In this chapter, we begin by looking at ways that you can plan and implement partnership work. Together, we will learn about a range of different approaches that we can use when planning on building relationships. We will examine strategies to determine the appropriate approach and come to an understanding on the importance of co-design in partnerships. This chapter will show you how to draw on the second premise of the TWINE Model of Partnership – to plan – so that you can prepare yourself to start implementating partnership work.

In this chapter, your thinking around leading partnerships will be challenged. Throughout this chapter, you will be supported to develop a critical understanding of how leadership can impact partnerships, both positively and negatively.In the chapters so far, you have explored conceptual and practical components of partnerships and the key aspects that lead to successful partnership work.We have explored how diversity can challenge us as well as enrich our understanding of family and ways of being. We have also delved deeply into developing approaches, actions, strategies and tools to implement effective partnerships whereby we can measure their success.Of critical importance is how we lead all these processes.The aim of this chapter is for you to identify when leadership approaches need to change in response to the positioning of partnership work and ensure a way forward for all parties.

In this chapter, we build on the legislative requirements, conceptual ideas and theories that underpin the way we see and understand partnership work in education settings. As you engage with the chapter, you will be introduced to theoretical modelling and ideas that can inform your work with families, professionals and community. The ideas presented in this chapter will also support you to understand the various practices and communication strategies that are introduced in the succeeding chapters in this book.

In this chapter, we discuss and illustrate the tools of evaluating partnership work as part of a continuous improvement that you will need to make when building partnerships. As you engage with the chapter, you will learn what evaluation is and how to use evaluation to ensure that partnerships work for the benefit of children and their families. Throughout the chapter, we will explore different methods for collecting data and analysis, including various strategies that will inform the improvement and change necessary for good work in partnership-building.

In this chapter, we begin by examining the term ‘family’ and how it is defined in different contexts. As we examine these different definitions you will come to understand the complexity of ‘family’ and the diverse ways in which families can be defined. We then explore some of the structural and functional definitions of the family before moving towards examining some of the underlying assumptions made about families within wider society, including how these assumptions might position families in educational contexts. Through this exploration, some of your own underlying assumptions may be challenged as you come to understand the importance of educators and families working together to achieve the best educational outcomes for children. The chapter continues by discussing the idea of a subjective definition of families and what this might mean for you as an educator. We then move toward the term ’partnership’ and explore some of the barriers and opportunities to partnership work and how they can be harnessed and/or overcome. The chapter concludes by introducing the notion of innovative partnership work, your role in it as an educator, and the importance of this work in the educational context.

In this chapter, we begin by examining the role of reflective practice in partnership work and in doing so highlight the importance of knowing yourself and who you are as person. In Chapter 7, we critically examine reflective practice and how it unfolds in partnership work. In this chapter, we introduce the notion of reflective practice and how it prepares educators to begin thinking about how to come together with families. As you commence this journey of self-reflection, you will come to understand the complexity of partnership work and the skills you might need to develop to effectively engage with difference. We then explore some of the key ideas underpinning the planning of partnership work, including the importance of communication and open and positive mindsets, as well as the idea of active engagement and development of intercultural knowledge and capabilities. Through this examination, you will come to understand the first premise of the TWINE Model of Partnership so that you can identify as well as learn how to draw on this premise of the model when planning for partnership work.

In this chapter, we extend the learnings from Chapter 4 to expand your knowledge and skills on reflective practice for building effective and dynamic relationships for partnerships. You will understand how further elements of the TWINE Model of Partnership inform your reflective practice in partnership work. You will also come to learn about tools of reflective practice and how these tools can be useful in helping you to build meaningful relationships that contribute to partnerships with families and communities. This chapter will invite you to challenge yourself by asking key questions that will help you to become a reflective practitioner who builds dynamic relationships with children, families and communities.

In this chapter, we begin by examining the importance of trust in partnership work. We will then discuss the final premise of the TWINE Model of Partnership - to adapt. Through this premise, we will explore concepts such as participatory action, mapping out timelines, funding and resourcing a partnership. We will also examine some of the common challenges that might be faced in partnership work and discuss the ways these challenges might be overcome in practice.

Problems involving mass, momentum and energy transport in one spatial direction in a Cartesian co-ordinate system are considered in this chapter. The concentration, velocity or temperature fields, here denoted field variables, vary along one spatial direction and in time. The ‘forcing’ for the field variables could be due to internal sources of mass, momentum or energy, or due to the fluxes/stresses at boundaries which are planes perpendicular to the spatial co-ordinate. Though the dependence on one spatial co-ordinate and time appears a gross simplification of practical situations, the solution methods developed here are applicable for problems involving transport in multiple directions as well.

There are two steps in the solution procedure. The first step is a ‘shell balance’ to derive a differential equation for the field variables. The procedure, discussed in Section 4.1, is easily extended to multiple dimensions and more complex geometries. The second step is the solution of the differential equation subject to boundary and initial conditions. Steady problems are considered in Section 4.2, where the field variable does not depend on time, and the conservation equation is an ordinary differential equation. For unsteady problems, the equation is a partial differential equation involving one spatial dimension and time. There is no general procedure for solving a partial differential equation; the procedure depends on the configuration and the kind of forcing, and physical insight is necessary to solve the problem. The procedures for different geometries and kinds of forcing are explained in Sections 4.4–4.7.

The conservation equations in Sections 4.2 and 4.4–4.7 are linear differential equations in the field variable—that is, the equations contain the field variable to the first power in addition to inhomogeneous terms independent of the field variable. For the special case of multicomponent diffusion in Section 4.3, the equations are non-linear in the field variable. This is because the diffusion of a molecular species generates a flow velocity, which contributes to the flux of the species. The conservation equation for the simple case of diffusion in a binary mixture is derived in Section 4.3, and some simple applications are discussed.

In Section 4.8, correlations for the average fluxes presented in Chapter 2 are used in the spatial or time evolution equations for the field variables.

Convection can be neglected when the Peclet number is small, and the field variables are determined by solving a Poisson equation ∇‡2Φ fv + S = 0 or a Laplace equation ∇‡2Φ fv = 0, subject to boundary conditions, where Φfv and S are the field variable and the rate of production per unit volume, respectively. It is necessary to specify two boundary conditions in each co-ordinate to solve these equations. The separation of variables procedure is the general procedure to solve these problems in domains where the boundaries are surfaces of constant co-ordinate. This procedure was earlier used in Chapters 4 and 5 for unsteady one-dimensional transport problems.

The procedure for solving the heat conduction equation in Cartesian co-ordinates is illustrated in Section 8.1. The ‘spherical harmonic’ solution for the Laplace equation in spherical co-ordinates is derived using separation of variables in Section 8.2, first for an axisymmetric problem of the heat conduction in a composite, and then for a general three-dimensional configuration. There are two types of solutions, the ‘growing harmonics’ that increase proportional to a positive power of r, and the ‘decaying harmonics’ that decrease as a negative power of r, where r is distance from the origin in the spherical co-ordinate system.

An alternate interpretation of the decaying harmonic solutions of the Laplace equation as superpositions of point sources and sinks of heat is discussed in Section 8.3. It is shown that the each term in the spherical harmonic expansions is equivalent to a term obtained by the superposition of sources and sinks in a ‘multipole expansion’. A physical interpretation of the growing harmonics is also provided.

The solution for a point source is extended to a distributed source in Section 8.4 by dividing the distributed source into a large number of point sources and taking the continuum limit. The Green's function procedure for a finite domain is illustrated by using image sources to satisfy the boundary conditions at planar surfaces.

Cartesian Co-ordinates

Consider the heat conduction in a rectangular block of length L and height H, in which the temperature is T0 at x = 0 and x = L, TA at y = 0 and TB at y = H, as shown in Fig. 8.1.