwittingly or intentionally describe a strategy different from the one used" (p. 703).
It is important to stress that the growth points are "big mathematical ideas or concepts", with many possible "interim" growth points between them. As a result, a child may have learned several important ideas or skills necessary for moving to the next growth point, but perhaps not of themselves sufficient to move there. Also, to achieve many of the growth points requires success on several tasks, not just one or some.
Of course, decisions on assigning particular growth points to children are based on a single interview on a single day, and a teacher’s knowledge of a child’s learning is informed by a wider range of information, including observations during everyday interactions in classrooms. However, teachers agree that the data from the interviews are revealing of student mathematical understanding and development, in a way that would not be possible without the opportunity for one-to-one interaction and much richer than a simple statement of growth points. It provides the teachers with a "window" into children’s thinking.
A key criteria for the framework to be successful is the extent to which it reports on the spread and development of children’s learning. Table 1 presents the percentages of children at each growth point in the March 2000 (beginning of the school year) interviews in reference schools for Addition and Subtraction Strategies.
Table 1
Percentage of Children at Each Counting Growth point, by Grade Level (%), March 2000
|
|
(~5 years)
|
(~6 years)
|
(~7 years)
|
|
|
0. Not apparent |
55.9 |
12.4 |
3.0 |
26.7 |
|
1. Count all |
37.8 |
51.3 |
22.7 |
37.6 |
|
2. Count on |
6.2 |
31.3 |
46.8 |
25.9 |
|
3. Count back/ down to/ up from |
0.2 |
4.3 |
13.7 |
5.4 |
|
4. Basic strategies |
0 |
0.2 |
10.4 |
3.0 |
|
5. Derived strategies |
0 |
0.5 |
3.3 |
1.1 |
|
0 |
0 |
0 |
0 |
There is a clear development from Prep (the first year of school) to Grade 1 to Grade 2. This part of the framework seems to allow description of a spread of development within the one grade level, and illustrates development across the grades.
The framework is designed to allow the quantification of the learning of the children. However, we are more interested in identifying factors that may contribute to such learning. To complement the data on the children’s learning, a range of other data is being collected, including detailed questionnaires on teachers’ beliefs and understandings about mathematics learning, regular journals kept by Early Mathematics Coordinators (the leaders of the professional learning teams in each school), as well as teacher and principal data on the effect of the project on teaching practice and student attitudes to mathematics.
In the third year of the project (2001), major emphasis will be given to studying those teachers and schools who have been shown to be particularly effective in building mathematics understanding.
Children’s Mathematical Development: Some Results
Interviews early in the school year and towards the end of the school year provide a measure of growth in student understanding. Such data enable a comparison between growth in understanding in trial and reference schools, thereby providing a measure of the effectiveness of the professional development program. These comparisons will not be discussed in detail here, but a measure of the positive effect of "full" involvement of teachers from trial schools in the ENRP is that in November, 63% of Prep children in trial schools exceeded March median growth points in all domains, compared with only 39% of children in reference schools. (Figures for Grades 1 and 2 were 49% and 30%, and 24% and 7% respectively).
Teachers Initial Response to the Interview
Following the completion of the first interviews in March 1999, teachers were asked to write about highlights and surprises that had emerged from the interview process. In compiling the responses, some insights into the power of the interview from the teachers’ perspective emerged.
Surprise at what many children were able to do. Many of the surprises related to children’s capacity to deal confidently with large numbers, and the wide variety of strategies used in solving the problems.
Working with a gifted five year-old who actually worked out the answers quicker than I did. Reading 24,746,154 on the calculator. Amazing! I have one grade 2 student in my P/1/2 class whom I know loves maths&endash;number in particular. He worked out the answer for 134 and 689 in his head. This child was able to articulate all the strategies he used.
My greatest surprise was that most children performed significantly better than I anticipated. Their thinking skills and strategies were more sophisticated than I expected.
Surprise at the difficulties that some children had with certain tasks. Tasks involving multiplicative reasoning (e.g., putting two teddies in each car or sharing teddies between "teddy mats") surprised teachers with the difficulty children displayed, as did some of the tasks relating to concepts of time.
Difficulty in counting backwards for some children was a surprise. I was amazed that many of my students had no logical system when naming the days of the week, or the months of the year. They just randomly stated days with no way of checking.
One child in grade 2 who was mathematically skilled with numbers and very aware of them, was completely floored when he had to share the teddies evenly on the teddy mats. He tried every possible way and could not work it out.
Some children tested could not explain their process of working "things" out and yet these children appeared to be very "able" in usual classroom activities.
The greatest surprise from the assessment interview process was discovering that the children who you thought had specific concepts, in fact couldn’t use these/didn’t have them in a one-to-one situation&endash;that they were good at "hiding" within the group.
I was surprised how few children had the ideas of early division.
The emergence of the quiet achievers, particularly girls. Several teachers commented that the interview with individual children "painted" a different picture from that which emerged during whole class and small group classroom activities.
Finding out the ability of some of my quieter students amazed me. Being given a chance to answer a question they knew without another student interrupting them was very rewarding for them. But it was also exciting for me to see what ability they had. Quiet achievers (especially girls).
In every grade there is that quiet child you feel that you never really "know"&endash;the one that some days you’re never really sure that you have spoken to. To interact one-on-one and really "talk" to them showed great insight into what kind of child they are and how they think.
The power of the interview data in informing teaching. Many teachers indicated that the information provided by the interview suggested "starting points" for instruction.
The greatest highlight of all is to be able to clearly see where the child is at and what maths work needs to be worked on to further enhance his skills. I was very surprised with how much many of the children knew and how many different, complex strategies were being utilised in order to work at the answers to the many open ended problems. It has been an eye-opener and I have since based a lot of my own teaching on the results gained.
My greatest surprise(s) was the wealth of information gained from the assessment interview&endash;how confident the children were in responding to a neverending supply of questions … and how I’ve been about to "use"(adapt) some of the ideas into my classroom practice.
The greatest highlight from the assessment interviews was having the one-to-one contact with my children which really enabled me to focus and see what they really knew and what I have to work on for them to enrich their learning.
The level of enjoyment and confidence displayed by the children during the interview. We can sometimes be preoccupied with children’s cognitive growth, with insufficient attention paid to affective aspects. The enjoyment showed by almost all children during the interview, whether their mathematical understanding was high or not so high was important. It emphasises that children appreciate the opportunity to show their understanding, particularly when they have the teacher all to themselves.
How enjoyable for both teacher and child. It gave the children the opportunity to spend individual time with the teacher and the children responded positively. Many couldn’t wait for their turn and chatted away during the interview. How adaptable/patient and flexible young children can be when working under difficult conditions, and how resilient teachers are after sitting on a small chair for two days!
I have a child with cerebral palsy who is in a wheelchair and has limited motor skills. He was determined to do all the tasks without help from his aide and he did. The look on his face when he completed each task was amazing.
My highlight was when the children tried to explain how they worked out their answer. Several said "my brain told me". The best one was "my mum told me the answer would be that".
[teacher in Specialist school] Staying power! Each child stuck it out without running away. Some children have a concentration span of 2 seconds, yet they sat quietly and really seemed to listen.
The greatest highlight was that no matter at what level the children were operating mathematically, all children displayed a huge amount of confidence in what they were doing. They absolutely relished the individual time they had with you; the personal feel, and the chance to have you to themselves. They loved to show what they can do.
On a humorous note, a favourite anecdote from the interviews was the following. Children had been asked to "draw a clock", for use as a basis of discussion of their understanding of how time and clocks work. The teacher takes up the story:
I asked the child ‘What are the numbers on the clock doing?’ The child looked strangely at me and said ‘the numbers are doing nothing, they are waiting for the arrows to come around. Don’t you know that? Are you stupid or something?’
Teachers’ Stated Professional Growth
Given the clearly successful efforts of trial school teachers in developing children’s mathematical skills and understandings in 1999, it becomes increasingly important for the research project to start to look at successful teachers’ practice to try to discern those aspects of "what the teacher does" that make a difference. After slightly more than one year’s involvement in the project, teachers were asked
1. Has your mathematics teaching changed in the last twelve months?2. If "yes", please give one example of the change.
These have not been analysed in detail, but several common themes have emerged:
Several of these themes are evident in the following quote from a teacher:
The assessment interview has given focus to my teaching. Constantly at the back of my mind I have the growth points there and I have a clear idea of where I’m heading and can match activities to the needs of the children. But I also try to make it challenging enough to make them stretch.
Teachers’ Observations of Children’s Growth
Teachers were also asked to comment on aspects of children’s growth that they had observed which were not necessarily reflected in movement through the growth points. Common themes were the following:
One teacher commented on her children’s positive attitudes to mathematics:
Children seem to be more enthusiastic, take more risks and have more confidence in their abilities. They can’t wait to participate. They’re excited about maths. For example, we brainstormed the combination of green or red lollies to make 10 and when the children opened their bag, they exploded with excitement! "I’ve got 3 and 7!" "I’ve got 2 and 8!" All this over adding to 10!!
This quote illustrates the importance that teachers are placing on the children’s development of a range and a sophistication of strategies.
Nature of Teacher Support
Given the considerable success of the ENRP to this point in terms of enhanced student learning, it is important for the purpose of this paper to summarise the nature of the support provided to teachers. This support can be considered as three "opportunities".
The experience of this project has led to some questioning of the content of preservice courses offered to prospective primary teachers. The value of gaining insight into children’s thinking as individuals has taken on greater importance. One recent change in my own work has been to provide greater exposure of preservice students to the mathematical thinking of individual children. This is achieved through viewing videotapes and through an assessment task that requires student teachers to undertake one-to one interviews with several children and to present an analysis of the children’s understanding, in relation to the growth points and children’s strategies.
In a conversation with a recent graduate who was four months into her teaching career, the teacher commented that she needed to develop more structure in her classroom. This is probably based on the observations of other classrooms and discussion with teachers. I encouraged her to think about whether she was imposing structure for structure’s sake. In the last two years, I have seen the reverse in ENRP classrooms of experienced junior primary teachers. Teachers are freeing up their routines to be responsive to children’s thinking. They are developing new routines and practices with a particular focus on children sharing strategies and thinking. The project did not attempt to impose a structure but recognised the teacher and the team as responsible for the decisions in the classroom. Some strategies were presented and teachers were encouraged to trial these, but the ultimate decision on classroom implementation was at the school and teacher level. One of the challenges in the preservice context is to provide students with exemplars of this type of teaching.
In the Australian setting, it is frequently the case that primary teachers (those teaching 5- to 11-year olds) lack confidence in their own mathematical understanding (Clarke & Clarke, 1996). Given that places in primary teacher training courses are not currently sought by many high attaining students, it is also the case that students with impressive results in high school mathematics form only a small proportion of those in such courses. However, understanding mathematics teaching involves more than understanding higher level mathematics.
Knowledge of mathematics must also be linked to knowledge of students’ thinking, so that teachers have conceptions of typical trajectories of student learning and can use this knowledge to recognise landmarks of understanding in individuals. (Carpenter & Lehrer, 1999, p. 31)
Teachers in the Early Numeracy Research Project are gaining the kinds of knowledge described above, have a clearer picture of the "typical trajectories of student learning", and can recognise "landmarks of understanding in individuals". Such a picture guides the decisions they make, in planning and in classroom interactions, as their knowledge of the understanding of individuals informs their practice.
The preparation of ENRP teacher for interviews also has the benefit of enhancing the confidence of the teacher in their own mathematical understanding, and places the teacher in a much stronger position to plan classroom experiences which may help children to increasingly acquire more sophisticated and efficient strategies. Consider the mathematical terms and structure involved in understanding the terms used in the addition and subtraction growth points, including "count down to", "count up from", "counting on", "modelling", "commutativity", "near doubles", and so on. I would argue that it is the engagement with the children and their thinking more so than the engagement in doing mathematics that is of value here, although these are not mutually exclusive. The former provides them with the knowledge they can use in the classroom and in their planning.
Returning to the idea of "knowing to" (Mason & Spence, 1999), clearly the ENRP framework has provided teachers with knowledge of children’s thinking both generally and specifically and that knowledge has been accessible and useful in practice. It has provided teachers with opportunities to work towards the type of teaching practice that is described by Australian teachers as excellent.
Excellent teachers of mathematics arouse curiosity, challenge students’ thinking, and engage them actively in learning. They initiate purposeful mathematical dialogue with and among students. As facilitators of learning, excellent teacher negotiate mathematical meaning and model mathematical thinking and reasoning. Their teaching promotes, expects and supports creative thinking, mathematical risk-taking in finding and explaining solutions, and involves strategic intervention and provision of appropriate assistance. (AAMT, 2000, p. 4)
Working together with teachers, we will continue to learn about the most appropriate forms of support in both inservice and preservice settings, but with a focus on children’s understanding I think we are likely to be making progress.
I have a vision of a mathematics classroom where the teacher has:
The support provided to teachers within the Early Numeracy Research Project appears to be making considerable progress towards making the vision a reality.
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